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TRANSCRIPT
Advanced Quantum Chaos andMesoscopics
SS 2010
∆J = uSz1z′1J z′2
z2
z′2J z2
ts tu
Priv. Doz. Dr. Heiner Kohler
————————————————————————————————–Latest Update: October 19, 2015
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Contents
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Chapter 1
Motivation and Introduction
The lecture’s goal is to provide an up to date introduction to some actualtopics of “chaotic quantum systems”, “mesoscopics” and “random matrixtheory”. Due to the rich nature of either of these topics we will not beable to present them comprehensively. We will not even try. At first glance“quantum chaos”, “mesoscopics” and “random matrix theory” seem to berather disjoint. An important goal of the lecture is to show what thesetopics have in common. In fact they have so much in common that it isjustified to cover them in one lecture. We have selected the material in anattempt to underpin these common features.
Random matrix theory (RMT) was introduced in the sixties of the lastcentury by Wigner and Dyson in nuclear physics. The foundation of RMTin mathematics dates back to the work of Wishard in the twenties of the lastcentury. The time from 1980 to 1995 were the heydays of RMT in physics,triggered by the theoretical and practical interest in chaotic quantum sys-tems and by the groundbreaking developments in mesoscopics. Historically,the three research fields have evolved slightly differently during the last twodecades since then. RMT is still a valuable tool in physics but the main ac-tivity has shifted more and more towards mathematics as an important fieldof probability theory. On the other hand quantum chaos and mesoscopichave merged more and more with respect to methods and objectives into onebigger research area.
This lecture is intended as a sequel of the “Quantum chaos I” lecture, soit’s an advantage to have attended it, but it is not necessary. The presentlecture notes are to a large extent self contained and only in very few occasionsreference to the “Quantum chaos I” is made.
Before we start with a detailed exposition let us give an initial preliminarydefinition of the above topics.
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Quantum chaos:
Usually when physicists speak of “Quantum chaos” they refer to the study ofquantum systems whose classic analogs behave chaotically, or more precisely,to quantum systems which behave like chaotic systems in the classical limitof large quantum numbers. Quantum mechanics is a “linear theory” andtherefore, as we will see in more detail briefly, it does not allow for chaoticbehavior in the classical sense. However, it is well accepted that “Quan-tum mechanics” is the more fundamental theory. This means that it shouldbe possible at least in principle to derive any kind of classical behavior, in-cluding “chaoticity” , from an appropriately chosen quantum system in theproperly chosen classical limit. A large bulk of research has been devoted tothis question. How can classical chaotic behavior be derived from quantumtheory, which is linear but the more fundamental theory?
A different question is, if and how we can detect in a quantum system,whether its classical analog is chaotic or not? This question is related withthe question of universality. Given a quantum mechanical spectrum, thereare many different types of dynamical systems, which can be reached in theclassical limit, depending on the choice of operators to become the classicalcanonical coordinates. So the question arises, which properties of the spec-trum a stable under such different choices and therefore have a certain degreeof universality.
Other questions in quantum chaos are related to the old critisism of Ein-stein of quantisation, who realised that quantisation in Bohr–Sommerfeldquantization is not possible for classical chaotic systems.
Mesoscopics:
An important trigger for the study of mesoscopics was the seminal work ofAnderson on localisation of a one dimensional particle in a disorder potential?. He found that this effect was due to quantum coherence, or in other wordsit was a purely quantum effect. Since then there has been a large activityin solid state physics to find other coherence effects for electrons or moregeneral for waves in disordered media.
Studying theoretically and experimentally these effects has become a re-search area which is known today as mesoscopics. Roughly speaking, for asystem of matter waves, like electrons there are two requisites to be meso-scopic. On the one hand temperatures should be low. For too high temper-atures an effect called dephasing occurs which spoils coherence and whichultimately accounts for the fact that we can often consider electrons as clas-sical particles.
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Figure 1.1: random motion of an elec-tron through a disordered conductor.After a few scattering events the initialdirection is randomized, the energy ofthe electron is unchanged.
The second requisite is the size of the probe, which is related to the firstrequisite via the decoherence time. Sooner or later any electron, if not exactlyat temperature zero, will loose its coherence. The length of the path it hastravelled until this happens is the decoherence length.
A probe is called in the mesoscopic regime if this length is larger thanthe size of the probe. This restricts the latter to a few microns ∼ 10−6m atmost. These are extensions, which are not macroscopic but are larger thanthe typical atomic lengths scale, which explains the name mesoscopic fromgreek: µεσoζ = “in between”.
An exact definition of mesoscopics is, as often happens in physics, missing.A good definition might be as follows: Study of quantum mechanic effectson length scales, which are large compared to atomic length scales, but notmacroscopic. In this lecture we adopt a slightly more restricted point ofview and define mesoscopics as the investigation of waves and interferencephenomena in disordered mediums.
Random matrix theory (spectral statistics):
Random matrix theory deals with the investigation of properties of matriceswith random numbers as entries. To illustrate the concept we consider a 2×2matrix as the simplest possible example of a random matrix
H =
[a c/
√2
c/√
2 b
], (1.1)
where a, b, c are normal distributed random variables.
a, b, c : p(x) =1√
2σ2πexp
(− x2
2σ2
), x = a, b, c . (1.2)
The two eigenvalues E1,2 of this matrix are rather complicated functions ofthe three entries
E1,2 =1
2
(a+ b±
√(a− b)2 + 2c2
). (1.3)
Although the three random numbers a, b, c are uncorrelated the eigenvaluesare not. We show in figure (??) the motion of the eigenvalues of the matrix
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H, when the diagonal elements are tuned from 1 to −1 and vice versa. Wesee that for non–vanishing off–diagonal element the levels avoid each otherat the degeneracy point. This is a general effect. Whenever a matrix has off-diagonal elements they induce a repulsion of the eigenvalues. This holds forrandom matrices as well. When we look at the joint probability distributionof the two eigenvalues a brief calculation reveals that it can be written as
p(E1, E2) ∝ |E1 − E2| exp
[− 1
2σ2
(E2
1 + E22
)], (1.4)
which vanishes when both eigenvalues are equal. Only in the case that c = 0the prefactor E1 − E2 disappears and a level crossing is allowed.
The joint probability distribution of eigenvalues of a given random matrixwith independent matrix entries is a typical object of interest in RMT. Manyother quantities were invented to measure the degree of correlation betweenthe eigenvalues. In particular interesting is the limit of infinite matrix di-mension. As we will see only in this limit smooth curves can be expected forthe correlation functions of interest.
The application of RMT in physics comes originally from the observationthat in many complicated systems, like nuclei avoided crossings, respectivelylevel repulsion, was observed. Many of theses systems are so complicatedthat they defied other conventional treatment. Therefore the approach ofWigner and Dyson and others was to investigate what properties these sys-tems share with random systems as described by random matrices. If itis proven that certain properties of complex nuclei are reliably reproducedby random systems one can then in a second step replace the complicatedmany–body Hamiltonian by a random Hamiltonian to investigate further thephysics of the system.
Relation between the areas
At first glance the three fields of research activity described before seem tobe rather disconnected. The link between them is established through thestatistics of energy levels. It turns out that certain statistical quantitiesof the spectra of chaotic quantum systems and of mesoscopic systems areperfectly described by random matrices. It required much effort of some ofthe best physicists of the late last century to explain this perfect equivalencefrom first principles. It is still not yet completely understood. The upshotof these efforts might be summarised in the following two statements
1. The relation between spectral statistics and quantum chaos is estab-lished by the Bohigas-Giannoni-Schmit conjecture: Spectral correla-
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-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
E
0.0 0.2 0.4 0.6 0.8 1.0t
c = 0c =√
2
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
E
0.0 0.2 0.4 0.6 0.8 1.0t
c = 0c = 0.1 ·
√2
Figure 1.2: Eigenvalue repulsion at an avoided crossing in a Landau Zenertransition, a(t) = 1−2t, b = −1+2t, t ∈ [0, 1] . For small off diagonal valuesc 1 the eigenvalues come close to each other at t = 0, for large c ' 1 theeigenvalues strongly repell each other.
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tions of quantum systems, whose classical analogs are chaotic, are iden-tical to the correlations of normal distributed random matrices.
2. Efetov’s non-linear sigma-model: The energy correlations of weak dis-ordered systems are identical to those of random matrices.
Symbolically they can be visualised as follows.
quantum chaos mesoscopics
spectral statistics
random matrix theory
1 2
There have been recent advances towards a proof of the conjecture byBohigas-Giannoni-Schmit, which will be presented in this lecture. On themesoscopics side a rigorous proof was provided by Efetov’s so–called non–linear sigma model.
Literature
1. Quantum chaos: A comprehesive account on the fundamentals andon recent developments is provided by the textbook by F. Haake,Quantum signatures of Chaos ?. The textbook by H. Stockmann,Quantum chaos, an Introduction ? is easier to read and covers ex-perimental aspects as well. The online book by P. Cvitanovic et al.,www.ChaosBook.org ? is probably the most comprehensive account onclassical and quantum chaos available in the moment.
2. Mesoscopics: A very good textbook on mesoscopic physics was releasedrecently by E Akkermans & G Montambaux, Mesoscopic Physics ofElectrons and Photons ?, which covers most of the subjects discussedhere. A milestone and standard reference is the textbook by K. Efetov,Supersymmetry in Disorder and Chaos?, which however is rather a workof research than an introductory textbook. Another good textbooks,which covers many aspects not discussed here is the book by Imry ?.
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3. Random matrix theory:The textbook by M. L. H. Mehta, Random Matrices ? is the standardreference on random matrix theory, which however does not serve fora first reading.
Good review articles on the application of RMT in quantum chaos, nuclearphysics and beyond were provided by Beenakker ? and in ?.
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Chapter 2
Classical mechanics
2.1 Hamiltonian Mechanics
In classical mechanics we study the time evolution of points in phase spaceunder symplectic transformations. Phase space is a symplectic manifold, i.e. a differentiable manifold Γ with a symplectic 2-form
ω(X, Y ) : TpΓ× TpΓ→ R , (2.1)
where X, Y are vector fields, i. e. elements of the tangent space TpΓ of themanifold at point p. The symplectic two–form ω as well as the vector fieldsare local objects. This means they depend on a given point p of the manifoldΓ as illustrated by the picture below.
X
Y
Γ
p
Figure 2.1: Two vectorfields X and Y emanat-ing from a point p ∈ Γ.
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2.1.1 Darboux coordinates and equation of motion
For the study of symplectic manifolds Darboux’s theorem is most important.It states that the symplectic 2–form can be written locally always as follows
ω =
f∑i=1
dqi ∧ dpi . (2.2)
The set of coordinates qi, pi is called “Darboux coordinates”, however inphysics the notion “canonical coordinates” is more common. From equation(??) it is seen that the dimension of symplectic manifolds is always evendim(Γ) = 2f and f is the number of degrees of freedom. For the wedgeproduct “∧” holds
dqi ∧ dpi = −dpi ∧ dqi (2.3)
dqi ∧ dqi = dpi ∧ dpi = 0 . (2.4)
Given the vector fields
X =
f∑i=1
(Xqi
∂
∂qi+Xpi
∂
∂pi
)(2.5)
Y =
f∑i=1
(Yqi
∂
∂qi+ Ypi
∂
∂pi
)(2.6)
in Darboux coordinates, we get for the action of the symplectic 2–form onthe vector fields
ω(X, Y ) =
f∑i=1
(XqiYpi −XpiYqi) . (2.7)
In many cases it is easier to imagine ω as a matrix
ω : I =
[0 1f
−1f 0
]. (2.8)
With the help of ω we can transform covariant vectors to contravariant vec-tors and vice versa in a procedure which is well known from special relativity
ω(X, ·) =∑
(−Xqidpi +Xpidqi) . (2.9)
Exercise 1: Verify equations (??) and (??) using the properties of the wedgeproduct.
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Equation (??) can be written as well in the more familiar matrix languageas [
0 1f
−1f 0
] [~Xq
~Xp
]=
[~Xp
− ~Xq
](2.10)
We can get an important class of vector fields from a potential function
H(~q, ~p) : Γ→ R . (2.11)
We can act with ω−1 on the differential one–form
dH =∑ ∂H
∂qidqi +
∂H
∂pidpi (2.12)
to get the associated vector field XH
XH =∑ ∂H
∂pi
∂
∂qi− ∂H
∂qi
∂
∂pi. (2.13)
Such a vector field is called Hamiltonian vector field. The derivative of di-rection of a phase space function f with respect to the vector field XH is thepoisson bracket
XH [f ] =∑ ∂H
∂pi
∂f
∂qi− ∂H
∂qi
∂f
∂pi= f,H . (2.14)
A phase space function will be constant along the vector field XH , if thepoisson bracket vanishes. In particularH itself must be a constant. Thereforea point in phase space
~z(t) =
[~q(t)~p(t)
](2.15)
which moves along a trajectory of constant H, must move along a Hamilto-nian vector field
T0Γ 3 ~z(t) = XH . (2.16)
These are nothing but the Hamiltonian equations of motion:
qi(t) =∂H
∂pi(2.17)
pi(t) = −∂H∂qi
. (2.18)
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2.1.2 Symplectic transformation
The time evolution in classical mechanics is propagated by a symplectic trans-formation, i. e. by a transformation, which keeps the symplectic 2-form in-variant
f∑i=1
dqi(t) ∧ dpi(t) =
f∑i=1
dqi ∧ dpi (2.19)
ω(t) = ω . (2.20)
To understand this, we write:
ω(t) =1
2[d~z(t)]T I d~z(t) (2.21)
d~z(t) = J (t, ~z(0))d~z(0) , (2.22)
where
J (t, z(0)) =
∂q1(t)
∂q1(0)· · · ∂q1(t)
∂qf (0)
∂q1(t)
∂p1(0)· · · ∂q1(t)
∂pf (0)...
∂pf (t)
∂q1(0)· · ·
(2.23)
is the Jacobi matrix. From equation (??) it follows that
ω(t) =1
2[d~z(0)]TJ T (t, ~z(0))IJ (t, ~z(0))d~z(0) (2.24)
= ω(0) , (2.25)
iff
J T (t, ~z(0))IJ (t, ~z(0)) = I . (2.26)
A matrix, which keeps the symplectic metric invariant according to J T IJ =I is called symplectic. It is obvious that the product of two symplecticmatrices J1 and J2 is symplectic as well, so the symplectic matrices form agroup (the so–called symplectic group) under matrix multiplication.
Even though the dependence of J on the initial values is highly nonlinear, it is possible to expand J (t, ~z(0)) for small times. For small times weexpect a linear dependence on the initial values. But before we do so let usfirst compare the evolution of time in classical and quantum mechanics.
H 3 ψ(t) = U(t, 0)ψ(0) (QM) (2.27)
T ∗Γ 3 d~z(t) = J (t, ~z(0))d~z(0) (classical mechanics) (2.28)
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The first equation is the well known time evolution of a quantum state ψ(0),element of a Hilbert space H, under the time evolution operator U . Thesecond equation describes the time evolution of an infinitesimal displacementof the phase space point ~z(t) under the action of the Jacobi matrix. Thereare two important differences
1. J symplectic ⇔ U unitary: Without going into details we mentionthat the symplectic group is a non–compact group. This means thatits action on a vector can increase the length of this vector by anarbitrary factor. On the other hand the unitary group is compact andits action onto a state vector leaves invariant the length of this vectoras required by conservation of probability.
2. J non–linear in ~z(0) ⇔ QM linear in ψ(0): Since the Jacobi matrixcan depend on the initial phase space point in an arbitrary way, thisdependence is in the generic case non–linear, in quantum mechanics thetime evolution operator U does not depend at all on the initial state.Thus time evolution depends linearly on the initial value.
We now perform the announced linearisation of J for small ∆t. We expand
qi(t) ≈ qi(0) +dqidt
∣∣∣∣t=0
∆t
≈ qi(0) +∂H
∂pi
∣∣∣∣t=0
∆t (2.29)
and likewise
pi(t) ≈ pi(0)− ∂H
∂qi
∣∣∣∣t=0
∆t . (2.30)
The Jacobi matrix reads for small times
J (∆t, z(0)) =
1 +
∂2H
∂q1∂p1
∆t · · · 1 +∂2H
∂p1∂p1
∆t
......
1− ∂2H
∂q1∂q1
∆t · · · 1− ∂2H
∂p1∂q1
∆t
......
(2.31)
= 1 + F0∆t , (2.32)
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etF0
~z(0)
~z(0) + d~z(0)
Figure 2.2: Whereas thepoint ~z(0) follows theoriginal trajectory. Aslightly displaced initialpoint follows a differenttrajectory, which gener-ically deviates from theoriginal one exponen-tially.
where F0 is given by
F0 =
∂2H
∂qi∂pj
∂2H
∂pi∂pj
− ∂2H
∂qi∂qj− ∂2H
∂pi∂qj
t=0
. (2.33)
With the help of the marix F0 we can calculate approximately the timeevolution of d~z(t). For small times t we write
d~z(t) = d~z(0) + F0t d~z(0) + . . .
≈ eF0td~z(0) . (2.34)
The matrix F0 is a measure for the stability of trajectory close to a pointin phase space. Therefore we call F0 stability matrix. The stability matrix iscrucial for characterising the behavior of a classical dynamical system. Wetake a closer look on its algebraic structure. From equation (??) it is seenthat F0 can be written symbolically as
F0 =
[A BC −AT
], (2.35)
where B = BT and C = CT . One can show that a matrix of the form asgiven in equation (??) fulfills
F T0 I + IF0 = 0 . (2.36)
Exercise 2: a) Verify that a matrix of the form (??) fulfills equation (??).b) Show that if F0 fulfills equation (??) the matrix U = exp(aF0), a ∈ C, issymplectic. c) Show that if λ is an eigenvalue of a matrix (??) so is −λ.
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The stability matrix F0 can be diagonalised by a matrix J0 which is itselfa symplectic matrix. Let us call the outcome of this diagonalisation Λ
Λ = J −10 F0J0 . (2.37)
Since F0 is non–Hermitean, Λ has in general 2f complex eigenvalues, whichcome in f pairs consisting of an eigenvalue λ and its negative. In principlewe can distinguish four cases
1. λ ∈ C,
2. Im(λ) = 0,
3. Re(λ) = 0,
4. Im(λ) = 0 and Re(λ) = 0 .
We want to obtain a deeper understanding of the physical meaning of thestability matrix and of its eigenvalues. In particular we want to see howchaoticity of a dynamical system can be characterised by the stability ma-trix. To this end we first look at regular (integrable) systems and try tocharacterise them by their stability matrix.
2.1.3 Canonical transformations and integrable systems
Canonical transformations are coordinate transformations that leave the sym-plectic 2-form invariant
(qi, pi) −→ (Qi, Pi) (2.38)∑dqi ∧ dpi −→
∑dQi ∧ dPi (2.39)
qi, pj = −δij −→ Qi, Pj = δij (2.40)
The Hamiltonian function transforms as
H( ~Q, ~P ) = H(~q( ~Q, ~P ), ~p( ~Q, ~P )) . (2.41)
A properly chosen coordinate transformation can solve a given problem im-mediately. As an example we solve the equations of motion for the classicalharmonic oscillator by canonical transformation. The Hamilton function ofthe harmonic oscillator is given by
H(q, p) =p2
2m+ω2m
2q2 . (2.42)
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Of course, it can be treated in the original coordinates without problems.We introduce a new pair of coordinates Q and P via
q =
√2P
mωsinQ , p =
√2Pmω cosQ , (2.43)
which likewise might be inverted
Q = arcsin(√
mω22Pq), P =
1
2mωp2 +
1
2mωq2 . (2.44)
Exercise 3: Show that the new coordinates Q and P have the same Poissonbracket as the original ones Q,P = 1.
In the new coordinates the Hamiltonian becomes particularly simple
H(Q,P ) = ωP . (2.45)
We notice that H does not depend on Q in the new coordinates. Sucha coordinate is called cyclic. The equations of motion for cyclic pairs ofcoordinates (Q,P ) become extremely easy to solve
Q =∂H
∂P⇒ Q(t) = ωt+Q0, ω =
∂H
∂PP = 0 ⇒ P (t) = const . . (2.46)
From the above example it is seen that in general canonical transformationsconnect the old coordinates in a not linear way with the new coordinates.
A systematic way to construct a canonical transformation is by definingmixed generating functions
F1(~q, ~Q), F2(~q, ~P ), F3( ~Q, ~p), F4(~p, ~P ) , (2.47)
which are functions of new and original coordinates. The symplectic 2-form’sinvariance requires
f∑i=1
dqi ∧ dpi =
f∑i=1
dQi ∧ dPi . (2.48)
For a generating function F1 this condition is fulfilled if we set
pi =∂F1
∂qi∨ Pi = −∂F1
∂Qi
. (2.49)
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With equation (??) we can write
dpi =∑j
∂F1
∂qi∂qjdqj +
∑j
∂F1
∂qi∂Qj
dQj
dPi =−∑j
∂F1
∂Qi∂Qj
dQj −∑j
∂F1
∂Qi∂qidqj . (2.50)
Using the two equations (??) the original and the transformed symplecticform can be written as
f∑i=1
dqi ∧ dpi =∑ij
(∂F1
∂qi∂qjdqi ∧ dqj +
∂F1
∂qi∂Qj
dqi ∧ dQj
)(2.51)
f∑i=1
dQi ∧ dPi = −∑ij
(∂F1
∂Qi∂Qj
dQi ∧ dQj +∂F1
∂Q∂qjdQi ∧ dqj
).
(2.52)
This is identical with equation (??), since due to the wedge product∑i,j
∂F1
∂qi∂qjdqi ∧ dqj =
∑ ∂F1
∂Qi∂Qj
dQi ∧ dQ = 0
∑i,j
∂F1
∂Qi∂qjdQi ∧ dqj =−
∑i,j
∂F1
∂qi∂Qj
dqi ∧ dQj . (2.53)
Likewise, we find for the other generating functions F2, F3 and F4 the equa-tions
pi =∂F2
∂qi, Qi =
∂F2
∂Pi, (2.54)
Pi =∂F3
∂Qi
, qi =∂F3
∂pi, (2.55)
qi =∂F4
∂pi, Qi = −∂F4
∂Pi. (2.56)
In principal one can choose an arbitrary function as generating function.In practice however one should use a generating function which generates asystem of Darboux coordinates, which render the new Hamiltonian functionas simple as possible. We might agree that the simplest possible function isa constant. Further elaboration on this idea leads to the theory of Hamilton-Jacobi.
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In this context F2 is very important. We denote this function with acapital W (~q, ~P ) and call it Hamilton characteristic function. Requiring thatH = const . in the new coordinates and using relation (??) for F2 yields
H
(q1, . . . , qf ;
∂W
∂q1
. . .∂W
∂qf
)= E , (2.57)
where we called the constant E and identify it with the energy. Equation(??) is the stationary Hamilton-Jacobi equation. It is in general a non linearpartial differential equation in f variables. In the general theory of partialdifferential equations (PDEs) a solution of a PDE of the form (??) dependson f initial conditions which we can chose as the momenta at given positionq
(0)i , i = 1 . . . f
∂W
∂q1
∣∣∣∣q1=q
(0)1
= p(0)1 , . . . ,
∂W
∂qf
∣∣∣∣qf=q
(0)f
= p(0)f . (2.58)
While this is a valid choice it is not the most convenient one, since the newmomenta depend on a rather arbitrary fashion on the initial points q
(0)i ,
i = 1, . . . f . For periodic motion we might choose
Pi =1
2πL
∮period
pi dqi =ω
L
T∫0
pi(t)qi(t) dt =IiL
(2.59)
as new coordinate, where L is the length of the period. This choice for Pi isdemocratic in the initial positions and most conveniently independent of theinitial condition. In the last equation we introduced the quantity Ii which canserve as constant of motion like Pi but with the dimension of an action. Notethat not all I1, . . . , If are independent but are connected by the condition(??). We introduce the quantity
I(E) =1
2π
∮period
~p(~q) d~q =1
2π
∮~∇W (~q, E)d~q
=
f∑i=1
Ii (2.60)
which is called action or action integral.The time dependent Hamiltonian principal function S(~q, ~P , t) is related to
the Hamilton characteristic function in a similar way as a solution of the time
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dependent Schrodinger equation to solutions of the stationary Schrodingerequation. Write
S(~q, ~P , t) =W (~q, ~P )− Et (2.61)
Then S(~q, ~P , t) fulfills the Hamilton Jacobi equation
H
(q1, . . . , qf ;
∂S
∂q1
. . .∂S
∂qf
)+∂S
∂t= 0 . (2.62)
The function S(~q, ~P , t) is called Hamilton’s principal function. Since by con-struction
dS
dt=
f∑i=1
∂S
∂dqiqi − E
dS
dt=
f∑i=1
piqi −H ⇒ S(t, t0) =
t∫t0
dt′L(t′) , (2.63)
where L(t) is the classical Lagrange function. Thus the Hamilton principalfunction is identical with the classical action integral. On the other hand weobtain for Hamilton characteristic function
W (~q, ~P ) =
t∫t0
(L+H)dt =
~q∫~q (0)
~p(~q)d~q . (2.64)
Thus, in particular for periodic motion at ~q = ~q (0) the characteristic functionbecomes independent of ~q and equal to the action integral W = 2πI.
Exercise 4: Derive the Hamilton Jacobi equation from the Schrodinger equa-tion
i~∂ψ(~q, t)
∂t= − ~2
2m∆ + V (~q) (2.65)
by making the ansatz ψ(~q, t) = exp(iS(~q, t)/~). In what limit the HamiltonJacobi equation arises?
We choose again the harmonic oscillator as example. In this case thestationary Hamilton–Jacobi equation reads
1
2m
(∂W
∂q
)2
+ω2m
2q2 = E . (2.66)
20
It can easily be integrated
∂W
∂q=√
2mE − ω2m2q2
⇒ W (q, P (E)) =
q∫0
√2mE − ω2m2q′2dq′ . (2.67)
Now the problem is in principle solved. We still have a freeness in the choiceof the transformed momentum P (E). Using the choice (??) wet get for theaction integral
I(E) =2
π
qmax∫0
√2mE − ω2m2q2 dq , qmax =
√2E
ω2m(2.68)
=4E
πω
1∫0
√1− q2 dq =
E
ω. (2.69)
We have therefore in this example a very simple relation between energy andaction.
I(E) =E
ω, P (E) =
I(E)
L=
√2Em
8(2.70)
The generating function of the canonical transformations, which transform
H → H(I) = ωI
→ H(P ) =32P 2
m(2.71)
is given by
W (q, I) =
q∫0
√2mωI − ω2m2q′ 2dq′ ,
W (q, P ) =
q∫0
√(8P )2 − ω2m2q′ 2dq′ (2.72)
Exercise 5: Show that for the harmonic oscillator the canonical transfor-mation induced by W (q, I) is the coordinate transformation (q, p) → (Q,P )as given in equations (??) and (??).
21
When the action integral is taken as momentum coordinate and its canon-ical conjugate coordinate Q is cyclic the latter is usually called angle variableand denoted by a lower case greek character.
Now we are able to give a definition of an integrable system: If one cantransform a Hamiltonian function
H(~q, ~p) → H(~I, ~θ) (2.73)
by a canonical transformation
(~q, ~p) → (~I, ~θ)
Ii, Ij = θi, θj = 0 , Ii, θj = δij , (2.74)
into a form such that the transformed Hamilton function does not dependon the angle coordinates
H(~I, ~θ)!
= H(~I) , (2.75)
the system is called integrable. The Darboux coordinates (~I, ~θ) in which allθi, 1 ≤ i ≤ f are cyclic are called angle action variables. By the actionangle variables the phase–space is naturally foliated in submanifolds of con-stant action. These submanifolds are products of one–dimensional manifolds,parametrised by the time–evolution of the angles
θi(t) = ωit+ θi(0) (2.76)
There are only two topologically distinct one–dimensional manifolds, whichare described by equation (??). The non–compact real line R and the com-pact circle S1. If one or more particles can escape to infinity the system isunbound and one or more one–dimensional submanifolds have the topologyof R. On the other hand if the state is bound the surfaces of constant actionI1, . . . , If are f−dimensional tori
S1 × S1 × . . . S1 . (2.77)
We add two remarks. Although in angle action coordinates the trajec-tory looks simple, the canonical transformation into angle action coordinatesmight be a highly complicated non–linear transformation. Therefore in theoriginal coordinates the trajectory can look very complicated. Second, forgeneric frequencies ωi = ∂H
∂Ii, 1 ≤ i ≤ f the motion is not periodic and for
long times the torus will be filled densely by a single trajectory. The ex-ist infinitely many periodic trajectories. They are also called Resonances.Therefore simply looking at the trajectories of a system and see if it behaves
22
Figure 2.3: The phasespace trajectories of abound integrable sys-tem twist around anf− dimensional torus.In the picture a shortperiodic orbit (thickblack) and a longer pe-riodic orbit (thin blue)are drawn.
“sufficiently complicated” or not is usually not enough to distinguish reg-ular and non–regular systems. Even regular systems can behave in a verycomplicated way.
We now can study the stability matrix F0 of an integrable system. Weassume, we have been able to transform it into angle action variables. Then
F0 =
∂2H
∂ϑi∂Ij
∂2H
∂Ii∂Ij
− ∂2H
∂ϑi∂ϑj− ∂2H
∂Ii∂ϑj
(2.78)
can be written as
F0 =
0∂2H
∂Ii∂Ij
0 0
, (2.79)
because by definition of a cyclic coordinate ∂H∂θi
= 0. We conclude that alleigenvalues of F0 are zero. This property holds for an arbitrary point inphase space. Moreover it holds for an arbitrary coordinate system, althoughit is only easy to see in angle–action coordinates. The fact that all eigevaluesof F0 are zero is a characteristic property of an integrable system.
23
d~zλ
~z0
d~z||
d~z−λ
Figure 2.4: Illustration of theprincipal axes at point p = ~z0.To every eigenvector to eigen-value λ corresponds an orthog-onal direction to eigenvalue −λ.
2.2 Chaotic systems
As pointed out at the end of section ?? in general the stability matrix F0
can have non–zero eigenvalues. These non–zero eigenvalues can be linked tonon–integrable or chaotic behavior. They are useful for the classification ofchaotic systems.
2.2.1 Characterisation of chaotic systems
The algebraic structure of F0 and their eigenvalues have already been studiedin section ??. We recall some of the results: All eigenvalues of the stabilitymatrix come in pairs. Accordingly we write the set of principal axes
J0 =[~eλ1 , ~e−λ1 , . . . , ~eλf , ~e−λf
]. (2.80)
Although in general F0 has non–zero eigenvalues, at least two eigenvalues arealways zero. They are obtained by a variation of ~z0 = ~z(0) into the directionof the Hamiltonian flow. For the non–zero eigenvalues one can distinguishfour different cases. They are called
• ioxodrom if λ(~z0) ∈ C,
• hyperbolic if λ(~z0) is real,
• elliptic if λ(~z0) purely imaginary:
• marginal if λ(~z0) = 0.
They are illustrated in figure ??.
24
d~z||
~z0 d~z−λ
d~zλ
Figure 2.6: Illustration of thebehavior for different displace-ment directions. The trajectoryof the undisplaced point ~z0 fol-lows in the beginning the direc-tion of ~z‖. A phase space pointwhich is displaced in the direc-tion ~eλ departs from the originaltrajectory. A point which is dis-placed in the direction of ~e−λ ini-tially comes closer to the originaltrajectory.
δp
δq
δp
δq
δp
δq
elliptic marginal hyperbolic
Figure 2.5: Illustration of an elliptic, a marginal and of an hyperbolic eigen-value.
Chaotic behaviour is identified with a real positive eigenvalue λ of thestability matrix. In this case an infinitesimal displacement of the initialpoint ~z(0) in the direction of ~eλ leads to an exponentially strong deviationof its trajectory from the original one
d~zλ(t) ≈ eλ(~z0)td~zλ(0) . (2.81)
However, one should notice that for every positive λ(~z0) there is a corre-sponding negative −λ(~z0). This means that an infinitesimal displacement ofthe initial point into to direction of ~e−λ has the opposite effect. The trajec-tory of this point comes closer and closer to the trajectory of the originalone
d~z−λ(t) ≈ e−λ(~z0)td~z−λ(0) . (2.82)
These different behaviors are illustrated in figure ??
25
We notice that λ(~z0) and e~λ are local quantities. They vary dependingon the point ~z0 in phase space. The positive λ(~z0) at a point ~z0 is called localLyapunov exponent or stretching rate. One can get the (global) Lyapunovexponent, if one calculates the mean of λ(~z0) over a trajectory with infinitelength
L(~z0) = limT→∞
1
T
T∫0
dt λ(~z(t)) . (2.83)
For many systems L(~z0) = L does not depend on the initial value ~z0. Suchsystems are called uniform hyperbolic systems. Hyperbolicity is the mostimportant characterisation of a chaotic system.
We enumerate two other generic properties of chaotic systems, which willbe needed in the sequel.
1. Ergodicity: Let f be a sufficently well behaved function of phase space
f : Γ → R . (2.84)
Then we can define a time average for this function evaluated along aphase space trajectory as follows:
〈f〉T =1
T
T∫0
f(~z(t))dt . (2.85)
Ergodicity means that for very large times this average can be replacedby a phase space average
〈f〉r = −∫Γ
f(~z)dµ(~z) , (2.86)
where
dµ(~z) =1
Ω
f∏i=1
dpidqiδ(H(~z)− E) (2.87)
is called Liouville measure and Ω is the volume of the energy shell. TheLiouville measure is the induced measure by restricting the equidis-tribution in Euklidean phase space to the energy shell. In formulaeergodicity means
limT→∞
〈f〉T = 〈f〉Γ . (2.88)
This property is crucial for the semiclassical approach to the proof ofthe conjecture of Bohigas, Gianonni and Schmit.
26
2. Mixing: The mixing property is stronger than ergodicity. Looselyspeaking it means that a trajectory after a sufficient long time haslost its memory on the initial state. More precisely it means that inthe long time limit a phase space average of two phase space functionsf and g, where f is a function of ~z(0) and g a function of ~z(T ) can bereplaced by the product of two independent phase space averages as
limT→∞
∫Γ
f(~z(0))g(~z(T ))dµ(~z(0)) =
∫Γ
f(~z)dµ(~z)
∫Γ
g(~z ′)dµ(~z ′) .
(2.89)
t0 t1 t2 ∞
t0 t1 t2
. . .
Figure 2.7: Illustration of the mixing property: The upper time evolution ismixing and ergodic, the time evolution in the lower row is ergodic but notmixing
Ergodic, mixing systems in which every point has at least one hyperbolicdirection are called fully chaotic.
2.2.2 Poincare section
Poincare surfaces are 2(f−1)dimensional Lagrangian submanifolds of Γ. Theprecise mathematical definition of a Lagrangian submanifold is involved andrequires additional mathematical concepts. The reader is referred for instanceto Ref. ?. A definition which is sufficient for our purposes sufficient goes asfollows: a Lagrangian submanifold is spanned by a position coordinate andby a momentum coordinate.
A Poincare section is constructed as follows.
27
qΣ
pΣ
Figure 2.9: For regu-lar systems the Poincaresection is pierced on aloop, corresponding toa deformed torus. Ifthe trajectory is res-onant the number ofpoints are finite, other-wise the loop will befilled densely.
qΣ
pΣ
Figure 2.10: For achaotic system thePoincare section isexpected to be filleduniformly, due to theergodicity and mixingassumption. Howeverthere might show upstable islands.
pΣ
qΣ
σ2σ1
σ3 σ3
σ1 σ2
pΣ
qΣ
Figure 2.8: Sketch of the construction of a Poincare plot. A trajectory piercesthe Poincare section, spanned by the momentum pΣ and by the position qΣ.Only the downwards piercings contribute to the Poncaree plot below.
Poincare sections are very useful to find chaotic behavior. In a regu-lar system the trajectory will pierce ΣR in a regular way, according to theinvariant tori. The Poincare section will look as indicated in figure ??.
On the other hand for a chaotic system the Poincaree section should befilled with piercing points densely.
28
θk
σ1
σ2s=0=L
0 Ls
1
−1
cos θkσ2
σ1
Figure 2.11: Sketch ofthe construction of aPoincare section for anirregular billiard. Thefirst two points are con-structed. By construc-tion the surface is peri-odic in the s–axis withperiod L.
2.2.3 Examples for classical chaos
We now give a few examples of classical chaotic systems. A more extensiveaccount is given in the Script to “Quantum Chaos I”.
• The prime and best studied example for a chaotic system are two di-mensional billards with an irregular border. Their phase space dimen-sion is dim Γ = 4. Here and in the following we refer to billiards alwaysas two–dimensional billiards if not stated otherwise. This is in agree-ment with our daily life intuition of a billiard table. We will use thedirection of the momentum cos θk at the boundary as the momentumaxis of the Poincare section. For the position axis we choose the dis-tance s of the point where the ball hits the boundary, measured froman arbitrarily chosen point s = 0 (see figure ??). One can indeedshow, that for billiards the two coordinates cos θk and s are canonicalconjugate phase space variables.
• Limacon billiard (Robnik billiard): The Limacon billiard is a goodexample for a dynamical system with a mixed phases space. The cir-cumference of the billiard is parametrised in polar coordinates by
ρ(ϕ) = 1− ε cosϕ , φ ∈ [−π, π] (2.90)
For ε = 0 it is identical with the integrable circular billiard and regular.As ε increases the tori will loose their shape according to the KAMtheorem, but the billiard remains regular. For a critical value εcrit thebilliard develops a cusp. It becomes more and more chaotic but stillhas integrable islands.
29
Exercise 6: Calculate the critical value εcrit from equation (??).
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-2.0 -1.5 -1.0 -0.5 0.0 0.5
ε = 0.2
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-2.0 -1.5 -1.0 -0.5 0.0 0.5
ε = 0.6
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-2.0 -1.5 -1.0 -0.5 0.0 0.5
ε = 0.8
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
-2.0 -1.5 -1.0 -0.5 0.0 0.5
ε = 1.0
Figure 2.12: Limaccon billiard for four different values of ε. For valuesε > εcrit a cusp emerges.
30
Figure 2.13: Poincare surface of section of the Limacon billard for parameterε = 0.3. We see that most of the phase space is chaotic, but some regularislands can clearly be seen.
• Henon-Heiles Hamiltonian: The Hamiltonian was introduced by the as-tronomer Michel Henon and a graduate student Carl Heiles as a math-ematical model to numerically investigate the so called third integralproblem of celestial mechanics ?. It is a celebrated and historically im-portant example for a chaotic system. The Henon-Heiles Hamiltonianis given by
H =1
2(p2x + p2
y) +1
2(x2 + y2) + x2y − y3
3
=1
2(p2x + p2
y) + V (x, y) (2.91)
In figure ?? the potential is plotted. It is seen that for small energiesthe potential felt by the particle is almost circular, the motion of theparticle is regular. As the energy of the particle increases the potential,which is felt by the particle has a more and more triangular shape. Themotion of the particle becomes chaotic. For a critical dimensionlessenergy ε = 1/6 the particle can escape.
Exercise 7: Show that for energies higher than ε = 1/6 the particlecan escape from the potential well.
31
-1.0 -0.5 0.0 0.5 1.0
-0.5
0.0
0.5
1.0
y
x
Figure 2.14: Plot of the equipotential lines of the potential V (x, y) = 12(x2 +
y2) + x2y − y3
3for the values V (x, y) = 0.01, 0.04, 0.1, 0.15, 0.2.
Figure 2.15: Poincare section for the Henon-Heiles Hamiltonian. We seethat as the energy ε increases and approaches the critical value the chaoticsea increases. Figures taken from http://mathworld.wolfram.com/Henon-HeilesEquation.html.
In the Poincare surfaces of section of the Limacon billiard ?? and of theHenon-Heiles Hamiltonian ?? it is nicely seen that the transition from regu-
32
lar to chaotic dynamics with increasing perturbation parameter ε is neitherabrupt nor smooth but of remarkable complexity. It is clear that for a com-pletely integrable system the Poincare surface of section is densly foliated byinvariant curves stemming from the tori for different values of the action vari-able. In the first panel of Fig. ?? we see that many of theses invariant curvespersist for small perturbation parameter ε. As the perturbation increasestheses invariant curves gradually disappear, yet certain resonant (periodic)orbits persist and develop secondary invariant curves around them. The in-variant curves are separated by the resonance through a small chaotic sea.The ensemble of an isolated resonance with a invariant curve separated bya chaotic sea is called stable island. Stable islands are abundantly seen inFig. ?? and also in the second and third panels of Fig. ??. Around theseparameterPoincare
Figure 2.16: Transition to chaos.
2.3 Classical perturbation Theory
Perturbation theory is the first and standard approach to any non–trivialquantum mechanical system. It gives a first intuition on what physical phe-nomena van be expected and serves as a valuable benchmark for other non–perturbative methods. Nevertheless in classical mechanics perturbation ismuch more complicated. We first focus on one degree of freedom.
2.3.1 Perturbation theory for f = 1
Let us consider the Hamilton function of a pendulum of length l
H(pφ, φ) =p2φ
2ml2+ml2ω2(1− cos(φ)) , (2.92)
where ω = g/l, and g the gravitational constant. A straightforward but naiveway to go beyond the harmonic approximation is to expand the cosine up to
33
fourth order and consider this fourth order term as a perturbation
H(p, q) =p2
2m+mω2q2 − εmω2
4!q4 . (2.93)
Here we made the substitutions
ml2 → m
pφ → p
φ→ q (2.94)
in order to stick with the usual notation in terms of position and momentumcoordinates. Moreover ε is formally a small parameter which is set ε = 1at the very end. Following the steps of quantum mechanical perturbationtheory as close as possible, we expand the solution to the equation of motionin powers of ε
q(t) = q0(t) + εq1(t) + ε2q2(t) + . . . . (2.95)
Plugging this into the equation of motion and comparing terms with the samepower in ε yields in zeroth order the solution of the unperturbed harmonicaloscillator
q0(t) = A cos(ωt) . (2.96)
In first order in ε the equation of motion for q1(t) reads
q(t) + ω2q1(t) =A3ω2
6cos3(ωt)
=A3ω2
24
(cos(3ωt) + 3 cos(ωt)
). (2.97)
The equation of motion for q1(t) is that of a driven oscillator. While thedriving frequency 3ω is harmless the second driving term is in resonancewith the oscillator and leads to the secular solution
q1(t) = − A3
192cos(3ωt) +
A3
64
(ωt cos(ωt) + 2 cos(ωt)
). (2.98)
The secular term ∼ t in the solution of q1(t) has the unpleasant consequencethat even for very small perturbation at a time scale tε ∼ ε the perturbativeterm dwarfs the unperturbed solution. The secular term is clearly unphysicaland leads to the conclusion that naive perturbation theory miserably fails inclassical mechanics.
34
The correct perturbative approach to classical mechanics is due to Poincareand von Zeipel. It makes use angle- action coordinates and of the fact thatthe unperturbed Hamilton function is a function of action coordinates only.We first focus on the one–dimensional case. Write the full Hamilton functionas
H(I, θ) = H0(I) + εH1(I, θ) . (2.99)
To zeroth order I(t) = I0, θ(t) = ω0t+β where ω0 = ∂H0/∂I0|I0 A canonicaltransformation is introduced by
W (I , θ) = Iθ + εW1 (2.100)
such that in lowest order in ε this transformation is just the identical trans-formation I → I and θ → θ. To first order in ε one obtains the relations
I = I + ε∂W1(I , θ)
∂θ
θ = θ − ε∂W1(I , θ)
∂θ(2.101)
between new and old coordinates. These can be plugged into the Hamiltonfunction
H(I , θ) = H0(I) + εH1(I , θ)
H1(I , θ) =∂H0(I)
∂I
∂W1(I , θ)
∂θ+H1(I , θ)
= ω0∂W1(I , θ)
∂θ+H1(I , θ) (2.102)
We still can choose W1 at will. We choose it in such a way that H1(I , θ) =H1(I), i. e. only depends on the action coordinate. If we write expandH1(I, θ)in Fourier components
H1(I, θ) =∞∑
n=−∞
H1,n(I)einθ
H1,n(I) =1
2π
2π∫0
dθe− inθH1(I, θ) (2.103)
then obviously
W1(I , θ) = −∞∑
n=−∞n 6=0
H1,n(I)
inωeinθ (2.104)
35
does the job. Thus the Hamilton function reads in the new action–anglecoordinates
H(I) = H0(I) +ε
2π
2π∫0
dθH1(I , θ) (2.105)
Let us treat with this method the former example. The Hamilton functionreads in the angel–action coordinates (??)
H(I, θ) = ω0I −εI2
6msin4(ωt) . (2.106)
Expanding sin4 θ = 38− 1
2cos(2θ) + 1
8cos(4θ) yields
H(I) = ω0I −εI2
16m
ε=1⇒ ω
ω0
= 1− I
8ω0m= 1− E
8ω20m
. (2.107)
The frequency becomes energy/amplitude dependent. This feature is notcaptured in naive perturbation theory. The frequency of the pendulum canbe calculated exactly
ω
ω0
=π
2K−1
(E/2mω2
0
), K(x) =
π/2∫0
dφ√1− x2 sin2(φ)
, (2.108)
for x < 1. Here K is the complete elliptic function of the first kind. Theseries expansion of K(x) recovers the result (??). One might conclude thatthe method of Poincare and von Zeipel is the correct perturbative approach toclassical mechanics. While this holds true in one dimension, unfortunately inhigher dimension, respectively for more than one degree of freedom, canonicalperturbation theory is plagued by singularities.
Exercise 8: Derive Eq. (??) from the Hamilton function of the pendulumEq. (??).
2.3.2 Canonical perturbation theory for f > 1
The method described in the previous section for one degree of freedom canin principle applied to systems with two or more degrees of freedom. Thegenerating function (??) can be generalized to f ≥ 2 by
W (I1, . . . , If , θ1, . . . , θf ) =
f∑n=1
(2.109)
36
ωω0
1
1EEcrit
Figure 2.17: Energy depen-dence of the frequency of thependulum. For small energiesfirst order canonical perturba-tion theory (red curve) yieldsa good approximation. As theenergy approaches its criticalvalue Ecrit = 2mω2
0 the timefor one period becomes infiniteT →∞.
-Π Π
-4
-2
2
4
Figure 2.18: Phase space dia-gram of the pendulum. Regionsof periodic motion (clear grey)are separated by regions of un-bound motion (dark grey) bythe so–called separatrix. At theseparatrix The energy at theseparatrix is Ecrit.
37
Chapter 3
Spectral statistics of quantumsystems
Quantum mechanics is more fundamental than classical mechanics. In prin-ciple it should therefore be possible to derive a theory of classical chaos fromquantum mechanics. In practice we choose to go the other way. We quantizea classical chaotic system and look at the properties of the quantum system.
In the statistical approach to quantum spectra the concept of universal-ity becomes important. We might consider many different classical chaoticsystems, leading to different quantum systems. We are only interested inquantities of the quantum systems which are shared by a wide class, whichwe will call universality class. In the classical limit all these quantum systemsshould have at least one common property, they should be chaotic.
3.1 Quantization
In a quantization procedure the classical phase space is coarse grained by di-viding it into volumes of magnitude hf . For a compact phase space a phasespace volume VΓ can be defined and we will get VΓ/h
f unit cells. This number
h
h
Γ
Figure 3.1: Coarsegraining of the phasespace into unit cells ofvolume hf
38
of unit cells is identified with the Hilbert space dimension of the quantummechanical system. In many cases Γ is not compact, then arbitrary high en-ergies are possible. In this case the Hilbert space of the quantum mechanicalsystem is infinite-dimensional. If some phase space function I(~q, ~p) is con-served under the Hamiltonian flow one might take surfaces of constant I forthe coarse graining to obtain a finite dimensional Hilbert space representationof the system.
Generically there are few fundamental guiding principles on how to choosethe form of the unit cells of volume hf and how to distribute them in thephase space. Should they all have the same shape? The same number ofneighboring cells? These questions usually can only be answered by ourphysical intuition and knowledge about the classical system.
An exception is the case of integrable systems. In this case the set ofaction variables I1, . . . If yields a natural foliation of the phase space bythe quantization condition In = ~n, where n ∈ N. A unit block is spannedby the f unit vectors ~en in the directions dIn. This is the Bohr–Sommerfeldquantization. However, as already observed by Einstein it works only forintegrable systems.
An unsatisfactory feature of the Bohr–Sommerfeld quantization is its de-pendence on the Hamiltonian. The prescription how phase space is foliatedis given by the Hamiltonian, i. e. by the problem under consideration. De-pending on what dynamical system we consider, we foliate one and the samephase space in many different ways. It would be more desirable to have aquantisation procedure which only depends on the geometry of the phasespace but not on any additional input.
The canonical quantization avoids this problems. The phase space is –independently of the Hamiltonian under consideration – divided in blocks de-fined by the infinitesimal volume element
∏fi=1 dqi ∧ dpi of volume h, where
qi, pi, 1 ≤ i ≤ f are the particle’s canonical position and momentum. Thismethod, however has other drawbacks. In more complicated Hamilton func-tions there might appear non–Hermitean products of non–commuting oper-ators like piqi, which leads to an ordering problem in the quantum Hamil-tonian. The canonical quantization works without modifications only in thecase that the phase space is topologically equivalent to R2f .
Any quantization scheme is flawed by the fundamental problem that ittakes the road into the wrong direction. A first principle approach should de-rive classical mechanics from quantum mechanics and make any quantizationscheme spurious.
39
3.2 Quantum mechanics
Lets us briefly recall the principles of quantum mechanics
1. A quantum mechanical system is determined by a Hilbert space Hof dimension N which is spanned by a complete set of state vectors|ψ〉 ∈ H. The completeness condition and orthogonality relation hold
〈ϕi|ϕj〉 = δij (3.1)
N∑i=1
|ϕi〉〈ϕi| = 1 . (3.2)
2. The time evolution of a state is governed by a time evolution operatorU(t) = exp(− i Ht/~).
3. Physical observables are self-adjoint operators A = A†. Only expecta-tion values 〈ψ|A|ψ〉 are amenable to measurement.
The spectral decomposition follows as a consequence of the first point above
H =∑
Ei|ϕi〉〈ϕi|
U(t) =∑
e− iEjt/~|ϕj〉〈ϕj| . (3.3)
Heisenberg uncertainty principle follows from the non–commutativity of twooperators A and B obeying the Heisenberg–algebra [A, B] = i~. Define thevariance of an operator
∆A = 〈ψ|(A− 〈A〉)2|ψ〉 . (3.4)
Then Heisenbergs uncertainty relation holds
∆A∆B ≥ ~2
4. (3.5)
States for which the equality ∆A∆B = ~2/4 holds are called coherent states.From the second point Heisenberg equations of motion follow for an arbitraryHeisenberg operator
d
dtA(t) =
1
i ~[A, H] . (3.6)
In particular Heisenberg operators fullfil the classical equations of motion.
40
In theory a quantum mechanical system is fully specified by the spectrumof the Hamilton operator and its eigenfunctions. In practice a quantum me-chanical system is complemented by a set of operators, for which we knowtheir classical limit, since they can be measured. This point is most impor-tant. Only with the knowledge of some operators, which we can in principlemeasure and by a measurement of this operator, respectively its expectationvalues, a quantum system becomes meaningful for us.
In many cases we are interested in a set of operators qi, pi, which fulfillthe Heisenberg–algebra. The set of expectation values
〈ψ|qi(t)|ψ〉, 〈ψ|pi(t)|ψ〉 → ~z(t) (3.7)
might then be interpreted as Darboux–coordinates of classical phases spacein the classical limit ~→ 0.
If we leave aside this additional input, a quantum mechanical system isnothing but a set of eigenvalues and eigenvectors and the time evolution ofany state vector is determined uniquely by this.
Two operators with the same spectrum
Spec(A) = Spec(B) (3.8)
but different eigenfunctions are called unitary equivalent. There are manyexamples of systems having the same spectrum but describing rather differentphysical situations. However it is natural to assume that unitary equivalentHamilton operators generate in the classical limit dynamical systems withsimilar global properties.
Therefore the spectrum of a Hamilton operator plays the most fundamen-tal role. It is the skeleton of the quantum system, flesh and blood is addedby the eigenfunctions and an appropriate set of operators. We then can askthe fundamental question: Can we detect already from the spectrum of aHamilton operator, whether it generates a dynamical system with chaoticbehavior or not in the classical limit?
3.3 Spectral Statistics
The basis of spectral statistics is an ergodicity argument. An unbound quan-tum mechanical spectrum is a large set N → ∞ of real numbers. Let RM
be a spectral quantity which is a function of a sequence of M eigenvaluestaken out of this infinite series. Here M might be very large, but since thespectrum is infinite N M it contains infinite many sequences of this sort.Denote them RMn, n = 1, . . . N/M . Each sequence can be considered a rep-resentative of an ensemble of spectra of length M . Thus the function RN
41
which depends on the whole spectrum is equivalent to an average over anensemble over smaller spectra.
limN→∞
RN(E1, . . . , EN) = limN→∞
N/M∑i=1
MRMi
N, (3.9)
Spectral quantities R for which equation (??) holds are called self–averaging.
3.3.1 Unfolding
To compare different quantum mechanical spectra we must compare them onthe same energy scale. Every quantum mechanical Hamilton operator has atypical energy scale, which is determined by the parameters of the operator.This energy scale corresponds to a length scale. The length scale gives anestimate of the extension of the bound states of the system. We give twoexamples:
1. In Gaussian units the Hydrogen atom is defined by the Hamilton oper-ator
HHydrogen =1
2me
p2 − e2
r. (3.10)
With the three parameters me, e and ~ a typical length scale can beconstructed, which is the celebrated Bohr’s atom radius a0
~2
mee2= a0 ≈ 5, 3 · 10−11m . (3.11)
With this a typical energy scale can be constructed as well
E(a0) =1
2mee
~2
a20
− e2
a0
=1
2
e4me
h2≈ 13, 4eV , (3.12)
which is the ionisation energy of a hydrogen atom in the ground state.
2. As a second example we consider a rectangular billard. Its Hamiltonoperator has as parameters the particle’s mass m and the length L andwidth B of the billiard. If the billiard is approximately quadratic, i. e.L and B are of the same order of magnitude, we can construct a typicalenergy scale of the billiard by
1
2m
~2
A, (3.13)
42
where A = BL is the area of the billiard. If we want for some reason comparethe spectra of the hydrogen atom and the rectangular billiard, we should dothis with the rescaled energies En/Etypical.
If we want to compare spectra of different quantum mechanical systems,their different energy scales are not the only problem we have to face. Evenwithin one and the same spectrum the number of energy levels per energyinterval many vary wildly. In fact in most systems with an unbound spectrumthe density of states increases with increasing energy.
We define the spectral density
%(E) =∞∑n=0
δ(E − En) (3.14)
and the integrated spectral density (aka staircase function)
N(E) =
E∫−∞
%(E ′)dE ′ =N∑n=0
θ(E − En) . (3.15)
From figure ?? it is seen that N(E) contains a smooth part and a fluctuatingpart
N(E) = N(E) +Nfl(E) , (3.16)
where N(E) is a smooth curve and Nfl(E) fluctuates around zero. The samedecomposition into a smooth and a fluctuating part is valid for the spectraldensity
ρ(E) = ρ(E) + ρfl(E) , %(E) =dN(E)
dE. (3.17)
If we want to calculate an expectation value of a spectral quantity, say f(E)
f(E) =N∑n=0
f(En) =
Emax∫Emin
f(E)%(E)dE ≈Emax∫Emin
f(E)%(E)dE (3.18)
we face the problem that %(E) might vary from intervall to intervall and fromsystem to system. One says %(E) is not universal. We want to suppress thisnon–universal part and have a look at spectral quantities where the meanspectral density is normalized to unity. We define the new variable
ξ = N(E) ⇒ dξ = %(E)dE . (3.19)
43
Then it follows for the averaged quantity f
f(E) ≈ξmax∫ξmin
f(N−1(ξ))dξ , (3.20)
and the influence of mean level density % has been eliminated. This procedureis called unfolding.
A problem arises in the calculation of the function N which is neededaccording to equation (??) for the unfolding procedure. Usually it is hard tobe determined from data of the complete spectrum.
44
0
2
4
6
8
10N
(E)
0.0 0.2 0.4 0.6 0.8 1.0E
-0.5
0.0
0.5
Nfl(E
)
0.0 0.2 0.4 0.6 0.8 1.0E
-0.5
0.0
0.5
Nfl(ξ
)
0 2 4 6 8 10ξ
Figure 3.2: Staircase function for the hydrogen atom (Coulomb potential)and the smooth integrated density of states.
Exercise 9: In figure ?? the staircase function of the hydrogen atom (Coulombpotential) is depicted. It is well known that the spectrum is given in this caseby En = −Ry 1
n2 . Calculate the smooth (integrated) density of states %(E).
In practice one proceeds as follows: One takes from the staircase functionof a spectrum an interval, which is large enough to contain sufficiently manylevels to obtain good statistics. On the other hand the interval is chosensmall enough that in the interval N(E) is well approximated by a straight
45
N(E)
E
Figure 3.3: Practicalway of unfolding: Notthe whole spectrum isunfolded but a regionwhere N(E) is approxi-mated well by a straightline.
line
N(E) ≈ E
D+ const . . (3.21)
Then it follows that in this interval the smooth density of states is constant% = D−1. Inverting N becomes trivial. We define as unfolded energy ξ = E
D
and obtain
f(E) ≈ξmax∫ξmin
f(Dξ)dξ (3.22)
D is called mean level spacing. After the unfolding one looks at spectralproperties on the scale of the mean level spacing. In figure ?? the spectra ofsome rather distinct systems taken from physics or other are compared onthe scale of the mean level spacing.
46
Figure 3.4: Examples for series of energy levels and of number series onthe unfolded scale. Observe that the number of sticks equals hundred in allexamples. From left to right the spectra are: a periodic array of evenly spacedlines; a random sequence; a periodic array perturbed by a slight random“jiggling” of each level; energy states of the erbium-166 nucleus, all havingthe same spin and parity quantum numbers; the central 100 eigenvalues of a300-by-300 random symmetric matrix; positions of zeros of the Riemann zetafunction lying just above the 1022nd zero; 100 consecutive prime numbersbeginning with 103,613; locations of the 100 northernmost overpasses andunderpasses along Interstate 85; positions of crossties on a railroad siding;locations of growth rings from 1884 through 1983 in a fir tree on Mount SaintHelens, Washington; dates of California earthquakes with a magnitude of 5.0or greater, 1969 to 2001; lengths of 100 consecutive bike rides. Taken from ?
As mentioned already, usually the smooth density of states can not bedetermined exactly over the whole spectrum. An exception are billiards. If Ais the area of the billiard, their typical energy scale is ~2
mA. This is the inverse
of the number of unit blocks with volume h2 fitting in the energy shell toenergy E of the phase space. It is the leading term in a power expansion ofthe spectral density in powers of E
N(E) =1
2π
(mA
~2E −
√m
2~2l√E + . . .
). (3.23)
The next leading term is governed by the circumference length of the billiardand the higher order terms contain more and more details about the shape
47
A
L
Figure 3.5: Billiard withsurface area A and cir-cumference length L.
0.0
0.2
0.4
0.6
0.8
1.0
P(s
)
0.0 0.5 1.0 1.5 2.0 2.5 3.0s
Figure 3.6: Wignerdistribution (fullline) versus Poissondistribution (dottedline).
of the billiard. Whether or not this series is unique is an old question goingback to the nineteenth century: can we hear the shape of a drum? Onlyrather recently examples were found of inequivalent billiards, which indeedhave the same expansion (??), which is called Weyl expansion. The first termof this expansion is often referred to as Weyl’s law.
3.3.2 Spectral quantities
Usually two different types of spectral quantities are distinguished. Shortrange spectral quantities probe the behavior of the nearest levels close to agiven eigenvalue. Long range spectral quantities probe the behavior of thespectrum in a larger interval. Both type of quantities, even the long rangeones, are local and do not probe global changes in the spectral density.
Nearest neighbor spacing distribution (NND)
The nearest neighbor spacing distribution p(s) is a short range spectral quan-tity. It gives the probability to find in a distance s (on the unfolded scale)of a given eigenvalue its neighboring eigenvalue. In general the distribution
48
has the following properties
∞∫0
p(s) ds = 1 ,
∞∫0
s p(s) ds = 1 (3.24)
The first condition is just the normalisation together with positive definite-ness, whereas the second condition is a consequence of the unfolding. Bydefinition unfolding requires that the mean distance between neighboringeigenvalues is one. In many complex systems one can observe that theNND-distribution vanishes for small distances. The “Wigner surmise” forthe NND-distribution describes this behavior with the following function
p(s) =π
2se−
π4s2 . (3.25)
Whenever a spectrum of a system has a NND-distribution, which is well ap-proximated by the Wigner distribution, one says the spectrum has “Wigner-Dyson” statistics.
On the other hand, if the energy levels are uncorrelated one expects aPoisson distribution
p(s) = e−s , (3.26)
i. e. a high probability of almost degenerate eigenvalues.
Exercise 10: To see that the Poisson distribution describes uncorrelatedeigenvalues, we consider the interval [0, L] and distribute randomly N levelsin the interval, i. e. with the distribution function
p(x) =
1L
0 ≤ x ≤ L0 otherwise.
(3.27)
These levels are obviously uncorrelated. The nearest neighbor spacing distri-bution between two neighboring levels, say x1 and x2 is given by the N−2-foldintegral
p(s) =1
LN−2
∫out
dx3 . . .
∫out
dxN . (3.28)
where the integration domain are the two intervals [0, x1] ∪ [x1 + s, L]. Cal-culate the integral and show that in the limit N → ∞ and L → ∞ andL/N = D finite
p(s) = exp(−s/D)
is obtained.
49
In practice the NND is obtained by a procedure as depicted in figure ??.
12345
1 4 23
6 5
12345
1 4 23
6 5s00 . . .0 s
s
1 0 1 2 . . .
Figure 3.7: Procedure of calculating numerically a NND: 1. A spectrum isobtained either numerically or from experiment. 2. A part of the spectrumis unfolded. 3. All energy differences of the unfolded part are measured. 4.The s–axis is splitted in small intervals of the same length(called bins). Toeach bin the number zero is assigned. 5. Whenever an energy–difference fallsinto a bin, the number of the bin increases by one. 6. After iterating theprocedure sufficiently often a histogramm is obtained.
50
Spectral Rigidity ∆3
Spectral rigidity is a long range observable which is constructed as follows.We look at the staircase function in an interval of length L on the unfoldedscale. Here L is large and can contain many levels but since the spectrum isinfinite we can split it into many intervals of length L. We label the intervalby an index m. In each interval the staircase function can be fitted by astraight line.
νm(ξ) ≈ Aξ +B (3.29)
The quantity
∆3m(L) =1
LminA,B
L∫0
(νm(ξ)− Aξ −B)2 dξ (3.30)
is a measure for the quality of the fit. The longer L the better should stayA = 1 due to the unfolding. Finally ∆3(L) is obtained by averaging overmany intervals
∆3(L) =1
M
M∑m=1
∆3m(L) . (3.31)
This average is replaced by an ensemble average in random matrix theory. Itis clear from the definition (??) that in general ∆3(L) will increase with thelength of the interval. But the way it increases is not so clear. The fasterthe spectral rigidity grows the lesser the energy states are correlated. It canbe shown that for an uncorrelated spectrum the spectral rigidity behaves inleading order in L as
∆3(L) ∼ L
15. (3.32)
Any behavior which differs from the linear dependence on the interval lengthis a signature of level correlations.
Energy-energy correlations
We next define the two–point correlation function
R2(E,E ′) =∑n,m
δ(E − En)δ(E ′ − Em) . (3.33)
51
Aξ +B
ν(ξ)
ξL
Figure 3.8: Construc-tion of spectral rigidityfrom an unfolded stair-case function.
Let R2(E,E ′) be the averaged quantity. The average is taken similarly as forthe NND. We take a large sequence of eigenvalues on the unfolded scale picka value for E and split the interval [E,∞] into a bins of a certain length andmeasure the distance between E and the eigenvalues of the spectrum. Weiterate this procedure for many different values of E until a smooth curve isobtained. This is allowed since on the unfolded scale R2 only depends on thedifference between both eigenvalues and not on the position of E alone. Wedefine
r =E − E ′
D. (3.34)
Then the averaged two–point correlation function can be written as
D2R2(E,E ′) −→ X2(r) = δ(r) + 1− Y2(r) . (3.35)
The delta–type contribution account for the singular case E = E ′ in thedefinition of R2(E,E ′). The connected part is expected to decay to zero forlarge energy separations. Usually higher energy correlations
Rk(E1, . . . , Ek) =∑n1...nk
δ(E1 − En1) . . . δ(Ek − Enk) (3.36)
respectively their averages cannot be measured in an experiment, but theycan be calculated in random matrix theory. The results predicted by randommatrix theory are plotted in figure ??. We see oscillatory behavior for thethree random matrix ensembles, which will be defiend in the next section.This is typical for correlated eigenvalues. The energy–energy correlationfunction of uncorrelated eigenvalues corresponds to a constant.
52
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.41−Y
2
0 1 2 3 4 5r = (E1 − E2)/D
GOE (β = 1)GUE (β = 2)GSE (β = 4)
Figure 3.9: Energy–energy correlation functions on the unfolded scale for thethree random matrix ensembles GOE, GUE, GSE.hhh
Spectral form factor
The spectral form factor is defined as the Fourier transform of the energy–energy correlator on the unfolded scale.
K(τ) =
∫e2π i rτX2(r)dr
=
∫e2π i rτ (δ(r) + 1− Y2(r)) dr
=1 + δ(τ) + b2(τ) . (3.37)
Like the energy–energy correlator it is a sum of three parts. A Dirac typedistribution a constant and a non–trivial part, which decays to zero in a char-acteristic way for large τ taking into account the spectral correlations. TheDirac contribution accounts for the fact that in the theoretical calculationof spectral correlations the finite width of spectral lines has been neglected.In real measurements typically cross–sections are measured and the peaks inthe cross section have a finite width. Taking into account this finite widthyields a broadening of the initial delta peak, which might be considerable.
53
0.0
0.5
1.0
1.5
2.0
b 2(t
)
0.0 0.5 1.0 1.5 2.0t/tH
GOE (β = 1)GUE (β = 2)GSE (β = 4)uncorr.
Figure 3.10: Spectral form factor K(τ) for the three energy correlation func-tions plotted in figure ??. The initial delta peak is not depicted.
We notice that τ is a dimensionless time, which is related to the inversemean level spacing via τ = ~/D. This is the typical time which is needed ina measurement to resolve neighboring energy–levels. It is called Heisenbergtime.
3.4 Conjecture of Bohigas, Giannoni &Schmit
In 1984 in a paper ? Bohigas, Giannoni and Schmit first formulated theircelebrated conjecture:
Spectra of time reversal invariant systems whose classical analogs are Ksystems show the same fluctuation properties as predicted by the GOE.
Here GOE stands for Gaussian orthogonal ensemble and will be definedprecisely in the next section. They investigated NND and spectral rigidity ofthe so–called Sinai billiard, which is one of the mathematically best studiedbilliards. It is known to be fully chaotic. They calculated numerically thespectrum of the billiard for four different values of the circle radius andanalysed the spectra over streches of 740 eigenvalues. For the NND theyfound nice agreement of their statistical analysis with the Wigner surmise.For the spectral rigidity they found logarithmic behavior. It was well knownat that time that both, the Wigner–surmise for the NND and the logarithmicbehavior of the spectral rigidity are results which are reproduced by the GOE.
The results of Bohigas, Giannoni and Schmit for the Sinai billiard andmany other experimental and numerical evidence backed their conjecture,which by then was rather a heuristical observation.
54
Figure 3.12: Spectral rigidityof the Sinai billiard. Thedashed line indicates the lin-ear law (??) for uncorrelatedlevels. The full line indicatesthe logarithmic behavior, pre-dicted by the GOE. The nu-merical values for the Sinai bil-liard are perfectly fitted by theGOE, taken from ?.
A similar observation was made earlier by Wigner in the spectra of com-plicated many–body systems like heavy nuclei. A chaotic quantum systemis a system with only a few degrees of freedom. However as for its spectralstatistics it behaves like a system with many interacting degrees of freedomlike a heavy nucleus.
Figure 3.11: Left: Nearest neighbor spacing distribution of the Sinai billiard,taken from ?. Right: Trajectory in the desymmetrized Sinai billiard con-sidered by Bohigas and coworkers. The Sinai billiard is a quadratic billiardwith a circle in the center. It is desymmetrized by confining the particlestrajectory to a single octant.
55
Chapter 4
Random matrix theory
4.1 Basic Definitions
In random matrix theory the spectral properties of matrix ensembles areanalyzed. We take a Hilbert space of finite dimension dimH = N and fixa set of basis vectors |1〉, . . . |N〉. A random matrix acts on an arbitrarystate vector randomly via
H|n〉 =N∑m=1
Hmn|m〉 , Hnn ∈ R, Hnm ∈ C, for n 6= m (4.1)
where the matrix entries Hnm of H are real or complex random numberswhich are chosen appropriately. For H to mimic the Hamilton operator ofa quantum system one has to require Hermitecity for H. This restricts thenumber of independent random numbers because we must require Hnm =H∗mn. The choice of the joint probability function for the random variablesHnm is governed by three principles
1. Simplicity
2. Independence of matrix entries, apart from Hermitecity and specificother symmetries to defined below.
3. Invariance under change of basis. Since the basis vectors were chosenat will, the probability distribution should not depend on this arbitrari-ness.
We introduce a notation, well known from statistics. We say a randomvariable X is distributed N(µ, σ) if its probability distribution function is
56
given by
X : p(x) =1
σ√
2πe−
(x−µ)2
2σ2 . (4.2)
We now present three joint probability distributions which comply the threerequirements formulated above and discuss their applications in the subse-quent sections.
1. Hnn is N(0, 1) distributed and Hnm is N(0, 1/√
2) for n < m. Thisjoint probability distribution function is called Gaussian orthogonalensemble (GOE). The number of independent random numbers is thesum of N diagonal and N(N − 1)/2 off–diagonal, i. e. N(N + 1)/2.
2. Hnn is N(0, 1/√
2) distributed and Hnm = Znm + iWnm for n < m.Here Znm and Wnm are both N(0, 1/2) distributed random numbers.This joint probability density function is called Gaussian unitary en-semble(GUE). The number of independent random numbers is N2.
3. The random matrix has a block structure
H =
[A BB† AT
](4.3)
where in addition the matrix B satisfies B = −BT . A matrix whichhas this form is called quaternionic self–dual. The matrix dimensionof H is dimH = 2N , the matrix dimensions of A and B are dimA= dimB = N . The matrix entries Ann are N(0, 1/
√2)distributed,
for n < m Anm = Znm + iWnm, where Znm and Wnm are N(0, 1/2)distributed. Likewise the entries Bnm = Xnm + iYnm, where Xnm andYnm are N(0, 1/2) distributed. Altogether the number of independentrandom variables is 2N2 − N . This joint probability distribution iscalled Gaussian symplectic ensemble (GSE).
The reason why the diagonal entries and the off–diagonal entries have dif-ferent variances is obscure at the moment. But it is exactly this differentvariances which convert the above joint probability distributions into randommatrix ensembles and which distinguishes them from ordinary multivariateprobability distributions.
57
To see this we look for instance at the joint probability function of theGaussian orthogonal ensemble
pN(H)d[H] =N∏
m≤n
p(Hmn)dHmn (4.4)
= (2π)−N(N+1)
4 2N(N−1)
2︸ ︷︷ ︸C
e−12
∑Nn=1H
2nn−
∑m<nH
2mn
∏m≤n
dHmn︸ ︷︷ ︸d[H]
= C exp
(−1
2trH2
)d[H] . (4.5)
Due to the at first glance strange looking variances of the individual matrixentries the joint probability distribution of the random matrix H can bewritten as a trace.
Exercise 11: Show that the joint probability function of the Gaussian uni-tary ensemble and the joint probability function of the Gaussian symplecticensemble can be written as
pN(H)d[H] = C exp(− trH2
)d[H] .
Calculate the constants C as well.
This results in an important invariance. Since the probability densityfunction pN(H) can be written as a trace it is invariant under unitary trans-formations
pN(H) −−−−−−→H→U†HU
p(U †HU)
exp
(−1
2trH2
)−−−−−→ exp
(−1
2trH2
), (4.6)
due to the cyclic invariance of the trace. The full measure comprises theinfinitesimal volume element d[H] as well. It turns out that this piece ofthe measure is only in the case of the GUE invariant under general unitarytransformations. However, for the GOE it is still invariant under orthogo-nal transformations. The orthogonal group is subgroup of the unitary group.Likewise the infinitesimal volume element of the GSE is invariant under trans-formations of the unitary symplectic group, which is another subgroup of theunitary group. These different invariance properties gave the ensembles theirname. To understand this deeper, some background on Lie algebrae will beprovided in section ??. We first make some general remarks on quantumsymmetries.
58
4.2 Symmetry and invariances
Many systems have obvious symmetries. In this cases one can state a self-adjoint operator, which commutes with the Hamilton operator. Well knownexamples are: parity, rotation, spin and isospin (in nuclear physics). If abasis is chosen in which this operator is diagonal H is block diagonal in thisbasis. We give some examples
1. Spin 1/2-Particle without spin dependent interaction. The Hamiltoniancan be written as[
H↑H↓
] [ψ↑(x)ψ↓(x)
], [Sz, H] = 0 . (4.7)
Obviously Sz commutes with the Hamiltonian.
2. In one–dimensional systems often parity is a good quantum number.A one dimensional potential function is splitted into a symmetric andinto an antisymmetric part
H =p2
2m+ Vsym(x)︸ ︷︷ ︸Hsym
+Vasym(x) . (4.8)
The Hamiltonian can be written in block form
Hψ(x) =
[Hsym Vasym
Vasym Hsym
] [ψsym(x)ψasym(x)
]. (4.9)
When Vasym vanishes the Hamiltonian becomes block diagonal and com-mutes with the parity operator. In contrast to the spin operator in thefirst example, parity has a clear classical meaning.
3. SO(3) - Symmetry: many Hamiltonians are invariant under a rotationin physical space. Then they commute with the angular momentumoperator
[~L2, H] = 0 . (4.10)
The Hamilton operator is block diagonal in a basis to fixed angularmomentum
H =
. . .
l = 2l = 1
l = 0
=∞∑l=0
H(l) . (4.11)
59
Likewise a wavefunction can be expanded in eigenfunctions of ~L2.
ψ(~x) =∞∑l=0
l∑m=−l
Ylm(ϑ, ϕ)alm(r) (4.12)
We see that a symmetry comes with an operator, which commutes withthe Hamiltonian and thereby singles out a certain basis. In general suchsymmetries are called unitary symmetries. The underlying hypothesis ofrandom matrix theory for the description of chaotic quantum systems is thatquantum system without further unitary symmetries is described by a fullyoccupied random matrix, where no distinguished basis exists.
One must be most careful at this point with the definition of symme-try, because every Hamilton operator can in principle be diagonalized by anappropriately chosen unitary transformation. So for every Hamiltonopera-tor there exist N linearly independent operators, which commute with H,if N = dim(H). However, these operators in general do not describe classi-cal conservation quantities. A classical system can have only f independentconserved quantities. The number of the degrees of freedom f is thereforeusually much smaller than the dimension N of the Hilbert space. This indi-cates that the concept of a degree of freedom is not completely well definedin quantum mechanics.
A necessaary condition that a quantum operator L which commutes withH yields a classical conserved quantity Il, is that the dimension of the L-invariant subspace scales like N towards infinity.
Apart from unitary symmetries there exist also anti–unitary operatorswhich might or might not commute with the Hamiltonoperator. An antiuni-tary operator T transforms vectors as
|ψ〉 −−−−→T
|Tψ〉 , |φ〉 −−−−→T
|Tφ〉 (4.13)
such that
〈Tψ|Tφ〉 = 〈φ|ψ〉 . (4.14)
One can write any antiunitary operator
T = J C , (4.15)
where C is the operator of complex conjugation and J is an arbitrary unitarymatrix, for a proof see for instance ?. A vector |ψ〉 and a matrix H transform
60
as
|ψ〉 −−−−→T
J |ψ∗〉
H −−−−→T
J CHCJ −1 = JHTJ −1 = HR (4.16)
The matrix HR is called the matrix dual to H. If H = HR the matrix isselfdual under the anitunitary transformation T . From the definition (??) itfollows that
T 2 = eiϕ . (4.17)
This implies for the matrix J
J CJ C = JJ ∗CC = JJ ∗ = eiϕ . (4.18)
We write the unitarity condition for J as J TJ ∗ = 1. Plugging this intoequation (??), we obtain an equation for ϕ
J = eiϕJ T = e2·ϕJ . (4.19)
This equation has obviously two solutions for ϕ
ϕ =
0⇒ JJ ∗ = 1 ⇒ J symmetricπ ⇒ JJ ∗ = −1 ⇒ J anti–symmetric
(4.20)
In the symmetric case we must choose J = 1, since any other choice wouldviolate our principle that H has no unitary symmetries. For the same reasonwe choose as antisymmetric unitary matrix J = i I, where we recall thedefinition (??)
I =
[1
−1
](4.21)
of the symplectic metric. The requirement of self duality H!
= HR leads tothe two conditions
H =
HT ⇒ H real symmetric (GOE)
IHT I−1 ⇒ H quaternion selfdual (GSE).(4.22)
The first condition yields the Gaussian orthogonal ensemble. The secondcondition yields the structure of a quaternion selfdual matrix
H = IHT I−1 ⇒ H =
[A BB† AT
](4.23)
61
where B =−BT . The ensemble with this kind of invariance is called Gausssymplectic ensemble (GSE). Notice: Every eigenvalue from H is two timesdegenerated (Kramers degeneracy).
The anitunitary operator T is in both cases called time reversal operator.Both ensembles GOE and GSE describe systems which are invariant undertime reversal symmetry but have no further unitary symmetries. The GOEdescribes systems with integer spin. The GSE describes systems with half-integral spin and time-reversal symmetry. To see that the operator T isindeed the the time–reversal operator, we look at the Schrodinger equationof a scalar particle. Under time reversal it transforms as
i ~∂
∂t|ψ〉 = H|ψ〉 −−−−−−−−−−−→
t→−t ,|ψ〉→T |ψ〉− i ~
∂
∂tT |ψ〉 = T H|ψ〉 (4.24)
Time reversal symmetry is conserved, if the time reversed wave function T |ψ〉satisfies the same Schrodinger equation as the original one in opposite timedirection. This holds if THT−1 = H. This is exactly the condition of selfduality. Applying the first condition of equation (??) leads to T = C andH = HR. Observe that complex conjugation transforms the operators in theexpected way under time reversal
C~xC = ~x , C~pC = −~p , C ~JC = − ~J . (4.25)
We next consider the Schrodinger equation of a spin–12
particle
i ~∂
∂t
[|ψ↑〉|ψ↓〉
]= H
[|ψ↑〉|ψ↓〉
]−−−−−→
t→−t− i ~
∂
∂tT
[|ψ↑〉|ψ↓〉
]= T H
[|ψ↑〉|ψ↓〉
](4.26)
In principle again the choice T = C would do the job again. However this
choice does not transform the spin ~S in the right way. Since the spin is anangular momentum it should transform under time reversal transformationlike an angular momentum
T ~ST−1 = − ~S (4.27)
Fortunately we have a second choice for T . This is the second possibility inequation (??), namely T = i IC. This choice transforms the spin operator inthe right way.
Exercise 12: Show that the operator T = i IC transforms the spin in thecorrect way. This means, show that
T SxT−1 = −Sx , T SyT
−1 = −Sy , T SzT−1 = −Sz , (4.28)
62
If time-reversal symmetry is broken, Hermitecity is the only requirementon the matrix H
H = H† (4.29)
which leads to the Gaussian unitary ensemble.
4.3 Algebraic background
Hermitean matrices H are defined by the condition H = H†. The set ofN × N Hermitean matrices can be view upon as Euklidian vector space ofdimension N2. We call this space HHerm(N). Like any vector space it can bedecomposed in two orthogonal subspaces in several ways.
4.3.1 Real symmetric Matrices
Real symmetric matrices are Hermitean matrices which fulfill the additionalcondition H = HT . Thus the space of real symmetric N× matrices canbe view as subspace of the space of Hermitean matrices with the reducedvector space dimension N(N + 1)/2, we call this space Hreal(N). Obviouslythere exists another subspace of HHerm(N), which is orthogonal to Hreal.It has vector space dimension N(N − 1)/2 and consists of all Hermiteanskewsymmetric matrices, i. e. matrices which fulfill the condition H = −HT .We call this space HO(N), thus we can write
HHerm(N) = Hreal(N) +HO(N) (4.30)
We choose T = C. Acting with the complex conjugation operator on anelement
H = P +K , H ∈ HHerm(N) , P ∈ Hreal(N) , K ∈ HO(N) (4.31)
yields
TH = P −K . (4.32)
The elements of HO change sign under time reversal symmetry whereas Hreal
is invariant. We calculate the commutators and find
[K1, K2] = iK3, [K,P1] = iP2 , [P1, P2] = iK , (4.33)
63
where Kn ∈ HO(N) and Pn ∈ Hreal(N). This means that the vector–spaceHO(N) is closed under the commutator. The matrices obtained by the ex-ponential map
K → exp(iK) , K ∈ HO(N) (4.34)
form a group. This means that there exists an element K3 ∈ HO(N) suchthat
exp(iK1) exp(iK2) = exp(iK3) , ∀ K1,2 ∈ HO(N) . (4.35)
Exercise 13: Show that equation (??) holds if a vector space is closed undercommutation. Find an expression for K3 expressed by multiple commutatorsof K1 and K2.
The group obtained by the exponential map K → exp(iK) is the orthogo-nal group O(N). Let U be an element of O(N) then, due to the commutationrelations (??) not onlyHO but alsoHreal are invariant under the group action
U−1P1U = P2 with P1,2 ∈ Hreal(N) ∀U ∈ O(N) . (4.36)
We can therefore decompose Hreal(N) in equivalence classes under the or-thogonal group. We have P1 ' P2, if there exists an orthogonal matrix suchthat equation (??) holds. For each equivalence class we choose its diagonalrepresentative. This leads to the spectral decomposition of a real symmetricmatrix
H = UTEU U ∈ O(N) , E = diag(E1, . . . , EN) , (4.37)
which will be most useful in the following. It means Hreal(N) = RN ×O(N).
The N eigenvalues are often called the radial coordinates (aka eigenvaluecoordinates).
4.3.2 Quaternionic selfdual matrices
Likewise the 4N2 dimensional space HHerm(2N) can be split into the vectorspace of 2N ×2N quaternion self–dual matrices HHsd(2N) ' R
2N2−N and itsorthogonal complement, which we call HUsp(2N) ' R
2N2+N . It consists ofmatrices of the form
H =
[A BB† −AT
], B = BT . (4.38)
64
Now HHsd(2N) is invariant under the second time–reversal invariance oper-ation in equation (??) and HUsp(2N) changes sign. We can exponentiate
K → exp(iK) , K ∈ HUsp(2N) , (4.39)
which yields the unitary symplectic group Usp(2N). It is the group of allunitary matrices, which in addition leave invariant the symplectic metric I,or equivalently the group of all symplectic matrices which are in additionunitary Usp(2N) = U(2N) ∩ SP(2N). The vector space HHsd(2N) can bedecomposed as RN × Usp(2N) leading to the spectral decomposition
H = U †EU , U ∈ Usp(2N) , E = diag(E1, E1, . . . , EN , EN) ,(4.40)
where every eigenvalue is two–fold degenerated (Kramers degeneracy).
4.3.3 Hermitean matrices
The case of Hermitean matrices is in some aspects simpler than the other twocases, in some aspects it is more complicated. The vector space HHerm(N)of Hermitean N × N matrices is itself closed under the Lie bracket. Thematrices obtained under the exponential map form for the same reasons asbefore a group. This group is the unitary group U(N) of matrices, whichleave invariant the Euklidean metric g = 1N as U †gU = g.
It is natural to assume that the vector space of Hermitean matricesHHerm(N) permits a spectral decomposition as HHerm(N) = R
N × U(N).However this is not true. This can be seen by a simple counting of dimen-sions. The vector space dimension of HHerm(N) is N2 but the vector spacedimension of RN × U(N) is N(N + 1). Indeed in U(N) there exists an Ndimensional subspace of matrices, which leaves invariant a diagonal matrix.This subspace consists of unitary diagonal matrices
U =
eiφ1
eiφ2
. . .
eiφN
, φn ∈ R . (4.41)
It is isomorphic to the N -torus. We call it U(1)×N . We have to divide outthis so–called stabilizer subgroup to obtain the correct spectral decompositionfor H ∈ HHerm
H = U †EU , U ∈ U(N)/U(1)×N , E = diag(E1, . . . , EN) . (4.42)
65
4.3.4 Invariant metric two–form
We now investigate in more detail the infinitesimal volume element d[H] forthe three Gaussian ensembles. Invariance of the probability measure requires
d[H] =
d[UTHU ] , U ∈ O(N) H ∈ GOEd[U †HU ] , U ∈ U(N) H ∈ GUEd[U †HU ] , U ∈ Usp(2N) H ∈ GSE .
(4.43)
To see this one must show that the determinant of the Jacobi matrix isequal to one. Instead of calculating the Jacobi matrix directly we follow adifferent route, which gives us additional insight into the structure of thematrix spaces. On the vector spaces HHerm, Hreal and on Hsdual an invariantlength element is defined, which can in all three cases be written as
tr dH dH =N∑i=1
dHiidHii + 2∑i<j
dHijdHji . (4.44)
For the GSE this expression is usally rescaled by a factor one half. From thiswe can read off the metric tensor g.
g =
1. . .
12
. . .
2
N βN(N−1)2
(4.45)
The metric is Euclidean but some directions have a different length scale asothers. The number of off–diagonal elements scales for all three ensembleswith N2 but with a different factor β = 1 for the GOE, β = 2 for the GUEand β = 4 for the GSE. In the following U is an element of either of thegroups O(N), U(N), Usp(N). Using the spectral decomposition of H we cantransform the invariant length element into eigenvalue - angle - variables
tr dH dH −−−−−→H=U†EU
tr d(U †EU)d(U †EU)
= tr((dU †)EU + U †dEU + U †EdU
)2
= tr ([dA,E]− dE)2 (4.46)
where we defined dA = UdU †. Recalling that U was obtained by the exponen-tial map (??) of one of the three spaces Hreal(2N), HHerm(2N) or HHsd(2N).
66
Exercise 14: Show that dA =−dA† also follows from UU † = 1 .
Since a commutator is always antisymmetric equation (??) simplifies to
tr dH dH −→ tr[dA,E]2 + tr dE2 (4.47)
We see that the N directions of eigenvalues dE and the βN(N−1)2
directionsof angles dA are orthogonal. To proceed further we have to write the matrixdA in a convenient orthogonal basis. For U ∈ O(N) we choose the basismatrices as follows
(ekl)mn =1√2
(δkmδln − δknδlm) , k < l , (4.48)
such that tr eijejk = −δilδjk and the basis matrices are orthogonal with re-spect to the trace as scalar product.
Exercise 15: Write down the matrix ekl. Find the appropriate basis matri-ces also for U ∈ U(N) and U ∈ Usp(2N).
Now the commutator [dA,E] can be calculated using the expansion
dA =∑ij
daijeij (4.49)
and the commutation relation [eij, E] = (Ej − Ei)eij we obtain immediately
tr[dA,E]2 = −(Ej − Ei)2daijdaij , (4.50)
for U ∈ U(N) and U ∈ Usp(2N) similar expressions are obtained. The metricis now given by
g =
1. . .
1. . .
−(Ei − Ej)2
. . .
N βN(N−1)2
.
(4.51)
It is still diagonal. In the lower right corner now squares of the N(N − 1)/2differences of eigenvalues appear. Every difference appears a number of timeswhich differs for every ensemble. This number is given by β defined earlier.
67
Now the determinant of g is easily calculated and the infinitesimal volumeelement transforms as
d[H] −→√
det(g)dµ(U)d[E]
= Cβ∏i<j
|Ei − Ej|βN∏i=1
dE dµ(U) , (4.52)
where dµ(U) is the Haar measure of the group under consideration, which ingeneral is complicated, but fulfills the following important invariance prop-erty. Let U be any arbitrary element of the group, then the Haar measuretransforms under multiplication with U as
dµ(U) −→ dµ(UU) = dµ(U) . (4.53)
It remains invariant. From the angle eigenvalue coordinates and the invariantof the group measure we obtain equation (??) for all three ensembles.
4.3.5 Universality classes
In angle eigenvalue variables it is easy to identify the differences between thethree universality classes.
GOE: d[H]→∏i<j
|Ei − Ej| d[E] dµ1(U) , U ∈ O(N) (4.54)
GUE: d[H]→∏i<j
|Ei − Ej|2 d[E] dµ2(U) , U ∈ U(N) (4.55)
GSE: d[H]→∏i<j
|Ei − Ej|4 d[E] dµ4(U) , U ∈ USp(2N) (4.56)
The main difference between the three symmetry classes is the exponentβ, which gives the power law with which the joint probability distributionvanishes, if two eigenvalues come close
∏i<j |Ei − Ej|β, it is called Dyson
index. With the help of the Dyson index one can write the joint probabilitydistribution for all three symmetry classes as
pN(H)d[H]→ p(β)N (E)d[E]dµβ(U)
p(β)N (E) = Cβ
∏i<j
|Ei − Ej|βe−β2
∑E2i . (4.57)
The normalisation is chosen such that p(β)N (E) itself is a probability density
function, i. e. normalised to one. With this equation it is easy to continue
68
the Dyson index analytically on R
β ∈ R+ . (4.58)
This was done in the past. A connection to fully occupied random matrices isonly given for β = 1, 2, 4. There exist more universality classes than the threediscussed here. In total there exist ten symmetry classes and ten randommatrix ensembles according to Cartan’s classification of the symmetric spacesderived from semisimple Lie algebras. They were discussed by Zirnbauer andAltland in ?.
4.4 Spectral quantities
The calculation of ensemble averages of spectral quantities is a notoriouslydifficult task. A a rule of thumb one can say that the calculations for theGUE are almost always easier as for the other two cases. In particular thecalculation of the nearest neighbor spacing distribution is highly complicatedand can not be covered here. We therefore focus on the calculation spectralcorrelations and of the spectral form factor, which can be measured by ex-periments and which is amenable to the semiclassical approach. For reasonsof simplicity we concentrate on the GUE case.
4.4.1 Energy correlation function of the GUE
The k–point energy correlation function is defined as
〈Rk(x1, . . . , xk)〉 =
∫d[H]pN(H)∑
n1...nk
δ(x1 − En1(H)) . . . δ(xk − Enk(H)) (4.59)
The spectral quantity Rk(x1, . . . , xk) only depends on the eigenvalues.Therefore it is most conveniently analyzed it in angle–eigenvalue coordinates.The integral over the unitary group becomes trivial and we can write
〈Rk(x1, . . . , xk)〉 = C2
∫d[E]
∑n1...nn
δ(x1 − En1) . . . δ(xk − Enk) (4.60)
∏n<m
(En − Em)2 exp(−N∑n=1
E2n) (4.61)
69
where C2 is a normalisation constant. From its definition we can see that〈Rk(x1, . . . , xk)〉 contains δ–like contributions for xi = xj, 1 ≤ i, j ≤ k. Toavoid these trivial contributions we require for the sequel that all argumentsof Rk are not equal
x1 6= x2 · · · 6= xk (4.62)
We choose n1 = 1, n2 = 2, . . . , nk = k to get rid of the sums from n1 to nk.A prefactor will keep track of the multiplicity
〈Rk(E1, . . . , Ek)〉 =N !
(N − k)!C2
∫dEk+1 . . . dEN
∏(En − Em)2e−
∑E2n .
(4.63)
Now the problem is to perform the integration over the remaining N − k− 1eigenvalues. We use the following important identity
∏n<m
(En − Em) = det
EN−11 EN−2
1 . . . 1...
EN−1N EN−2
N . . . 1
︸ ︷︷ ︸
Vandermonde determinant
. (4.64)
The determinant on the right hand side is called Vandermonde determinant.It is a crucial quantity in random matrix theory. Due to the properties ofdeterminants we can blow up the right hand side as
∏n<m
(En − Em) ∝ det
HN−1(E1) HN−2(E1) . . . 1...
HN−1(EN) HN−2(EN) . . . 1
, (4.65)
where Hn(x) is the Hermite-Polynom of degree n. Hermite polynomials arewell known from the quantum theory of the harmonic oscillator. We recallthe orthogonality relation of Hermite polynomials∫
e−x2
Hn(x)Hm(x)dx = δnm√
2πn!2n (4.66)
and the fact that it solves a second order differential equation
d2
dx2Hn(x)− 2xHn(x) + 2nHn(x) = 0 (4.67)
Now the integrand can be expressed through oscillator wave functions
ϕn(x) =1√√π2nn!
e−x2
2 Hn(x) . (4.68)
70
They are the solutions of the stationary Schrodinger equation of the one–dimensional harmonic oscillator to eigenvalue n+ 1/2. Now we can write
〈Rk(E1, . . . , Ek)〉 ∝∫dEk+1 . . . dEN det
ϕN−1(E1) . . . ϕ0(E1)...
...ϕN−1(EN) . . . ϕ0(En)
2
(4.69)
∝∫dEk+1 . . . dEN det[KN(En, Em)]n,m=1...N . (4.70)
The quantity
KN(x, y) =N−1∑n=0
ϕn(x)ϕn(y) (4.71)
is called kernel. We notice that the oscillator wave functions appear in thekernel. This results from the fact that hermite polynomials are orthogonalwith respect to the Gaußian measure. The method presented here, however,is more general. It will work for any kind of probability measure of the form
C2
∏i<j
|Ei − Ej|2N∏i=1
w(Ei)dEi , (4.72)
where w(x) can be an arbitrary probability density. Multivariate probabil-ity distributions The normal distribution w(x) ∝ exp(−x2) is the only onewhich is simultaneously unitary invariant. The remaining integrals can becomputed with the help of a theorem from Dyson.
Theorem (Dyson, 1970): Let K(x, y) be a function of two real variables,which satisfies the following conditions
1. K(x, y) = K(y, x) ,
2.∞∫−∞
K(x, y) K(y, z) dy = K(x, z)
3.∞∫−∞
K(x, x) dx = c
Then the following integral identity holds∞∫
−∞
det[K(xn, xm)]n,m=1...NdxN = (c−N + 1) det[K(xn, xm)]n,m=1,...,N−1
(4.73)
71
Exercise 16: Prove Dyson’s theorem. The idea of the proof is: Laplaceexpansion of the determinant and use of property 2 of the function K(x, y).
Using Dyson’s theorem we can immediately write down the k–point func-tion
〈Rk(x1, . . . , xk)〉 = det[KN(xn, xm)]n,m=1...k . (4.74)
The most important k–point functions are the one–point function, i. e. thespectral density, and the two–point function. For the spectral density weobtain
〈%(x)〉 = KN(x, x) =N−1∑n=0
ϕn(x)ϕn(x) (4.75)
In figure ?? the spectral density KN(x, x) is plotted for various values ofthe matrix dimension N . We see that the number of wiggles of the functionKN(x, x) equals the matrix dimension N .
0.0
0.5
1.0
1.5
2.0
2.5
%(x
)
-10 -5 0 5 10x
N = 30N = 10N = 3
Figure 4.1: Spectral density for three different matrix dimensions N =3, 10, 30. We see that for increasing matrix dimension %(x) approaches asmooth curve of the form of a semicircle
For N →∞ the spectral density approaches a smooth function which isnot constant, but a semicircle, called Wigner semicircle
〈%(x)〉 −−−→N→∞
1
π
√2N − x2 . (4.76)
72
This not really a physical result. As stated above, it is common for real sys-tems that their energy density is a strictly monotonically increasing function.However, there exist some important model systems, e. g. lattice models,with compact spectrum (i. e. with a maximum energy). Those models oftenhave a semicircle–like energy density. The most common one is known asHubbard model.
Anyway, we are not really interested in the spectral density because we donot expect universal behavior. We are rather interested in higher correlationson the unfolded scale. That means with constant %. In principle it is possibleto use every position to unfold the spectrum but it is easiest to use the centerof the semicircle (see figure ??) .
%(x = 0) =√
2Nπ
= 1D
√2N−
√2N
Figure 4.2: Unfolding in the center of the semicircle
In order to calculate the higher energy correlation functions on the un-folded scale, we have to find the asymptotics of the kernel KN(x, y) in thelimit N →∞ and (x− y)→ 0 keeping
x− yD
=
√2N(x− y)
π≡ r (4.77)
finite. For the calculation of DKN(x, y) in this limit we need two resultsfrom the theory of orthogonal polynoms. The first is the Christoffel-Darbouxformula :
N−1∑n=0
ϕn(x)ϕn(y) =
√N
2
ϕN(x)ϕN−1(y)− ϕN(x)ϕN−1(x)
x− y(4.78)
It relates the sum of products of oscillator wave functions to an expressionwhere only the oscillator wave functions of highest and of second highestorder appear. The second result concerns the asymptotics of oscillator wavefunctions. Define x = Dξ, then oscillator wave functions have the following
73
asymptotic large N–limit
limN→∞x→0
√π2
2DϕN(x) =
cos(πξ) N even
sin(πξ) N uneven(4.79)
Using the Christoffel–Darboux formula and equation (??) we obtain
DKN(x, y) −−−→N→∞
D
√N
2
2D
π2
sin(πξ) cos(πη)− sin(πη) cos(πξ)
D(ξ − η). (4.80)
From this we get the remarkably simple result
DKN(x, y) −−−−−−−−→N→∞ ,r finite
sin πr
πr. (4.81)
Using this result we can immediately write down an arbitrary k–point corre-lation function on the unfolded scale. For example the two–point correlationfunction
〈R2(x1, x2)〉 = KN(x1, x1)KN(x2, x2)− (KN(x1, x2))2 (4.82)
becomes
D2〈R2(x1, x2)〉 −−−−−−−−→N→∞ ,r finite
X2(r) = 1−(
sin πr
πr
)2
. (4.83)
The function X2(r) is plotted in figure ??. As mentioned before, to obtainthe same analytical result for the GOE and for the GSE is much harder.
Exercise 17: Write down the three–point correlation function on the un-folded scale.
4.4.2 Spectral form factor
In order to obtain the full spectral from factor K(τ) it is sufficient to Fouriertransform the connected part of the energy–energy correlator on the unfoldedscale. It is given by
b2(τ) =
∞∫−∞
dr e2π i rτY2(r) =
∞∫−∞
dr e2π i rτ
(sin πr
πr
)2
(4.84)
74
In case of the GUE. To get rid of the denominator we calculate the secondderivative
d2
dτ 2b2(t) =
1
2
∞∫−∞
−4π2r2
π2r2e2π i rτ (1− cos(2πr))dr
= − 1
π
∞∫−∞
ei rτ (1− cos(r))dr
= −2δ(τ) + (δ(τ + 1) + δ(τ − 1)) . (4.85)
This expression can be integrated again
d
dτb2(τ) = − sgn(τ) +
1
2sgn(τ + 1) +
1
2sgn(τ − 1)
b2(τ) = −|τ |+ 1
2|τ + 1|+ 1
2|τ − 1| , (4.86)
which is the final result for the GUE. The calculations for the GOE and forthe GSE are more complex and we only present the result here
b2(τ) =
1− 2τ + τ ln(1 + 2τ) τ ≤ 1
τ ln∣∣2τ+1
2τ−1
∣∣− 1 τ > 1GOE
b2(τ) =
1− τ τ ≤ 1
0 τ > 1GUE
b2(τ) =
1− τ2
+ τ ln2τ + 1
2τ − 1τ ≤ 2
0 τ > 2 .GSE (4.87)
We finally present the first terms in a small τ < 1 expansion of the spectralform factor K(τ)
K(τ) =
2τ − 2τ 2 + 2τ 3 + . . . GOE
τ GUEτ2
+ τ2
4+ τ3
8+ . . . GSE .
(4.88)
This series expansion of K(τ) will be recovered in a semiclassical analysis.
4.5 Relation to Fermionic systems
There exists a close connection between the behavior of the eigenvaluesof gaussian random matrices and one dimensional particles which interact
75
among each other via a certain interaction potential. This connection hasbeen exploited extensively in the past. We focus on GUE, which is thesimplest one. We show that the eigenvalues of the GUE behave like non–interacting one–dimensional Fermions. In particular the energy–energy corre-lation function X2(r) coincides with the density–density correlation functionof a non–interacting fermion gas at zero temperature.
4.5.1 Free Fermions
The Hamilton operator of one–dimensional Fermions is given in second quan-tisation by
H =∑k
a†a =
L∫0
dx ψ†(x)ψ(x) (4.89)
where ak and a†k are fermionic annihilation and creation operators
ak, a†k′ = δkk′ . (4.90)
Define the N−particle ground state |0〉. Then the static density–densitycorrelation function is given by
〈0|%(x)%(0)|0〉 = 〈0|ψ†(x)ψ(x)ψ†(0)ψ(0)|0〉 (4.91)
=1
L2
∑k,k′,k′′,k′′′
e− i(k−k′)x〈0|a†kak′a†k′′ ak′′′ |0〉 (4.92)
The expectation value yields only a contribution if a creation and an anni-hilation operator have the same index. There are two contributions to thedensity–density correlator according to the two contractions
a†kak′a†k′′ ak′′′ , a†kak′a
†k′′ ak′′′ (4.93)
The density–density correlator becomes
〈0|%(x)%(0)|0〉 =1
L2〈0|N2|0〉+
1
L2
∑kk′
e− i(k−k′)x〈0|a†kak′ a†k′ ak|0〉 (4.94)
where N is the particle number operator. The action of the creation andannihilation operators on the ground state is depicted in figure ??.
76
−kF kFak a†k
Fermi sea
Figure 4.3: Action of the creation and annihilation operators onto the Fermisea.
The action of the creation operator a†k yields only a non–zero result if|k| is larger than the Fermi momentum kF , the action of the annihilationoperator ak is only non–zero if |k| is smaller than the Fermi momentum. Wefind
〈0|%(x)%(0)|0〉 = %2 +1
L2
∑kk′
e− i(k−k′)xθ(kF − |k|)θ(|k′| − kF ) (4.95)
where % = N/L is the particle density. In the thermodynamical limit N →∞, L → ∞ and % constant the sum is converted into an integral 1
L
∑k →∫
dk2π
and for the density–density correlstion function one obtains
〈0|%(x)%(0)|0〉 = %2 +
kF∫−kF
dk
2πe− i kx
∞∫kF
dk′
2π
(ei k′x + e− i k′x
)(4.96)
= %2 −(
sin kFx
πx
)2
(4.97)
Using the relation kF = π% between Fermi momentum and particle densitythis leads to
〈0|%(x)%(0)|0〉 = %2
(1−
(sin π%x
π%x
)2)
, (4.98)
which is the same result as found for the energy–energy correlator of theGUE.
Exercise 18: Prove the relation kF = π% between Fermi momentum andparticle density. Does this relation also hold for interacting systems?
77
4.5.2 Interacting Fermions
A relation to Fermionic systems can be found for the GOE and for the GSE aswell. We look at the joint probability density in angle–eigenvalue coordinatesas given in equation (??). We can take the square root of this expressionand interpret it as a Bosonic or Fermionic wave function
p(β)N (x) = Cβ
∏i<j
|xi − xj|βe−β2
∑x2i
=|ψ(β)N (x)|2 . (4.99)
The wavefunction
ψ(β)N (x) =
√Cβ∏
i<j |xi − xj|β/2e−β4
∑x2i Bosonic√
Cβ∏
i<j(xi − xj)|xi − xj|β/2−1e−β4
∑x2i Fermionic
(4.100)
is nowhere negative. It is therefore the ground state wavefunction of a one–dimensional many–body Hamiltonoperator. It is given by
H(β) =− 1
2
N∑n=1
∂2
∂x2n
+ω2
0
2
N∑n=1
x2n +
∑n<m
β2
(β2− 1)
(xn − xm)2, (4.101)
for ω0 = β/2. The ground state energy is given by
E0 =ω0
2(N + β(N − 1)N) . (4.102)
The Hamiltonian (??) is called Sutherland–Hamiltonian. It is one of the fewinstances of a many–body Hamiltonian, where the ground state is known ex-actly. It describes a one–dimensional many body system of particles (Fermionsor Bosons), which move in an oscillator potential and interact via an inter-action potential ∝ 1/(xi − xj)2.
The Operator (??) is defined for any value of the Dyson index β. But fromequation (??) we can see that only positive values are allowed. For negativeβ the ground state energy becomes −∞ in the thermodynamical limit. Thewave function collapses. nevertheless for 0 < β < 2 the interaction of theparticle is attractive, with the maximal possible value at β = 1. For β = 2and β = 0 the interaction vanishes. For β = 2 we obtain the equivalencebetween the GUE and non–interacting Fermions.
4.5.3 Different interpretations of the Dyson index β
In the last two sections we pointed out the interpretation of the probabilitydensity
78
Chapter 5
Semiclassics
In the semiclassical approach to chaotical systems one tries to obtain quan-tum mechanical properties in the semiclassical limit having knowledge of theclassical quantities. The zeroth order approximation in a semiclassical ex-pansion is the classical limit and semiclassics is equivalent to an expansionwhere ~ is the small parameter.
As mentioned before it would be desirable to go the other way, but thisis far more difficult. The quantum mechanical system is fully defined by itsspectrum and its eigenfunctions an does apriori not refer to any classicallimit. To derive a classical theory we need a set of operators whose expec-tation values become phase space coordinates in the classical limit. In thisrespect the conjecture of Bohigas, Gianonni & Schmit:
The spectra of quantum systems, whose classical analogues are chaoticshow the same correlations as the GOE. is somewhat vague. They do notspecify what exactly the classical analog is. What they mean might besketched by the following diagramm
clas. mechanics QM
semi-classical
quantize
First a classical Hamilton function is quantized. The spectrum of thequantum Hamiltonian can then be analysed. If the quantum system is anal-ysed in a semiclassical approximation, the zeroth order terms of the analysedquantities coincide with the ones of the original classical system.
79
5.1 Path integrals
We are going to express the energy density (spectral function)
%(E) =∞∑n=0
δ(E − En) , (5.1)
as a path integral. We assume that En = Spec(H) is discrete, i. e. westudy bounded systems. In order to achieve this goal we introduce a complexspectral density via
%(c)(E) = − 1
πtr
1
E − H + i ε. (5.2)
Using the important identity
1
x+ i ε= P
1
x− i πδ(x) (5.3)
one verifies easily that %(E) = Im %(E). The operator
GR(E) =1
E − H + i ε(5.4)
is called resolvent. It is crucial for the semiclassical approach that the classicalHamilton function is known. We will assume this in the following. We willbe even more specific and will only consider Hamilton operators of the form
H =~p2
2m+ V (~q) . (5.5)
The trace can be written as an integral over eigenfunctions of the positionoperator
tr1
E − H + i ε=
∫d~q 〈~q| 1
E − H + i ε|~q〉 (5.6)
=
∞∫0
dt
i ~
∫d~q 〈~q| exp
(i(E − H + i ε)t
~
)|~q〉 (5.7)
=
∞∫0
dt
i ~
∫d~q exp
(i(E − H + i ε)t
~
)K(~q, ~q, t) . (5.8)
80
In the last line we have introduced the function
K(~q, ~q ′, t) = 〈~q | exp
(− i Ht
~
)|~q ′ 〉 . (5.9)
It is nothing but the time evolution operator in position space. It is calledthe propagator in the semiclassics literature. Likewise we introduce for itsFourier transform the notation
GR(~q, ~q ′, E) = 〈~q | 1
E − H + i ε|~q ′ 〉 , (5.10)
and call it retarded Green’s function. It is the resolvent in the position repre-sentation. If we find a semiclassical expression for the propagator or for theGreen’s function we will obviously have a semiclassical expression for %(E)as well. We split the time evolution operator for time t
U(t, 0) = e− i Ht/~ (5.11)
into an M–fold product of time evolution operators for time ∆t such thatM∆t = t
U(t, 0) =M−1∏n=0
U((n+ 1)∆t− n∆t)
= (U(∆t))M . (5.12)
For small times we can approximate the time evolution operator as
e− i H∆t = exp
(− i
~p2
2m~∆t− iV (~q)
∆t
~
)
≈ exp
(− iV (~q)
∆t
~
)exp
(− i
~p2
2m
∆t
~
)+O(∆t2) . (5.13)
We can therefore approximate K(~q, ~q ′, t) for infinitesimal time intervals as
〈~q|U(∆t)|~q ′〉 ≈ 〈~q|e− i ~p2
2m~∆t|~q ′〉e− iV (~q ′) ∆t~
=
∫d~p〈~q|~p〉〈~p|~q ′〉e
− i
(~p2
2m+V (~q ′)
)∆t~
=1
(2π~)f
∫d~p e
i~ ~p(~q−~q
′)−i ~p2
2m∆t~ −V (~q ′) ∆t
~
=
√m
2π i ~τ
f
eim
2~∆t(~q−~q ′)2−V (~q) ∆t
~ (5.14)
81
This expression can for U(∆t) be iterated M times:
〈~q|U(∆t)M |~q ′〉 =
∞∫−∞
dx1 . . .
∞∫−∞
dxM−1〈~q|U(∆t)|xM−1〉
× 〈xM−1|U(∆t)|xM−2〉 . . . 〈x1|U(∆t)|~q ′〉 (5.15)
=( m
2π i ~∆t
)Mf2
∞∫−∞
d~x1 . . .
∞∫−∞
d~xM−1
× exp
(M∑n=1
i
~S(~xn, ~xn−1)
). (5.16)
In the last line we defined ~xM = ~q and ~x0 = ~q ′. We observe that the function
S(~x, ~y) =m
2∆t(~x− ~y)2 − V (~x)∆t (5.17)
has the dimension of an action.
~q ′
~q
t = 0
∆t
2∆t
t
Figure 5.1: Three paths connecting the initial point ~q ′ with the end point ~q.
Now we could perform the continuous time limit M → ∞, ∆t → 0,and M∆t = t finite. One obtains the path integral representation of thepropagator
K(~q, ~q ′, t) =
~q(t)∫~q ′(0)
D[~x(t)] exp
i
~
t∫0
dt′L(~x(t), ~x(t))
, (5.18)
82
where
L =m
2~x2 − V (~x) (5.19)
is the Lagrange function. There is a caveat in the form (??) which must behandled with care. It concerns the integration measure
D[~x(t)] = limM→∞∆t→0
M−1∏n=1
( m
2π i ~∆t
)f/2dxn . (5.20)
This measure is not well defined in the limit ∆t → 0. In some cases it istherefore recommendable to keep M finite. We will do so in the following.Nevertheless also in the form (??) the path integral representation is wellsuited for the semiclassical approximation for the propagator K(~q, ~q ′, t) andhas been frequently used.
Exercise 19: Verify that the integration over the measure (??) really cor-responds to an integral over all different path as indicated in figure ??. Whydo crossings not count. Does the same identification hold for a lattice with afinite number of lattice points as well?
5.2 Tabor and Van-Vleck propagator
In order to keep the calculations transparent and to avoid the problems withthe singular path integral measure we will use the discrete version as statedin equation (??). The discrete action
∑Mn=1 S(~xn, ~xn−1) will be minimized by
solutions of the discrete classical equations of motion
m
∆t2(~qn+1 − 2~qn + ~qn−1) = −∇V (~qn) . (5.21)
We set here ~qM = ~q and ~q0 = ~q ′. We vary around this solution and write
~xn = ~qn + δ~xn . (5.22)
Plugging this into the expression for the discrete action, we obtain its semi-classical approximation
M∑n=1
S(~qn + δ~xn, ~qn1 + δ~xn−1) ≈M∑n=1
S(α)(~qn, ~qn−1)
+m
2∆t
M∑n=1
(δ~x(α)
n − δ~x(α)n−1)2 − (δ~x(α)
n )T∂2V
∂~qn∂~q Tnδ~x(α)
n
(∆t)2
m
.
(5.23)
83
The first term on the l.h.s. of equation (??) is the action evaluated alongthe classical path. We had to introduce an additional upper index α whichdistinguishes the different solutions of the classical equations of motion, ifthere are any. The integrals of the fluctuating quantities δ~xi are all Gaussianintegrals of the type
1√i π
∞∫−∞
eiλx2
dx =1√|λ|
1 λ > 0
e− iπ/2 λ < 0 .(5.24)
Performing the Gaussian integrals we obtain the semiclassical approximationof the discrete time propagator
〈q|U(∆t)M |q′〉 =∑α
( m
i 2π~∆t
)f/2| detG
(α)M |−1/2e
i~S
(α)M (~q,~q ′)− i
2ναπ .
(5.25)
The quantities entering in this formula are
S(α)M (~q, ~q ′) =
M∑i=1
S(α)(~qi, ~qi−1) , (5.26)
which is the classical action along the path α and the integer number να,which counts the number of negative eigenvalues of the matrix G
(α)M .
The f(M −1) dimensional matrix G(α)M can be obtained by looking at the
quadratic form in the second line of equation (??). In the following we limitourselves to the case f = 1. Thus we can drop the vector notation. We find
G(α)M =
dα1 −1−1 dα2 −1
−1. . .
dαM−1
, dαi = 2− ∆t2
mV ′′(qαi ) . (5.27)
Laplace expansion of the determinant with respect to the last line yields therecurrence formula
detG(α)M+1 = d
(α)M detG
(α)M − detG
(α)M−1
=
(2− ∆t2
mV ′′(q
(α)M )
)detG
(α)M − detG
(α)M−1 . (5.28)
This can be rewritten asm
∆t2
(detG
(α)M+1 − 2 detG
(α)M + detG
(α)M−1
)= −V ′′(q(α)
M ) detG(α)M .
(5.29)
84
We observe the similarity with the discrete classical equation of motion (??)in one dimension
m
∆t2
(q
(α)M+1 − 2q
(α)M + q
(α)M−1
)= −V ′(q(α)
M ) . (5.30)
Iterating the classical equation of motion starting with the initial values q0
and q(α)1 one can see that any q
(α)M can be calculated recursively from q0
and q(α)1 and is therefore a function of these two initial values q
(α)M (q
(α)1 , q0).
We now differentiate the equation of motion with respect to q(α)1 keeping q0
constant
m
∆t2
(∂q
(α)M+1
∂q(α)1
− 2∂q
(α)M
∂q(α)1
+∂q
(α)M−1
∂q(α)1
)∣∣∣∣∣q0
= −V ′′(q(α)M )
∂q(α)M
∂q(α)1
∣∣∣∣∣q0
. (5.31)
This can be compared with equation (??). We find
detG(α)M =
∂qM
∂q(α)1
∣∣∣∣∣q0
=m
∆t
∂qM
∂p(α)0
∣∣∣∣∣q0
. (5.32)
Here, in the second equation we defined p(α)0 = m
∆t(q
(α)1 −q0) or p
(α)0 = − ∂
∂q0S
(α)M .
Therefore we can write the determinant of G(α)M in terms of derivatives of the
action with respect to pn at time n = 0 and with respect to qn at time n = Musing the following sequence of equalities
∂2
∂q0∂qMS
(α)M = − ∂
∂qMp
(α)0 = −
(∂qM
∂p(α)0
)−1
=m
∆t detG(α)M
. (5.33)
Therefore we find in one dimension (f = 1) for the discrete time propagatorin the semiclassical limit
〈q|U(∆t)M |q′〉 =∑α
√1
2π i ~∂2S
(α)M
∂q∂q′exp
(i
~S
(α)M (q, q′)− i π
2να).
(5.34)
This result can be extended to arbitrary dimensions in a straightforward butsomewhat cumbersome calculation
〈q|U(∆t)M |q′〉 =∑α
√√√√ 1
(2π i ~)f
∣∣∣∣∣det∂2S
(α)M
∂~q T∂~q ′
∣∣∣∣∣× exp
(i
~S
(α)M (~q, ~q ′)− i π
2να). (5.35)
85
2
31
~q
~q ′
Figure 5.2: Exampleof three different pathsleading from ~q to ~q ′ in abilliard
In this form the discrete time semiclassical propagator was first deduced byTabor. We observe that the problematic integration measure has disappearedand no singularities occur in the limit ∆t→ 0. The transition to continuoustime is now particularly easy and transparent. The discrete action becomes
S(α)M −−−−→
∆t→0S(α)(t, ~q, ~q ′) =
t∫0
L(~q (α)(t), ~q (α)(t))dt (5.36)
where ~q (α)(t) is the solution to the classical equation of motion to the bound-ary problem ~q (α)(0) = ~q ′ and ~q (α)(t) = ~q. Therefore the continuous timesemiclassical propagator reads
Ksk(q, q′, t) =∑α
1√
2π i ~f
√∣∣∣∣det∂2S(α)(t)
∂~q T∂~q ′
∣∣∣∣× exp
(i
~S(α)(~q, ~q ′, t)− i π
2να). (5.37)
This is the van Vleck propagator. It was derived by van Vleck much earlierthan the discrete time version by Tabor.
5.2.1 Some Geometrical Considerations
One should notice that the principal function ( and therefore the propagator)depend on the initial value, on the end value and on time but not on mo-mentum. The conclusion is that there are different possible classical pathsfrom position ~q ′ to ~q .
The different paths have different lengths, so in general the energies aredifferent for each path. Looking at figure ?? one can easily imagine that itis possible to reach ~q from ~q ′ with an arbitrary initial momentum, if oneonly waits long enough. This should hold especially for chaotic systems dueto their ergodicity property. To visualise the ermergence of new paths, we
86
image the point ~q ′ as a f−dimensional submanifold of phase space, whichevolves under the Hamiltonian flow.
q′ = 0 q
t larget > 0t = 0
q′ = 0
0
p
0 0 0
Figure 5.3: Time evolution of the manifold q′ = 0 for a 2–dimensional phasespace.
The points along the perpendicular line q′ = 0 may follow arbitrary com-plex trajectories. The original straight line becomes a complex figure, inparticular the mapping
q → p(q) (5.38)
is in general multi–valued. The number of the branch cuts gives the numberof possible classical trajectories from q to q′.
The Morse index να is defined as the number of negative eigenvalues ofthe matrix G
(α)M . For a vanishing potential it is immediately seen that all
eigenvalues are positive. Due to the influence of the potential negative eigen-values can occur. This influence is small at the beginning of the trajectorybut might become important during time evolution. To be more specific, letthe eigenvalues ofG
(α)M be all larger than zero then it follows that detG
(α)M > 0.
If the determinant of G(α)M+1 is negative the new eigenvalue must be neg-
ative (the old eigenvalues of G(α)M+1 are shifted by the appearance of the new
one but they cannot change their sign). In the continuous time limit ∆t→ 0,M →∞ and ∆tM = t the change in sign of the determinant corresponds toa divergence of the second derivative of the action with respect to the initialvalue q′ and with respect to the end point q
∂2S(α)(t)
∂q∂q′=∞ . (5.39)
Let us remember that we can write this expression as
∂S(α)(t)
∂q(t)∂q(0)= −
(∂q(p′, t)
∂p′
)−1
. (5.40)
87
q′ = 0 qc
L′ L
p 1
2
3
Figure 5.4: The submanifold L′
is defined by fixing q′ at t = 0.From all point of L′ emanate tra-jectories. Trajectory 1 is right atqc, which is a conjugate point toq′, trajectory 2 has already passedthrough qc and trajectory 3 willreach it immediately.
This means the Morse index changes whenever at a time t a point q(t) isreached where the derivative with respect to the initial momentum ∂q(t)/∂p′
vanishes. Such a point is called is conjugated to q′. The function ~p(~q(t))obtains a branch cut at a conjugate point and a new classical trajectory isborn. As an example we chose a one dimensional particle in a potential with“almost” hard walls.
A similar mechanism occurs for higher dimensional systems. In figure ??the trajectoryies of a particle in a circular billiard are shown for three differentinitial values. We see that a trivial caustic occurs at the boundary r = 1.Another less trivial caustic occurs at some value r = r0 due to the centrifugalbarrier for particles with non–vanishing angular momentum. To every initialpoint in phase space correspond two two–dimensional submanifolds whichhave the shape of a circle in ~q–space.
The example of the circular billiard illustrates the slightly sloppy defini-tion of a caustic as the boundary to a classically forbidden region.
88
p
t smallt large
Figure 5.5: Conjugatepoint of a particle in analmost hard wall poten-tial. For small times theparticle moves like a freeone. The submanifoldq′ = 0 rotates clockwisearound the point q′ = 0,p = 0 (red line). Forlarge times the subman-ifold q′ = 0 obtains theshape as indicated by theblue curve. At the turn-ing point of the particlenew trajectories arise.
89
Figure 5.6: Caustics of a circular billiard of radius one for three initial pointsin phase space. Let α be the angle enclosed by the initial momentum and thex–axis and r the distance from the center. Then the initial points are fromleft to right (α, r) = (π/10, 0.9), (π/10, 0.5) and (π/10, 0.1). The plot showssome trajectories of intermediate length. For very long trajectories the spacebetween the two caustics will be filled densely.
5.3 Gutzwiller trace formula
Using the van-Vleck propagator we can find a semiclassical expression for theenergy density
%(c)(E) =− i
π~
∞∫0
dt
∫d~qei(E+i ε)t/~Ksk(~q, ~q, t) (5.41)
First we have a closer look at the time integral. We calculate this integralwith the approximation of the stationary phase. The imaginary part of theexponent is exactly the classical action integral I(α)(E) along the path αdefined earlier in section ??
S(α)(~q, ~q ′, t) + Et =I(α)(~q, ~q ′, t)
I(α)(~q, ~q ′, t) =
t∫0
dt′~q (t′)~p(α)(~q, ~q ′, t′) . (5.42)
Taking the derivative with respect to time
d
dt
(S(α)(~q, ~q ′, t) + Et
)= 0 (5.43)
yields the saddle point equation
d
dtS(α)(~q, ~q ′, t) = −E
∣∣∣∣t=tα
(5.44)
90
The saddle point times tα depend on the energy E, on the end point ~q and onthe initial point ~q ′. It is intuitively clear that the saddle point time is exactlythe time a particle with energy E needs to travel from point ~q ′ to point ~qalong path α. Fluctuations around this time can be taken into account by astationary phase approximation.
Since the t–integral runs over the semi–compact interval [0,∞], in a sta-tionary phase approximation we have to treat the integral at t = 0 separately.The exact form of the asymptotic expansion is as follows
limN→∞
∞∫0
dt A(t)eiNf(t) =iA(0)
Nf(0)+ saddle point contribution .
(5.45)
We therefore write the semiclassical Green’s function as
GR(~q, ~q ′, E) =1
i ~
∞∫0
dt Ksk(~q, ~q ′, t)ei(E+i ε)t/~
= GR
(~q, ~q ′, E) +GRosc(~q, ~q
′, E) , (5.46)
where GR
is the contribution to the integral stemming from the lower bound-ary and GR
osc the contribution of the saddle points. We first look at the saddlepoint contributions.
We expand the integrand to second order around t = tα
GRosc(~q, ~q
′, E) =1
i ~∑α
√∣∣∣∣det∂2Sα
∂~q T∂~q ′
∣∣∣∣ 1√
2π i ~f∞∫
0
dt exp
(i
~I(α)(~q, ~q ′, t(α)) +
i
2~Sα(~q, ~q ′, tα)(t− tα)2
).
(5.47)
The Gaussian integral can be performed using the rule (??)
GRosc(~q, ~q
′, E) =1
i ~∑α
1√
2π i ~(f−1)
√∣∣∣∣det∂2Sα
∂~q T∂~q ′
∣∣∣∣ 1
|Sα(tα)|
exp
(i
~Iα(~q, ~q ′, tα)− i
π
2(να + µα)
), (5.48)
91
where, due to equation (??)
µα =
−0 , Sα(tα) > 0
+1 , Sα(tα) < 0 .(5.49)
One can simplify the determinant further and make the energy dependencyexplicit. In a straightforward calculation one finds
− 1
Sαdet
(∂2Sα
∂~qT∂~q ′
)= detAα(~q, ~q ′, E)
Aα(~q, ~q ′, E) =
∂2Sα
∂~qT∂~q ′∂2Sα
∂~qT∂E∂2Sα
∂~q ′∂E
∂2Sα
∂E2
. (5.50)
Exercise 20: Prove equation (??).
Finally we need to calculate
%(c)osc(E) =
1
i ~π1
√2π i ~f−1
∑α
∫d~q√
detAα(~q, ~q, E)
exp
(i
~Iα(~q, ~q, tα)− i π
2(να + µα)
)(5.51)
to get the oscillating part oft the energy density. Since ρ(c)(E) is the traceof the resolvent, only orbits contribute which are periodic in position space.
We will now show that the only orbits which contribute in the semiclas-sical limit to the energy density are the orbits which are not only periodicin position but also in momentum. We solve the remaining integral againwithin the stationary phase approximation. We vary the action with respectto the end point ~q and with respect to the initial point ~q ′
Iα
∂~qd~q
∣∣∣∣~q=~q ∗ +∂Iα
∂~q ′d~q ′∣∣∣∣~q ′=~q ∗ = 0 (5.52)
and require this variation to vanish at the saddle point ~q ∗. Since the classicalaction is defined by
Iα(~q, ~q ′, E) =
~q∫~q ′
~p(~x)d~x , (5.53)
92
we find the saddle point equation
~p(~q ∗)− ~p ′(~q ∗) = 0 , ~p ′ = ~p(0) . (5.54)
From this equation we conclude that in the saddle point approximation onlyorbits contribute to %osc(E) which are periodic in ~q as well as in ~p . Thoseare exactly the classical periodic orbits.
⇔
Figure 5.7: Comparison of two orbits: the orbit on the r. h. s. is periodic inspace but not in momentum. The orbit on the r. h. s. is periodic in space andin momentum. Only this orbit contributes in the semiclassical approximationto the spectral density.
The further procedure is similar to the saddle point approximations wedid before. We expand the action Iα along the classical periodic orbits incomponents parallel and perpendicular to the orbit
Iα = Iα(~q‖, ~q‖, E) +1
2
(~q T⊥, ~q
′T⊥)Bα
(~q⊥~q ′⊥
), (5.55)
where the matrix Bα contains second derivatives of the action with respectto the initial point and with respect to the end point
Bα =
∂2Iα
∂~q T⊥∂~q ⊥
∂2Iα
∂~q T⊥∂~q′⊥
∂2Iα
∂~q ′T⊥ ∂~q⊥
∂2Iα
∂~q ′T⊥ ∂~q′⊥
. (5.56)
Exercise 21: Verify that Iα(~q‖, ~q‖, E) is indeed the action at the saddlepoint, defined by equation (??) and that the quadratic variation around thissaddle point solution can be written like in equations (??) and (??).
We can calculate the Gaussian integrals
%(c)osc(E) = − 1
i ~π∑α
∫d~q||
√detAα
detBαexp
(i
~Iα(E)− i π
2λα
), (5.57)
93
d~q ′⊥
d~q ′‖
d~q(T ) Figure 5.8: Evolution of a smalldeviation of the initial point ~q ′
during one period. The distanceto the original orbit is for chaoticsystems well described by an ex-ponential exp (λT ).
where we have set Iα(~q||, ~q||, E) = Iα(E), since the classical action along a pe-riodic orbit does apparently not depend on the the starting point. Moreoverwe introduced the symbol λα
λα = να︸︷︷︸conjugated point
+ µα︸︷︷︸turning point
+ κα︸︷︷︸negative EV of Bα
, (5.58)
which is the sum of the three phases picked up in the three saddle pointapproximations, due to the integration rule (??). The last integer κα de-notes the number of negative eigenvalues of the matrix Bα. The ratio of thetwo determinants can be further simplified. To this end we introduce themonodromy matrix
M =
∂~q⊥
∂~q ′T⊥
∂~q ⊥∂~p ′T⊥
∂~p ⊥∂~q ′T⊥
∂~p ⊥∂~p ′T⊥
, where ~q = ~q(T ) , ~p = ~p(T ) . (5.59)
The monodromy matrix is similar to the stability matrix introduced inchapter ??. The stability matrix describes the variation of the trajectory~z(t) with respect to a variation of the initial values for infinitesimal times,while M does this for the finite time T . It is important to notice that oneand the same initial point ~q ′ may belong to different periodic orbits withdifferent periods and for any of these orbits the monodromy matrix might bedifferent. Therefore the monodromy matrix will be equipped with the indexα labelling the periodic orbit it belongs to.
In a straightforward but somewhat lengthy calculation one finds√detAα
detBα=
1√det(Mα − 1)
1
|~q||. (5.60)
94
Finally, we can perform the integration over ~q||∮d~q||
|~q|||= Tαprim , (5.61)
where Tαprim is the period of a primitive orbit. A primitive orbit is an orbitwhere every phase space point of the orbit has been reached exactly onceby the trajectory. It is clear that any iteration of a primitive orbit is againa periodic orbit with period nTαprim, n ∈ N. These non–primitive orbits areincluded in the sum over the orbits as well and have to be taken into accountproperly.
Exercise 22: Verify equation (??).
Now we have all ingredients to state the main result of this section. Tostate it in a most compact form we define
Γα =Tαprim√
det(Mα − 1). (5.62)
The oscillatory part of the real and the complex spectral functions %osc and%osc can now be written as
%osc(E) =−1
i ~π∑α
Γα exp
(i
~Iα(E)− i π
2λα
)%osc(E) =
1
~π∑α
Γα cos
(Iα(E)
~− π
2λα
). (5.63)
The lower of these two equations is known as Gutzwillers trace formula.To get a complete expression for the energy density we need to calculate
the contribution to the Green’s function stemming from the lower boundaryof the time integral
G(~q, ~q ′, E) = − i
~
∞∫0
dtK(~q, ~q ′, t)ei(E+i ε)t/~ . (5.64)
We recall the approximation of the propagator for small times ∆t.
K(~q, ~q ′,∆t) =( m
2π i ~∆t
)f/2e
im2∆t~ (~q−~q ′)2− i ∆t
~ V(~q+~q ′
2
). (5.65)
Here in contrast to section ?? we evaluated the potential at the point (~q +~q ′)/2 instead at ~q ′.
95
Exercise 23: Convince yourself that for small times
V
(~q + ~q ′
2
)≈ V
(~q ′)≈ V (~q) (5.66)
holds. Therefore it does not matter at which point between ~q ′ and ~q thepotential is evaluated.
Introducing ~q+ = (~q + ~q ′)/2 and ~q− = ~q − ~q ′ the infinitesimal time prop-agator can be Fourier transformed with respect to the difference ~q−. Thisyields
K(~q, ~p,∆t) =
∫d~q−K(~q−, ~q+,∆t) exp
(− i
~~p ~q−
)= exp
(− i
~H(~q, ~p)∆t
). (5.67)
This can be plugged into equation (??) and the Green’s function can beevaluated further
G(~q−, ~q+, E) = − i
~
∆t∫0
dt
∫d~p
(2π~)fei ~p ~q−~ei(E−H(~q,~p)+i ε)t/~
G(~q, ~q, E) = − i
~
∆t∫0
dt
∫d~p
(2π~)fei(E−H(~q,~p)+i ε)t/~
=
∫d~p
(2π~)f1
E −H(~q, ~p) + i ε. (5.68)
The last equation was obtained using the asymptotic formula (??) for thetime integral. For the smooth part of the spectral function we obtain
%(E) = − 1
π
∫d~q G(~q, ~q, E)
=
∫d~qd~p
(2π~)fδ(E −H(~q, ~p)) (5.69)
This is the phase space volume of the energy shell ΓE divided by a power ofPlanck’s constant.
%(E) =ΓE
(2π~)f(5.70)
We divide the energy shell into small unit cells of volume hf , see figure ??.The inverse of the number of unit cells which fit into the energy shell gives thesmooth part of the energy density. This is Weyl’s law, which we encounteredalready in section ??.
96
h h hh h h
h
h hΓ
h
Figure 5.9: Division of a givenenergy shell into unit cells of vol-ume hf . The number of unit ele-ments equals the degeneracy fac-tor of the the eigenvalue E of theHamilton operator
5.4 Spectral correlations
Gutzwillers trace formula can now be used to acquire semi–classical expres-sions for spectral correlations. The essence of the conjecture of Bohigas,Gianonni and Schmit is that these should reproduce the RMT results andvice versa. The first step towards a proof was made by Berry in ?.
5.4.1 Berry’s diagonal approximation
The semiclassical energy–energy correlation function is defined by
X2(r) =
⟨1
%2(E)%
(E +
r
2%(E)
)%
(E − r
2%(E)
)⟩∆E
. (5.71)
Now the average〈. . . 〉∆E is over an energy interval which is large enough toyield a smooth function for X2(r) but which so small that changes in %(E)can be neglected. The mean level spacing is given by D = %(E)−1. Thesmooth energy density itself is obtained from Weyl’s law
%(E) =Ω(E)
(2π~)f=
tH2π~
, (5.72)
where tH is the Heisenberg time, which is the time scale related to the meanlevel spacing tH = h/D. According to equation (??) the energy density iswritten as
%(E) = %(E) + %fl(E) . (5.73)
This implies for the connected part of the semiclassical energy-energy corre-lator
Y2(r) =4π2~2
t2H
⟨%osc
(E + π
r~tH
)%osc
(E − πr~
tH
)⟩∆E
+ δ(r) . (5.74)
97
For convenience we use Gutzwillers trace formula in its complex form
%osc(E) =−1
i ~π∑α
Γαei~ Iα(E)− iπ
2λα (5.75)
Expanding the action for large Heisenberg times tH yields
Iα(E ± π~r
tH
)= Iα(E)± ∂Iα
∂E
π~rtH
(5.76)
The derivative of the action integral with respect to the energy is nothingbut the period of the periodic orbit ∂Iα
∂E= Tα.
Exercise 24: Prove that the derivative of the action integral
I(E) =1
2π
∮period
~p d~q =1
2π
∮∇S(~q, E) · d~q (5.77)
with respect to the energy is the period of the periodic orbit.
We define the slightly more general two–point correlator by
Y σ2 (r) =
1
t2H
⟨∑α
∑α′
Γα′Γαei~
(Iα(E)+σIα
′(E))
ei~ (Tα+σTα
′)π~rtH e−
iπ2
(λα+σλα′ )
⟩∆E
. (5.78)
From Y σ2 one can recover Y2 by
Y2(r) =4(ReY +
2 (r)− ReY −2 (r))
+ δ(r) (5.79)
From this we can obtain an expression for the semiclassical spectral formfactor
K(τ) = 1−∫dr e2π i rτY2(r)
= 4∑σ=±
∫dr e2π i rτReσY σ
2 (r) (5.80)
where τ is time measured in units of Heisenberg time τ = t/tH .
K(τ) =4
tH
⟨∑α,α′
δ
(τtH ±
Tα ∓ Tα′
2
)ΓαΓα′
e±(Iα(E)∓Iα′ (E)
)e∓
iπ2
(λα∓λα′ )⟩
∆T
(5.81)
98
where now the energy average has been converted into a time average. Thetime interval ∆T is on the one hand side large enough to contain a sufficientnumber of periodic orbits to obtain a smooth function. On the other handit is small compared to the period Tα of the periodic orbits itself.
In Berrys diagonal approximation only orbits with equal action contributeto the double sum∑
α′α
≈∑α
δαα′ . (5.82)
With this approximation the expression for the spectral form factor simplifiesdrastically
K(τ) =k
tH
⟨∑α
|Γα|2δ(τ − Tα/tH)
⟩∆T
. (5.83)
Here the factor k counts the number of orbits with the same action. Thisfactor is one in the case that time reversal symmetry is broken but it is twofor conserved time reversal invariance. In the latter case the orbit and itstime reversed orbit contribute to the sum
k =
1 time-reversal invariance broken
2 time-reversal invariant.(5.84)
The remaining average can be calculated further by observing that the timewindow of averaging is small compared to the mean time of the periodic orbitsin the window. This leads to the convenient but contraintuitive simplificationthat we can replace the δ distribution in equation (??) by one⟨∑
α
|Γα|2δ(τ − Tα/tH)
⟩∆T
≈
⟨∑α
|Γα|2⟩
∆T
. (5.85)
The expression on the r. h. s. can be evaluated with the help of the Hannay-Ozorio de Almeida sum formula to⟨∑
α
|Γα|2⟩
∆T
≈ τ tH . (5.86)
This yields for the spectral form factor
K(τ) =
τ time-reversal invariance broken
2τ conserved time-reversal invariance.(5.87)
99
A comparison with the Taylor expansion (??) of the spectral form factoras obtained by random matrix theory reveals that in the cases of brokentime reversal invariance the RMT result is recovered for times smaller thanHeisenberg time. In the cases of a time-reversal invariant system the firstterm in the taylor expansion (??) is recovered correctly. However, it wasnot clear how to get the higher terms in powers of τ within the semiclassicalapproximation for a couple of years.
There must be some non-diagonal terms, which contribute to the formfactor in a remarkable way. They must be a result of periodic orbits, whoseactions only differ in a small amount. This is hard to belief, because neigh-boring points run expontentially away from each other, i. e. the orbits arenot stable. The solution to this puzzle will be presented in section ??. Beforewe do so we derive the sum rule of Hannay and de Almeida.
5.4.2 Hannay-Ozorio de Almeida sum formula
We state the sum rule for long unstable periodic orbits derived by Hannayand de Almeida ?
1
2∆T
∑t−∆T
2<Tα<t+ ∆T
2
1
| det(Mα − 1)|≈ t . (5.88)
Here the sum runs over all periodic orbits with a period falling in an intervalof length ∆T around time t and Mα is the monodromy matrix of the periodicorbit.
We sketch the proof of this formula for the case that the time steps arediscrete. The main ingredient is an ergodicity asssumption for long periodicorbits. In discrete time the ergodicity condition reads
limt→∞
1
t
t∑n=1
f(~zn) =
∫d~z
Ωδ(E −H(~z))f(~z) (5.89)
Choosing f(~zn) = δ(~z − ~zn) one obtains
limt→∞
1
t
t∑n=1
δ(~z − ~zn(~z0)) =1
Ω. (5.90)
In the limit t →∞ the average over t can be replaced by an average over atime window δt
limt→∞
1
∆T
t+∆T/2∑t−∆T/2
δ(~z − ~zn(~z0)) =1
Ω, 1 ∆T t . (5.91)
100
Such an assumption for ergodicity should hold for long long periodic orbits,too. The ergodicity hypothesis for long periodic orbits requires
limt→∞
1
∆T
t+∆T/2∑t−∆T/2
δ(~z0 − ~zΓα(~z0)) =1
Ω. (5.92)
Here δ(~z0 − ~zΓα(~z0)) is a sum of δ’s at every possible point of the orbit withlength Tα. We can split this sum
δ(~z0 − ~zΓα(~z0)) =Period Tα∑
α
Tαprim∑i=1
∣∣∣∣det
(∂~zTα
∂~z0
− 1
)∣∣∣∣−1
δ(~z0 − ~zαi ) (5.93)
Tαprim is length of the primitive orbit Tα = vTαprim, v ∈ N. Phase spaceintegration yields⟨∑
α
Tαprim
| det(Mα − 1)|
⟩∆T
= 1 ≈ t
⟨∑α
1
|det(Mα − 1)|
⟩∆T
, (5.94)
which si the discrete version of the sum rule of Hannay and ozorio de Almeida.For the continuous version see Hannay-Ozorio de Almeida J Phys. A 17, 3429(1984) ?. Using equation (??) we find equation (??).
Exercise 25: Derive equation (??) from equation (??).
5.4.3 Corrections for the diagonal approximation
This section follows closely the PhD thesis by S. Muller ?. To keep thediscussion simple we will only study the case with 2 degrees of freedom,which is the smallest dimension where chaos can occur f = 2 and phasesspace dimension is dim Γ = 4. The monodromy matrix Mα for a periodicorbit α has a stable and an unstable eigenvalue
Spec(Mα) = eλαTα , e−λαTα , (5.95)
where λα is the Lyapunov exponent for the periodic orbit α under considera-tion. An explicit expression for the prefactor Γα in Gutzwillers trace formulacan be found
Γα =Tαprim
2 sinhλαTα/2(5.96)
101
P
s
stable
unstable
q⊥
p⊥
u Figure 5.10: Sketchof the stable and theunstable direction ina Poincare sectionspanned by a momen-tum and a positioncoordinate (p⊥, q⊥)
In principle it is hard to imagine that the overlap of two distinct orbits cancontribute substantially to the spectral form factor. By definition of a chaoticsystems neighboring points depart exponentially in time. One should noticethat this statement is only valid in position space. In phase space it is possibleto have stable and unstable orbits. In general the unstable direction and theposition direction of the Poincare section are not orthogonal ~eu · ~eq⊥ 6= 0 upto a set of measure zero.
Sieber and Richter found pairs of periodic orbits (α, α′) for time-reversalinvariant systems, whose difference in action is so small that the terms∑
αα′
ΓαΓα′ cos(Iα(E)− Iα′(E))/~ (5.97)
contribute to the form factor. These are periodic orbits and their so–calledpartner orbits which have the structure as depicted in figure ??. Whenevertwo points of an orbit come close to each other in phase space this gives riseto a partner orbit which along most part of the trajectory coincides with itspartner but differs in the region where points of the partner orbit come close.Such a region is called encounter.
102
∆J = uS
z1
z′1
J z′2z2
z′2
J z2
ts tu
Figure 5.11: Sketch of a periodic orbit with an encounter and of its partnerorbit
To understand the occurrence of a partner orbit better we look at aPoincare section P inside the encounter. It is pierced by the original orbitat point ~z1 and by its time reversed orbit at point ~z2. The small differencebetween both points can be decomposed in a stable and an unstable direction
T ~z2 − ~z1 = s~es + u~eu . (5.98)
An encounter begins when the stable component s falls below a certain valuec and it ends when the unstable component u reaches c. The duration ofan encounter is therefore obtained by adding the time tu until the unstablecomponent reaches c and the time that has passed since the stable componenthas fallen below c. The constant c may be chosen arbitrarily. Due to theexponential behavior, the times tu and ts are given by
tu ∼1
λαln
c
|u|, ts ∼
1
λαln
c
|s|(5.99)
The duration of the encounter is therefore given by
tenc = tu + ts =1
λαln
c2
|us|(5.100)
Exercise 26: The constant c should be chosen such that the time evolutionfollows a linearised treatment as described in section ??. Show that withinthis linearised treatment the duration of the encounter does not depend onthe exact position of the Poincare section. In other words: show that tenc isconstant.
Now start the partner orbit α′ at the position ~z ′1 as indicated in figure ??.The starting point is shifted from ~z 1 along the unstable direction therefore
103
~z1
T ~z2
~z ′1
T ~z ′2
u
s
unstable
stableFigure 5.12: Piercingpoints of the originalorbit α and its part-ner orbit α′ through aPoincare section, whichis parametrised by thestable and the unstablecoordinates. The areaof the rectangle coin-cides with the actiondifference ∆I.
it will rapidly depart from the original orbit. On the other hand it differsfrom T ~z 1 only in the stable coordinate and therefore in the course of thetrajectory it will approach more and more the time reversed orbit of α. Thetime reversed partner orbit will start its trajectory at ~z ′2 it differs from thestarting point of the original orbit ~z only in the stable coordinate. Thereforeit will approach more and more the original orbit α. In conclusion we haveconstructed a partner orbit α′ which coincides almost everywhere with theoriginal orbit α and which only differs from α in the small region of theencounter. One can show that the difference between the partner orbit’saction is given by the area of the Poincare section in the encounter
∆I = Iα − Iα′ = u · s (5.101)
It is plausible and can be proven that both orbits have the same Maslovindex. One can also show that the partner orbits have the same stabilityamplitude
Γα = Γα′ . (5.102)
We have to obtain an estimate of the number of the encounters, or moreprecisely on the rate of encounters. This rate depends on the constant c via∆I. We define
PTα(∆I)d∆I (5.103)
as the rate of encounters within orbit α of period Tα with difference of action∆I. We express this rate in terms of an auxiliary density wTα(s, u) of stableand unstable separations inside the encounter
PTα(∆I) =
c∫−c
ds
c∫−c
du wTα(s, u)δ(∆I − su) (5.104)
104
Figure 5.13: Piercing of the orbit tra-jectory throught the u − s plane thepiercing points are equidistributed
Ergodicity of long periodic orbits gives
wTα(s, u)ds du =Tα(Tα − 2tenc)
2Ωtenc
ds du (5.105)
with Ω as the volume of the energy shell. Now we can write the correctionto the diagonal approximation as
KSR(τ) =2
tHRe
⟨∑α
|Γα|2δ(τ − Tα/tH)
∫d∆IPτtH (∆I)ei ∆I/~
⟩∆T
.
(5.106)
This can be simplified by invoking the Hannay–Ozorio de Almeida sum rule
KSR(τ) = 2τ
⟨ c∫−c
ds
c∫−c
du wτtH (s, u)ei su/~
⟩∆T
(5.107)
The period of the orbit Tα is of order of the Heisenberg time whereas theduration of the encounter is of order of the Ehrenfest time. The expressionfor wttH can be split into two contributions which scale with different powersof Heisenberg time.
wTα(s, u) =(Tα)2
2Ωtenc︸ ︷︷ ︸∼t2H
− Tα
Ω︸︷︷︸∼tH
(5.108)
where the first term depends on s, u via tenc. One can show that the firstterm does not contribute anything within the semi classical limit. Using theexpression (??) for tenc the integral of the first term reads
c∫−c
ds
c∫−c
duλα
ln c2 − ln |su|ei su/~ = 2λα~ sin
(c2
~
). (5.109)
105
The expression on the right hand side is a rapidly oscillating function whichvanishes upon averaging. The desired correction to the diagonal approxima-tion is provided by the second term in equation (??). Replacing Tα by τtHand tH by = Ω
2π~ it reads
− 2τ 2
2π~
c∫−c
ds
c∫−c
du ei s·u/~ = −2τ 2 . (5.110)
This is the second order term in the Taylor expansion (??) of K(τ). By thesame method as outlined here, taking into account more and more encountersall higher order terms in the expansion (??) of K(τ) were obtained.
Exercise 27: Prove the following asymptotic formula for integral
c∫−c
ds
c∫−c
du ei s·u/~ −−→~→0
2π~ (5.111)
106
Chapter 6
Mesoscopics
Often it is possible to describe the motion of small quantum mechanicalparticles with classical equations of motion. This is possible due to an processwhich is called dephasing. The particle interacts with its surroundings andtherefore loses its quantum mechanical properties. For this to happen thesurrounding must be dynamic, i. e. it must be able to absorb or emit energy.Dephasing is not the topic of mesoscopics. Mesoscopics takes place in aparameter range where dephasing can be omitted. None of the less we takea closer look at the concept with a simple example.
6.1 Dephasing
We study the dephasing of a particle which is coupled to its surroundings viaits position space operator. Here we assume harmonic oscillators.
H =p2
2m+ q
∑k
λk
(ak + a†k
)+∑k
~ωka†kak . (6.1)
The state of the whole system is described by the density matrix %. Weassume that at at time t = 0 it is a tensor product of a density matrix %S ofthe system and of a density matrix %B of the bath
% = %S ⊗ %B , %B =∏k
1
Zkeβωka
†kak . (6.2)
For the concept of dephasing (or decoherence) the reduced density matrix isimportant
%red = trB % , (6.3)
107
where %B denotes the partial trace with respect to the bath degrees of free-dom. It is defined by
trB %ik1k2...kN ,jk′1k′2...k
′N
=∑k1...kN
%ik1...kN j,k1...kN = (%red)ij . (6.4)
We notice, that at time t = 0 we have
%red = %S , tr %2red = tr %2
S = 1 , (6.5)
i. e. the particle is in a pure state at time t = 0. The evolution of time forthe full density matrix is given by the Neumann equation
˙% =1
i ~[H, %] . (6.6)
At time t > 0 the reduced density matrix of the system is in general no purestate any more. This means
tr %2red(t) < 1 . (6.7)
The particle gets entangled with its surroundings due to the interaction withit. This entangling with the environment is called decoherence or dephasing.Even within this simple example it is not easy to calculate the exact evolutionof time of %red. We therefore state the result and discuss its consequences.The time evolution of the reduced density matrix is given in the so–calledBorn Markov approximation by
%red(q, q′, t) = %S(q, q′, 0) exp
(−2γkBT
(q − q′)2tm
~2
). (6.8)
Here γ is a constant which includes the interaction effects. It depends on allcoupling constants λk. More precisely one can show that the effect of thebath is completely described via the effective spectral function
J(ω) =∑
λ2kδ(ω − ωk) (6.9)
In many situations it is plausible to assume a linear behavior of the spectralfunction for small frequencies
J(ω) = m~γω . (6.10)
When an effective spectral function has this behavior for small frequenciesit is called Ohmic. The phenomenological factor γ has the dimension of adiffusion constant [γ] = 1
sec.
108
The reduced density matrix becomes diagonal on a time scale, which isgiven by the so–called decoherence time tdec
tdec =1
γ
(λdB
∆q
)2
, λdB =
√~2
2mkBT, (6.11)
where λdB is the thermal de-Broglie wave length. For typical macroscopicsituations are kBT ∼ 1Joule, λdB is very small and the dephasing time foraccessible distances like ∆q ∼ 1nm is gigantically small tdec ∼ 10−34. This isalso true for relatively small values of γ, where γ has a classical meaning. Itcorresponds to the classical relaxation time.
One can show that the reduced density matrix, once diagonal, describesa heavy mixed system
tr %2red(t > tdec) 1 . (6.12)
So it will be hard to observe coherence in a solid state at room temperature,because a conduction electron has enough possibilities to interact with itssurroundings. This is the reason why for many applications we can treatelectrons as tiny classical particles and quantum effects are small.
conduction electron
other electrons
phonons
Figure 6.1: Possibilities of interactions for a conduction electron
6.2 The mesoscopic regime
If we inject an electron into a metal at very low temperatures the phononsare freezed and the conduction band electrons are stuck in the Fermi sea andcannot be excited.
109
valence band
conduction band
donor
acceptor
EG kBT EG
An individual electron in the conduction band will move freely and co-herently through the probe. Its time evolution is described by the Hamiltonof a free particle in space or on a lattice. For a metal the Fermi energy liesin an only partially filled energy band. For an insulator the Fermi energylies in the gap between to bands. In general a solid state is not pure crystal.It has impurities on which the electron can scatter. They turn an insulatorinto a semiconductor. If there are many impurities the donator and acceptorlevels are able to fill the complete energy gap almost densely forming a newband, called the impurity band.
impurity band
Important is the difference between elastic scattering and inelastic scat-tering. At a static scattering center the particle changes its momentum butnot its energy
110
Es =const
|kout, E〉
|kin, E〉
Es =
|kout, Eout〉
|kin, Ein〉
Ein − Eout
We summarize the important length scales:
• The decoherence length or inelastic scattering length Lφ is the distance aparticle travels until it has lost its coherence it is given by vFtdec, wherevF is the fermi velocity and tdec is the decoherence time introduced insection ??. The decoherence length grows with decreasing temperatureand usually one assumes that it diverges for T → 0. On this latter pointthere has been some debate during the past years.
• The system size L is usually assumed to be smaller than the decoher-ence length Lφ. For reasonable temperatures like T ' 100mK thismeans a typical system size L ∼ 1µm.
• The mean free path for elastic scattering le is called the elastic scat-tering length. We distinguish two regimes. In the diffusive regime theelastic scattering length is much smaller than the extesnion of the probele L. As we will see, in this regime the electron propagation is welldescribed by a diffusion equation. In the ballistic regime l & L the elec-tron moves ballistically through the probe. A Gaussian wave packetwill loose its original shape by coherent scattering at the boundaries.If the boundaries are irregular this regime is intimately connected withthe theory of chaotic quantum billiards.
As mentioned earlier mesoscopics effects are effects due to the coherent na-ture of electron waves propagation through disordered medium. They takeplace at low temperature and at spatial dimensions which are larger than theatomic scale, but smaller than the decoherence length L < Lφ. We now givea few examples of typical mesoscopic effects.
111
ΦFigure 6.2: Sketch ofthe experiment of Wash-burn and Webb. Theelectrons scatter muliplybut coherently in theAharonov–Bohm ring.
6.2.1 Aharonov-Bohm effect (magnetoresistance)
Webb and Washburn demonstrated in an influencial experiment coherentelectonic transport in an Aharanov–Bohm device as depicted in figure ??.They transmitted electrons at low temperatures through the device andmeasured the linear conductance G = dI/dV |V=0, respectively the resistanceR = 1/G as a function of the magnetic flux Φ. From theory for non–coherenttransport a constant resistance G0 is expected. If transport through the ringis coherent the conductance should be an oscillatory function of the magneticfield strength
G(Φ) = G0 + δG cos(δ + 2πΦ
Φ0
) , (6.13)
where the period is given by the quantum resistance Φ0 = he, also called
fluxon. The results of the experiment (see figure ??) show clearly that thecurrent through the ring is coherent.
6.2.2 Universal conductance fluctuations
The classical mean electric conductance is given by Ohm’s law
〈G〉 ∝ σLd−2 (6.14)
where σ is the conductivity and 〈. . . 〉 denotes a disorder average. If thelinear dimension of the probe is much larger than the decoherence length,the fluctuations of 〈G〉 are expected to become small as as the size of theprobe increases. We split the sample in N subsystems. Coherence mightbe conserved inside each subsystem but is not conserved between differentsubsystems. Altogether we have
N =
(L
L1
)d(6.15)
112
Figure 6.3: Measure-ment of resistanceas a function ofthe magnetic fieldstrength H. Regularoscillations of theresistance as a func-tion of H are clearlyseen. This becomeseven more evident inFourier space: it con-tains a peak at thequantum resistanceh/e ≈ 130Tesla−1.A second smallerpeak at 2h/e canbe distinguished aswell.
L
L
L Figure 6.4: Self averag-ing of conductance. Ahuge block is splitted intosmaller ones, each of whichhas its well defined conduc-tance.
113
independent subsystems ( L1 ' Lφ). Then, due to the law of large numbers⟨δG2
⟩=⟨(G− 〈G〉)2
⟩(6.16)
should become small compared to 〈G〉√〈δG2〉〈G〉
∝ 1√N
=
(L1
L
) d2
(6.17)
If such an argument holds true, one says the system is self-averaging. Thefluctuations themselves behave as
〈δG2〉 ∝ Ld−4 , (6.18)
i. e. for dimension d = 3 the fluctuations should vanish for growing systemsizes if the conductance were a self–averaging quantity. Instead one findsexperimentally in the mesoscopic regime
〈δG2〉 = const . (6.19)
In other words 〈δG2〉 does not depend on the system size (universal). Thisphenomena is a result of the quantum coherence in mesoscopic systems. Onesays the system is not self-averaging. In general fluctuations of an observableX behave with system size as
〈δX2〉〈X〉2
∝
1 ⇒ X is not self averaging
N−1 ⇒ X is self averagingN−p ⇒ X is weakly self averaging (0 < p < 1).
6.2.3 Weak localization
The mean conduction can be classically calculated via Drude theory. Cor-rections to the classical mean conductance due to quantum coherence havearoused a lot of interest from the theoretical sied and from the experimentalside. They were calculated theoretically and are of the form
G = Gcl(1− pwl) , (6.20)
where the positive correction term pwl is a result of coherence effects in themesoscopic sample.
114
6.3 Models for disordered metals
From a theoretical point of view mesoscopics deals with the calculation ofquantum mechanical quantities for a one particle Hamiltonian with disorder,i. e. one or more parameters of the Hamiltonian are random. So in a certainsense mesoscopics deals with random matrices as well, however with very spe-cial ones. A certain basis, usually the position basis is clearly distinguishedand the measure of the random matrix has no unitary invariance.
Anderson model
The Anderson model is defined by
H =∑
εic†i ci +
∑〈ij〉
Tij c†i cj , (6.21)
where c†i , ci creates, respectively annihilates, particles at lattice position i.The sum 〈ij〉 runs over neighboring lattice sites. The onsite energies arerandom numbers
εi : p(εi) =
1W
if εi ∈ [−W/2,W/2]
0 otherwise.(6.22)
This is called diagonal disorder. An alternative is the model with off–diagonaldisorder, where the hopping matrix elements Tij are chosen randomly.
Edwards model
The Edwards model is a continuum single particle model with N disordercenters ~qj, j = 1, . . . N
H =~p 2
2m+
N∑j=1
V (~q − ~qj) . (6.23)
The scattering centers are equidistributed random numbers in Rd
~qj : pk(qjk) =
1L
0 ≤ qjk < L
0 otherwise, k = 1, 2, . . . d . (6.24)
In the limit of many scattering centers N → ∞ and infinitely large volumeL3 →∞ keeping n = N/Ld finite we can introduce a disorder density
n(~q) =N∑j=1
δ(~q − ~q j) , (6.25)
115
with mean value 〈n(~q)〉 = n. Then we can write
H =~p 2
2m+
∫d~q ′n(~q ′)V (~q − ~q ′) . (6.26)
The disorder potential usually is chosen such that the first moment vanishes
〈~V (~q)〉 = n
∫d~q ′ V (~q − ~q ′) = 0 . (6.27)
The second moment is defined as
〈V (~q1)V (~q2)〉 =
∫d~q ′d~q ′′V (~q ′ − ~q1)V (~q ′′ − ~q2)〈n(~q ′)n(~q ′′)〉 (6.28)
Gaussian model
The Gaussian model is defined by
H =~p 2
2m+ V (~q) (6.29)
where only the second moment is given by
〈V (~q)V (~q ′)〉 = B(~q − ~q ′) (6.30)
The Gaussian model is a special case of the Edwards model for the limitof infinite density and infinite small perturbation. In the following we willmainly work with a general potential in some situations for simplicities sakewe specialise on a δ–correlated potential
B(~q, ~q ′) = Bδ(~q − ~q ′) . (6.31)
This becomes particularly simple in Fourier space
B(~k) =
∫B(~q, ~q ′)ei~k(~q−~q ′)d(~q − ~q ′) = B . (6.32)
The constant B with the unit [B] = Joule2md measures the strength of thedisorder potential in the Gaussian model.
116
6.4 Diagramatic perturbation theory
In the following we are going to investigate the Schrodinger equation for aparticle in a disorder potential
i ~ψ(~q) =
(~p 2
2m+ V (~q)
)ψ(~q) (6.33)
=(H0 + V
)ψ(~q) (6.34)
In difference to usual quantum mechanics we have only imperfect knowledgeabout the potential. Therefore we can only calculate ensemble averages withrespect a given distribution of disorder potential presented in the previoussection. The disorder average can be taken with different degrees of sophis-tication.
6.4.1 Greens function in disordered systems
The free retarded resolvent (Green’s) operator is
GR0 (E) =
1
E − H0 + i ε. (6.35)
It becomes most simple in momentum space
GR0 (~k) =
1
E − ~2k2
2m+ i ε
. (6.36)
This expression can be transformed into coordinate representation by
GR0 (~q, ~q ′) =
1
Ld
∑~k
e− i~k(~q−~q ′)G(~k) (6.37)
=2mΩd
(2π)d
∞∫0
dkkd−1
π∫0
dϑ(sinϑ)d−2 e− i k|~q−~q ′| cosϑ
2mE − ~2k2 + i ε, (6.38)
where we converted the sum into an integral and expressed it in d–dimensionalpolar coordinates and the constant Ωd accounts for the integration over thed− 1 dimensional sphere. The integral over cosϑ is a special function, calledzonal spherical function. It can be expressed in terms of the more familiarν–th order Bessel function Jν(z)
GR0 (~q, ~q ′) =
2m√
2πd
∞∫0
dk kd/2
|~q − ~q ′|(d−2)/2
J(d−2)/2
(k|~q − ~q ′|
)2mE − ~2k2 + i ε
, (6.39)
117
The remaining integral can be evaluated in any dimension. Only thedimensions d = 1, 2, 3 are interesting. We find
GR0 (~q, ~q ′) =
− im
~2kexp
(i k|~q − ~q ′|
)− i
m
2~2H
(1)0
(k|~q − ~q ′|
)−i m
2π~2|~q − ~q ′|exp
(i k|~q − ~q ′|
) , (6.40)
where k =√
2mE/~.
Exercise 28: Derive the coordinate Green’s function
GR0 (~q, ~q ′) =− i
πm
~2
1√
2πd
k(d−2)/2
|~q − ~q ′|(d−2)/2H
(1)(d−2)/2(k|~q − ~q ′|) (6.41)
in arbitrary dimension from equation(??). Deform the integration contourand use the method of residues for the k–integral. Moreover use the factthat the Bessel function Jν(z) can be expressed in terms of Hankel functions
H(1,2)ν (z) of the first and second type as
Jν(z) =1
2
(H(1)ν (z) +H(2)
ν (z)), (6.42)
and the asymptotic behavior of the Hankel functions
H(1,2)ν (z) ∼
√2
πzexp
(± i(z − 1
2νπ − 1
4π)
)(6.43)
Likewise the exact retarded Green’s operator is given by
GR(E) =1
E − H + i ε. (6.44)
It can be expanded in a full set of (unkown) eigenfunctions φn(~q) as
GR(~q1, ~q2, E) =∞∑n=0
φ∗n(~q1)φn(~q2)
E − En + i ε. (6.45)
Using this expression we can define the local density of states as the diagonalpart of the Green’s function
%(~q, E) =∞∑n=0
φ∗n(~q)φn(~q)δ(E − En)
=− 1
πImGR(~q, ~q, E) (6.46)
118
The density of states %(E) is expressed through the local density of states as
%(E) =
∫d~q %(~q, E) . (6.47)
We look at the expressions for the coordinate Greens function (??) and ob-serve that the imaginary part is finite. Employing the asymptotic limit ofthe Bessel functions for small arguments we find for %(E)
%(E) =mLd
~2
Ωd
(2π)d
(2mE
~2
)(d−2)/2
. (6.48)
This is again Weyls law, which we already encountered in Sec. ?? for billiardsand in Sec. ?? in equation (??). We recall that Ωd is the volume of the d−1-dimensional unit sphere.
Exercise 29: Show that the expressions (??) and (??) are identical.
6.4.2 Feynman diagrams
We now perturbatively calculate the Green’s function for a mesoscopic dis-ordered system. Combining equation (??) and (??) we find
(E − H + i ε)GR(E) = 1 (6.49)(1
GR0
− V
)GR = 1 . (6.50)
or likewise
GR = GR0 + GR
0 V GR . (6.51)
Here and in the following to unburden notation we suppress the energy argu-ment of the Green’s function. We use this as a starting point for a diagramaticexpansion. We iterate the series
GR = GR0 + GR
0 V GR0 + GR
0 V GR0 V G
R0 + . . . (6.52)
and write it in coordinate basis as
GR(~q, ~q ′) = GR0 (~q, ~q ′) +
∫d~q0 G
R0 (~q, ~q1)V (~q1)GR
0 (~q1, ~q′)
+
∫d~q1d~q2G
R0 (~q, ~q1)V (~q1)GR
0 (~q1, ~q2)V (~q2)GR0 (~q2, ~q
′)
+ . . . (6.53)
119
The higher the perturbation’s order, the more complicated it gets. So itmakes sense to visualize the series. To this end we introduce a graphicalrepresentation of the series via Feynman diagrams:
~q ~q ′
vertex~q ~q ′
~q1
= GR0 (~q, ~q ′)
=
∫d~q1G
R0 (~q, ~q1)GR
0 (~q1, ~q′)V (~q1)
~q ~q ′= GR(~q, ~q ′) (6.54)
A directed full line connecting two space points denotes the free Green’sfunction. The joint of a full line with a dotted line is a vertex. The vertexcoordinate ~q1 is integrated over. The dotted line connects the vertex with across, which indicates the disorder potential at the vertex position ~q1. A dou-ble line denotes the full Green’s function. With these rules one can visualizethe series expansion for GR(~q, ~q ′) as
= + +
+(6.55)
Upon averaging two or more disorder crosses are joined according to the ruleof disorder averaging. Since the expectation value of the potential vanishesall graphs with solitary crosses vanish as well
= + +
+
〈GR〉
+
1 2
3
0
4
. (6.56)
In equation (??) the Feynman diagrams contributing to the averagedGreen’s function up to fourth order are depicted. We assumed that the oddmoments of the disorder potential vanish. When looking at the four lastdiagrams we observe a difference between diagram 1, 3, 4 and diagram 2.Diagram 2 can be cut in two disjoint graphs with one cut. Diagrams forwhich this is possible are called reducible. Graphs which cannot be severedlike this are called irreducible. Therefore diagram 1, 3 and 4 are irreducible.
120
Moreover we see that diagram 2 is the junction of two diagrams 1. It ispossible in general to compose a reducible graph as a sequence of irreduciblegraphs. The complete Green’s function itself can be written as indicatedin equation (??). This is the diagramatic version of Dyson’s equation. Thecircle named Σ denotes the sum of all possible irreducible diagrams, Thissum is called self energy.
=〈GR〉
Σ+GR
0 GR0 〈GR〉
(6.57)
The diagramatic equation (??) translates into the following integral equationfor the averaged Green’s function
〈GR(~q, ~q ′)〉 = GR0 (~q, ~q ′) +
∫d~q1d~q2 G
R0 (~q, ~q1)Σ(~q1, ~q2)〈GR(~q2, ~q
′)〉 .
(6.58)
This is the Dyson equation in configuration space. It becomes an algebraicequation in momentum space
〈GR(~k)〉 =GR
0 (~k)
1− GR0 (~k)Σ(~k)
, (6.59)
where
GR(~k) =
∫Ld
ei~k(~q−~q ′)GR(~q, ~q ′)d(~q − ~q ′) (6.60)
is the Green’s function in momentum space. Now an explicit expression forGR(~k) can be given. We recall the expression for the free retarded Green’sfunction in momentum space (??) and write the full averaged Green’s func-tion as
〈GR(~k)〉 =1
E − ~2k2
2m+ i ε− Σ(~k)
. (6.61)
This is an exact expression for the full averaged propagator. However westill did not solve anything but just reformulated the problem. The difficultproblem of calculating the averaged Green’s function has been shifted tothe problem of calculating the self energy. This calculation is in principalas difficult as the former one because we still have to calculate an infinite
121
number of graphs. We can to this perturbatively by calculating only thelowest order self energy diagramms.
Σ = + + + . . . (6.62)
Here we only calculate the first graph, keeping in mind that with the cal-culation of one self–energy diagram we have summed up already an infiniteamount of diagrams. We look at the following diagram
=
∫d~q1d~q2G
R0 (~q, ~q1)B(~q1, ~q2)GR
0 (~q2, ~q′)GR
0 (~q1, ~q2)(6.63)
and convert it into momentum space
=1
Ld
∑~k1
GR0 (~k)B(~k1)GR
0 (~k − ~k1)GR0 (~k) . (6.64)
The self energy itself is the above diagram but without external legs. In thefirst order approximation applied here, it is given by
Σ(~k,E) =1
Ld
∑~k1
B(~k1)GR0 (~k − ~k1) (6.65)
This expression simplifies further for a δ−correlated potential. Using equa-tions (??) we get
Σ(~k,E) =B
Ld
∑~k
1
E − ~2k2
2m+ i ε
. (6.66)
Exercise 30: Derive equation (??) from equation (??). Show that equationare equivalent. Use and the Fourier transform of the Kronecker delta
δ~k1,~k2=
1
Ld
∫d~q ei(~k1−~k2)~q . (6.67)
and the Fourier backtransformation of the Greens function (??)
The self energy has a real part and an imaginary part. The meaning ofthe real part is a global shift in the spectrum. Especially important is theimaginary part of the self energy, which describes the life time of a particlewith momentum ~k. We look at real part and imaginary part separately. The
122
imaginary part of the free Green’s function is equal to a δ distribution andis related to the density of states via
%(E) =∑k
δ(E − E(~k)) . (6.68)
Therefore the imaginary part of the self–energy can be written as
Im Σ(E) = −πBLd
%(E) (6.69)
in terms of the density of states. Notably in this approximation the self–energy is momentum independent. We observe the relation to RMT and tosemiclassics. Like in RMT and in semiclassics an energy scale is suppliedby the mean density of states around some fixed energy. In RMT we chosethe center of the semicircle, in semiclassics it was an arbitrary energy in thecenter of the energy window of averaging. Here we choose E = EF as theFermi energy. If we look on phenomena on the scale of the mean level spacingwe can neglect the energy dependence of Σ and assume %(EF ) = %F. Thevalue of %F depends on the geometry and in particular on the dimension ofthe probe through Weyl’s law. As before we can associate with the energydensity a Heisenberg time tH = ~%F as a typical time scale. Here we see acharacteristic of mesoscopic systems. Whereas for macroscopic systems theprobe is assumed to be infinite and the energy spectrum is a continuum inmesoscopics the dimension of the probe is assumed finite in the calculations.As a consequence Heisenberg time might be large but is finite.
One finds for the first order approximation of Green’s function
〈GR(~k)〉 =1
E − ~2k2
2m+ i ~
2te
, (6.70)
where we defined a second characteristic time by
1
2te=π%FB
Ld~. (6.71)
This time, which measures the strength of the disorder is called mean elasticscattering time, it can be considered as well as the life time of a plane wavein the medium. This can be seen with Fermi’s golden rule. Second orderperturbation theory gives a simple formula for the decay rate of a quantumstate in terms of the matrix elements of the perturbation
1
te=2π
∑~k
|〈~k|V |~k′〉|2δ
(E~k − E~k
′), (6.72)
123
where the states |~k 〉 are plane waves. Assuming for V delta correlated dis-order, we find for the disorder average⟨
|〈~k|V |~k′〉|2⟩
=B
Ld, (6.73)
which shows that equations (??) and (??) are identical. We can relate te totH through a dimensionless constant
te =tHγ, γ =
2πρ2FB
Ld. (6.74)
We finally consider the real part of the self–energy
Re Σ(~k,E) =B
Ld
∑~k
1
E − ~2k2
2m
. (6.75)
Converting the sum into an integral we see that the integral diverges withpower d − 2 in the ultraviolet. The integral has to be regularised by someupper momentum, which is usually chosen to be the band width. We willnot go into more detail here.
In order to find the averaged Green’s function in position space we haveto Fourier transform 〈GR(k)〉. The integration can be performed with thehelp of the residue theorem in the same way as discussed in exercise ??. Theresult is⟨
GR(~q1, ~q2, E)⟩
= GR0 (~q1, ~q2, E)e−|~q1−~q2|/2le , (6.76)
or explicitly in three dimensions⟨GR(~q1, ~q2, E)
⟩= − i
m
2π~2
1
|~q1 − ~q2|ei√
2mE|~q1−~q2|/~e−|~q1−~q2|/2le (6.77)
where the elastic scattering length is defined as
le =
√~τem
. (6.78)
The Green’s function become short range due to multiple scattering.
6.5 Quantum diffusion
The averaged Green’s function alone is not sufficient to describe many phys-ical quantities. To calculate transition probabilities it is necessary to takethe average of products of two Green’s functions.
124
We consider the following situation, which is very close to a real exper-imental setup. At time t = 0 we inject an electron at position ~q1 with amean energy equal to the Fermi energy of the solid EF. The wave functionis given by wave packet centered around ~q1 in position and moving with avelocity vF =
√2EF/m. We want to calculate the probability to measure
the electron at time t at position ~q2.
~q1
~q2
We assume the wave function to be composed of eigenstates |φn〉 of oneof the disorder Hamiltonians of Sec. ?? with a spread in energy of σ2, suchthat we can write the initial wave function in position representation as
〈~q|ψ~q1〉 = A∞∑n=0
〈~q |φn〉〈φn|~q1〉e−(En−EF)2/4σ2
, (6.79)
where A is a normalisation constant. The probability to measure the electronat time t at point ~q2 is given by
P (~q1, ~q2, t) =⟨|〈~q2|e− i Ht|ψ~q1〉|2
⟩(6.80)
Fourier transforming the time evolution operator and using the expansion ofthe Green’s function in eigenfunctions (??) yields
P (~q1, ~q2, t) =A2
∫dE dE
(2π)2e−(E+E/2−EF)2/4σ2−(E−E/2−EF)2/4σ2−iEt⟨
ImGR(~q1, ~q2, E + E/2)ImGA(~q2, ~q1E − E/2)⟩
(6.81)
To simplify this expression further we make some assumptions. We assumethat the Green’s function only depends weakly on E. Further let be E σ.
125
Then the integral over E can be performed in a saddle point approximationσ →∞
P (~q1, ~q2, t) =1
2π
∞∫−∞
dE P (~q2, ~q1, E)e− iEt (6.82)
P (~q1, ~q2, E) =1
2π%F
⟨ImGR(~q1, ~q2, EF + E/2)ImGA(~q2, ~q1, EF − E/2)
⟩(6.83)
Exercise 31: Show that the normalisation constant A is given by A =√√2π%Fσ.
We assume only weak disorder, kle 1 with kF =√
2mEF/~. Theenergy E is measured on the scale of the average level spacing, like in RMT.We will see that P (~q1, ~q2, E) consists of three parts
P (~q1, ~q2, E) = PDB(~q1, ~q2, E) + Pdiff(~q1, ~q2, E) + Pcoop(~q1, ~q2, E) . (6.84)
The first contribution is the transition probability in the so–called DrudeBoltzmann approximation
PDB(~q1, ~q2, E) =1
2π
1
%F
⟨GR(~q1, ~q2, EF + E/2)
⟩ ⟨GA(~q2, ~q1, EF − E/2)
⟩,
(6.85)
where the average of the product of two Green’s functions has been replacedby the product of two averages of Green’s functions. In this approximation allcorrelations between the forward path and the backward path are neglected.Using the result for the averaged Green’s function (??) and the fact thatGA(~q1, ~q2) = GA(~q2, ~q1) = [GR]∗(~q2, ~q1) we find for PDB(~q1, ~q2, E)
PDB(~q1, ~q2, E) =1
4πvF
1
|~q1 − ~q2|2eiE|~q1−~q2|/(vF~)−|~q1−~q2|/le . (6.86)
This can be transformed into the time domain. We find
PDB(~q1, ~q2, t) =δ(|~q1 − ~q2| − vFt)e
−t/τe
4π|~q1 − ~q2|2(6.87)
For τe → ∞ this corresponds to the probability to find the particle at ~q2 ifit has started at ~q1 with velocity vF into a random direction and then hasmoved ballistically, i. e. collision free, to ~q2. For a finite τe this probability isreduced by an exponential factor. We observe that PDB is not normalised∫
PDB(~q1, ~q2, t) d~q2 = Θ(t)e−t/τe < 1 . (6.88)
126
This means that the probability of collision free propagation from ~q1 to ~q2
decreases exponentially with time at a rate 1/τe given by the inverse of theelastic scattering time. The “missing probability” will be restored by the twocorrections Pdiff and Pcoop, we are going to investigate in the next section.
6.5.1 Diffusons and Cooperons
To find the correction to the Drude–Boltzmann approximation, we look atthe disorder average of the product of an advanced an a retarded Green’sfunction⟨
GR(~q1, ~q2, EF + E/2)GA(~q2, ~q1, EF − E/2)⟩
(6.89)
and define the Feynman diagram for free propagators
GR0
GA0
~q1 ~q2 = 〈GR0 (~q1, ~q2, EF + E/2)GA
0 (~q2, ~q1, EF − E/2)〉(6.90)
The free advanced Green’s function has the same Feynman diagram as theretarded Green’s function but with the reversed arrow direction. For theadvanced propagator the arrow shows in the direction from the second argu-ment of GA
0 to the first argument. Since in equation (??) the arguments of
~q ~q ′= GR
0 (~q, ~q ′)
~q ~q ′= GA
0 (~q, ~q ′)
~q~q ′= GA
0 (~q ′, ~q)
~q~q ′= GR
0 (~q ′, ~q)
Figure 6.5: Feynmangraphs of retarded andadvanced Green’s function.
the retarded and of the advanced Green’s functions are swapped, as a resultin the Feynman diagramm (??) both arrows show into the same direction.Using the perturbative expansion of both Green’s function
GR = GR0 + GR
0 V GR0 + GR
0 V GR0 V
GA = GA0 + GA
0 V GA0 + GA
0 V GA0 V (6.91)
we get a perturbative expansion of their product. Up to second order in Vwe find⟨
GRGA⟩
=⟨GR
0 GA0
⟩+⟨GR
0 V GR0 G
A0 V G
A0
⟩+⟨GR
0 V GR0 V G
A0
⟩+⟨GR
0 GA0 V G
A0 V⟩
+ . . . (6.92)
127
The three second order contributions are represented by the following threegraphs
+ + + . . . (6.93)
Averaging yields
+ + + . . . (6.94)
We notice that there are two kinds of graphs. The first graph connectsthe forward propagator with the backward propagator, the second and thethird do not. Summing up to all orders the graphs which do not connectthe forward and the backward propagator yield exactly the contributions ofthe Drude-Boltzmann approximation. They are described by the Feynmandiagram
PDB = = 〈GR〉〈GA〉 (6.95)
We next study the remaining terms which couple the forward and backwardpropagator. In the same way as we introduced the self energy for the Green’sfunction we can write this contribution as
P (~q1, ~q2, E) = PDB+~q1 ~q2
vertex functionΓE(~q, ~q ′)
(6.96)
where we replaced the free Green’s function connecting the starting pointwith the point where the first connection between forward and backwardpropagator occurs by the full Green’s function. The same we do for theGreens function’s emanating from the last connection between forward prop-agator and backward propagator and ending at the final point ~q2. Everythingbetween the first connection between forward and backward propagator andtheir last connection is subsumed in the shaded box, which we will call vertexfunction.
The diagramatic expression (??) is stille exact. However the problemhas not been solved but only was shifted into the calculation of the vertexfunction.
128
To see which graphs contribute to the vertex function we write down allFeynman diagrams which contribute to the average (??) up to third order inthe perturbation V
+ +
+ + +
+++
1 2 3
4 5 6
987
(6.97)
One can show in detail that in leading order in (kF le)−1 the only graphs
which will contribute to the vertex function are the graphs 1, 2, 3, i. e. thefirst order term and both second order terms and the two third order graphsnumber 4 and 9. We will not go into the details of the calculation but willtry to understand in a more heuristic way why only these graphs contributein the limit (kF le)
−1 1.
,~q1
~q2
,
~q ~q ′
~q ~q ′
,~q ~q ′
~q2~q1
~q1 ~q2
~q2~q1
~k
~k′
~q1
~q2
~k
~k′
~q1
~q2
~k
~k′
~q1
~q2
~k
~k′
~q1
~q2
~k
~k′
129
~q1
~q2
Figure 6.6: Forward andbackward trajectorieswith five scatteringevents.
Let
S(~k,~k′) =∑~q1,~q2
f(~q1, ~q2)ei(~k~q1−~k′~q2) (6.98)
be the complex amplitude of an electron which is injected in the disorderedmedium with momentum ~k and reemitted from the medium with momentum~k′. The function f(~q1, ~q2) is the complex amplitude corresponding to thepropagation of the wave between two scattering event at ~q1 and ~q2. Thisamplitude may be expressed as a sum of the form
f(~q1, ~q2) =
paths∑j
aj(~q1, ~q2)ei δj , (6.99)
where each path represents a sequence of scatterings joining the points ~q1
and ~q2. The intensity is given by
|S(~k, ~k′)|2 =
∑~q1,~q2,~q3,~q4
f(~q1, ~q2)f ∗(~q3, ~q4)ei(~k~q1−~k′~q2)−i(~k~q3−~k
′~q4) (6.100)
The function f can be written as a sum over trajectories. Every trajectoryyields a scattering phase and a scattering amplitude. This leads to
f(~q1, ~q2)f ∗(~q3, ~q4) =∑jj′
aj(~q1, ~q2)aj′(~q3, ~q4)ei(δj−δj′ ) (6.101)
If at each scatterer the phase is assumed random only trajectories contribute
to the average of |S(~k, ~k′)|2, where the forward path and the backward path
have the same sequence of scattering events, either in the same or in oppositedirections. Such trajectories are schematically represented in Fig. ??, andcorrespond to the sequences
~q1 → ~ra → ~rb → . . .→ ~ry → ~rz → ~q2 , (Diffuson)
~q2 → ~rz → ~ry → . . .→ ~rb → ~ra → ~q2 , (Cooperon) (6.102)
130
In particular with view on Eq. (??) this can only happen if ~q1 = ~q3 and~q2 = ~q4 or ~q1 = ~q4 and ~q2 = ~q3. The intensity can be written as a sum of twocontributions
|S(~k, ~k′)|2 = 2Re
∑~q1,~q2
(|f(~q1, ~q2)|2 + f(~q1, ~q2)f ∗(~q2, ~q1)ei(~k+~k′)(~q1−~q2)
).
(6.103)
The second term in relation ?? contains a phase factor. It depends on thepoints ~q1 and ~q2. The sum over these points in the averaging makes this termvanish in general, with two notable exceptions.
• If the outbound direction is exactly opposite to the direction of inci-
dence ~k + ~k′, the intensity is twice the classical value. The classical
contribution has no angular dependence on average, and the secondterm gives an angular dependence to the average intensity reflected bythe medium which appears as a peak in the albedo.This phenomenonwas observed first in optics and is known as coherent backscattering
• The terms for which ~q1 = ~q2 are special. They correspond to closedmultiple scattering trajectories. Their contributions to the averagedinterference term survive even when it is impossible to select the direc-
tions ~k and ~k′.This is the case for metals or semiconductors for which
the interference term affects the average transport properties such asthe electrical conductivity. This is the origin of the phenomenon ofweak localization.
Exercise 32: Convince yourself that in the limit of weak disorder kFle 1indeed only the diffuson and the cooperon contribution are important . Cal-culate to this end the following three Feynman diagrams
a) b) c)
Write down the corresponding integral expressions in position space andtransform them into momentum space. The resulting k integral can be eval-uated by the residue theorem.
131
~k
~q1
~k′
~q2
diffusion
~k′
~k
~k
cooperon
Figure 6.7: L.h.s Feynman graphs for the diffuson and for the cooperon. Onthe r.h.s. a typical trajectory of a diffuson and of a cooperon are sketched.
We first look at the vertex function of the diffuson. We call it ΓdiffE . We
depict in the upper line of equation (??) the first three contributions to ΓdiffE .
We see that these and all other contributions can be summarized again byΓdiffE itself. We obtain the second equation of equation (??)
ΓE
= + + + . . .
= +
ΓE
(6.104)
This is a self consistency equation for ΓdiffE similar to the Dyson equation
for the self energy. It translates into the following integral equation for thevertex function Γdiff
E (~q1, ~q2)
ΓdiffE (~q1, ~q2) =Bδ(~q1 − ~q2) +B
∫ΓdiffE (~q1, ~q
′)⟨GR(~q ′, ~q2, EF + E/2)
⟩⟨GA(~q2, ~q
′, EF − E/2)⟩d~q ′ (6.105)
Integral equations of this kind are called Bethe-Salpeter-equations.
132
Exercise 33: Translate all diagrams depicted in equation (??) into the cor-responding integrals.
Since in equation (??) the product of two averaged Green’s functionsappears in the integral it can conveniently be expressed through the Drude-Boltzmann contribution to the transition probability (??)
ΓdiffE (~q1, ~q2) = Bδ(~q1 − ~q2) +
1
τe
∫d~q ′Γdiff
E (~q1, ~q′)PDB(~q ′, ~q2, E) .
(6.106)
The equation is still a complicated integral equation which can in general notbe solved in a closed form. For homogeneous systems it can be converted intoan algebraic equation by Fourier transformation in the same fashion as thecorresponding equation (??) for the averaged Green’s function. This yields
ΓdiffE (~k) =
B
1− PDB(~k,E)/τe. (6.107)
From this an expression in Fourier space for Pdiff(~k, E) can be worked out,which then can approximately transformed back into real space.
We do not follow this route here, but choose a different way to find anapproximate solution of (??). An integral equation of the type (??) canalways be transformed into a differential equation of infinite order by Taylorexpanding Γdiff
E with respect to its second argument
ΓdiffE (~q1, ~q
′) = ΓdiffE (~q1, ~q2) + (~q ′ − ~q2)∇~q2Γdiff
E (~q1, ~q2)
+1
2
[(~q ′ − ~q2)∇~q2
]2ΓdiffE (~q1, ~q2) + . . . . (6.108)
If we assume that ΓdiffE (~q1, ~q2) depends only weakly on the endpoint ~q2 and re-
calling that PDB decays exponentially with distance on the length scale of le,we can confine us to the first two terms. This is called diffusion approxima-tion. Plugging this equation into equation (??) shows that we transformed itinto a second order partial differential equation. The remaining integrationscan be performed easily∫
d~q ′PDB(~q ′, ~q2, E) =τe
1− iEτe/~≈ τe −
E
~τ 2e∫
d~q ′PDB(~q ′, ~q2, E)(~q ′ − ~q2)2 ≈ 2l2eτe (6.109)
133
Altogether one finds for ΓdiffE(
− iE
~−Dd∆~q2
)ΓdiffE (~q1, ~q2) =
B
τeδ(~q1 − ~q2) (6.110)
This is a diffusion equation in frequency space with diffusion constant Dd =vFled
=v2Fτed
. We next turn to the diffusion part of the transition probabilityPdiff and show that Pdiff fulfills a diffusion equation as well. We look at theFeynman diagramm representing Pdiff
Pdiff =~q1 ~q2 . (6.111)
This translates into the integral
Pdiff(~q1, ~q2, E) =1
2π%F
∫d~qd~q ′Γdiff
E (~q, ~q ′)⟨GR(~q1, ~q, E0 + E/2)
⟩ ⟨GA(~q, ~q1, E0 − E/2)
⟩⟨GR(~q ′, ~q2, E0 + E/2)
⟩ ⟨GA(~q2, ~q
′, E0 − E/2)⟩,(6.112)
which can again be expressed through the Drude–Boltzmann contributionPDB calculated in equation (??)
Pdiff(~q1, ~q2, E) = 2π%F
∫PDB(~q1, ~q, E)Γdiff
E (~q, ~q ′)PDB(~q ′, ~q2, E)d~qd~q ′
(6.113)
We now act with the Laplace operator in the second argument on Pdiff . SincePDB(~q ′, ~q2, E) depends only on ~q2−~q ′ we can integrate by parts with respectto ~q ′. As a result Pdiff(~q1, ~q2, E) fulfills a diffusion equation as well(
− iE
~−Dd∆~q2
)Pdiff(~q1, ~q2, E) = δ(~q1 − ~q2) . (6.114)
This becomes upon Fourier transformation the standard heat kernel equationin d dimensions(
∂
∂t−Dd∆~q2
)Pdiff(~q1, ~q2, t) = δ(~q1 − ~q2)δ(t) (6.115)
In free space the solution is given by
Pdiff(~q1, ~q2, t) =1
(4πDdt)d/2exp
(− 1
4Ddt|~q − ~q ′|2
). (6.116)
134
We next look at the diagramm of the Cooperon part Pcoop. It is depicted inequation (??).
~q1 ~q2Pcoop =
(6.117)
Here the maximal crossing of all connections between forward and backwardpropagator in the original diagramm are converted into the twisted loop inin the backward propagator line. From the picture it is clear that in thisform the cooperon vertex function Γcoop
E equals the diffuson vertex functionΓcoopE = Γdiff
E . This statement can also be proven rigorosly. The Feynmandiagramm corresponds to the following integral
Pcoop(~q1, ~q2, E) =1
2π%F
∫d~qd~q ′Γcoop
E (~q, ~q ′) (6.118)⟨GR(~q1, ~q, EF + E/2)
⟩ ⟨GR(~q ′, ~q2, EF + E/2)
⟩⟨GA(~q ′, ~q1, EF − E/2)
⟩ ⟨GA(~q2, ~q, EF − E/2)
⟩Since vertices are here interchanged as compared with the diffuson graph theintegrand can not be expressed in terms of Drude–Boltzmann contributions.As a consequence the Cooperon contribution PCoop is not a solution to adiffusion equation. An approximation, similar to the diffusion approximationyields the following result, which we state without derivation
Pcoop(~q, ~q ′, E) =Pdiff(~q, ~q, E)sin2 kF|~q − ~q ′|k2
F|~q − ~q′|2
e−|~q−~q′|/le (6.119)
where kF is the Fermi wave vector. For a detailed derivation see ?. In thetime domain the cooperon part of the transition probability reads in threedimensions
Pcoop(~q, ~q ′, t) =1
(4πDdt)d/2sin2 kF|~q − ~q ′|k2
F|~q − ~q′|2
e−|~q−~q′|/le . (6.120)
The diffuson and cooperon contribution to the transition probability are plot-ted in figure ?? for three times.
135
0.0
0.1
0.2
0.3
0.4
0.5
0.6P
(|~q−~q′ |,t)
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
|~q − ~q ′|
t = 1t = 0.5t = 0.2
Figure 6.8: Diffuson and Cooperon contribution to the transition probabilityon the scale of the diffusion constant Dd = 1 for three different times. Weset ~ = m = 1. The other free parameter is the initial velocity vF of theparticle. It is taken as vF = 10Dd.
The cooperon contribution to the transition probability is a genuine quan-tum effect and therefore a prime example of a mesoscopic phenomenon. Fromfigure ?? it is seen that its visibility is restricted to a small area around theinitial point. Most notably it leads to an enhanced probability of return Z(t)
Z(t) =
∫d~qP (~q, ~q, t) = 2
L3
(4πDdt)3/2. (6.121)
6.6 Energy correlations
With the diagramatic perturbation theory we have a tool to calculate ap-proximately the energy–energy correlation function of a disordered system.Recalling the definition of the energy–energy correlation function first givenin Sec. ?? in Eq. (??) we write
〈R2(E,E ′)〉 =1
π2
⟨Im tr GR(E)Im tr GA(E ′)
⟩. (6.122)
In order to calculate this we need not only 〈tr GRtr GA〉 but 〈tr GR tr GR〉 aswell then
〈R2(E,E ′)〉 =1
2π2Re(⟨
tr GR(E)tr GA(E ′)⟩−⟨
tr GR(E)tr GR(E ′)⟩)
.
(6.123)
136
We will see that this causes no additional calculation effort. We introduceEF = (E +E ′)/2 and ω = (E−E ′)/D, E+ = EF +Dω/2, E− = EF−Dω/2where D = ρ(EF) as usual. We proceed along the same perturbative linesas in the calculation of the diffuson and of the cooperon. We now have tocalculate the averages of the following product of Greens functions
LRA(~q1, ~q2) =⟨GR(~q1, ~q1, E
+)GA(~q2, ~q2, E−)⟩
LRR(~q1, ~q2) =⟨GR(~q1, ~q1, E
+)GR(~q2, ~q2, E−)⟩
(6.124)
in the random phase approximation. This corresponds to Feynman graphsof the types
~q
~q2
~q1
~q ′
Γs(~q1, ~q2)
a)
(6.125)
The integral corresponding to the diffuson Feynman diagram a) is
LRA(~q, ~q prime) = 〈GR(~q, ~q1, E+)〉〈GR(~q, ~q1, E
+)〉〈GR(~q, ~q1, E+)〉〈GR(~q, ~q1, E
+)〉(6.126)
GR (r, r1)GR (r2, r)GA (r1, r )GA (r, r2)(r1, r2) dr1 dr2.
6.7 Finite size effects and Anderson localisa-
tion
As mentioned frequently before, mesoscopic systems are distinguished bytheir finite extension. The size of the probe is smaller than the coherencelength. The Green’s function of the diffusion equation is therefore not a
137
simple Gaussian but a complicated superposition of all eigenmodes of thediffusion operator. The variance 〈|~q(t) − ~q ′|2〉 which increases linearly forfree diffusion as
〈|~q(t)− ~q ′|2〉 = 2Ddt (6.127)
can not increase ad infinitum but is bound from above by the limiting valueL2. This corresponds to an equidistributed wavefunction across the probe.The time scale at which the variance starts to deviate from the linear behav-ior, corresponds to the mean time the particle requires to explore the probediffusively τD = L2/Dd. It is called the Thouless time. The energy scaleassociated with the Thouless time is called Thouless energy EC = ~/τD. Wecan compare the Thouless time with the Heisenberg time. Their ratio is adimensionless quantity, which scales as
tHτD∼ Ld−2 (6.128)
with the size of the system. We see that, if we increase the system size by sayL, this ratio increases for d = 3 by a factor 2, for d = 2 it remains invariantand for d = 1 it decreases by a factor 2. We can write a differential equationfor g = tH/τD as a function of s = lnL
dg(s)
ds∼ (d− 2)g(s) (6.129)
or with β = d ln g/(d lnL)
β(g) ∼ d− 2 , (6.130)
which is called β–equation and the quantity g is the called dimensionlessconductance or Thouless energy on the scale of mean level spacing. Eqs. ??and ?? have most remarkable consequences. For d = 3 a finite initial valueof g becomes bigger and bigger as system size increases. The time the parti-cle needs to diffuse through the probe measured in units of Heisenberg timedecreases more and more and the system becomes a better and better con-ductor. One the other hand in one dimension as system size increases theThouless time increases more and more as compared to Heisenberg time andeventually becomes infinite. The particle never happens to diffuse throughthe whole probe. It is stuck somewhere in the probe. Its wave function isessentially localized to a small region of the probe. The system becomes aninsulator. This is the celebrated Anderson localisation in one dimension. Thecase d = 2 is the marginal case β = 0.
138
Figure 6.9: Sketch of the β–function as a function of the dimen-sionless conductance g.
These considerations do not imply that in two or three dimensions noAnderson localization can occur. If disorder is strong enough indeed manyeigenfunctions of the disordered system are localized also in three dimensions.A wave function, which is localized at a point ~q0 in the probe is of the form
ψ~q0(~q) ∼ exp
(−|~q − ~q0|
ξ
). (6.131)
Eigenfunctions of this form lead to an exponential decay of the conductanceG(L) ∼ exp(−L/ξ) which is reflected in a β–function β(g) = ln g. Theone–paramter scaling hypothesis asserts that β(g) is a smooth function ofthe dimensionless conductance as depicted in Fig. ??. Metallic behavior isreflected by regions, where β > 0 and where the conductance increases withthe size of the probe. Insulating behavior is reflected by β < 0, where theconductance decreases with system size. The critical unstable fixed point isat β(g∗) = 0. At this point extended states and localized states coexist inthe probe.
Figure 6.10: The mobility edge ofthe metal insulator transition in a3d disordered metal. The extendedstates and the localized states areseparated by a critical energy. Atthe critical point the localizationlength diverges. Taken from ?
139
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