fermi liquids and luttinger liquids

8/3/2019 Fermi Liquids and Luttinger Liquids

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2 Fermi Liquids and Luttinger Liquids

Heinz J. Schulz, Gianaurelio Cuniberti, and Pierbiagio Pieri

2.1 Introduction

In these lecture notes, corresponding roughly to lectures given at the sum-mer school in Chia Laguna, Italy, in September 1997, an attempt is madeto present the physics of three-dimensional interacting fermion systems (veryroughly) and that of their one-dimensional counterparts, the so-called Lut-

tinger liquids (in some more detail). These subjects play a crucial role in anumber of currently highly active areas of research: high temperature andorganic superconductors, quantum phase transitions, correlated fermion sys-tems, quantum wires, the quantum Hall effect, low-dimensional magnetism,and probably some others. Some understanding of this physics thus certainlyshould be useful in a variety of areas, and it is hoped that these notes will behelpful in this.

As the subject of these lectures was quite similar to those delivered at LesHouches, some overlap in the notes [1] was unavoidable. However, a numberof improvements have been made, for example a discussion of the Kleinfactors occurring in the bosonization of one-dimensional fermions, and newmaterial added, mainly concerning spin chains and coupled Luttinger liquids.

Some attempt has been made to keep references up to date, but this certainlyhas not always been successful, so we apologize in advance for any omissions(but then, these are lecture notes, not a review article).

2.2 Fermi Liquids

Landaus Fermi liquid theory [24] is concerned with the properties of amany-fermion system at low temperatures (much lower than the Fermi en-ergy) in the normal state, i.e. in the absence or at least at temperatures aboveany symmetry breaking phase transition (superconducting, magnetic, or oth-erwise). The ideal example for Landaus theory is liquid helium 3, above itssuperfluid phase transition, however, the conceptual basis of Landaus theory

is equally applicable to a variety of other systems, in particular electrons inmetals. Quantitative applications are however more difficult because of a va-riety of complications which appear in real systems, in particular the absenceof translational and rotational invariance and the presence of electron-phononinteractions, which are not directly taken into account in Landaus theory.


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10 Heinz J. Schulz et al.

Subsequently, I will first briefly discuss the case of a noninteracting many-fermion system (the Fermi gas), and then turn to Landaus theory of theinteracting case (the liquid), first from a phenomenological point of view, andthen microscopically. A much more detailed and complete exposition of thesesubjects can be found in the literature [59].

2.2.1 The Fermi Gas

In a noninteracting translationally invariant systems, the single-particle eigen-states are plane waves

|k = 1

eikr (2.1)

with energy

k = k2

2m, (2.2)

where is the volume of the system, and we will always use units so thath = 1. The ground state of an N-particle system is the well-known Fermisea: all states up to the Fermi wavevector kF are filled, all the other statesare empty. For spin-1/2 fermions the relation between particle number andkF is

N = k3F

32. (2.3)

The energy of the last occupied state is the so-called Fermi energy EF =k2F/(2m), and one easily verifies that

EF =E0(N)

N= (T = 0) (2.4)

i.e. EF is the zero-temperature limit of the chemical potential (E0(N) in theformula above is the ground state energy).

It is usually convenient to define the Hamiltonian in a way so that the ab-solute ground state has a well-defined fixed particle number. This is achievedsimply by including the chemical potential in the definition of the Hamil-tonian, i.e. by writing

H =k

knk , (2.5)

where nk is the usual number operator, k = k, and the spin summationis not written explicitly (at finite temperature this of course brings one tothe usual grand canonical description where small fluctuations of the parti-cle number occur). With this definition of the Hamiltonian, the elementaryexcitations of the Fermi gas are


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2 Fermi Liquids and Luttinger Liquids 11

addition of a particle at wavevector k (nk = 1). This requires |k| > kF,and thus the energy of this excitation is k = k > 0,

destruction of a particle at wavevector k (nk =

1), i.e. creation of a

hole. This requires |k| < kF, and thus the energy is k = k > 0.The dispersion relation of the elementary particle and hole excitation is shownin Fig. 2.1a.

k particlesholes

Fk(a)F2k

E

k(b)

Fig. 2.1. (a) the energy-momentum relation for the elementary particle (k > kF)and hole (k < kF) excitations; and (b) the particle-hole continuum

These excitations change the total number of particles. Construction ofstates at constant particle number is of course straightforward: one takes oneparticle from some state k, with

|k|

< kF, and puts it into a state k, with

|k| > kF. These particle-hole excitations are parameterized by the two quan-tum numbers k, k and thus form a continuum, as shown in Fig. 2.1b. Therestriction on the allowed values ofk, k insures that all particle-hole stateshave positive energy. Higher excited states, i.e. states with many particlesand many holes, are straightforwardly constructed, the only restriction beingimposed by the Pauli principle.

Thermodynamic quantities are easily obtained and are all determined bythe density of states at the Fermi energy. For example, the specific heat obeysthe well known linear law C(T) = T, with

=22

3N(EF)k

2B (2.6)

and similarly the (Pauli) spin susceptibility and the compressibility aregiven by

= 2N(EF)2B (2.7)

= 2N(EF)/2 . (2.8)


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Here for the quadratic dispersion relation (2.2) the density of states (perspin) at the Fermi energy is given by N(EF) = mkF/(2

2), but it should beemphasized that eqs. (2.6) to (2.8) are valid for an arbitrary density of states,in particular in solids where bandstructure effects can change the electronicdispersion relation quite drastically. Thus, for noninteracting electrons oneexpects the so-called Wilson ratio

RW =2k2B32B

(2.9)

to be unity, independently of details of the bandstructure. Any deviationfrom unity is necessarily an indication of some form of interaction effect.

2.2.2 Landaus Theory of Fermi Liquids

Basic hypothesis. Landaus theory is to a large extent based on the idea ofa continuous and one-to-one correspondence between the eigenstates (groundstate and excited states) of the noninteracting and the interacting system.For this to be an acceptable hypothesis it is crucial that the interactions donot lead to any form of phase transition or symmetry-broken ground state.

In particular one can consider a state obtained by adding a particle (withmomentum |p| > kF) to the noninteracting ground state:

|p, N + 1 = a+p |0, N . (2.10)

Here a+p is a fermion creation operator for momentum state p, and |0, N isthe N-particle ground state of the noninteracting system. Now we add someform of particle-particle interaction. In a translationally invariant system, in-

teractions conserve total momentum, and thus even after switching on theinteraction the state still has total momentum p. However, the interactionof the added particle with the filled Fermi sea, as well as the interaction ofthe particles in the sea amongst themselves, will change the distribution ofparticles in k-space, and of course also modify the energy of our state. Thecomplex formed by the particle added at p and the perturbed distribution ofthe other particles is called a Landau quasiparticle. The Pauli principle im-plied |p| > kF in the absence of interactions, and by the continuity hypothesisthe same restriction remains valid in the interacting case. In particular, thevalue of kF, which imposes a lower limit on the allowed momentum of thequasiparticle, is unchanged by the interactions.

Analogous considerations can be performed for a state obtained by de-struction of a particle (e.g. creation of a hole):

|p, N 1 = ap|0, N . (2.11)

Note that due to the momentum p the total momentum of this state isindeed p.


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The quasi-particle concept has a certain number of limitations, mainly dueto the fact that, as will be discussed below, the lifetime of a quasi-particleis finite. However, for excitations close to kF one has 1/

(

EF)

2, i.e.the lifetime becomes much longer than the inverse excitation energy, andthe quasi-particles therefore are reasonably well defined. In practice, thismeans that Landaus theory is useful for phenomena at energy scales muchsmaller than the Fermi energy, but inapplicable otherwise. In metals, whereEF 3 . . . 5 eV, this restriction is not too serious when one is concernedwith thermodynamic or transport properties. One should also note that theground state energy itself has important contributions from states well belowEF, and therefore is not accessible to Landaus theory.

Equilibrium properties. In order to derive physical quantities from thepicture of the low-energy excitations, we need some information about the en-ergetics of the quasiparticles and of their interactions. To be specific, startingfrom the ground state quasiparticle distribution

n0(k) = 1 if |k| < kF= 0 if |k| > kF (2.12)

one considers changes in quasiparticle occupation number of the form n0(k) n0(k) + n(k), i.e. n(k) = 1 represents an excited quasi-particle, n(k) = 1an excited quasi-hole (with the notation k = (k, ), and =, the spinindex). The corresponding change in energy is

E =k

0kn(k) +1

2

kk

f(k, k)n(k)n(k) , (2.13)

where the first and second term represent the energy of a single quasi-particleand the interaction between quasiparticles, respectively. To be more precise,we assume that the chemical potential is included in the Hamiltonian, as in(2.5). Consequently, 0k vanishes on the Fermi surface, and, given that we aremainly interested in phenomena in the vicinity of kF, it is sufficient to retainthe lowest order term in an expansion around |k| = kF. One thus writes

0k =kFm

(|k| kF) , (2.14)

thus defining the effective mass m which is different from the bare massm due to interaction effects that could in principle be calculated from amicroscopic theory of the system.

The energy of a quasi-particle added to the system is easily obtained from

(2.13) by calculating the difference in E between a state with n(k) = 1 anda state with n(k) = 0. One finds

k = 0k +

1

k

f(k, k)n(k) , (2.15)


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i.e. the energy of an added quasi-particle is not just the bare quasi-particleenergy 0k but also depends, via the interaction term, on the presence of theother quasi-particles. Given that the non-interacting particles obey FermiDirac statistics, the quasi-particles do so too, and consequently, the occupa-tion probability of a quasi-particle state is given by

n(k) =1

ek + 1. (2.16)

Note that the full and not the bare quasi-particle energy enters this expres-sion. In principle, n(k) thus has to be determined self-consistently from (2.15)and (2.16).

For the subsequent calculations, it is convenient to transform the quasipar-ticle interaction f(k, k). First, spin symmetric and antisymmetric f-functionsare defined via

f(k ,k ) = fs(k,k) + fa(k,k)f(k ,k ) = fs(k,k) fa(k,k) . (2.17)

Moreover, given the implicit restrictions of the theory, one is only interestedin processes where all involved particles are very close to the Fermi surface.Under the assumption that the interaction functions are slowly varying asa function of k, one then can set |k| = |k| = kF. Because of rotationalsymmetry, the f-functions then can only depend on the angle between k andk, called . One can then expand the f-function in a Legendre series as

fa,s(k,k) =L=0

fa,sL PL(cos ) , cos =k k

k2F, (2.18)

where the PL are the Legendre polynomials. Finally, one usually puts thesecoefficients into dimensionless form by introducing

Fa,sL =kFm

2fa,sL . (2.19)

We are now in a position to calculate some equilibrium properties. Thefirst one will be the specific heat at constant volume

C =1

U

T, (2.20)

where U is the internal energy. The temperature-dependent part of U comes

from thermally excited quasi-particles, as determined by the distribution(2.16). In principle, in this expression k is itself temperature-dependent,because of the temperature dependent second term in (2.15). However, onecan easily see that this term only gives contributions of order T2, and there-fore can be neglected in the low-temperature limit. Consequently, one can


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indeed replace k by 0k, and then one only has to replace the bare mass bym in the result for a non-interacting system to obtain

C =mkF

3k2BT . (2.21)

The spin susceptibility (at T = 0) is related to the second derivative ofthe ground state energy with respect to the (spin) magnetization M:

=

2E0M2

1. (2.22)

Spin magnetization is created by increasing the number of spin particlesand decreasing the number of spins (M = B(N N)), i.e. by changingthe Fermi wavevectors for up and down spins: kF kF + kF for = andkF

kF

kF for =

.

By calculating with (2.13) the corresponding change of the ground stateenergy, we obtain from (2.22):

=1

1 + Fa0

2BkFm

2. (2.23)

Note that here, and contrary to the specific heat, interactions enter not onlyvia m but also explicitly via the coefficient Fa0 , which is the only coefficientthat appears here because the distortion of the Fermi distribution functionis antisymmetric in the spin index and has spherical symmetry ( L = 0). TheWilson ratio is then

RW =1

1 + Fa0. (2.24)

Following a similar reasoning, one can calculate the compressibility ofa Fermi liquid:

= 1

P=

2E02

1=

mkF22(1 + Fs0 )

. (2.25)

It is also interesting that in a translationally invariant system as we haveconsidered here, the effective mass is not independent of the interaction co-efficients. One can show indeed, by exploiting the Galilean invariance of thesystem, that

mm

= 1 + Fs1 /3 . (2.26)

Nonequilibrium properties. As far as equilibrium properties are con-cerned, Landaus theory is phenomenological and makes some important


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qualitative predictions, the most prominent being that even in the presenceof interactions the low-temperature specific heat remains linear in tempera-ture and that the spin susceptibility tends to a constant as T

0. However,

Landaus theory has little quantitative predictive power because the crucialLandau parameters have actually to be determined from experiment. The sit-uation is different for non-equilibrium situations, where the existence of newphenomena, in particular collective modes, is predicted. These modes are an-other kind of elementary excitations which, contrary to quasiparticles, involvea coherent motion of the whole system. We shall not enter into the detailsof the treatment of non-equilibrium properties of Fermi liquids (see [5,8,9])and just briefly sketch the general conceptual framework and some of themore important results. To describe non-equilibrium situations, one makestwo basic assumptions:

Deviations from equilibrium are described by a Boltzmann equationfor aspace- and time- dependent quasiparticle distribution function n(k,r, t),which describes the density of quasi-particles of momentum and spin(k, ) at point r and time t. At equilibrium, n is of course given by(2.12). The fact that in the distribution function one specifies simultane-ously momentum and position of course imposes certain restrictions, dueto the quantum-mechanical nature of the underlying problem. More pre-cisely, spatial and temporal variations of the distribution function have tobe slow compared to the typical wavelength and frequency of the quasi-particles. We have then the conditions vF|q|, || < EF, where q and ||set the scale of the spatial and temporal variations of n(r, t).

Because of the r-dependent n, the quasiparticle energy is itself, via (2.15),r-dependent. One then assumes the following quasi-classical equations ofmotion

r = kk(r)

k = rk(r) . (2.27)Note in particular that a space-dependent distribution function gives rise,via the f(k, k) function, to a force acting on a quasi-particle.

By linearizing the Boltzmann equation and studying the collisionless regi-me, where the collision term in the Boltzmann equation can be neglected, onefinds collective mode solutions which correspond to oscillations of the Fermisurface. The most important one is the longitudinal symmetric mode which,like ordinary sound, involves fluctuations of the particle density. This kindof sound appears, however, in a regime where ordinary sound cannot exist

(the existence of collisions is indeed crucial for the propagation of ordinarysound waves) and is a purely quantum effect. Since collisions can always beneglected at very low temperatures, this new kind of sound has been calledby Landau zero sound. The collision term of the Boltzmann equation is onthe contrary essential to calculate the quasi-particle lifetime . One can find


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indeed, for a quasiparticle of energy p

1

m3

(T)2 + 2p1 + ep . (2.28)

The most important result here is the divergence of the lifetime for low ener-gies and temperatures as max(p, T)2, so that the product p in factdiverges as the Fermi surface is approached. This shows that the quasiparticlebecomes a well-defined (nearly-) eigenstate at low excitation energies, i.e. inthe region where Landaus theory is applicable. On the other hand, at higherenergies the quasiparticle becomes less and less well-defined. One may notethat initially we had assumed that a quasiparticle is an exact eigenstate ofthe interacting system, which was obtained from a noninteracting eigenstateby switching on the interaction, and therefore should have infinite lifetime.We now arrive at the conclusion that the lifetime is not strictly infinite, but

only very long at low energies. In the following section we will try to clarifythis from a microscopical point of view.

2.2.3 Microscopic Basis of Landaus Theory

At our current knowledge, it does not seem generally possible to derive Lan-daus theory starting from some microscopic Hamiltonian, apart possibly inperturbation theory for small interactions. It is however possible to formulatethe basic hypotheses in terms of microscopic quantities, in particular one- andtwo-particle Green functions. This will be outlined below.

Quasiparticles. As far as single particle properties are concerned it is suf-

ficient to consider the one-particle Green function

G(k, ) = Tak()a+k (0) , (2.29)

where is the usual (Matsubara) imaginary time. In this quantity, interac-tion effects appear via self-energy corrections in the Fourier transformedfunction

G(k, ) = 1i 00k (k, )

. (2.30)

Here 00k is the bare particle energy, without any effective mass effects. Exci-tation energies of the system then are given by the poles of G(k, ). In these

terms, Landaus assumption about the existence of quasiparticles is equiva-lent to assuming that (k, ) is sufficiently regular close to the Fermi surfaceas to allow an expansion for small parameters. Regularity in (k, ) space im-plies that in real space the self-energy has no contributions that decay slowlyin time and/or space. Given that the self-energy can be calculated in terms


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of the effective interaction between particles this is certainly a reasonable as-sumption when the particleparticle interaction is short-range (though thereis no formal prove of this). For Coulomb interactions, screening has to beinvoked to make the effective interaction short ranged.

One can further notice that (kF, 0) just renormalizes the chemical po-tential. Given that we want to work at fixed particle number we can absorbthis term in the effective . Expanding then to first order around the Fermisurface, the Green function takes the form

G(k, ) = zi 0k

, (2.31)

where 0k has the form (2.14) of the phenomenological approach, with

m = m1

1 +m

kF

k 1

, (2.32)

and the quasiparticle renormalization factor is

z =

1

1. (2.33)

All derivatives are to be taken at the Fermi surface and at = 0. One shouldnotice that a sum rule imposes that the frequency-integrated spectral density

A(k, ) = 1

Im G(k,i + i) (2.34)

equals unity. Consequently, in order to fulfill the sum rule, if z < 1 there has

to be a contribution in addition to the quasiparticle pole in (2.31). This isthe so-called incoherent background from single and multiple particleholepair excitations which can extend to rather high energies but becomes smallclose to the Fermi surface.

The form (2.31) gives rise to a jump in the momentum distribution func-tion at kF of height z, instead of unity in the noninteracting case (Fig. 2.2). Inaddition to the jump, the incoherent background gives rise to a contributionwhich is continuous through kF.

A finite quasiparticle lifetime arises if the expansion of (k, ) is carriedto second order. Then (2.31) generalizes to

G(k, ) = zi 0k + isign()()1

, (2.35)

where () is typically given by an expression like (2.28).

Quasiparticle interaction. The quasiparticle interaction parametersf(k, k) are expected to be connected to the two-particle vertex function.


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Fk k

nk

Fig. 2.2. The momentum distribution function nk = a+k ak in the interacting (fullline) and noninteracting (dashed line) cases

This function, which we will denote (2)(P1, P2; K) describes the scatteringof two particles from initial state P1, P2 to the final state P1 K, P2 + K,and the notation is Pi = (p

i, i, i). The contribution of first and second

order in the interaction potential V(k) are shown in Fig. 2.3. Let us nowstudy the case of small transfer K, but arbitrary P1,2, only restricted to beclose to the Fermi surface. One then notices that diagram 2.3b (part c) givesrise to singularities, because the poles of the two intervening Green functionscoalesce. On the other hand diagrams 2.3b (part a) and 2.3b (part b) remainnon-singular for small K. This motivates one to introduce a two-particle ir-reducible function (2) which is the sum of all contributions which do notcontain a single product G(Q)G(K+ Q). This function then is non-singularfor small K, and consequently the total vertex function is determined by theintegral equation

(P1, P2; K) = (P1, P2) +1

(2)4

dd3q(P1, Q)G(Q)G(K+ Q)(Q, P2, K) . (2.36)

For simplicity, the spin summation is omitted here. The singular contributionnow comes from small K and Q in the vicinity of the Fermi surface. Inthis area the Q-dependence of the s in (2.36) is non-singular and can beneglected. The energy and radial momentum integral over Q can then bedone, leading to

(P1, P2; K) = (P1, P2) +z2k2F(2)3

d2q(P1, Q)

q k

vFq k(Q, P2, K) , (2.37)

where q is a vector on the Fermi surface, and d2q is the correspondingangular integration. Here only the quasiparticle pole in G has been takeninto account. The contribution from the incoherent parts can in principle beabsorbed into the definition of .


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2

=1(P ,P ;K) 2

(2)

2P

1P 1P -K

2P +K

= V(k) + V(k-p +p )1

(a)

2

1

2

1

2

1

(c)

(b)

(a)K+Q

P

Q

PQ

P

PP

P

Q

1 2P +P -Q

1P -K

1P -KP -K1

P +K2

2P +K

P +K2

21P -P -K+Q

(b)

Fig. 2.3. The lowest (a) and second order (b) contributions to the two-particlevertex (2). Note that the external lines do not represent actual Green functionsbut only indicate the external connections

The expression (2.37) is clearly singular because it has radically differentbehavior according to whether one first sends k or to zero. In the first case,the limit of can be related to the Landau f-function, while the second oneis relevant for the calculation of the transition probabilities determining thelifetime of quasiparticles. Here we will consider only the former case. Sendingk to zero first one finds straightforwardly

lim0

(limk0

) (P1, P2) = (P1, P2) . (2.38)

Closer inspection then shows that in this case the poles of the two Greenfunctions in (2.36) are always on the same side of the real axis and con-sequently the singular term in (2.36) vanishes. To make the identification

between (P1, P2) and the Landau f-function we notice that the densityresponse function at energy-momentum K, whose poles give the collective(zero-sound) modes, contains the interactions via (P1, P2, K). In particu-lar, the existence of a pole in the response function implies a pole in . Thecomparison between the equations for this pole and those which can be ob-


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tained through the Boltzmann equation within Landaus theory allows thenthe identification

f(k, k) = z2(k, k) . (2.39)

2.2.4 Summary

The basic assumption of Landaus theory is the existence of low-energy quasi-particles with a very long lifetime, and their description in terms of a rathersimple energy functional, (2.13). From this a number of results for thermo-dynamic properties is obtained. At this level, the theory is of little quantita-tive power because the Landau parameters are not determined. Qualitatively,however, the predictions are important: the low-temperature thermodynamicproperties of an interacting fermion system are very similar to those of anoninteracting system, the interactions only lead to quantitative renormal-

izations. Actual quantitative predictions are obtained when one extends thetheory to nonequilibrium properties, using the Boltzmann equation [5]. A newphenomenon predicted (and actually observed in 3He [10]) is the existence ofcollective excitations, called zero sound. This approach also allows the cal-culation of the quasiparticle lifetime and its divergence as the Fermi energyis approached, as well as the treatment of a number of transport phenomena.

As already mentioned, the ideal system for the application of Landaustheory is 3He, which has both short-range interaction and is isotropic. Theapplication to electrons in metals is more problematic. First, the interactionsare long-ranged (Coulombic). This can however be accommodated by prop-erly including screening effects. More difficulties, at least at the quantitativelevel, arise because metals are naturally anisotropic. This problem is not of

fundamental nature: even when the Fermi surface is highly anisotropic, anexpansion like (2.13) can still be written down and thus interaction param-eters can be defined. However, a simple Legendre expansion like (2.18) isnot in general possible and the description of the quasiparticle interaction interms of a few parameters becomes impossible. An exception case, with a verynearly spherical Fermi surface, are the alkali metals, where a determinationof Landau parameters can indeed be attempted [5]. It should be noticed thatthe difficulties with the Landau description of metals are not of conceptualnature and in particular do not invalidate the quasiparticle concept but arerather limitations on the usefulness of the theory for quantitative purposes.

Landaus theory can be interpreted in terms of microscopic quantitieslike Green functions (the quasiparticle pole) and interaction vertices, as dis-cussed above. It should however be emphasized that these arguments do pro-

vide a microscopic interpretation of Landaus picture, rather than provingits correctness. Similar remarks apply to the calculated diverging quasiparti-cle lifetime: this at best show that Landaus picture is internally consistent.Considerable progress towards a deeper formal understanding of Fermi liquidtheory has been made in recent years [11,12].


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2.3 Renormalization Group for Interacting Fermions

In this chapter, we will consider properties of interacting fermions in theframework of renormalization group theory. This will serve two purposes:first, the treatment of one-dimensional interacting fermions, which will beconsidered in considerable detail in the following chapters, gives rise to diver-gences which can only be handled by this approach. Results obtained in thisway will be an essential ingredient in the subsequent discussion of Luttingerliquids. More generally, the renormalization group method will clarify thestatus of both Landaus Fermi liquid theory and the Luttinger liquid pictureas renormalization group fixed points, thus establishing a link with a num-ber of other phenomena in condensed matter physics. We will formulate theproblem in terms of fermion functional integrals, as done by Bourbonnais inthe one-dimensional case [13] and more recently for two and three dimensionsby Shankar [14]. For the most part, I will closely follow Shankars notation.

Before considering the interacting fermion problem in detail, let us brieflyrecall the general idea behind the renormalization group, as formulated byKadanoff and Wilson: one is interested in the statistical mechanics of a sys-tem described by some Hamiltonian H. Equilibrium properties then are de-termined by the partition function

Z =

configurations

eH =

configurations

eS , (2.40)

where the second equality defines the action S = H. Typically, the actioncontains degrees of freedom at wavevectors up to some cutoff , which isof the order of the dimensions of the Brillouin zone. One wishes to obtainan effective action containing only the physically most interesting degrees

of freedom. In standard phase transition problems this is the vicinity of thepointk = 0, however, for the fermion problem at hand the surface |k| = kFis relevant, and the cutoff has to be defined with respect to this surface. Inorder to achieve this one proceeds as follows:

1. Starting from a cutoff-dependent action S() one eliminates all degreesof freedom between and /s, where s is a factor larger than unity. Thisgives rise to a new action S( = /s).

2. One performs a scale change k sk. This brings the cutoff back to itsoriginal value and a new action S() is obtained. Because of the degreesof freedom integrated out, coupling constants (or functions) are changed.

3. One chooses a value of s infinitesimally close to unity: s = 1 + , andperforms the first two steps iteratively. This then gives rise to differentialequations for the couplings, which (in favourable circumstances) can beintegrated until all non-interesting degrees of freedom have been elimi-nated.


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2.3.1 One Dimension

The one-dimensional case, which has interesting physical applications, will

here be mainly used to clarify the procedure. Let us first consider a nonin-teracting problem, e.g. a one-dimensional tight-binding model defined by

H =k

kakak , k = 2t cos k , (2.41)

where t is the nearest-neighbour hopping integral. We will consider the metal-lic case, i.e. the chemical potential is somewhere in the middle of the band.Concentrating on low-energy properties, only states close to the Fermi pointskF are important, and one can then linearize the dispersion relation to ob-tain

H = k,r=

vF(rk

kF)a

krakr , (2.42)

where vF = 2t sin kF is the Fermi velocity, and the index r differentiatesbetween right- and left-going particles, i.e. particles close to kF and kF. Tosimplify subsequent notation, we (i) choose energy units so that vF = 1, (ii)translate k-space so that zero energy is at k = 0, and (iii) replace the k-sumby an integral. Then

H =r=

dk

2rkar(k)ar(k) . (2.43)

For the subsequent renormalization group treatment we have to use afunctional integral formulation of the problem in terms of Grassmann vari-

ables (a detailed explanation of this formalism is given by Negele and Orland[15]). The partition function becomes

Z() =

DeS() , (2.44)

where D indicates functional integration over a set of Grassmann variables.The action is

S() =

0

d

r=

dk

2r(k, )r(k, ) + H(

, )

, (2.45)

where the zero-temperature limit

has to be taken, and H(, )

indicates the Hamiltonian, with each a replaced by a , and each a replacedby a . Fourier transforming with respect to the imaginary time variable

r(k, ) = Tn

r(k, n)ein (n = 2(n + 1/2)T) (2.46)


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and passing to the limit T 0 one obtains the noninteracting action

S0() =r=

d

2

dk

2 r(k, )[i + rk]r(k, ) . (2.47)

We notice that this is diagonal in k and which will greatly simplify thesubsequent treatment. Because of the units chosen, has units of (length)1

(which we will abbreviate as L1), and then r(k, ) has units L3/2.We now integrate out degrees of freedom. More precisely, we will integrate

over the strip /s < |k| < , < < . The integration over all keepsthe action local in time. One then has

Z() = Z(,/s)Z(/s) , (2.48)

where Z(, /s) contains the contributions from the integrated degrees of

freedom, and Z(/s) has the same form of (2.44). The new action is thenS0(/s) = S0(/s). Introducing the scale change

k = ks , = s , = s3/2 (2.49)

one easily finds that S0() = S0(). The action does not change thereforeunder scale change (or renormalization): we are at a fixed point. One shouldnotice that the scale change of k implies that k is quantized in in units ofk = 2s/L, i.e. eliminating degrees of freedom actually implies that we areconsidering a shorter system, with correspondingly less degrees of freedom.This means that even though the action is unchanged the new Z() is thepartition function of a shorter system. To derive this in detail, one has totake into account the change in the functional integration measure due to

the scale change on .Before turning to the problem of interactions, it is instructive to considera quadratic but diagonal perturbation of the form

S2 =r=

d

2

dk

2(k, )r(k, )r(k, ) . (2.50)

We assume that (k, ) can be expanded in a power series

(k, ) = 00 + 10k + 01i + . . . (2.51)

Under the scale change (2.49) one then has

nm s1

n

m

nm . (2.52)

There now are three cases:

1. a parameter nm grows with increasing s. Such a parameter is calledrelevant. This is the case for 00;


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2. a parameter remains unchanged (10,01). Such a parameter is marginal;

3. finally, all other parameters decrease with increasing s. These are calledirrelevant.

Generally, one expects relevant parameters, which grow after elimination ofhigh-energy degrees of freedom, to strongly modify the physics of the model.In the present case, the relevant parameter is simply a change in chemicalpotential, which doesnt change the physics much (the same is true for themarginal parameters). One can easily see that another relevant perturbationis a term coupling right- and left-going particles of the form m(12+

21).

This term in fact does lead to a basic change: it leads to the appearance of agap in the spectrum.

Let us now introduce fermionfermion interactions. The general form ofthe interaction term in the action is

SI =k

u(1234)(1)(2)(3)(4) . (2.53)

Here (3) is an abbreviation for r3(k3, 3), and similarly for the other fac-tors, while u is an interaction function to be specified. The integration mea-sure is

k

=

4i=1

di2

dki2

(k1 + k2 k3 k4)

(1 + 2 3 4) . (2.54)We now note that the dimension of the integration measure is L6, and the

dimension of the product of fields is L

6

. This in particular means that ifwe perform a series expansion of u in analogy to (2.51) the constant termwill be s-independent, i.e. marginal, and all other terms are irrelevant. Inthe following we will thus only consider the case of a constant (k- and -independent) u.

These considerations are actually only the first step in the analysis: in factit is quite clear that (unlike in the noninteracting case above) integrating outdegrees of freedom will not in general leave the remaining action invariant.To investigate this effect, we use a more precise form of the interaction term:

SI =

k

ss

{g1s+(1)s(2)s+(3)s(4)

+g2s+(1)

s

(2)s

(3)s+(4)

}. (2.55)

Here we have reintroduced spin, and the two coupling constants g1 and g2denote, in the original language of (2.42), backward ((kF, kF) (kF, kF))and forward ((kF, kF) (kF, kF)) scattering. Note that in the absence ofspin the two processes are actually identical.


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Now, the KadanoffWilson type mode elimination can be performed via

eS

= D eS , (2.56)

where D denotes integration only over degrees of freedom in the strip /s 0. Then one has the fixed point V = 0, F(1

2) arbitrary.What is the underlying physics of this fixed point? One notices that here3 = 1, 4 = 2, i.e. the marginal term in the action is

1

212 . Inthe operator language, this translates into

Hint

d1d2n1n2 . (2.79)

We now can recognize this as an operator version of Landaus energyfunctional, (2.13). The fixed point theory is thus identified as LandausFermi liquid theory.

The generalization of the above to three dimensions is rather straightfor-ward. In addition to the forward scattering amplitudes F, scattering wherethere is an angle 12;34 spanned by the planes (1,2) and (3,4) is also

marginal. For 12;34 = 0 these processes are the ones contributing to thequasiparticle lifetime, as discussed in Sect. 2.2.2, however they do not affectequilibrium properties. The (zero temperature) fixed point properties thusstill only depend on amplitudes for 12;34 = 0, i.e. the Landau f-function.

2.4 Bosonization and the Luttinger Liquid

The Fermi liquid picture described in the preceding two sections is believedto be relevant for most three-dimensional itinerant electron systems, rangingfrom simple metals like sodium to heavy-electron materials. The best under-stood example of non-Fermi liquid properties is that of interacting fermionsin one dimension. This subject will be discussed in the remainder of these

lecture notes. We have already started this discussion in Sect. 2.3.1, wherewe used a perturbative renormalization group to find the existence of onemarginal coupling, the combination g1 2g2. This approach, pioneered bySolyom and collaborators in the early 70s [16], can be extended to strongercoupling by going to second or even third order [17] in perturbation theory.A principal limitation remains however the reliance on perturbation theory,which excludes the treatment of strong-coupling problems. An alternativemethod, which allows one to handle, to a certain extent, strong-interactionproblems as well, is provided by the bosonization approach, which will bediscussed now and which forms the basis of the so-called Luttinger liquid de-scription. It should be pointed out, however, that entirely equivalent resultscan be obtained by many-body techniques, at least for the already highlynontrivial case of pure forward scattering [18,19].

2.4.1 Spinless Model: Representation of Excitations

The bosonization procedure can be formulated precisely, in the form of op-erator identities, for fermions with a linear energy-momentum relation, as


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discussed in Sect. 2.3.1. To clarify notation, we will use a+-(a-)operatorsfor right-(left-)moving fermions. The linearized noninteracting Hamiltonian(2.42) then becomes

H0 = vFk

(k kF)a+,ka+,k + (k kF)a,ka,k

, (2.80)

and the density of states is N(EF) = 1/(vF). In the Luttinger model [2022], one generalizes this kinetic energy by letting the momentum cutoff tend to infinity. There then are two branches of particles, right moversand left movers, both with unconstrained momentum and energy, as shownin Fig. 2.7. At least for weak interaction, this addition of extra states farfrom the Fermi energy is not expected to change the physics much. However,this modification makes the model exactly solvable even in the presence ofnontrivial and possibly strong interactions. Moreover, and most importantly,

many of the features of this model carry over even to strongly interactingfermions on a lattice.We now introduce the Fourier components of the particle density operator

for right and left movers:

(q) =k

a,k+qa,k . (2.81)

The noninteracting Hamiltonian (and a more general model including inter-actions, see below) can be written in terms of these operators in a rathersimple form and then be solved exactly. This is based on the following facts:

k

EF

k

Fig. 2.7. Single-particle energy spectrum of the Luttinger model. Occupied statesare shown in grey, the dark grey area represents the states added to make the modelsolvable


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1. the density fluctuation operators , with = , obey Bose type com-mutation relations:

[(q), (q)] = qq qL2 . (2.82)The relation (2.82) for q = q or = can be derived by straightforwardoperator algebra. The slightly delicate part is (2.82) for q = q. One easilyfinds

[+(q), +(q)] =k

(nkq nk) , (2.83)

where nk is an occupation number operator. In a usual system with afinite interval of states between kF and kF occupied, the summationindex of one of the n operators could be shifted, giving a zero answer in(2.83). In the present situation, with an infinity of states occupied belowkF

, this is not so. Consider for example the ground state and q > 0. Theneach term in (2.83) with kF < k < kF + q contributes unity to the sum,all other terms vanish, thus establishing the result (2.82). More generally,consider a state with all levels below a certain value k0 (< kF) occupied,but an arbitrary number of particle hole pairs excited otherwise. Onethen has, assuming again q > 0,

k

(nkq nk) =kk0

+k


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i.e. states created by +(q) are eigenstates of H0, with energy vFq. Con-sequently, the kinetic part of the Hamiltonian can be re-written as a termbilinear in boson operators, i.e. quartic in fermion operators:

H0 =2vF

L

q>0,=

(q)(q) . (2.87)

This equivalence may be made more apparent noting that +(q) cre-ates particlehole pairs that all have total momentum q. Their energy isk+q k, which, because of the linearity of the spectrum, equals vFq,independently of k. Thus, states created by +(q) are linear combina-tions of individual electron-hole excitations all with the same energy, andtherefore are also eigenstates of (2.80).

3. The above point shows that the spectra of the bosonic and fermionicrepresentations of H0 are the same. To show complete equivalence, one

also has to show that the degeneracies of all the levels are identical. Thiscan be achieved calculating the partition function in the two representa-tions and demonstrating that they are equal. This then shows that thestates created by repeated application of on the ground state form acomplete set of basis states [23,24].

We now introduce interactions between the fermions. As long as only for-ward scattering of the type (kF; kF) (kF; kF) or (kF; kF) (kF; kF) isintroduced, the model remains exactly solvable. The interaction Hamiltoniandescribing these processes takes the form

Hint =1

2L

q,=

(g2(q)(q)(q) + g4(q)(q)(q)) . (2.88)

Here, g2(q) and g4(q) are the Fourier transforms of a real space interaction po-tential, and in a realistic case one would of course have g2(q) = g4(q) = g(q),but it is useful to allow for differences between g2 and g4. For Coulombinteractions one expects g2, g4 > 0. In principle, the long-range part of theCoulomb repulsion leads to a singular q-dependence. Such singularities in thegi can be handled rather straightforwardly and can lead to interesting physi-cal effects as will be discussed below. Here I shall limit myself to nonsingularg2, g4. Electron-phonon interactions can lead to effectively attractive interac-tions between electrons, and therefore in the following I will not make anyrestrictive assumptions about the sign of the constants. One should howevernotice that a proper treatment of the phonon dynamics and of the resultingretardation effects requires more care [25].

Putting together (2.87) and (2.88), the complete interacting Hamiltonian,the TomonagaLuttinger model, then becomes a bilinear form in boson op-erators that is easily diagonalized by a Bogolyubov transformation. A first


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consequence is the expression for the excitation spectrum

(q) = |q|vF + g4(q)2 2

g2(q)2 2

. (2.89)

The diagonal boson operators are linear combinations of the original op-erators, and consequently, these elementary excitations are collective densityoscillations, their energy being determined both by the kinetic energy termand the interactions.

We note here that in order for the Bogolyubov transformation to be awell-defined unitary transformation, g2(q) has to decrease at large q at leastas |q|1/2 [23]. On the other hand, the large-q behavior of g2 is unimportantfor the low-energy properties of the model. We therefore in the following willalmost always use a q-independent g2 and g4. An approximate and frequentlyused way to cure the divergences arising due to this procedure is to keep a

parameter in subsequent formulae as a finite short-distance cutoff, of theorder of a lattice spacing. One can then also include the backward scat-tering (kF; kF) (kF; kF), because for spinless electron this is just theexchange analogue of forward scattering and does not constitute a new typeof interaction. It is worthwhile emphasizing here that the solution is valid forarbitrarily strong interactions, no perturbative expansion is needed!

Up to this point, the construction does not allow for a direct calculationof correlation functions like the one-particle Green function or more gener-ally any function involving individual creation or destruction operators. Thistype of correlation function becomes tractable by representing single particleoperators in terms of the boson operators. To this end, we introduce the fieldoperators

(x) = iLq=0

1q e|q|/2iqx (+(q) + (q)) NxL , (2.90)

(x) =1

L

q=0

e|q|/2iqx (+(q) (q)) + JL

, (2.91)

N = N+ + N , J = N+ N . (2.92)Here N are the numbers of particles added to the ground state on the right-and left-moving branch i.e.

N = (q = 0) =k

(n,k n,k0) . (2.93)

Because addition of a particle changes both N and J, one has the selection

rule (1)N = (1)J; is a cutoff parameter which (at least in principle,see the discussion above) has to be set to zero in the end of any calculation.The fields and then obey canonical boson commutation relations:

[(x), (y)] = i(x y) , (2.94)


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and is related to the local particle density via

x =

((x)

0) , (2.95)

where 0 is the average particle density in the ground state. More precisely,in a lattice model this would represent the slowly varying components (q 0) of the density, whereas components with q 2kF would correspond tocrossproducts between .

The expression for the single fermion operators then is

(x) = lim0

12

U exp[ikFx i(x) + i(x)] , (2.96)

where the upper and lower sign refer to right- and left-moving electrons re-spectively, and

(x) = x

(x) dx

=i

L

q=0

1

qe|q|/2iqx[+(q) (q)] + Jx

L. (2.97)

The U-operators (sometimes referred to as Klein factors; they are non-Hermitian: U = U) decrease the total particle number on one of thebranches by unity and are necessary because the boson fields all conservethe total particle number. These operators also insure proper anticommu-tation between right- and left-going operators: one easily checks that thechiral components + = and = + commute with each other,one therefore needs the anticommutation relations [U, U ]+ = 0, etc. In the

thermodynamic limit L , the fact that the Us change the particle num-ber is of minor importance because this represents a shift ofkF by a quantityof order 1/L, and one then can replace the Us by Majorana (Hermitian)fermion operators obeying [, ]+ = 0, [26] as discussed in more detail inthe appendix. In single-chain problems these effects play a minor role, how-ever in the many chain systems to be discussed at the end of these notes,proper account of anticommutation is crucial.

A detailed derivation of the important (2.96) as an operator identity isgiven in the literature [23,24]. However, a simple plausibility argument can begiven: creating a particle at site x requires introducing a kink of height in ,i.e. has to be shifted by at points on the left ofx. Displacement operatorsare exponentials of momentum operators, and therefore a first guess would be(x)

exp(i x (x

)dx). However, this operator commutes with itself,instead of satisfying canonical anticommutation relations. Anticommutationis achieved by multiplying with an operator, acting at site x, that changessign each time a particle passes through x. Such an operator is exp(i(x)).The product of these two factors then produces (2.96).


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The full Hamiltonian can also be simply expressed in terms of and .In the long-wavelength limit, neglecting the momentum dependence of thegi, one can express the total Hamiltonian in the field phase formalism

H = H0 + Hint =

dx

uK

2(x)2 +

u

2K(x)

2

. (2.98)

This is obviously just the Hamiltonian of an elastic string, with the eigen-modes corresponding to the collective density fluctuations of the fermionliquid. It is important to notice that these collective modes are the only (low-energy) excited states, and that in particular there are no well-defined singleparticle excitations, nor are there the incoherent particlehole pair excitationstypical of a Fermi gas. The parameters in (2.98) are given by

u =

vF +

g42

2 g2

2

2, K =

2vF + g4 g22vF + g4 + g2

. (2.99)

The energies of the eigenstates are (q) = u|q|, in agreement with (2.89).From the continuity equation, the expression (2.95) for the local particledensity and the equation of motion of the (number) current density is

j(x) = uK(x) . (2.100)

Note in particular that for g2 = g4 one has uK = vF, i.e. the expression forthe current density is interaction-independent. The relation uK = vF holdsin particular for systems with full (Galilean) translational invariance. On theother hand, in the continuum limit of lattice systems this relation is in generalnot true.

The most remarkable result here is the collectivization of the dynam-ics: there are no quasiparticle-like excitations. In fact there is a rather simple

physical picture explaining this: imagine accelerating one particle a little bitin one direction. Very soon it will hit its neighbour and transmit its momen-tum to it, and the neighbour will in turn transmit its momentum to a furtherneighbour, and so on. Quite quickly, the initial localized motion will havespread coherently through the whole system. This picture can be formalizednoting that in one dimension the difference between a solid and a fluid isnot well-defined: whereas is higher dimensions solids and fluids are differen-tiated by the presence or absence of long-wavelength transverse modes, notransverse modes can exist in a system with movement along only one direc-tion. The long-wavelength modes thus can equally well be considered as thephonons of a one-dimensional crystal [27,28]. Note that on the contrary in di-mensions larger than one the neighbours of any given particle can be pushedaside, giving rise to a backflow that allows the particle to move through the

system more or less freely.The exclusive existence of collective excitations, at least at low energies,

is one of the typical properties of the Luttinger liquid. Rather than discussingthe physics of the spinless case in detail, we will turn now to the more inter-esting case of fermions with spin.


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2.4.2 Model with Spin: the Concept of the Luttinger Liquid

In the case of spin-1/2 fermions, all the fermion operators acquire an addi-

tional spin index s. Following the same logic as above, the kinetic energy thentakes the form

H0 = vFk,s

(k kF)a+,k,sa+,k,s + (k kF)a,k,sa,k,s

=2vF

L

q>0,=,s

,s(q),s(q) , (2.101)

where density operators for spin projections s =, have been introduced:,s(q) =

k

a,k+q,sa,k,s . (2.102)

There are now two types of interaction. First, the backward scattering(kF, s; kF, t) (kF, s; kF, t) which for s = t cannot be re-written as aneffective forward scattering (contrary to the spinless case). The correspondingHamiltonian is

Hint,1 =1

L

k,p,q,s,t

g1a+,k,sa

,p,ta+,p+2kF+q,ta,k2kFq,s . (2.103)

And, of course, there is also the forward scattering, of a form similar to thespinless case

Hint,2 =1

2L

q,,s,t

g2(q),s(q),t(q) + g4(q),s(q),t(q)

.

(2.104)

To go to the bosonic description, one introduces and fields for the twospin projections separately, and then transforms to charge and spin bosonsvia , = ( )/

2, , = ( )/

2. The operators and

obey Bose-like commutation relations:

[(x), (y)] = i(x y) , (2.105)and single fermion operators can be written in a form analogous to (2.96):

,s(x) = lim0

,s2

expikFx i(( + s) + ( + s))/

2

,

(2.106)

where (x) = x (x)dx.

The full Hamiltonian H = H0 + Hint,1 + Hint,2 then takes the form

H = H + H +2g1

(2)2

dx cos(

8) . (2.107)


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Here is a short-distance cutoff, and for = ,

H =

dxuK

2

2

+

u

2K (x)

2, (2.108)

with

u =

vF +

g4,

2 g

2

2, K =

2vF + 2g4, + g2vF + 2g4, g , (2.109)

and g = g12g2, g = g1, g4, = g4, g4, = 0. The choice of sign (which is theconventional one) for the cosine-term in (2.107) corresponds to a particulargauge choice, as discussed in the appendix.

For a noninteracting system one thus has u = vF (charge and spin ve-locities equal!) and K = 1. For g1 = 0, (2.107) describes independent long-wavelength oscillations of the charge and spin density, with linear dispersion

relation (k) = u |k|, and the system is conducting. As in the spinless case,there are no single-particle or single particlehole pair excited states. Thismodel (no backscattering), usually called the TomonagaLuttinger model, isthe one to which the bosonization method was originally applied [20,21,29].

For g1 = 0 the cosine term has to be treated perturbatively. We havealready obtained the corresponding renormalization group equations in theprevious Section (2.62). In particular, for repulsive interactions (g1 > 0), g1is renormalized to zero in the long-wavelength limit, and at the fixed pointone has K = 1. The three remaining parameters in (2.107) then completelydetermine the long-distance and low-energy properties of the system.

It should be emphasized that (2.107) has been derived here for fermionswith linear energy-momentum relation. For more general (e.g. lattice) models,

there are additional operators arising from band curvature and the absenceof high-energy single-particle states [23]. One can however show that all theseeffects are, at least for not very strong interaction, irrelevant in the renor-malization group sense, i.e. they do not affect the low-energy physics. Thus,(2.107) is still the correct effective Hamiltonian for low-energy excitations.The lattice effects however intervene to give rise to higher harmonics inthe expression for the single-fermion operators, i.e. there are low energy con-tributions at wavenumbers q (2m + 1)kF for arbitrary integer m [27].

The Hamiltonian (2.107) also provides an explanation for the physics ofthe case of negative g1, where the renormalization group scales to strongcoupling (2.62). In fact, if |g1| is large in (2.107), it is quite clear that theelementary excitations of will be small oscillations around one of theminima of the cosine term, or possibly soliton-like objects where goes

from one of the minima to the other. Both types of excitations have a gap,i.e. for g1 < 0 one has a gap in the spin excitation spectrum, whereas thecharge excitations remain massless. This can actually investigated in moredetail in an exactly solvable case [30]. We will subsequently concentrate onthe case g1 > 0, so that there is no spin gap, however investigations of spectral


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functions as described below have also been recently performed for the casewith a spin gap [31].

Spin-charge separation. One of the more spectacular consequences of theHamiltonian (2.107) is the complete separation of the dynamics of the spinand charge degrees of freedom. For example, in general one has u = u,i.e. the charge and spin oscillations propagate with different velocities. Onlyin a noninteracting system or if some accidental degeneracy occurs one doeshave u = u = vF. To make the meaning of this fact more transparent,let us create an extra particle in the ground state, at t = 0 and spatialcoordinate x0. The charge and spin densities then are easily found, using(x) = (2/)x (note that (x) is the deviation of the density from itsaverage value) and z(x) = (

2/)x:

0

|+(x0)(x)

+(x0)

|0

= (x

x0) ,

0|+(x0)z(x)+(x0)|0 = (x x0) . (2.110)Now, consider the time development of the charge and spin distributions. Thetime-dependence of the charge and spin density operators is easily obtainedfrom (2.107) (using the fixed point value g1 = 0), and one obtains

0|+(x0)(x, t)+(x0)|0 = (x x0 ut) ,0|+(x0)z(x, t)+(x0)|0 = (x x0 ut) . (2.111)

Because in general u = u, after some time charge and spin will be localizedat completely different points in space, i.e. charge and spin have separatedcompletely.

Here a linear energy-momentum relation has been assumed for the elec-trons, and consequently the shape of the charge and spin distributions istime-independent. If the energy-momentum relation has some curvature (asis necessarily the case in lattice systems) the distributions will widen withtime. However this widening is proportional to

t, and therefore much smaller

than the distance between charge and spin. Thus, the qualitative picture ofspin-charge separation is unchanged.

Physical properties. The simple form of the Hamiltonian (2.107) at thefixed point g1 = 0 makes the calculation of physical properties rather straight-forward. The specific heat now is determined both by the charge and spinmodes, and consequently the specific heat coefficient is given by

/0 =1

2(vF/u + vF/u) . (2.112)

Here 0 is the specific heat coefficient of noninteracting electrons of Fermivelocity vF.


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The spin susceptibility and the compressibility are equally easy to obtain.Note that in (2.107) the coefficient u/K determines the energy necessaryto create a nonzero spin polarization, and, as in the spinless case, u/K fixesthe energy needed to change the particle density. Given the fixed point valueK = 1, one finds

/0 = vF/u , /0 = vFK/u , (2.113)

where 0 and 0 are the susceptibility and compressibility of the noninter-acting case. From (2.112) and (2.113) the Wilson ratio is

RW =

00

=2u

u + u. (2.114)

The quantity (x) is proportional to the current density. As before, theHamiltonian commutes with the total current, one thus has

() = 2Ku() + reg() , (2.115)

i.e. the product Ku determines the weight of the dc peak in the conductiv-ity. If the total current commutes with the Hamiltonian reg vanishes, howevermore generally this part of the conductivity varies as 3 at low frequencies[32].

The above properties, linear specific heat, finite spin susceptibility, and dcconductivity are those of an ordinary Fermi liquid, the coefficients u, u, andK determining renormalizations with respect to noninteracting quantities.However, the present system is not a Fermi liquid. This is in fact alreadyobvious from the preceding discussion on charge-spin separation, and canbe made more precise considering the single-particle Green function. Usingthe representation (2.106) of fermion operators one finds (at the fixed pointg1 = 0)

GR(x, t) = i(t)

+,s(x, t), +,s(0, 0)

+

= (t)

eikFxRe

1

(ut x)(ut x)

2

( + iut)2 + x2

/2

(2.116)

where the presence of the Heaviside function ensures the retarded nature ofGR and [, ]+ denotes the anticommutation of fermion operators. Note that this

expression factorizes into a spin and a charge contribution which propagatewith different velocities. Fourier transforming (2.116) gives the momentumdistribution function in the vicinity of kF:

nk nkF const. sign(k kF)|k kF| , (2.117)


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and for the single-particle density of states (i.e. the momentum-integratedspectral density) one finds:

N() || . (2.118)In both cases = (K + 1/K 2)/4. Note that for any K = 1, i.e. forany nonvanishing interaction, the momentum distribution function and thedensity of states have power-law singularities at the Fermi level, with a van-ishing single particle density of states at EF. This behavior is obviously quitedifferent from a standard Fermi liquid which would have a finite density ofstates and a step-like singularity in nk. The absence of a step at kF in themomentum distribution function implies the absence of a quasiparticle polein the one-particle Green function. In fact, a direct calculation of the spectralfunction A(k, ) from (2.116) [33,34] shows that the usual quasiparticle poleis replaced by a continuum, with a lower threshold at min(u(k kF)) andbranch cut singularities at = up and = up:

A(k, ) ( u(k kF))1/2 u(k kF)(1)/2 (u > u) ,

(2.119)

A(k, ) ( u(k kF))(1)/2 u(k kF)1/2 (u < u) .

(2.120)

The coefficient K also determines the long-distance decay of all other cor-relation functions of the system: Using the representation (2.106) the chargeand spin density operators at 2kF are

OCDW(x) =

s,s(x)+,s(x)

= lim0

e2ikFx

ei

2(x) cos[

2(x)] , (2.121)

OSDWx(x) =s

,s(x)+,s(x)

= lim0

e2ikFx

ei

2(x) cos[

2(x)] . (2.122)

Similar relations are also found for other operators. It is important to notehere that all these operators decompose into a product of one factor depend-ing on the charge variable only by another factor depending only on the spinfield. Using the Hamiltonian (2.107) at the fixed point g1 = 0 one finds forexample for the charge and spin correlation functions1

n(x)n(0) = K/(x)2 + A1 cos(2kFx)x1K ln3/2(x)+ A2 cos(4kFx)x

4K + . . . , (2.123)

S(x) S(0) = 1/(x)2 + B1 cos(2kFx)x1K ln1/2(x) + . . . , (2.124)1 The time- and temperature dependence is also easily obtained, see [35].


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with model dependent constants Ai, Bi. The ellipses in (2.123) and (2.124)indicate higher harmonics of cos(2kFx) which are present but decay fasterthan the terms shown here. Similarly, correlation functions for singlet (SS)and triplet (TS) superconducting pairing are

OSS(x)OSS(0) = Cx11/K ln3/2(x) + . . . ,OTS(x)OTS(0) = Dx11/K ln1/2(x) + . . . . (2.125)

The logarithmic corrections in these functions [36] have been studied in detailrecently [3740]. The corresponding susceptibilities (i.e. the Fourier trans-forms of the above correlation functions) behave at low temperatures as

CDW(T) TK1| ln(T)|3/2 , SDW(T) TK1| ln(T)|1/2 , (2.126)SS(T) T1/K1| ln(T)|3/2 , TS(T) T1/K1| ln(T)|1/2 , (2.127)

i.e. for K < 1 (spin or charge) density fluctuations at 2kF are enhanced

and diverge at low temperatures, whereas for K > 1 pairing fluctuationsdominate. The phase diagram, showing in which part of parameter spacewhich type of correlation diverges for T 0 is shown in Fig. 2.8.

These correlation functions with their power law variations actually de-termine experimentally accessible quantities: the 2kF and 4kF charge cor-relations lead to X-ray scattering intensities I2kF TK , I4kF T4K1,and similarly the NMR relaxation rate due to 2kF spin fluctuations varies as1/T1 TK . The remarkable fact in all the above results is that there is onlyone coefficient, K, which determines all the asymptotic power laws.

We here re-emphasize the two important properties of spin-1/2 interactingfermions in one dimension: (i) correlation functions show power-law decay,with interaction-dependent powers determined by one coefficient, K; and

(ii) spin-charge separation: spin and charge degrees of freedom propagatewith different velocities. Both these properties are typical of the Luttingerliquid and invalidate the Landau quasiparticle concept in one dimension.

SS(CDW)(SS)

CDWCDW SS

SDW(CDW)

TS(SS)

21/2 K

1g

Fig. 2.8. Phase diagram for interacting spin-1/2 fermions


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A nice experimental example of different spin-charge separation (differentvelocities for the spin and charge modes) is provided by Raman scatteringexperiments on single-channel quantum wires [4143]. On the other hand, thesituation in quasi-onedimensional conductors [44] is less clear: in compoundsof the TTFTCNQ series, the observation of strong 4kF CDW fluctuations[45] seems only to be explainable in the Luttinger-liquid picture. On theother hand, in the much studied (TMTSF)2X family of compounds, the NMRdata can be interpreted in the Luttinger liquid framework [46], but much ofthe magnetotransport work of Chaikin and collaborators is explained rathersatisfactorily using a Fermi liquid like picture.

Long-range interactions: Wigner crystallization. The above calcula-tions can be straightforwardly generalized to the case of long-range interac-tions. Of interest is the case of unscreened Coulomb interactions (V(r) =e2/r) which for example is of relevance for the physics of an isolated quan-tum wire. The short-distance singularity has to be cut off, and for examplein a wire of diameter d an approximate form would be V(r) = e2/

x2 + d2,

leading to a Fourier transform V(q) = 2e2K0(qd). The long-range nature ofthe interaction is only of importance for the forward-scattering processes, andthese appear only in the charge part of the Hamiltonian which is now givenby

H =vF2

dx

2(1 + g1)2 + (1 g1)(x)2

+

1

2

dxdxV(x x)xx , (2.128)

where g1 = g1/(2vF). The elementary excitations then are found to becharge oscillations (plasmons), with energy-momentum relation

(q) = vF|q|

(1 + g1)(1 g1 + 2V(q) ) (2.129)

where V(q) = V(q)/(vF). The long-wavelength form (q 0), (q) |q2 ln q|1/2, agrees with RPA calculations [47,48], however, the effect of g1,which is a short-range exchange contribution, are usually neglected in thosecalculations. The spin modes are still given by (q) = u|q|, with u =vF

1 g21 .In the evaluation of correlation functions, the charge averages lead to

((x) (0))2 = 0

dq 1 cos qxq

1 + g1

1 g1 + 2V(q) c2

ln x , (2.130)


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with c2 =

(1 + g1)vF/e2. One thus obtains for example

(x)(0)

= A1 cos(2kFx)exp(

c2

ln x)/x

+A2 cos(4kFx)exp(4c2ln x) + ... , (2.131)where A1,2 are interaction dependent constants, and only the most slowlydecaying Fourier components are exhibited. The most interesting point hereis the extremely slow decay (much slower than any power law!) of the 4 kFcomponent, showing an incipient charge density wave at wavevector 4kF.This slow decay should be compared with the power-law decay found forshort-range interactions (2.123). The 4kF oscillation period is exactly theaverage interparticle spacing, i.e. the structure is that expected for a one-dimensional Wigner crystal. Of course, because of the one-dimensional na-ture of the model, there is no true long-range order, however, the extremelyslow decay of the 4kF oscillation would produce strong quasi-Bragg peaks ina scattering experiment. It is worthwhile to point out that this 4k

Fcontri-

bution arises even if the Coulomb interaction is extremely weak and dependsonly on the long-range character of the interaction. On the other hand, any2kF scattering is considerably weaker, due to the 1/x prefactor in (2.131)which has its origin in the contribution of spin fluctuations. The 2kF spincorrelations equally contain a 1/x factor and thus have the same asymptoticdecay as the 2kF charge correlations, (2.131). On the other hand, correlationfunctions that involve operators changing the total number of particles (e.g.the single particle Green function and pairing correlation functions) decaylike exp[cst. (ln x)3/2], i.e. faster than any power law. This in particularmeans that the momentum distribution function nk and all its derivativesare continuous at kF, and there is only an essential singularity at kF.

It is instructive to compare the above result (2.131), obtained in the limit

of weak Coulomb interactions, with the case of strong repulsion (or, equiv-alently, heavy particles). The configuration of minimum potential energy isone of a chain of equidistant particles with lattice constant a, and quantumeffects are expected to lead only to small oscillations in the distances betweenparticles. The Hamiltonian then is

H =l

p2l2m

+1

4

l=m

V(ma)(ul ul+m)2 , (2.132)

where ul is the deviation of particle l from its equilibrium position. In thelong-wavelength limit, the oscillation of this lattice have energy (q) =

2/(ma)eq| ln(qa)|1/2. The most slowly decaying part of the density-density

correlation function then is

(x)(0) cos(2x/a)exp

4(2me2a)1/2

ln x

. (2.133)

Noticing that kF = /(2a), one observes that the results (2.131) and (2.133)are (for g1 = 0) identical as far as the long-distance asymptotics are con-


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cerned, including the constants in the exponentials. Equation (2.131) wasobtained in the weak interaction limit, whereas (2.133) applies for strongCoulomb forces. Similarly, the small-q limit of the charge excitation energiesis identical.

2.4.3 Summary

In this section we have developed the basic bosonization formalism for one-dimensional interacting fermions and seen some elementary and direct appli-cations to the calculation of some physical properties, in particular correlationfunctions. We have seen that the properties of the one-dimensional interactingsystem, the Luttinger liquid, are fundamentally different from two- or three-dimensional Fermi liquids. In particular the elementary excitations are notquasiparticles but rather collective oscillations of the charge and spin density,propagating coherently, but in general at different velocities. This gives rise to

the interesting phenomenon ofspin-charge separation. Finally, and again con-trary to the Fermi-liquid case, most correlation functions show non-universalpowerlaws, with interaction-dependent exponents. However, all these expo-nents depend only on one parameter, K, the spin analogue of which, K,being fixed to unity by spin-rotation invariance (K = 1 is possible if spinrotation invariance is broken). Beyond K, the only parameters that inter-vene in the low-energy physics of a Luttinger liquid are the velocities of thespin and charge modes, u,. In the spinless case only two parameters, K andu are involved. Finally, we have seen that long-range (Coulomb) interactionscan profoundly modify these properties.

For lack of time and space, we have not touched here upon a number ofinteresting uses and generalizations of the bosonization method, in partic-

ular the Kondo effect [49] and applications to Fermi systems in more thanone dimension [50,51]. An extensive recent review on Luttinger liquids andbosonization has been given by Voit [52].

2.5 Applications

We will subsequently discuss some results from applications of the formalismdeveloped to interesting physical questions: transport and the effect of dis-order, and the physics of antiferromagnetic spin chains. Here, we will comeacross the two cases where Luttinger liquid behavior is probably best estab-lished experimentally: the physics of quantum Hall edge states, upon whichwe will only touch briefly, and quantum spin systems, in particular the spin-

1/2 antiferromagnet, which we will discuss in some detail, with some referenceto experiment.


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2.5.1 Transport

Conductivity and conductance. In the previous section we were con-

cerned with equilibrium properties and correlation functions, in order tocharacterize the different phases possible in a one-dimensional system of in-teracting fermions. Here, we will investigate transport, in particular the dcconductivity. Finite-frequency effects have also been investigated, and thereader is referred to the literature [32,53].

To clarify some of the basic notions, let us first consider a Luttinger modelin the presence of a weak space- and time-dependent external potential .The interaction of the fermions with is described by the extra term

Hext = e

dx(x)(x, t) (2.134)

in the total Hamiltonian. We will assume that the external field is slowly

varying in space, so that in the particledensity operator only productsof either two right- or two left-going fermion operators appear but no crossterms. Standard linear response theory tells us that the current induced bythe potential is given by

j(x, t) = e2

h

t

dt

dxDj(x x, t t)(x, t) , (2.135)

where the retarded current-density correlation function is given by

Dj(x, t) = i(t)[j(x, t), (0, 0)]= uK

(t) ((x ut) + (x + ut)) . (2.136)

The second line is the result for spin-1/2 electrons. For spinless fermions onehas to make the replacement uK uK/2.

Let us now first consider the situation where we adiabatically switch ona potential of frequency and wavenumber q along the whole length ofthe system. From (2.136) one then straightforwardly obtains the q- and -dependent conductivity as

(q, ) =4e2

huK

i( + i0+)

( + i0+)2 u2q2. (2.137)

In particular, the real part of the conductivity for constant applied field is

Re (0, ) = 2e2

huK() , (2.138)

in agreement with (2.115) (where units with e2 = h = 1 were used).


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Applying on the other hand a static field over a finite part of the sample,one obtains a current j = 2e2KU/h, where U is the applied tension. Theconductance thus is

G =2e2

hK , (2.139)

and depends on K only, not on u. For the noninteracting case K = 1 thisis Landauers well-known result [54]. Note that interactions affect the valueof the conductance. The conductance here is independent on the length overwhich the field is applied. Noting that in dimension d the conductance isrelated to the dc conductivity via G = Ld2, a length-independent conduc-tance implies an infinite conductivity in one dimension, in agreement with(2.136). The fact that u does not appear in the expression for G can beunderstood noting that applying a static field over a finite (but large) partof the sample, one is essentially studying the wavenumber-dependent con-ductivity at strictly zero frequency, which from (2.137) is given by (q, 0) =2e2K(q)/h, indeed independent on u. On the other hand, applying a fieldof finite frequency over a finite length , one can see that one measures theconductivity (q 0, ) if > u/.

The situation of a finite static potential drop over only a finite part of awire is clearly difficult to realize experimentally. However, the result (2.139),or more precisely its spinless analogue G = Ke2/h, applies to the chiralLuttinger liquid as realized on the edge of quantum Hall systems [5557]. Onthe other hand, for a quantum wire connected to measuring contacts whichimpose a potential difference over the whole length of the wire, one obtains[58,59,56,60]

G =2e2

h

, (2.140)

independent of K, i.e. the momentum-conserving interactions play no rolein the DC conductance, as to be expected. On the other hand, the value ofK plays an important role for the effects of random and contact potentials[61,62]. These results permit a consistent explanation of some experimentson quantum wires [63], but leave open questions in other cases [64].

Persistent current. The Luttinger model description can be used straight-forwardly to obtain the current induced in a strictly one-dimensional ringthreaded by a magnetic flux [65]. The argument can in fact be made verysimply: in the one-dimensional geometry, the vector field can be removedentirely from the Hamiltonian via a gauge transformation, which then leads

to the boundary condition (L) = exp(2i/0)(0) for the fermion fieldoperator. Here L is the perimeter of the ring, and 0 = hc/e. For spinlessfermions, this is achieved by replacing

(x) (x) + 2L0

(2.141)


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in the bosonization formula, (2.96). The total J-dependent part of the Hamil-tonian then becomes

HJ = uK2L

(J + 2/0)2 , (2.142)

giving rise to a number current

j =02

E

=

uK

L

J +

2

0

. (2.143)

At equilibrium, J is chosen so as to minimize the energy. Given that atconstant total particle number J can only change by two units, one easilysees that the equilibrium (persistent) current has periodicity 0, and reachesis maximum value uK/L at = 0/2, giving rise to the familiar sawtoothcurve for the current as a function of flux.

For fermions with spin, as long as there is no spin gap ( g1 > 0), theabove results can be taken over, with the simple replacement uK 2uK,the factor 2 coming from the spin degeneracy. Note in particular that thepersistent current, an equilibrium property, is given by the same combinationof parameters as the Drude weight in the conductivity. This is an illustrationof Kohns result [66] relating the Drude weight to the effect of a magneticflux through a ring.

In the case of negative g1, electrons can be transferred from the rightto the left-going branch only by pairs, Consequently, the periodicity of thecurrent and the ground state energy is doubled to 20, and the maximum cur-rent is equally doubled. This behavior has actually been found in numericalcalculations [67,68].

Quantum Hall edge states. The Luttinger liquid picture has an interestingapplication to the physics of the fractional quantum Hall effect, as discoveredand discussed by Wen [69,70]. To see how this comes about, consider thestates available in the different Landau levels in the vicinity of the edge ofthe quantum Hall device, as shown in Fig. 2.9 [71]. It is clear that low-energystates only exist at the edge (the bulk quantum Hall state is well-known tobe characterized by a finite excitation gap), and close to the Fermi energy(i.e. the edge) the states have a linear dispersion relation. This can be madeparticularly clear if one assumes a disk-shaped sample: the states have awavefunction

zk

(ei)k , (2.144)

with k increasing linearly with radial position. The angular momentum quan-tum number k thus plays a role very similar to linear momentum in the lineargeometry we have assumed up to now. One thus can linearize the dispersionin Fig. 2.9 and obtains essentially the spinless model discussed in Sect. 2.4.1,


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angular momentum

E

EF

2nd Landau level

1st Landau level

r

Fig. 2.9. Quantum states in the lowest Landau level in the vicinity of the edge ofa quantum hall device. The spatial variation of the confining potential is assumedto be slow on the scale of the magnetic length, so that the energies of the different

quantum states are determined by the local value of the confining potential. Fora circular device the angular momentum of a state increases proportionally to itsdistance r from the origin

the only difference being that here only right-going particles exist. This dif-ference is the origin of the term chiral Luttinger liquid (in fact, the left goingbranch is to be found on the opposite edge of the device). Because there areno left-going particles (or at least they can be thought of as being at a macro-scopic distance), there also is no right-left interaction, and consequently oneexpects the noninteracting value K = 1. Moreover all the left-going compo-nents of the fields have to be projected out, for example one has to replace

+ = (

)/2.

However, straightforward adoption of this scheme leads to trouble: fromthe preceding subsection we know that K = 1 leads to a conductance (whichin the present case is the Hall conductance) of G = e2/h, different from thewell-known

G = e2

h(2.145)

valid for a fractional quantum Hall state ( = 1/m is the filling factor).To repair this problem one makes the hypothesis that instead of (2.95) and(2.100) one has

(x) =

+

x

, j(x) = u

+(x) , (2.146)

where the subscripts indicate projection on right-going states. With these def-initions following the calculations of the previous subsection one now straight-forwardly reproduces the correct result, (2.145). The appearance of the fac-tors

in (2.146) indicates that the objects occupying the states in Fig. 2.9


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are not free electrons but rather strongly affected by the physics of the bulkof the samples. A more detailed derivation, starting from a ChernSimonsfield theory for the bulk physics, has also been given by Wen [70].

Beyond reproducing the correct value of the Hall conductance, the abovehypothesis leads to a number of interesting conclusions. Consider first thecreation operator for a real electron (charge e) on the edge. Following thearguments of Sect. 2.4.1, because of (2.146), the bosonized version of theelectron operator now must create a jump of of height /

, rather than

of height . This leads to

+(x) ei+(x)/ , (2.147)

x being the coordinate along the perimeter of the sample. Now, these opera-tors obey the relation

+

(x)+

(x) = ei/

+(x)

+(x) . (2.148)

But the real electron is still a fermion, i.e. + must obey anticommutationrelations. Thus m = 1/ has to be an odd integer. One thus reproduces oneof the fundamental facts of the fractional quantum Hall effect. From (2.147)one also finds a decay of the single-electron Green function as

G(x, t) 1(x ut)1/ . (2.149)

Another fundamental property of the quantum Hall state appears whenone considers the fractionally charged elementary excitation of charge e atthe edge. A charge-e object is created by

+(x) ei+(x) , (2.150)

leading to a slow decay of the corresponding Green function, with exponent, instead of 1/ in (2.149). One now has the relation

+(x)+(x) = e

i+(x)+(x), (2.151)

i.e. exchanging the fractionally charged objects one obtains nontrivial (= 1)phase factors and these quasiparticles thus also obey fractional statistics.

A single hypothesis, the insertion of the factors

in (2.146), thus repro-duces two of the fundamental facts about the fractional quantum Hall effect!In addition one obtains a basis for the treatment of transport phenomena

mentioned at the end of Sect. 2.5.2 and results for the asymptotics of Greenfunctions.


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2.5.2 Disorder

Effects of isolated impurities. The infinite conductivity in the ideally

pure systems considered up to here is a natural but hardly realistic result:any realistic system will contain some form of inhomogeneity. This in gen-eral leads to a finite conductivity, and in one dimension one can anticipateeven more dramatic effects: in a noninteracting system any form of disor-der leads to localization of the single-particle eigenstates [72,73]. How thisphenomenon occurs in interacting systems will be discussed in this and thefollowing section.

Following Kane and Fisher [74], consider first the case of a single inho-mogeneity in an otherwise perfect one-dimensional system. The extra termin the Hamiltonian introduced by a localized potential v(x) is (for spinlessfermions)

Hbarrier dx(x)(x) . (2.152)Decomposing the product of fermio

fermi liquids and luttinger liquids

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