Band structure theory

Objective. Although typically covered in undergraduate solidstate physics curricula, it is still useful to review the generalproperties of a quantum mechanical particle in a periodic ex-ternal potential. The second half of this part is devoted to asketch of the notion of Fermi liquid, explaining why the in-dependent particle picture has been so successful. A sketchyreview of Green’s function techniques is provided.

Key concepts: Bloch theorem; reciprocal lattice; Brillouinzone; nearly-free electron gas; Wannier function; k.p perturba-tion; Fermi liquid; quasiparticle; self energy; Green’s function;spectral function

Reading: Girvin & Huang, Chapter 7.

References1. Modern quantum mechanics (revised edition). J.J. Saku-rai.2. Quantum theory of many-particle systems. Fetter &Walecka.3. Effects of electron-electron and electron-Phonon interac-tions on the one-electron states of solids. Hedin & Lundquist,in Solid State Physics (vol. 23).

1

1 An electron on a lattice

1.1 Bloch theorem

Consider an electron moving in a periodic potential so its po-tential energy is unaltered by a translation R “ <iniai,

Upr ` Rq “ Uprq (1)

If ψprq is a stationary state, then ψpr ` Rq is also a solutionof the Schrodinger equation, describing the same state. Thenψpr ` Rq “ eiφpRqψprq. But since all translations commute,it is demanded that φpR ` R1q “ φpRq ` φpR1q., i.e., φ is alinear scalar function of R:

ψpr ` Rq “ eik¨Rψprq (2)

which is the Bloch theorem. Equivalently, a Bloch functionlabeled by k can be written as

ψnkprq “ eik¨runkprq (3)

And an electron in a periodic potential is often dubbed a Blochelectron. We use the following normalization convention:

xψnk|ψn1k1y “ż

d3r ψ˚nkprqψn1k1prq “

p2πq3

Vcδnn1δpk ´ k1q

(4)

It follows then that k is not uniquely defined: k is equivalentto k ` G, if G ¨ R “ n2π, that is, G “ <ihibi is a reciprocal

2

lattice vector. Here, the primitive reciprocal lattice vectors aredefined as

bi ¨ aj “ 2πδij (5)

The parallelepiped b1 ˆ b2 ˆ b3 is called the Brillouin zone.The quantity !k is called the quasimomentum of an electron ina periodic potential. It is like replacing the conserved momen-tum of a free particle by a constant vector. The non-uniquenessk leads to some of the most subtle discussions in recent con-densed matter physics. A convenient (by no means necessary)choice is to view k and k ` G as identical, that is, ψnk isperiodic in k

|ψnk`Gy “ |ψnky (6)

The energy is also periodic, εnk`G “ εnk. With this choice ofgauge, the Brillouin zone is not a parallelepiped but a torus.

1.2 Band structure

The one-electron Hamiltonian H “p2

2m` Uprq will be trans-

formed to

Hk “ e´ik¨rHeik¨r “1

2mpp ` !kq2 ` Uprq (7)

which shows up in the new Schrodinger equation

Hk|unky “ εnk|unky (8)

3

Expanding unkprq as unkprq “ÿ

G1

cnkpG1qeiG1¨r, we obtain

1

Vc

ż

uc

d3re´iG¨rHkunkprq

“ÿ

G1

1

Vc

ż

uc

d3re´ipG´G1q¨r

„

!2

2mpG1 ` kq2 ` Uprq

ȷ

cnkpG1q

“!2

2mpG ` kq2cnkpGq `

ÿ

G1‰G

UG´G1cnkpG1q

Here, the Fourier component of the lattice potential is

Uk “ δk“G1

Vc

ż

uc

d3re´iG¨rUprq

and we set UG“0 “ 0. Thus we have the eigenvalue matrixequation (secular equation)

!2

2mpG ` kq2cnkpGq `

ÿ

G1‰G

UG´G1cnkpG1q “ εnkcnkpGq

For a simple illustration, consider a 1D weak periodic poten-tial, Upxq “ U0 cos x. We will focus on the components of cwith G “ 0,˘1, and the Hamiltonian matrix is

¨

˚

˚

˚

˚

˝

E0pk ´ 1qU0

20

U0

2E0pkq

U0

2

0U0

2E0pk ` 1q

˛

‹

‹

‹

‹

‚

4

Diagonalizing, we obtain the band structure of the lowest fewbands

In the case of so-called covalent solids, where valence elec-trons are more localized around the atoms than itinerant. Thereare also other situations in which a localized basis set is advan-tageous. In fact, there is a natural localized basis set, calledWannier functions

|χnRy “ Vc

ż

BZ

rdks|ψnky e´ik¨R (9)

where we use the notationż

rdks Ñż

ddk

p2πqd(10)

It can be verified that the Wannier functions are orthonormal

xχnR|χn1R1y “ δnn1δRR1 (11)

One could write the effective Hamiltonian for one Bloch bandin a second-quantized form

H “ż

rdksεka:kak (12)

5

where we have suppressed the band index, and a:k means putting

an electron in the Bloch state |ψky. Then the transformationto the Wannier basis means

a:k “

ÿ

R

a:Re

ik¨R, ak “ÿ

R

aRe´ik¨R (13)

For example, take a 1D band structure εk “ ´2t cos k fork P r´π, πs, we obtain

H “8

ÿ

x“´8

p´tqpa:xax`1 ` a:

x`1axq

This kind of Hamiltonian is usually called a tight-binding Hamil-tonian. The matrix elements (like ´t) are called the transferintegrals or hopping integrals, which are the amplitudes fortransferring one electron from orbital βR1 on to αR

xχαR|H|χβR1y (14)

Examples: graphene and Su-Schrieffer-Heeger model. See7.5–7.6 of Girvin&Huang.

Now we study the Bloch theorem when the spin-orbit inter-action is also included:

H “p2

2m` V prq `

!4m2c2

σ ¨ ∇V prq ˆ p. (15)

which is the non-relativistic approximation to the Dirac equa-tion, without the kinetic energy correction and the Darwin

6

term. The Pauli matrices, σ, are defined as follows

σx “

„

0 11 0

ȷ

, σy “

„

0 ´ii 0

ȷ

, σz “

„

1 00 ´1

ȷ

. (16)

The quantity

s “!2σ, (17)

is the spin of Schrodinger equation. If we take a central field,V prq “ V prq, such that ∇V prq “ rdV {dr, we have

!4m2c2

σ ¨ ∇V prq ˆ p “!

4m2c21

r

dV prqdr

σ ¨ pr ˆ pq

“1

2m2c21

r

dV prqdr

s ¨ L,

showing that this term corresponds to the spin-orbit interac-tion in atomic physics.

The spin-orbit interaction Hamiltonian of Eq. (15) is in-variant under lattice translation, since V prq possesses the fullsymmetry of the crystal. Therefore, its eigen wavefunctionscan also be written in a form of irreducible representation ofthe T group. On account of the spinor form of wavefunctions,we write

ψkprq “ eik¨rukprq “ eik¨r

„

φkprqϕkprq

ȷ

(18)

where φkprq and ϕkprq are cell periodic functions. The two-component cell periodic function ukprq is again the eigenstate

7

of the k-dependent Hamiltonian,

Hk “pp ` !kq2

2m`V prq`

!4m2c2

σ ¨∇V prqˆpp`!kq. (19)

This fact is later used to develop the perturbative treatmentof spin-orbit interactions in band systems.

What can be said of the wavefunctions if the crystal hassymmetry other than the translations? We will leave generaldiscussions to other courses (like group theory). Here we ex-emplify an analysis of this sort for inversion and time-reversalsymmetries.

The action of spatial inversion, i, on a wavefunction at k

iψkprq “ e´ik¨rukp´rq ” e´ik¨ru´kprq (20)

we find that the transformed function is a Bloch function of´k. Since

iHki´1iukprq “ iεkukprq

ñ

ˆ

pp ´ !kq2

2m` V prq `

!4m2c2

σ ¨ ∇V prq ˆ pp ´ !kq

˙

ˆu´kprq “ εku´kprq

ñ H´ku´kprq “ εku´kprq.

we haveε´k “ εk. (21)

8

We conclude that when the system has inversion symmetry,when i operates on an eigen wavefunction of k it generates thedegenerate eigen wavefunction at ´k, to within a position-indepenendent phase factor.

The Hamiltonian with spin-orbit interaction in Eq. (15) isalso invariant under time-reversal. The time-reversal opera-tion for a spin-1/2 system is represented by an antiunitary,antilinear operator

Θ “ ´iσyK, (22)

whereK takes complex conjugate of the numbers it acts upon. i

The time-reversal operation has the following properties

ΘrΘ´1 “ r;ΘpΘ´1 “ ´p;ΘσΘ´1 “ ´σ;Θ2 “ ´1. (23)

If a one-electron Hamiltonian commutes with Θ, and |ψy isan eigen state, then

HΘ|ψy “ ΘH|ψy “ εΘ|ψy;

that is, Θ|ψy is also a degenerate eigenstate. This is knownas the Kramers theorem. The degeneracy resulted from time-reversal symmetry is often called Kramers degeneracy. Thepair of eigen states related by Θ is called a Kramers pair,which are also mutually orthogonal, as can be seen from

xψ|Θψy “ xΘ2ψ|Θψy “ ´xψ|Θψy “ 0

iSee Modern Quantum Mechanics (Revised Edition), Sakurai (1993).

9

Acting Θ on an eigenstate of the crystal Hamiltonian ψkprq,we have

´iσyKψkprq “ e´ik¨r

„

´ϕ˚kprq

φ˚kprq

ȷ

” e´ik¨ru´kprq. (24)

The last equality ” comes from the fact thatΘψkprq is anothereigenstate which is seen to belong to ´k. Take the expectationof σz

xΘψk|σz|Θψky “ ´xψk|σz|ψky.

Thus, for a Bloch band, the Kramers pair has opposite mo-menta and spin. Therefore the Kramers degeneracy is writtenas

εÒ,k “ εÓ,´k (25)

If both inversion and time-reversal symmetries are present,we have

εÒ,k “ εÓ,k (26)

andψk,Ó “ eiθkΘiψk,Ò. (27)

Let’s now look at a concrete example. Consider a 2-dimensionalsystem with the following Hamiltonian

H “p2

2m`

!eE4m2c2

σ ¨ z ˆ p. (28)

Physically, this corresponds to 2-dimensional electron gas sub-ject to an electric field in the perpendicular direction, ∇V “eEz. Evidently, rH,ps “ 0, so that the energy eigenstates

10

can also be momentum eigenstates. The wavefunction can begenerally written as ψkprq “ eik¨ruk. This leads to the neweigenvalue equation

ˆ

!2k2

2m`

!2eE4m2c2

pkxσy ´ kyσ

xq

˙

uk “ εkuk

Define k “ qeE{4mc2, and we have in energy units!2{m

peE{4mc2q2

p12q2 ` qxσ

y ´ qyσxquq “ εquq.

Thus, for each k the Hamiltonian is a 2 ˆ 2 matrix

Hk “

„

12q

2 ´iqe´iθ

iqeiθ 12q

2

ȷ

diagonalization of which leads to

ε˘q “ 1

2q2 ˘ q

The band structure is just a revolution of parabola about thez-axis, as shown below. Only one degeneracy appears at q “ 0.

11

What is the symmetry of the Hamiltonian in Eq. (28)? Itclearly has translational invariance. Spatial inversion is brokenby the imposed electric potential, V prq ‰ V p´rq, which canalso be explicit verified by noting that σ Ñ σ and p Ñ ´punder parity. Since both σ and p are odd under time rever-sal, the Hamitonian commutes with Θ. Therefore, each bandis expected to have no generic spin degeneracy, but Kramersdegeneracy is expected.

u˘q “

1?2

ˆ

e´iθ

˘i

˙

And we find

xsy˘pqq “ ˘!2q ˆ z,

confirming the spin reversal for the Kramers pairs.

Using Eq. (19), we have an eigenvalue equation for the nthband

ˆ

p2

2m` V prq `

!mk ¨ p `

h2k2

2m

˙

unkprq “ Enkunkprq.

(29)Define εnk “ Enk ´ !2k2{2m. Suppose we have solved theproblem at a special point k0; that is, we have the completeset of periodic functions at k0

Hk0|nk0y “

ˆ

p2

2m` V `

!mk0 ¨ p

˙

|nk0y “ εnk0|nk0y.

(30)

12

It follows then, by setting q “ k ´ k0ˆ

Hk0 `!mq ¨ p

˙

|nk0 ` qy “ εnk0`q|nk0 ` qy. (31)

Then provided that we are interested in the bands close tok0, the problem can be treated as a perturbation to Hk0. Ifspin-orbit interaction is included, from Eq. (19), we have

ˆ

Hk0 `!mq ¨ π

˙

|nk0 ` qy “ εnk0`q|nk0 ` qy, (32)

where

π “ p `!

4mc2σ ˆ ∇V. (33)

When there is no degeneracy at k0 (except for maybe spindegeneracy), we carrying out perturbation theory to the secondorder to obtain

Enq “ εnk0 `!2k2

2m`

!mq ¨ xnk0|π|nk0y

`!2

m2

ÿ

n1‰n

q ¨ xnk0|π|n1k0y xn1k0|π|nk0y ¨ qEnk0 ´ En1k0

(34)

Further analysis of the expansion will require the knowledgeof the symmetry of k0. It often happens we perform the expan-sion at the Γ point, which has the full symmetry of the crystalHamiltonian. And we will assume that the system has time-reversal and inversion symmetry. Under inversion symmetry ,the first-order term vanishes since

xnk0|π|nk0y “ ´λ2xnk0|π|nk0y,

13

where λ is the parity eigenvalue of |nk0y. It then follows thatthe effective mass tensor at k0, to second order of perturbationis given by

pm´1n qµν “

δµν

m`

2

m2

ÿ

n1‰n

xnk0 |πµ|n1k0y xn1k0 |πν|nk0yEnk0 ´ En1k0

,

(35)which is sometimes referred to as the f -sum rule at k0.

The point of Eq. (34) is that we have now a Hamiltonian interms q “ k ´ k0, whose coefficients are determined by thematrix elements

πnn1 “ xnk0|π|n1k0y . (36)

Oftentimes, it suffices to know if πnn1 is zero from a symmetrypoint of view. Then the non-zero matrix elements are deter-mined by fitting physical parameters, such as effective massand band gap, or from band structure calculated from othermethods.

When degeneracy is present, the usual perturbation theoryfails. Here, we go through a simple example which does notinvolve spin-orbit interaction. Suppose k0 has D4h symmetry,whose character table is given in the previous section. We willfocus on three bands: one band lying below the band gap thattransforms as A1g, and a pair of bands that are degenerate atk0 and transform as Eu. The degenerate pair can be taken tobe px- and py-like, whereas the one that belongs to the totallysymmetric representation is taken to be s-like.

14

The general problem of deriving a second-order perturbationinvolving degeneracy is outlined here. We have two subspaces,the valence and conduction bands, that are separated by theband gap ∆g. We would like to derive an approximate Hamil-tonian where the coupling between the conduction and valencebands vanish. Let’s suppose

H “ H0 ` V (37)

where V is the perturbation. Impose a unitary transformationon H

H “ eiSHe´iS, (38)

where S is Hermitian and is viewed as first order in V . We re-quire that the first order term vanish by an appropriate choiceof S, which is the solution to an operator equation

V “ irH0, Ss. (39)

We then find

iS “ÿ

ab

|ayVab xb|Ea ´ Eb

, (40)

15

where a, b are eigenstates of H0. When Eq. (38) is expandedto second order of perturbation, it is found that

H “ H0 `i

2rS, V s ` OpV 3q. (41)

Combining Eqs. (40, 41), we obtain

Hab “ ´12

ÿ

n

ˆ

1

En ´ Ea`

1

En ´ Eb

˙

VanVnb. (42)

This process will effectively decouple the two weakly couplednon-degenerate subspaces separated by a large energy gap.

To see how this works for the present problem, let’s look atthe following hamiltonian (compare this with Problem 5.12 ofSakurai? )

Hq “

»

–

Es Vsx Vsy

Vxs Ep 0Vys 0 Ep

fi

fl

where the band gap is ∆g “ Ep´Es, and the matrix elements

Vsj “!mq ¨ xsk0|p|jk0y, j “ x, y,

are the coupling arising from the k ¨ p perturbation. Simi-lar coupling between p states vanishes by parity. It should benoted that although k0 has C4 as a symmetry, the k ¨ p per-turbation does not have the symmetry (otherwise, the matrixelements Vsx, Vsy vanish). By the prescription of the pertur-

16

bation theory above, we will have a new set of basis functions

|sy “

ˆ

1 ´VsxVxs ` VsyVys

2∆2g

˙

|sy ´1

∆gpVsx|xy ` Vsy|yyq ,

|xy “

ˆ

1 ´VsxVxs

2∆2

˙

|xy `Vxs

∆g|sy,

|yy “

ˆ

1 ´VsyVys

2∆2g

˙

|yy `Vys

∆g|sy.

In this new basis, we now compute the diagonal matrix ele-ments

xs|H|sy “ Es ´VsxVxs ` VsyVys

∆g,

xj|H|jy “ Ep `VsjVjs

∆g, j “ x, y.

The off-digonal elements are now computed

xs|H|xy “ 0 ` OpV 3q,

xx|H|yy “VxsVsy

∆g.

We then arrive at a k ¨ p Hamiltonian to second order,

Hq “

»

–

E0s ´ Apq2x ` q2yq 0 0

0 E0p ` Aq2x Aqxqy

0 Aqxqy E0p ` Aq2y

fi

fl

17

2 Key ideas of Fermi liquid

We have so far been limited to single-electron pictures, whichlead to band structures from the one-particle Schrodinger equa-tions in periodic systems. As convenient as it may seem, thisapproach begets the question: since electrons carrying charge´e interact through the long-ranged Coulomb potential, howcan one expect the single-particle theories to work at all? Theresolution of the dilemma is provided by Lev Landau, who hy-pothesized that the excitations of an interacting electron liquid,i.e., Fermi liquid, is very similar to the energy spectrum of aFermi gas. The hypothesis was later justified more formally.

Instead of wallowing in the complicated proof, we shall bebegin by finding out the key ideas of the Fermi liquid theory.An ideal Fermi gas follows the Fermi-Dirac distribution,

f “1

eβpε´µq ` 1. (43)

At T “ 0, all energy levels are occupied if ε ă µp0q, andempty if ε ą µp0q. The quantity µp0q is called the Fermienergy, which is for a metal the highest occupied level. Themomentum of of the Fermi level is given by µp0q “ p2F{2m, andcorrespondingly, the Fermi wavevector is kF “ pF{!. With2-fold spin degeneracy, the electron density is twice the thenumber of states in the Fermi sphere,

n “ 2 ˆ4

3πk3F{p2πq3 “

k3F3π2

. (44)

18

At finite temperatures, the sharp step at µ gets smeared outto an energy window of width kBT ; that is, some the electronsreceive energy of order kBT and escape the Fermi sphere. Onewould think of the completely filled Fermi sphere as a “vac-uum”, and the electrons outside as particles and holes inside asantiparticles. With the Fermi vacuum, we only need to thinkof the low-lying excitations, or the quasiparticles, whose energymust be counted from µ

ηppkq “ εk ´ µ « !vF ¨ pk ´ kF q (45)

ηhpkq “ µ ´ εk « !vF ¨ pkF ´ kq (46)

where the expansion is taken for k ! kF .

A key insight of Landau is the hypothesis that the quasipar-ticle spectrum of an isotropic Fermi liquid (i.e. with strong in-teraction between electrons) can be constructed the same wayas for an ideal Fermi gas. There is a kF which is connected tothe density through some relation similar to Eq. (44), which al-lows the definition of quasiparticles for k ą kF and quasiholes

19

for K ă kF . For k ! kF

ηp « !vpk ´ kF q; ηh « !vpkF ´ kq (47)

where v is just an undetermined coefficient with the same di-mension as velocity. Equivalently, we can introduce an alter-native parameter by v “ !kF{m˚. m˚ is called effective mass.

An essential idea of the quasiparticle picture is that the ele-mentary excitations decay, which departs from the expectationfor a non-interacting system

ψk „ e´iηkt{!´γkt{! (48)

We may speak of quasiparticles for time when γk ! |ηk|. Thesource of finite γk, i.e., damping of the excitation arises fromscattering, here by allowing electrons to interact. Clearly, γis proportional to the rate of transition to other states. Let’sconsider the following transition

The process described by the diagram is as follows. There isan electron at k1 outside the Fermi distribution. This particle

20

is then scattered to k11 by interacting with an electron initially

at k2 inside the Fermi sea. After the transition, the secondparticle is located at k1

2. We wold like to compute the rate oftransition. Momentum conservation k1 ` k2 “ k1

1 ` k12 along

with energy conservation means

W „ż

δpε1 ` ε2 ´ ε11 ´ ε1

2qd3k1

1d3k2

There is no integration over k12 because k1 is given so k1

2 isalready fixed by momentum conservation.

At T “ 0, if k1 “ kF then all other states must all lie onthe Fermi surface. Thus, the phase volume of the scatteringprocess is zero and the scattering rate vanishes. Therefore, thelifetime for an electron on the Fermi level is infinite at zerotemperature.

When k1 « kF , then all other three wavevectors must alsobe close to kF in magnitude: k1`k2 « k1

1`k12. Since k

12 ą kF ,

it must be true that k11 ă k1 ` k2 ´ kF . We have

kF ´ k1 ă k2 ´ kF ă 0 ă k11 ´ kF ă k1 ´ kF ` k2 ´ kF

Roughly, the phase space is confined to a thin momentum shellaround the Fermi surface,

γ „ W „ż

d3k2d3k1

1 „ pk1 ´ kF q2 (49)

Note that this is proportional to δk2, which is of the same

21

order as pε1 ´ εF q2. Thus, at T “ 0

1

τ“

γ

!“ aη2 (50)

where !a is of the order 1{εF .

At a finite temperature T , the quasiparticles are always in theenergy range η „ kBT , so the scattering rate is proportionalto pkBT q2. Combining these analyses we have

1

τ“ aη2 ` bT 2 (51)

It would seem that when 1{τ is small, the low-lying excitationis sufficiently long-lived to be treated as a particle. This condi-tion may be satisfied when η ! εF , kBT ! εF . In this regime,electron-electron interaction does not appear to invalidate theindependent particle approximation.

Is there a single-particle-like Schrodinger equation for quasi-particles? To answer this question, we will use the Green’sfunction defined as

Gp1, 2q “ ´i xN |T ψp1qψ:p2q|Ny (52)

where |Ny is theN -electron ground state, T the time-orderingoperator. The Dyson equation reads

ˆ

i!B

Bt1´ H0p1q

˙

Gp1, 2q ´ż

Σp1, 3qGp3, 2qdr3s

“ !δp1 ´ 2q (53)

22

here we use the notation 1 Ñ r1t1, 2 Ñ r2t2, etc. Considera wavepacket of an electron added to an N -electron groundstate

|N ` 1,ϕy “ż

d3rϕprqψ:prq|Ny (54)

We write the following one-particle wavefunction

Ψpr, tq “ż

d3r1Gprt, r10qϕpr1q (55)

which is the probability amplitude of finding an electron at rtif the wavepacket is introduced to the system at t “ 0. Inprinciple, the N `1-electron system is destined for three typesof states after long evolution:

• elastically scattered sector: ψ:prq|Ny

• inelastically scattered sector: ψ:prq|N, jy

• possible bound states: |N ` 1, jy

Then Ψ defined above should correctly describe the incomingand elastically scattered waves.

Using the Dyson equation, we obtainˆ

i!B

Bt1´ H0p1q

˙

Ψp1q ´ż

dr2sΣp1, 2qΨp2q “ 0 (56)

The self energy, Σ, plays the role of a single-particle potential,which can be dynamical and nonlocal. Let’s incorporate the

23

quasiparticle picture into the formulation, and let the quasi-particle wavefunction assume the form

|Ψptqy „ expp´iεt{! ´ γ|t|{!q|Ψy (57)

We obtain the quasiparticle equation

H0prqΨprq `ż

d3r1 Σpr, r1, εqΨpr1q “ εΨprq (58)

Here, ε is identified as the quasiparticle energy. The self en-ergy, Σpr, r1; εq, is a complex, nonlocal and frequency(energy)-dependent potential, sometimes called an optical potential. Inessence, the usual single-particle approximation is introducedby making the self energy local, frequency-independent (staticin time domain) and Hermitian.

Define

ψsprq “

"

xN |ψprq|N ` 1, sy, for εs “ EN`1,s ´ EN ą µ

xN ´ 1, s|ψprq|Ny, for εs “ EN ´ EN´1,s ă µ

(59)

This describe the amplitude of a quasiparticle at position r.The quasiparticle can be either hole-like (εs ă µ) or particle-like (ε ą µ). Then the single-particle Green’s function incoordinate representation is written

Gprt, r10q “ ´ixN |T ψprtqψ:pr10q|Ny

“ ´ixN |ψprqe´ipH´EN qt{!ψ:pr1q|Nyθptq

`ixN |ψ:pr1qe`ipH´EN qt{!ψprq|Nyθp´tq.(60)

24

Here, we suppress spin indices, EN is the energy of the N -particle ground state. Inserting |N ` 1, sy xN ` 1, s| into thefirst term, and |N ´ 1, sy xN ´ 1, s| the second term, we have

Gprt, r10q “ ´iÿ

s

ψsprqψ˚spr

1qe´iεst{!

ˆ rθptqθpεs ´ µq ´ θp´tqθpµ ´ εsqs . (61)

Performing Fourier transform

Gpr, r1; εq “1

!

ż 8

´8dtGprt, r10qeiεt{!

“ ´i

!ÿ

s

ψsprqψ˚spr

1q

„

θpεs ´ µqż 8

0

dt eipε´εsqt{!

´θpµ ´ εsqż 0

´8dt eipε´εsqt{!

ȷ

. (62)

Introducing a convergence factor by

εs Ñ εs ` sgnpµ ´ εsq ˆ i0`, (63)

we obtain

Gpr, r1; εq “ÿ

s

ψsprqψ˚spr

1qε ´ εs

. (64)

We note that this is nothing but the Lehmann representa-tion of the Green’s function. We see that the functions ψsprqplay the role of wavefunctions of a non-interacting system. Butthere are fundamental differences, in that ψsprq’s are not nor-malized or linearly independent. But at least they are complete

ÿ

s

ψsprqψ˚spr

1q “ δpr ´ r1q (65)

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The spectral function of the single-particle Green’s functionis then defined as a Hermitian matrix

Apr, r1; εq “ÿ

s

ψsprqψ˚spr

1qδpε ´ εsq (66)

Comparing Eq. (66) with Eq. (62), we find

Gpr, r1; εq “ż

C

dzApr, r1; zq

ε ´ z(67)

which is theHilbert transform. The contourC is shown below.

Let’s consider the case of non-interacting electrons. We maygeneralize Green’s function to an arbitrary representation,

Gss1ptq “ ´i xN |T ψsptqψ:s1|Ny (68)

If one-particle Hamiltonian is diagonalized in this representa-tion, then

Gss1ptq “ ´ie´iεst{!”

θptqxN |ψsψ:s1|Ny ´ θp´tqxN |ψ:

s1ψs|Nyı

“ ´ie´iεst{! rθptqp1 ´ nsq ´ θp´tqnss δss1 (69)

The Green’s function is also diagonal. Keeping only the diago-nal elements of the Green’s function, and in energy (frequency)domain

Gspεq “1

ε ´ εs ` sgnpεs ´ µqi0`(70)

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Considering an interacting system, let ψs describe a decayingparticle, i.e., a quasiparticle,

|ψsptqy “

"

|ψsy e´iεst{! e´γst{!, t ą 0

0, t ă 0(71)

for γs ą 0. This describes a particle-like excitation, c.f.the quasiparticle hypothesis Eq. (57). The diagonal part ofGreen’s function becomes

Gspεq “|ψsy xψs|

ε ´ εs ` i2γs. (72)

Thus the trace of spectral function for a quasiparticle initiallyin the state |ψsy is

Aspεq “ ´1

πIm

ż

d3rGspr, r; εq “1

π

Γ

pε ´ εsq2 ` Γ2(73)

where Γ “ 2γs. Thus, the spectral weight of a quasiparticlemimics the shape of a Lorentzian, whose width is proportionalto its decay rate.

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