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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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