R?ssler U. Solid State Theory: An Introduction

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198 7 Correlated Electrons

(0)

(α, ω)=



+∞

−∞

¯h

(¯hω+iδ)τ

(0)

(α, τ)dτ, (7.15)

we ﬁnd the characteristic form of the Green function (with E =¯hω)

(0)

(α, E) = lim

δ→0

E − ǫ

+iδ

, (7.16)

where the small δ has been intro duced to yield, in the back transform again,

the retarded Green function. It has a pole in the lower complex energy (or

frequency) plane close to the real axis at the single-particle energy ǫ

.(Com-

pare with the response function for the optical phonons in Sect.

3.5 or with

the inverse dielectric function in Sect.

4.5 which exhibit similar structures.)

Making use of the relation

lim

δ→0

E − ǫ

+iδ

= P

E − ǫ

− iπδ(E − ǫ

), (7.17)

one ﬁnds the density of states

D(E)=



ImG

(0)

(α, E)





δ(E − ǫ

). (7.18)

Let us include now the electron–electron interaction for the sp ecial case of

electrons in a partially ﬁlled energy band. For this case, the band index can

be dropped, and α is speciﬁed by the wave vector k and the spin σ of the

electron (see Problem 6.1). The commutator of c

kσ

with the interaction term

int



′

σσ

′

V (k, k

′

, q)c

†

k+qσ

†

′

−qσ

′

kσ

(7.19)

can be calculated as Problem 7.1 and yields

kσ

, H

int



′

qσ

′

V (k

′

, k + q, q)c

†

′

+qσ

′

k+qσ

. (7.20)

The equation of motion for the Green function is now

i¯h

∂

∂t

− ǫ

kσ

G(kσ; t, t

′

)=δ(t − t

′



′

V (k

′

, k + q, q)Γ(k σ, k

′

; t, t

′

)

(7.21)

where

Γ(kσ, k

′

; t, t

′

)=−

¯h

θ(t − t

′

){c

†

′

+qσ

′

(t)c

′

(t)c

k+qσ

(t),c

†

kσ

′

)}

(7.22)

7.1 Retarded Green Function for Electrons 199

is a higher order Green function. It is convenient to introduce, by means of



′

V (k

′

, k + q, q)Γ(k σ, k

′

; t, t

′



Σ(kσ; t, t

′′

)G(kσ; t

′′

′

)dt

′′

, (7.23)

the single-particle self-energy Σ(kσ; t, t

′′

). If the Hamiltonian do es not depend

on t, we can exploit the homogeneity of the time, because the Green functions

depend only on the time diﬀerence and perform the Fourier transformation

to obtain

G(α, E) = lim

δ→0

E − ǫ

− Σ(α, E)+iδ

. (7.24)

This is the same structure as for the Green function without interaction, but

now in addition to the single-particle energy there is a self-energy correction,

which in general depends on E and changes the pole structure. Thus, (

7.24)

indicates the structure of the Green function. However, the problem of includ-

ing the electron–electron interaction is still to be accomplished by calculating

the self-energy and to ﬁnd the poles of the Green function. For the concepts,

how this can be done in a systematic way, we refer to the literature [

64].

The Green function of the noninteracting particle (

7.16) is related to that

of the full Green function (

7.24)bytheDyson

equation

G = G

(0)

+ G

(0)

ΣG, (7.25)

known from scattering theory. It can be obtained also by taking the Fourier

transform of (

7.21) together with (7.23) (Problem 7.2).

We may look for approximate calculations of the self-energy and do this

for T = 0, for which the thermal expectation value reduces to the expectation

value in the system ground state (see Problem 2.2). The Green function (

7.22)

can be approximated with the following replacements

†

′

+qσ

′

(t)c

′

(t)c

k+qσ

(t) →c

†

′

+qσ

′

(t)c

′

(t)c

k+qσ

(t)

−c

†

′

+qσ

′

(t)c

k+qσ

(t)c

′

(t), (7.26)

which is a factorization of the three-fermiontermintoanexpectationvalue

and a single-fermion term. The former yields the occupation number n

′

q =0andn

k+qσ

provided k

′

= k,andσ

′

= σ, while the latter combines

with the rest of (

7.22) to the Green function G(α; t, t

′

). The correspon ding

expressions for the self-energy, which do not depend on the time arguments,

are

(kσ)=



′

V (k, k

′

, 0)n

′

, (7.27)

Freeman John Dyson *1923.

200 7 Correlated Electrons

kk’

k’

Fig. 7.1. Graphs of the Hartree and Hartree–Fock self-energies

which is the Hartree self-energy,and

(kσ)=−



′

V (k, k

′

, q)n

′

+qσ

, (7.28)

which is the Hartree–Fock self-energy. There is a quite intuitive graphical

representation of these self-energies (Fig.

7.1): Both graphs are obtained from

the interaction graph (see Fig.

4.10) by connecting incoming a nd outgoing

fermion lines, which is possible in just two distinct ways, leading to th e Hartree

and Hartree–Fock self-energy.

For free electrons, the potential matrix element simpliﬁes according to

′

→ v

,wherev

is the Fourier transform of the bare Coulomb poten-

tial and we may compare with the results obtained in the jellium model of

Chap.

4. In this case, the Hartree self-energ y vanishes, because the contribu-

tion for q = 0 is exempt from the summation in the interaction term due to

compensation with the jellium term and the Hartr ee–Fock self-energy becomes

the one calculated in Sect.

4.4.

Let us ﬁnally reformulate the denominator of the Green function for inter-

acting particles close to the chemical potential with E ≃ μ. Making use of the

expansion

Σ(α, E) ≃ Σ(α, μ)+

∂Σ

∂E

E=µ

(E −μ), (7.29)

the denominator of the retarded Green function (

7.24) takes the form

E − ǫ

− Σ(α, E) ≃ (E − μ)

1 −

∂Σ

∂E

E=µ

−



− μ +Σ(α, μ)



(7.30)

and the Green function can be written as

G(α, E)=

Z(α)

E − μ − ˜ǫ

− iγ

. (7.31)

Here, we introduce

−1

(α)=1−

∂Σ

∂E

E=µ

(7.32)

˜ǫ

= Z(α)(ǫ

− μ +ReΣ(α, μ)) (7.33)

≃ Z(α)ImΣ(α, ˜ǫ

). (7.34)

7.2 The Hubbard Model 201

Due to the interaction, the pole of the Green function is now at the energy

˜ǫ

+iγ

and has the spectral weight Z(α), which equals 1 if the self-energy

does not depend on the energy, as is the case for the Hartree and Hartree–

Fock approximation. The pole is in the lower (upper) half plane of complex

energies depending on the sign of E − μ and corresponds to a quasi-particle

(quasi-hole). We shall come back to this result in Sect.

7.3. Here, we note only

that the Fourier back transformation leads to a time-dependent propagator

of the form (see [

122], Sect. 10)

G(α; t − t

′

)=−

¯h

Z(α)θ(t − t

′

−(i˜ǫ

+γ

)(t−t

′

)/¯h

. (7.35)

The imaginary part of the self-energy indicates a ﬁnite lifetime of the quasi-

particle or quasi-hole due to particle–particle interaction. A closer inspection

shows a decrease of the lifetime with increasing distance fr om the chemical

potential μ (see Sect.

7.3).

7.2 The Hubbard Model

After having gained some insight into the Green function concept and the

calculation of self-energies, we turn now toaspecialmodel,forwhichthecorre-

lation eﬀects can be evaluated in closed form. This model has been designed to

explain the observation that tra nsition metal oxides, which according to their

electronic conﬁguration should be metals due to partially ﬁlled single-particle

energy bands, turn out to be insulators. Take as an example CoO, which crys-

tallizes in the rocksalt structure: the conﬁguration of the va l ence electrons,

Co 4s

and O 2s

, tells us that there is an odd number of electrons

for each lattice point (of the fcc lattice), which in the independent particle

picture leads to half-ﬁlling of the topmost band and means metallic behavior.

This topmost band is a narrow d band (see Sect.

5.4),whichisevennarrower

here than for Co (Fig.

5.14), because due to the larger spacing between neigh-

boring Co atoms in CoO the overlap of the d orbitals is smaller. Under this

condition the electronic correlation leads to a signiﬁcant modiﬁcation of the

electronic structure, which – as we shall see – allows one to understand the

experimental fact that CoO is not a metal but an insulator [

169].

Let us consider electrons in an energy band deriving from one atomic (d or

f) orbital. The corresponding Bloch function can be expr e ssed (see (6.7)) by

kσ

(r)=

√



ik·R

(r − R), (7.36)

where we have dropped the band index and kept only the spin index. We may

switch with

kσ

√



ik·R

Rσ

(7.37)

202 7 Correlated Electrons

φ(r)

R–dRR+d

Fig. 7.2. Schematic of atomic orbitals at lattice points of a linear chain and their

overlap

to a formulation with fermion operators (see Problem 6.2) and write for the

single-particle term as in Sect.

6.1



kσ

†

kσ



′



ik·(R−R

′

)



 

′

†

′

Rσ



′

†

′

Rσ

. (7.38)

The dispersion of the energy band ǫ

is now expressed in terms o f the ho pp i n g

or transfer matrix elements t

′

, R = R

′

, while the term with R = R

′

gives

the atomic energy level from which the band derives. For small overlap of the

atomic orbitals, a s is the case here and illustrated in Fig.

7.2, the tight-binding

approximation applies and we may write



kσ

†

kσ

≃ ǫ



Rσ

†

Rσ

+ t



Rdσ

†

R+dσ

Rσ

(7.39)

with the atomic level energy ǫ

and the hopping or transfer matrix element

between nearest neighbors R − R

′

= d (both assumed not to depend on σ)

t =



∗

(r)v(r)φ(r − d)d

r. (7.40)

Likewise, the electron–electron interaction takes the form



′

†

′

(7.41)

with the interaction matrix element now being the four-center integral:

′



′

∗

(r − R

)φ

∗

′

− R

′

)

4πε

|r − r

′

φ(r

′

− R

′

)φ(r − R

). (7.42)

7.2 The Hubbard Model 203

With the same arguments as used in Sect.

5.4, it is conceivable that the

dominating contributions come from two-center terms with R

= R

and R

′

= R

′

= R

′

, which give the approximate form

int

≃



′

σσ

′

†

Rσ

†

′

Rσ



′

σσ

′

Rσ

′

. (7.43)

The last expression results by identifying the number operator n

Rσ

= c

†

Rσ

and applying the anticommutation rules for (Rσ) =(R

′

). If only the term

with R = R

′

is considered, which describes double occupancy of the site

R with electrons of opp osite spins, we arrive at the single-band Hubbard

Hamiltonian [207, 208]

H = ǫ



Rσ

†

Rσ

+ t



Rdσ

†

R+dσ

Rσ

+ U



R↑

R↓

. (7.44)

Here U = V

, called the Hubbard–U, weighs the strength of the correlation

in case of two electrons at the same site. This Hamiltonian contains two com-

pet ing mechanisms: the hopping term is responsible for band formation and

delocalization, while the correlation term, with its energy increase for double

o ccupancy o f sites with electrons of opposite spin, favors a ground state with

localized magnetic moments of uncompensated spins at each site.

In the following, we replace R by the site index i and consider the Hubbard

Hamiltonian (

7.44)intheform

H = H

+ H



ijσ

†

iσ

jσ



iσ

i−σ

, (7.45)

where t

is the hopping matrix element between two sites (which need not be

the nearest neighbors). In order to calculate the spectrum of H,wecalculate

the Green function describing the propagation of an electron with spin σ from

site j at time t

′

to site i at time t

G(ijσ; t, t

′

)=−

¯h

θ(t − t

′

){c

iσ

(t),c

†

jσ

′

)}. (7.46)

This is done by solving the equation of motion

i¯h

∂

∂t

G(ijσ; t, t

′

)=δ(t − t

′

)δ

−

¯h

θ(t − t

′

){[c

iσ

(t), H],c

†

jσ

′

)}. (7.47)

The commutator of c

iσ

with H can be evaluated with

iσ

†

jσ

′

mσ

′

]=c

mσ

σσ

′

(7.48)

and

John Hubbard 1931–1980.

204 7 Correlated Electrons

iσ

jσ

′

j−σ

′

]=δ

iσ

i−σ

(δ

σσ

′

+ δ

σ−σ

′

) (7.49)

giving

iσ

, H]=



mσ

+ Un

i−σ

iσ

. (7.50)

Thus, the equation of motion (

7.47)reads

i¯h

∂

∂t

G(ijσ; t, t

′

)=δ(t − t

′

)δ



G(mjσ; t, t

′

)+UΓ(iiijσ; t, t

′

) (7.51)

with a higher order Green function of the general form

Γ(ilmjσ ; t, t

′

)=−

¯h

θ(t − t

′

){c

†

i−σ

(t)c

l−σ

(t)c

mσ

(t),c

†

jσ

′

)}. (7.52)

Instead of truncating the hierarchy by factorizing the higher order Green

function in (

7.51)(which would be the Hartree–Fock approximation to be

considered later), we continue by writing its equation of motion as

i¯h

∂

∂t

Γ(iiij; t, t

′

)= n

i−σ

δ

δ(t − t

′

)

−

¯h

θ(t − t

′

){[n

i−σ

(t)c

iσ

(t), H],c

†

jσ

′

)} . (7.53)

The commutator with H gives the two contributions (Problem 7.3)

i−σ

iσ

, H





i−σ

mσ

+ c

†

i−σ

m−σ

iσ

− c

†

m−σ

i−σ

iσ



(7.54)

and

i−σ

iσ

, H

]=Un

i−σ

iσ

(7.55)

which leads to



i¯h

∂

∂t

− U



Γ(iiijσ; t, t

′

)=n

i−σ

δ

δ(t − t

′





Γ(iimjσ; t, t

′

)

+Γ(imijσ; t, t

′

) − Γ(miijσ; t, t

′

)



. (7.56)

This equation of motion simpliﬁes for the case t

= ǫ

, i.e., for a band

without dispersion, known also as the atomic limit,to

i¯h

∂

∂t

− ǫ

− U

Γ(iiijσ; t, t

′

)=n

i−σ

δ

δ(t − t

′

). (7.57)

After Fourier transformation, it yields the solution

Γ(iiijσ; E)=

n

−σ



E − ǫ

− U

, (7.58)

7.2 The Hubbard Model 205

ε +U

–σ

(E)

l –<n

–σ

Fig. 7.3. Density of states for the Hubbard model in the atomic limit and

comparison with the HF result

with n

−σ

 = n

i−σ

 independent of the site index i due to the translation

invariance of the system. For the same situation, (

7.51) takes a form, which

after Fourier transformation leads to

G(iiσ; E)=

E − ǫ

− U(1 −n

−σ

)

(E −ǫ

)(E − ǫ

− U)

(7.59)

and can be written in the form

G(iiσ; E) = lim

δ→0

1 −n

−σ



E − ǫ

+iδ

n

−σ



E − ǫ

− U +iδ

. (7.60)

The surprising result is that G(iiσ ; E) has two p oles, one at E = ǫ

− iδ

with the weight Z =1−n

−σ

 and one at E = ǫ

+ U − iδ with the weight

Z = n

−σ

 (Fig.

7.3). This pole structure is a consequence of the energy

depen d ence of the self-energy. The second term vanishes for n

−σ

→0and

the rema ining ﬁrst term reduces for U = 0 to the Green function of the

noninteracting electron

(0)

(iiσ; E) = lim

δ→0

E − ǫ

+iδ

. (7.61)

The eﬀect of considering the correlation U is the splitting of the single-particle

level into two spin independent quasi-particle levels (separated by U )whose

weights depend on the occupati on. This is clearly seen also in the density of

states

(E)=



ImG(iiσ; E)





1 −n

−σ





δ(E − ǫ

)+n

−σ

δ(E − ǫ

− U) (7.62)

shown in Fig.

7.3. The new quality of the result can be emphasized also by

comparing with the Hartree–Fock approximation in (

7.51), which means to

replace Γ(iiijσ)byn

−σ

G(ijσ). The result for the case t

= ǫ

206 7 Correlated Electrons

(iiσ; E)=

E − ǫ

− Un

−σ

+iδ

, (7.63)

which has a single pole only with weight Z =1atE = ǫ

+ U n

−σ

 as

indicatedinFig.

7.3.

Let us now consider the band formation by relaxing the approximation

∼ δ

. Then, (

7.56) can be factorized with (for i = m)

Γ(iimjσ) →n

−σ

G(mjσ)

Γ(imijσ) →c

†

i−σ

m−σ

G(ijσ)

Γ(miijσ) →c

†

m−σ

i−σ

G(ijσ),

where the last two terms compensate each other in (

7.56) due to t

= t

and one obtains with τ = t − t

′

i¯h

∂

∂τ

− ǫ

− U

Γ(iiijσ; τ)=n

−σ





δ(τ)+



m=i

G(mjσ; τ )



. (7.64)

The Fourier transform of this equation

Γ(iiijσ; E)=

n

−σ



E − ǫ

− U





m=i

G(mjσ; E)



(7.65)

can be use d to write, instead of (

7.51), the equation

(E −ǫ

)G(ijσ; E)=





m=i

G(mjσ; E)



Un

−σ



E − ǫ

− U

. (7.66)

Finally, we transform into the Bloch representation, which for the one-

dimensional model with lattice constant a means

G(kσ; E)=



−ik(i−j)a

G(ijσ; E), (7.67)

to ﬁnd

(E − ǫ

)G(kσ; E)=



1 − ǫ

G(kσ; E)



Un

−σ



E − ǫ

− U

(7.68)

with ǫ



n=m

exp (−ik(n − m)a). This can b e solved to yield

G(kσ; E)=

E − ǫ

− Σ(σ; E)

(7.69)

with the self-energy

Σ(σ; E)=Un

−σ



E − ǫ

− U(1 −n

−σ

)

. (7.70)

7.2 The Hubbard Model 207

ε +U

insulator metal t

Fig. 7.4. Spectrum of the Hubbard model in dependence on the hopping matrix

element t to indicate the insulator–metal transition

Due to the energy dependence of the self-energy, the Green function (

7.69)

has two separate poles with diﬀerent spectral weights (as for the atomic limit

(Problem 7.4)) for each k. The spectrum, obtained by projecting the eigen-

values onto the energy axes, exhibits two bands whose widths are determined

by that of the single-particle energy dispersion ǫ

(note that the self-energy

(

7.70) does not depend on k). In tight-binding approximation, the dispersion

of the energy band b ecomes ǫ

= ǫ

+2t cos ka with t being the hopping

matrix element between nearest neighbors.

In Fig.

7.4, the spectrum of the Hubbard model is depicted for ﬁxed U in

dependence on t. Two bands evolve with increasing t from the two levels of

Fig.

7.3, the upper and lower Hubbard band: they are separated by a gap as

long as the band width 4t is smaller than U, but they overlap for 4t>U.

For a system with one electron per lattice site, a single band can be com-

pletely ﬁlled. Thus for 4t>U the overlapping Hubbard bands are partially

ﬁlled (the Fermi energy is in a region with ﬁnite density of states) and the

model describes a metal, while for 4t<U the Hubbard bands are separated

by a gap, the lower (upper) band being completely ﬁlled (empty), and the

model describes an insulator. This metal–insulator transition is the important

result o f the Hubbard model. Systems showing this behavior are called Mott

–

Hubb ard insulators. For the case of half-ﬁlling, the Hubbard Hamiltonian can

be transformed into a Hamiltonian of the Heisenberg type (Problem 7.5), that

allows one to describe the magnetic prop erties of these systems. However, the

Hubbard model is not correct for small values of U , because it always gives

the two separate eigenvalues for each k, while for small correlation we expect

the single band solution.

Sir Nevill Mott 1905–1996, Nobel prize in Physics 1977, together with Philip

W. Anderson and John H. Van Vleck.