Slave-boson field fluctuation approach to the extended Falicov-Kimball model: charge, orbital, and excitonic susceptibilities

Abstract

Based on the SO(2)-invariant slave-boson scheme, the static charge, orbital,
and excitonic susceptibilities in the extended Falicov-Kimball model are
calculated. Analyzing the phase without long-range order we find instabilities
towards charge order, orbital order, and the excitonic insulator (EI) phase.
The instability towards the EI is in agreement with the saddle-point phase
diagram. We also evaluate the dynamic excitonic susceptibility, which allows
the investigation of uncondensed excitons. We find qualitatively different
features of the exciton dispersion at the semimetal-EI and at the
semiconductor-EI transition supporting a crossover scenario between a BCS-type
electron-hole condensation and a Bose-Einstein condensation of preformed bound
electron-hole pairs.

Full text

Page 1

arXiv:1102.4887v1 [cond-mat.str-el] 24 Feb 2011
Slave-boson field fluctuation approach to the extended Falicov-Kimball model:
charge, orbital, and excitonic susceptibilities
B. Zenker1, D. Ihle2, F. X. Bronold1, and H. Fehske1,3
1Institut f¨ ur Physik, Ernst-Moritz-Arndt-Universit¨ at Greifswald, D-17489 Greifswald, Germany
2Institut f¨ ur Theoretische Physik, Universit¨ at Leipzig, D-04109 Leipzig, Germany
3School of Physics, University of New South Wales, Kensington 2052, Sydney NSW, Australia
(Dated: February 25, 2011)
Based on the SO(2)-invariant slave-boson representation the static charge, orbital, and excitonic
susceptibilities in the extended Falicov-Kimball model are calculated. Analyzing the phase without
long-range order we find instabilities towards charge order, orbital order, and the excitonic insulator
(EI) phase. The instability towards the EI is in agreement with the saddle-point phase diagram.
We also evaluate the dynamic excitonic susceptibility, which allows the investigation of uncondensed
excitons. We find qualitatively different features of the exciton dispersion at the semimetal-EI and at
the semiconductor-EI transition supporting a crossover scenario between a BCS-type electron-hole
condensation and a Bose-Einstein condensation of preformed bound electron-hole pairs.
PACS numbers: 71.30.+h, 71.35-y, 71.35.Lk, 71.28.+d
I. INTRODUCTION
At low temperatures electronic correlations can cause
anomalies at the semimetal-semiconductor (SM-SC)
transition.1Half a century ago, Mott2argued that in
a SM with a very low carrier density the Coulomb at-
traction between electrons and holes should lead to the
spontaneous formation of electron-hole bound states (ex-
citons), and the system would become insulating. Shortly
afterwards, Knox3noticed that a SC is unstable against
the spontaneous formation of excitons if the exciton bind-
ing energy overcomes the gap energy separating valence
and conduction band. Both arguments suggest a new
distorted phase, an exciton condensate known as the ex-
citonic insulator (EI), to be the crystal ground state. The
SM-EI transition is mathematically similar to the BCS
theory of superconductivity, while the SC-EI transition
can be treated as a Bose-Einstein condensation (BEC) of
preformed excitons. Hence, the EI is discussed in view of
a BCS-BEC crossover scenario in a solid.4–7
Whilst theoretically predicted a long time ago,8(for re-
cent reviews see Ref. 9) no conclusive experimental proof
of the existence of the EI has been achieved yet. How-
ever, there are a few promising candidates. In the mixed
valence compound TmSe0.45Te0.55detailed studies of the
pressure-induced SC-SM transition suggest that excitons
are created in a large number and condense below 20
K.10More recently, several transition-metal dichalco-
genides were reported to exhibit an EI phase. Angle-
resolved photo emission spectra (ARPES) measurements
of Ta2NiSe5 traced back the extreme valence band top
flattening at low temperature to an EI ground state.11
ARPES data of 1T-TiSe2indicate that the EI is the driv-
ing force for the charge-density-wave (CDW) transition
in this material.12
From a theoretical point of view, the description of the
EI with a Falicov-Kimball-type model seems promising.
The original Falicov-Kimball (FKM) model13contains
itinerant c-electrons (with bandcenter Ec and hopping
FIG. 1: (Color online) Hartree-Fock ground-state phase dia-
gram of the EFKM in two dimensions for Coulomb strength
U = 2. The difference between CDW and SOO (staggered
orbital order) is explained in the text. The black solid line
represents the second-order transition from an EI to a BI
(band insulator), the dashed line represents the first-order
CDW/SOO-EI transition.
amplitude tc) that interact via a local Coulomb repul-
sion U with localized f-electrons (with energy level Ef),
where the spin is neglected. Since the local f-electron
number is strictly conserved in the FKM, f-c-coherence
cannot be established.14One way to overcome this short-
coming is to include an f-c hybridization.15As shown in
Refs. 16,17, the extension by a finite f-bandwidth also
induces f-c coherence. The model with a direct f-f
hopping (with hopping amplitude tf) is called the ex-
tended Falicov-Kimball model (EFKM) and has previ-
ously been used to describe different properties of the
EI phase.6,7,18,19The ground-state phase diagram of the
EFKM was determined with a constraint path Monte
Carlo (CPMC) technique for one and two dimensions
(1D and 2D) in the strong16and intermediate coupling

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2
regime17as well as in the Hartree-Fock (HF) approx-
imation for 2D,203D,20and infinite dimensions.21All
approaches yield a qualitatively similar phase diagram.
Figure 1 displays the HF ground-state phase diagram for
U = 2 in 2D, depicting the generic situation. It was
shown previously that Fig. 1 agrees with the CPMC data
even quantitatively.20Besides full c- and full f-band in-
sulator (BI) regions, the EFKM ground-state phase di-
agram exhibits three symmetry broken phases: the EI,
a CDW, and a staggered orbital order (SOO). The EI
is characterized by a nonvanishing average ?c†f?. The
CDW is described by a periodic modulation in the total
electron density comprising both f- and c-electrons. The
SOO is characterized by a periodic modulation in the
difference between the f-electron and the c-electron den-
sity, which may be accompanied by a CDW. The SOO
(CDW) establishes the ground state for the symmetric
case (Ef = Ec) for all ratios of −tf/tc (for the CDW
the point |tf| = |tc| has to be excluded, see below).
These phases are rapidly suppressed in favor of the EI if
Ef ?= Ec. Between the uniform EI phase and the CDW
or SOO phase there is a first-order phase transition. The
EI-BI transition is of second order. For tf= 0 the FKM
is recovered, and the EI phase cannot be realized.
For the investigation of electron correlation effects the
Gutzwiller approximation22is an established technique.
Kotliar and Ruckenstein introduced a scalar slave-boson
(SB) scheme which reproduces the Gutzwiller solution
of the Hubbard model as a saddle-point.23A manifestly
spin-rotation invariant form of the SB representation has
been worked out for the Hubbard model24and for multi-
band Hubbard models.25We have developed an SO(2)-
invariant SB approach for the EFKM (Ref. 19) that re-
produces the HF result for the EI phase boundary at
T = 0, but leads to a substantial reduction of the crit-
ical temperature. It is the aim of this work to include
Gaussian fluctuations around the saddle-point26–28in the
SO(2)-invariant SB scheme at zero and finite tempera-
ture. This offers an opportunity to calculate suscepti-
bilities for investigating instabilities against long-range
ordered phases and the formation of excitons.
The paper is organized as follows. In Sec. II the model
Hamiltonian and the SB scheme are introduced. More-
over the saddle-point approximation is given and the cal-
culation of response functions within the SB scheme is
explained. In Sec. III we present numerical results for
the instabilities toward the CDW, the SOO, and the EI
phase. Finally we investigate the formation of excitons
in the phase without long-range order. Section IV sum-
marizes our results.
II. THEORY
A.Model Hamiltonian
Expressing the orbital flavor by a pseudospin variable
σ =↑,↓, where c(†)
i↑≡ f(†)
i
and c(†)
i↓≡ c(†)
i, the EFKM can
be written as an asymmetric Hubbard model,
H =
?
i,σ
(Eσ− µ)c†
iσciσ−
?
?i,j?,σ
tσc†
iσcjσ+ U
?
i
ni↑ni↓,
(1)
where c(†)
Wannier site i and niσ= c†
number operator. Eσ denotes the bandcenter of the σ-
electron band, µ gives the chemical potential, tσ is the
hopping amplitude, and U measures the Coulomb inter-
action strength. In what follows we consider E↓ = 0,
E↑ < 0, t↓ = 1, and t↑ < 0.
sured in units of t↓. We restrict ourselves to t↓t↑ < 0,
i.e., the valence band top and the conduction band min-
imum are located at the Brillouin zone center. Moreover
we exclusively investigate the half-filled band case, i.e.,
1
N
?
iσannihilates (creates) a σ-band electron at the
iσciσis the corresponding
All energies are mea-
i,σ?niσ? = 1, where N is the number of lattice sites.
B.Slave-boson functional integral representation
Following Refs. 19,24 the Hilbert space is enlarged by
introducing auxiliary bosons: e(†)
site, d(†)
i, related to a double occupied site, and the ma-
trix operator p(†)
i, related to a single occupied site,
i, related to an empty
p(†)
i
=1
2
?
p(†)
p(†)
i0+ p(†)
ix+ ip(†)
iz
p(†)
p(†)
ix− ip(†)
i0− p(†)
iy
iy
iz
?
. (2)
The fermionic degrees of freedom are captured by the
pseudofermions ˜ c†
Unphysical states of the extended fermion-boson Fock
space are excluded by two sets of local constraints,
i= (˜ c†
i↑,˜ c†
i↓) and ˜ ci= (˜ ci↑,˜ ci↓)T.
C(1)
i
C(2)
i
= e†
iei+ 2Trp†
= ˜ ci⊗ ˜ c†
ipi+ d†
ipi+ d†
idi− 1 = 0,
idiτ0− τ0= 0,
(3)
i+ 2p†
(4)
where τ0denotes the unit matrix and ? τ = (τx,τy,τz)T
is the vector of the Pauli-matrices.
Since the bosonic occupation number of one site is
coupled to the fermionic occupation, the bosons have to
change simultaneously when an electron is created or an-
nihilated. This is achieved by introducing the bosonic
hopping operator zi,
ciσ=
?
ρ
ziσρ˜ ciρ. (5)
The choice of ziis not unique. We choose19,24
zi= Lie†
iMipiNi+ Li˜ p†
iMidiNi
(6)
with
Li= [(1 − d†
Ni= [(1 − e†
Mi= [1 + e†
idi)τ0− 2p†
iei)τ0− 2˜ p†
iei+ d†
ipi]−1/2,
i˜ pi]−1/2,
(7)
(8)
idi+ 2Trp†
ipi]1/2, (9)

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and ˜ p(†)
free-fermion result on the mean-field level. The Hubbard
interaction term is bosonized via ni↑ni↓→ d†
The resulting coupled fermion-boson system is evalu-
ated within a functional integral representation. Then,
the bosons become complex fields and the fermions are
given by complex Grassmann fields. The Lagrange mul-
tipliers λ(1)
enforce the constraints (3) and (4). Exploiting the gauge
symmetry of the action and transforming the Lagrange
multipliers into real time-dependent Bose fields we can
remove the phases of pi0, piz, and ei. Using the Grass-
mann integration formula, we obtain the grand canonical
partition function given by a functional integral over Bose
fields only,
iρρ′ = ρρ′p(†)
i−ρ′−ρ, which guarantees the correct
idi.
i, λ(2)
i0, λ(2)
ix, λ(2)
iy, and λ(2)
izare introduced to
Z =
?
D[e]D[p0]D[p∗
x,px]D[p∗
y,py]D[pz]D[d∗,d]
D[λ(1)]D[λ(2)
0]D[?λ(2)]e−S
(10)
with the effective bosonic action (µ = 0,x,y,z)
S =
β
?
0
dτ
??
i
?
− λ(1)
i
+ λ(1)
ie2
i+
?
− i?λ(2)
i0)|di|2
iy∂τpiy+ d∗
µ
(λ(1)
i
− λ(2)
i0)|piµ|2
− pi0(? p∗
+ (λ(1)
i+ ? pi)?λ(2)
i
i(? p∗
i× ? pi)
i
+ U − 2λ(2)
ix∂τpix+ p∗
+ p∗
i∂τdi
??
− Tr ln
?
− G−1
?ij?,ρρ′(τ,τ′)
?
,(11)
where ? pi= (pix,piy,piz) and?λ(2)
inverse Green propagator is given by
i
= (λ(2)
ix,λ(2)
iy,λ(2)
iz). The
G−1
?ij?,ρρ′(τ,τ′) =
??− ∂τ+ µ − λ(2)
−E↑
+ (z∗
itzj)ρρ′,ττ′(1 − δij),
?t↑ 0
time, space, and spin variables. For the half-filled band
case Eqs. (11) and (12) are an exact representation of the
partition function of the EFKM. One obtains zi= ziτ0.
i0
?δρρ′
2(τ0+ τz)ρρ′ −?λ(2)
i? τρρ′
?
δijδ(τ − τ′)
(12)
where t =
0 t↓
?
. The trace in Eq. (11) extends over
C.Saddle-point approximation
To proceed we approximate all bosonic fields by their
time-averaged values (static approximation), i.e., the
bosonic fields are taken to be real. Moreover, we look
for uniform solutions, that is, the Bose fields are taken
to be independent of the lattice site.
We restrict ourselves to the phase without long-range
order, which we denote as paraphase. The saddle-point
equations for the paraphase (px= py= λ(2)
are
1
2(n↑− n↓) ,
1
2+
1
2z2
x
= λ(2)
y
= 0)
p0pz =
(13)
p2
0=
?
?
?
?z2
n↑n↓(1 − z2) ,(14)
d2=
z2?2 − p2
z2(2 − p2
0− p2
z
?+ 2p2
z) + z4p2
0
−2p0
0− p2
1
0− p2
z
z2ǫ(0) − 2λ(2)
z+ p2
0
?
, (15)
λ(2)
z
= −pz
U = −2d2− p2
1
N
p0
2d2−
0+ z2p2
p2
p2
z
?
z2ǫ(0) , (16)
0d2
z
pz
p0
, (17)
nσ =
?
?
k
nkσ,(18)
ǫ(0) =
1
N
k
(t↑γknk↑+ t↓γknk↓) , (19)
where
nkσ = [exp(βEkσ) + 1]−1,
Ekσ = Eσ+ σλ(2)
˜ µ = µ − λ(2)
D-dimensional
l=1coskl. The chemical potential is determined by
the condition
1
N
k,σ
(20)
z
− ˜ µ − z2tσγk, (21)
0
. (22)
On
2?D
a hypercubic lattice,γk
=
?
nkσ= 1.(23)
The quasiparticle gap Egindicates the splitting of the
↑- and ↓-band (in the paraphase), which is caused by the
correlation-induced quasiparticle bandshift λ(2)
D-dimensional hypercubic lattice, Egis given by
z . For a
Eg= |E↑| + |2λ(2)
For a SM, Eg≤ 0 and for a SC, Eg> 0.
We obtain the EI phase boundary by solving the SB
gap equation,
z| − 2Dz2(|t↑| + |t↓|) .(24)
1 =
1
p0pzλ(2)
z
1
N
?
k
nk↑− nk↓
Ek↑− Ek↓
,(25)
resulting from Eqs. (63) and (65) of Ref. 19.
The gap equation (25) captures both the BCS and the
BEC situation, but it cannot discriminate between them.
To this end, we follow an idea from Ihle et al. (Ref. 6) and
investigate the excitonic susceptibility in the paraphase.
D.Gaussian fluctuations
In order to study response functions, we take into
account Gaussian fluctuations around the saddle point

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for the paraphase, that is, Φia =¯Φa+ δΦia, where
Φia= {ei,pi0,pix,piy,piz,di,λ(1)
the action is given by
i,λ(2)
ix,λ(2)
iy,λ(2)
iz}. Then,
S =¯S +
?
q,a,b
δΦa(−q)Sab(q)δΦb(q) , (26)
where the bar denotes the saddle-point value.
In order to achieve comparability with the saddle-point
results, we start the fluctuation calculation from the same
level of approximation as for the saddle-point calculation,
i.e., we first perform the static approximation and con-
sider only the fluctuations of the 11 real-valued fields Φia.
The fluctuation matrix can be calculated according to
Sab(q,q′)
1
2Nβ
=
?
Ri,Rj
e−iqRi
∂2S
∂Φia∂Φjb
?????Φi= Φj=¯ Φ
δ(τ − τ′)
e−iq′Rj
= Sab(q)δq,−q′ .
Here, we use the shorthand notation Ri = (Ri,τ) and
q = (q,ωm), where τ is the imaginary time, ωm= 2πm/β
denote the bosonic Matsubara frequencies, Riis the po-
sition vector, and q is the wave vector.
The response functions can be expressed in terms of
the SB field fluctuations using the local constraints (3)
and (4). The charge susceptibility reads
(27)
χc(q) = ?δ [n↑(−q) + n↓(−q)]δ [n↑(q) + n↓(q)]?
= 4?e2?δe(−q)δe(q)? − 2ed?δe(−q)δd(q)?
+d2?δd(−q)δd(q)??
The orbital susceptibility is given by
. (28)
χo(q) = ?δ [n↑(−q) − n↓(−q)]δ [n↑(q) − n↓(q)]?
= 4?p2
+p2
z?δp0(−q)δp0(q)? + 2pzp0?δp0(−q)δpz(q)?
0?δpz(−q)δpz(q)??
Considering the creation operator of an onsite electron-
hole pair6
. (29)
b†
i= c†
i↓ci↑,b†
q=
1
√N
?
k
c†
k+q↓ck↑, (30)
the electron-hole susceptibility, hereafter denoted as ex-
citonic susceptibility, is given by
χX(q) = ?δbqδb†
= p2
q?
0[?δpx(−q)δpx(q)? + ?δpy(−q)δpy(q)?
−i?δpy(−q)δpx(q)? + i?δpx(−q)δpy(q)?] .
(31)
The correlation functions may be expressed as func-
tional integrals over Bose fields:
?δΦa(−q)δΦb(q)? =1
Z
?
D[Φ] δΦa(−q)δΦb(q) e−S(q).
(32)
Hence, the correlation functions are related to the inverse
fluctuation matrix by
?δΦa(−q)δΦb(q)? =1
2S−1
ab(q) . (33)
It turns out that for the paraphase the 11×11 fluctu-
ation matrix decomposes into a 7 × 7 matrix containing
the charge fluctuations (δe, δp0, δd, δλ(1), δλ(2)
orbital fluctuations (δpz, δλ(2)
containing the electron-hole pair fluctuations (δpx, δλ(2)
δpy, δλ(2)
saddle-point equations (13)–(19) self-consistently.
The description of the CDW and SOO requires the in-
clusion of inhomogeneous solutions with a periodic mod-
ulation in the densities, ?niσ? = nσ+δσcos(QRi), where
the order vector in 3D is given by Q = (π,π,π). The
CDW and SOO order parameters are δCDW=1
and δSOO =
the CDW and SOO describe the same symmetry broken
state. We can investigate the formation of both phases
without generalizing the SB formalism to a bipartite lat-
tice by calculating the static (ω = 0) charge and orbital
susceptibility with order vector q = Q, given by
0) and the
z ) and into a 4 × 4 matrix
x ,
y ). The SB fields are obtained by solving the
2(δ↑+δ↓)
1
2(δ↑− δ↓), respectively.29If |δ↑| ?= |δ↓|,
χc = χc(Q,0)
= 2?e2(S−1)ee+ d2(S−1)dd− 2ed(S−1)ed
χo = χo(Q,0)
= 2?p2
?
,(34)
z(S−1)p0p0+ p2
0(S−1)pzpz− 2p0pz(S−1)p0pz
?
.
(35)
Since the analytical inversion of a 7 × 7 matrix is a
formidable task, we perform it numerically.
After analytic continuation (iωm→ ω +i0+) the exci-
tonic susceptibility (31) yields
χX(q,ω) =
χ(0)
X(q,ω)
χ(0)
X(q,ω) + 1
−Spxpx
p2
0
, (36)
with
χ(0)
X(q,ω) =
1
N
?
k
nk↑− nk+q↓
ω + Ek↑− Ek+q↓
(37)
and
Spxpx=
?1
p2
0
−1
2
p2
0− p2
p2
z
0d2z2
?
z2ǫ(0) +pz
p0λ(2)
z
. (38)
For the BI at T = 0 the random phase approximation
result6is recovered, −Spxpx
0
To determine the EI phase we compute the static exci-
tonic susceptibility χX(q,0). The direct band gap situa-
tion gives the order vector of the EI phase as q = 0. Using
Eq. (16) the fluctuation matrix element Spxpx[Eq. (38)]
reduces to
Spxpx=p0
p2
= U.
pzλ(2)
z
.(39)

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It is easy to see that the condition for the divergence of
χX(0,0) equates to the gap equation (25).
The poles of Reχ(0)
X(q,ω) [Eq. (37)] give the continuum
of electron-hole excitations, i.e., ωk(q) = Ek+q↓− Ek↑.
Excitonic pairing of electrons and holes is described by
the pole of ReχX(q,ω) [Eq. (36)] outside the electron-
hole continuum,6i.e., by
Reχ(0)
X(q,ωX) =
p2
0
Spxpx
, (40)
with
0 < ωX(q) < ωC(q) , (41)
where ωC(q) = ωk(q)|min. The exciton binding energy is
given by
EB
X(q) = ωC(q) − ωX(q) .(42)
We want to emphasize that ωX, ωCand EB
q-dependent in contrast to Ref. 6, where only excitons
with q = 0 are considered, and Ref. 5, where the exciton
binding energy is assumed to be independent of q.
Xare explicitly
III.NUMERICAL RESULTS
A.Instabilities against CDW and SOO
To obtain results for the 3D EFKM we transform the
k-summation into an energy integral using the tight-
binding density of states (DOS) for a simple cubic lattice.
From the charge and orbital susceptibility we derive
information about the CDW and SOO formation, respec-
tively. For asymmetric bands (|t↑| ?= |t↓|) the charge and
orbital susceptibility diverge at the same critical E↑, as
shown in Fig. 2, implying |δ↑| ?= |δ↓|. The analogy be-
tween CDW and SOO vanishes if the bandwidths are
-1
0
1
χc
-0.5 -0.4-0.3-0.2
E↑
-50
0
50
χo
t↑ = -0.4
t↑ = -0.8
t↑ = -1.0
FIG. 2: (Color online) Static charge and orbital susceptibility
of the 3D EFKM for T = 0 and U = 4 as a function of E↑.
equal, as can be seen for t↑ = −1.0 in Fig. 2. In this
case, the orbital susceptibility diverges contrary to the
charge susceptibility, thus, a CDW will not develop and
δ↑= −δ↓. We conclude that the density inhomogeneity
δσis largely affected by the bandwidth.
-1
0
1
χc
-2
-1.5 -1
-0.50
E↑
-100
0
100
χo
U = 2
U = 4
U = 6
FIG. 3: (Color online) Static charge and orbital susceptibility
of the 3D EFKM for T = 0 and t↑ = −0.8 as a function of
E↑.
Figure 3 shows χoand χcfor t↑= −0.8. The suscepti-
bilities diverge at the same critical E↑. With increasing
strength of the Coulomb interaction the critical |E↑| for
CDW (SOO) formation increases, because for a larger
interaction the charge (orbital) order becomes more fa-
vorable. Figure 3 clearly shows that the CDW and SOO
region is confined close to the symmetric case E↑= 0.
For small band splitting either the CDW (SOO) or
the EI, separated by a first-order phase transition, can
be realized, and one has to compare the free energies to
identify the true ground state. Hence, to determine the
SB ground-state phase diagram (analogous to the HF
case shown in Fig. 1) the generalization of the saddle-
point equations to a bipartite lattice is inevitable, which
is beyond the scope of this work. To investigate the EI in
the following, we choose the band-structure parameters
E↑= −2.4 and t↑= −0.8, where a CDW (SOO) is not
realized (see Fig 3).
B.Instability against EI
Figure 4 shows that the EI phase boundary in the
weak-coupling as well as in the strong-coupling regime
is reproduced by poles of the uniform static excitonic
susceptibility, as demonstrated analytically in Sec. IID.
To determine the region where free excitons can exist,
we evaluate the condition for exciton formation (40) sub-
jected to the constraint (41). The exciton binding en-
ergy has to be positive. For numerical reasons we set
the threshold to min(EB
X) = 10−6. For the 3D case we