Resonant Raman scattering effects in a nesting-driven charge-density-wave insulator: Exactanalysis of the spinless Falicov-Kimball model with dynamical mean-field theory

O. P. Matveev,1 A. M. Shvaika,1 and J. K. Freericks2

1Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, Lviv 79011, Ukraine2Department of Physics, Georgetown University, Washington, DC 20057, U.S.A.

Received 10 June 2010; revised manuscript received 13 August 2010; published 8 October 2010

We calculate the total electronic Raman scattering spectrum for a system with a charge density wave on aninfinite-dimensional hypercubic lattice. The problem is solved exactly for the spinless Falicov-Kimball modelwith dynamical mean-field theory. We include the nonresonant, mixed, and resonant contributions in threecommon experimental polarizations, and analyze the response functions for representative values of the energyof the incident photons. The complicated scattering response can be understood from the significant tempera-ture dependence of the many-body density of states and includes a huge enhancement for photon frequenciesnear the charge-density-wave gap energy.

DOI: 10.1103/PhysRevB.82.155115 PACS numbers: 71.10.Fd, 71.45.Lr, 78.30.j

I. INTRODUCTION

Inelastic light scattering is a powerful probe of the chargefluctuations in a strongly correlated material.1 By using po-larizers on the incident and the reflected light, one can ex-amine different symmetry channels for the charge excita-tions, and how easily they can scatter light. Using inelasticlight scattering, one can learn about the symmetry of under-lying order, such as the d-wave superconductivity in thehigh-temperature superconductors. Here, we will focus onthe effects of static charge-density-wave CDW order on theinelastic light scattering of a strongly correlated material.

The field of inelastic light scattering has been increasingin interest. When x-rays are used for the light source, one canexamine resonant inelastic x-ray scattering, where both en-ergy and momentum are exchanged between the light and thecharge excitations of the solid. Here, we focus on the zero-momentum limit, where only energy is exchanged, becausewe will be using optical light. Hence we will be examiningresonant effects in electronic Raman scattering. The dynami-cal mean-field theory DMFT approach to this problem wascompleted a few years ago2–5 in the normal state. One of theinteresting results from that work was that one could see ajoint resonance of low-energy features with higher energyfeatures when the photon energy was on the order of theinteraction strength U between the electrons. When one hascharge-density-wave order, there are two additional compli-cations that arise: i the density of states DOS has signifi-cant temperature dependence below Tc, where excitationswith energies smaller than the gap energy will be depleted asT→0 and ii the DOS develops sharp, singular peaks asT→0 that arise at the gap edge. One would hence expect theRaman response to have much more temperature dependencethan what was seen in the normal state and to have morestriking resonant effects because of the sharp peaks whichdevelop due to a pileup of the DOS at the gap edge. Indeed,the nonresonant Raman response, in the CDW phase, showsdramatic effects due to the singularity in the DOS in some ofthe symmetry channels.16

CDW order is also interesting because there are a numberof strongly correlated materials that display this behavior.

The most prevalent class of such materials are the transitionmetal dichalchogenides and trichalchogenides, which displayeither quasi-one-dimensional NbSe3 or quasi-two-dimensional TaSe2 or TbTe3 CDW order.6–8 In addition,there are known three-dimensional systems such as BaBiO3and Ba1−xKxBiO3 which display charge-density-wave ordervia nesting on a bipartite lattice at half-filling.9 This latterexample is particularly relevant to our work, since theDMFT is more accurate as the dimensionality increases. Oneof the longstanding questions in the field is the question ofwhether the order is driven electronically, with a lattice in-stability following the electronic instability, or vice versa.We would not have any direct answers to that question in thiswork since we are not examining time-resolved phenomenabut we will note that experimental light scattering work hasalready examined the phonon softening phenomena that isassociated with the lattice distortion.10 Here we focus onelectronic effects, which would be the obvious next genera-tion of experimental probes on these systems.

We will be varying the photon energy over a wide rangeof different values. We will see the most remarkable resonanteffects when the photon energy is equal to the gap energy, asone might naively expect. For many CDW systems, this gapenergy is at most a few hundred millielectron volt, which ismuch below the optical photon energies. Hence, the experi-mentally most relevant results will rely on examining jointresonant effects, such as what was observed in the normalstate in previous calculations. But we also will focus someattention on the most dramatic resonant effects under thehope that such CDW systems, made from strongly correlatedelectronic systems, might be found in the future, and thatthey can be studied with electronic Raman scattering.

We use the Falicov-Kimball model in our analysis be-cause it is one of the simplest models11 which possessesstatic CDW ordering and has an exact solution within DMFTRef. 12 for a review see Ref. 13. In particular, the irre-ducible charge vertex is known exactly and that is needed toexamine the charge screening effects. Our work also extendsrecent results on transport, optical conductivity, and nonreso-nant x-ray scattering in CDW systems14–16 to the realm ofresonant inelastic light scattering. A brief report on resonant

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Raman scattering has also appeared as a conferenceproceeding.17

The organization of this paper is as follows: in Sec. II, weintroduce the model and briefly review the dynamical mean-field theory approach in the ordered phase; in Sec. III, wedescribe the general formalism for inelastic light scattering;in Sec. IV, we focus on the detailed formulas for the mixedand resonant contributions to Raman scattering; in Sec. V,we present our numerical results and we analyze the Ramanscattering response for two different cases; and in Sec. VI,we present our conclusions.

II. ORDERED PHASE DYNAMICAL MEAN-FIELDTHEORY

Historically, the Falicov-Kimball model11 was introducedin 1969 to describe metal-insulator transitions in rare-earthcompounds and transition-metal oxides involving a simpli-fied two-band model with localized heavy electrons and itin-erant light electrons which hop between sites. The mobileelectrons hop to neighboring sites with a hopping integral −tand they interact with the localized particles at the same sitewith the Coulomb energy U. The mobile electron creation

annihilation operator at site i is denoted by di†di and the

local electron creation annihilation operator at site i is

f i† f i. We perform our calculations at half-filling because, in

this case, there is an insulating CDW phase at low tempera-ture for all values of U. The explicit formula for the Hamil-tonian appears in Eqs. 1 and 2.

An algorithm to determine the period-two ordered-phaseGreen’s functions within DMFT was developed by Brandtand Mielsch18 shortly after Metzner and Vollhardt19 intro-duced the idea of the many-body problem simplification inlarge dimensions. The CDW order parameter displaysanomalous behavior at weak coupling,20,21 and higher periodordered phases are possible, and have been examined on theBethe lattice.22 In previous works,14–16 the transport proper-ties and nonresonant inelastic light and x-ray scattering wereexamined in the commensurate CDW phase. A detailed de-scription of the DMFT solution for the CDW phase of theFalicov-Kimball model has also appeared in our previouspapers15,16 so we restrict ourselves to a brief summary inorder to establish our notation.

We work on an infinite-dimensional hypercubic latticewith nearest-neighbor hopping. This lattice is bipartite, im-plying that it can be divided into two sublattices, denoted Aand B, with the hopping being nonzero only between thedifferent sublattices. In this case, the Falicov-Kimball modelhas particle-hole symmetry, and the noninteracting Fermisurface is nested at half-filling with an ordering wave vectorat the zone boundary along the diagonal, which implies theCDW order will lie on the sublattice structure, with the den-sity of the light and of the heavy electrons being uniform oneach sublattice, but different on the different sublattices. Thisdifference in electron filling serves as the order parameter forthe CDW phase. Keeping this in the mind, we introducesublattice indices into the Falicov-Kimball model Hamil-tonian

H = ia

Hia −

ijab

tijabdia

† djb, 1

where i and a=A or B are the site and sublattice indices,respectively, and tij

ab is the hopping matrix, which is nonzeroonly between different sublattices tij

AA= tijBB=0. The local

part of the Hamiltonian is equal to

Hia = Unid

a nifa − d

anida − f

anifa 2

with the number operators of the itinerant and localized elec-

trons given by nid= di†di and nif = f i

† f i, respectively. For com-putational convenience, we have introduced different chemi-cal potentials for different sublattices, which allows us towork with a fixed order parameter, rather than iterating theDMFT equations to determine the order parameter which issubject to critical slowing down near Tc. The systemachieves its equilibrium state when the chemical potential isuniform throughout the lattice d

A=dB and f

A= fB.

The first step of the DMFT approach is to scale the hop-ping matrix element as −t=−t /2D we use t=1 as the unitof energy and then take the limit of infinite dimensionsD→.19 The self-energy is then local

ijab = i

aijab 3

and in the case of two sublattices has two values A andB. As a result, the DMFT equations become matrixequations for the CDW phase. Hence, we can write the so-lution of the Dyson equation in momentum space in a ma-trix form

Gk = z − tk−1, 4

where the irreducible part z and the hopping term tk arerepresented by the following 22 matrices:

z = + dA − A 0

0 + dB − B

,

tk = 0 k

k 0 5

with the band structure k satisfying k=−t limD→i=1

D cos ki /D. Then we can represent the localGreen’s function on sublattice a

Gaa =1

N

kGk

aa 6

in terms of the local dynamical mean field a, via

Gaa =1

+ da − a − a

. 7

Finally, we close the system of equations for a anda by finding the local Green’s function from the solutionof an impurity problem in the dynamical mean field a.For the Falicov-Kimball model such a problem can be solvedexactly and the result is equal to

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Gaa =1 − nf

a

+ da − a

+nf

a

+ da − U − a

, 8

where nfa is the average concentration of the localized elec-

trons on the sublattice a. In the CDW phase, the total con-centration of localized electrons is fixed nf

A+nfB=const and

the difference of the concentrations on each sublattice nf=nf

A−nfB is the order parameter of the CDW phase and is

defined from the equilibrium condition on the sublatticechemical potentials: f

A− fB=0.

Numerical solutions of these equations are given in Ref.15, where the evolution of the DOS in the CDW-orderedphase is shown. At T=0, a real gap develops of magnitude Uwith square root singularities at the band edges even on thehypercubic lattice which has infinite tails to the DOS in thenormal state. As the temperature increases, the system de-velops substantial subgap DOS which are thermally activatedwithin the ordered phase. Additional plots of the DOS can befound in Ref. 15. Note that the singular behavior occurs forone of the “inner” band edges on each sublattice and that thesubgap states develop very rapidly as the temperature risesand completely fill in the CDW gap at the critical tempera-ture Tc.

III. FORMALISM FOR INELASTIC LIGHT SCATTERING

The interaction of a weak external transverse electromag-netic field A with an electronic system with nearest-neighborhopping is described by the Hamiltonian23,24

Hint = −e

c

kjk · A− k

+e2

22c2kk

A− k, k + kA − k , 9

where the current operator and stress tensor for itinerantelectrons are equal to

jq = abk

tabkk

da†k + q/2dbk − q/2 10

and

, q = abk

2tabkk k

da†k + q/2dbk − q/2 , 11

respectively. Here tabk are the components of the 22hopping matrix in Eq. 5. The general formula for the in-elastic light scattering cross section

Rq, = 2i,f

e− i

Z f − i −

gkigk fei e

f f M qi212

is expressed23,24 through the square of the scattering operator

f M qi = f , qi + l f j k fllj− kii

l − i − i

+f j− killj k fi

l − i + f , 13

which contains both nonresonant and resonant contributions.Here =i− f and q=ki−k f are the transferred energy andmomentum of the photons, respectively, eif is the polariza-tion of the initial final states of the photons and if denotesthe electronic energies for the initial i and final f electroniceigenstates. The quantity gq= hc2 /Vq1/2 is called the“scattering strength” with q=cq, and Z is the partitionfunction for the electronic system. The nonresonant part of

the scattering operator Mq is constructed from the stresstensor and the resonant one is constructed from the square ofthe current operators. After substituting the expression forthe scattering operator into the formula for scattering crosssection, one obtains three terms in the response functionq ,: a nonresonant term; a mixed term; and a pure reso-nant term. The result is

Rq, =2g2kig2k f1 − exp−

q, , 14

where

q, = Nq, + Mq, + Rq, . 15

In Ref. 3, we have described in detail how to extract thecomponents of the cross section from the appropriate corre-lation functions in the normal phase: we must calculate cor-responding multitime correlation functions for imaginaryMatsubara frequencies and then analytically continue to thereal axis. Inelastic light scattering examines charge excita-tions of different symmetries by employing polarizers onboth the incident and scattered light. The A1g symmetry hasthe full symmetry of the lattice and is primarily measured bytaking the initial and final polarizations to be ei=e f

= 1,1 ,1 ,1 , . . .. The B1g symmetry involves crossed polariz-ers: ei= 1,1 ,1 ,1 , . . . and e f = −1,1 ,−1 ,1 , . . .; while theB2g symmetry is also using crossed polarizers, but with thepolarizers rotated by 45°; it requires the polarization vectorsto satisfy ei= 2,0 ,2,0 , . . . and e f = 0,2,0 ,2, . . ..Note in previous work we used the wrong normalization forthe B2g polarization vectors resulting in a resonant response afactor of four smaller. For Raman scattering q=0, it iseasy to show that for a system with only nearest-neighborhopping and in the limit of large spatial dimensions, the A1gsector has contributions from nonresonant, mixed and reso-nant scattering, the B1g sector has contributions from non-resonant and resonant scattering only, and the B2g sector ispurely resonant.25,26 These results continue to hold in theordered phase.

IV. MIXED AND RESONANT CONTRIBUTIONS TO THESCATTERING RESPONSE

Since the nonresonant contributions to Raman scatteringin the ordered phase have already been determined,16 we

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focus here on the modifications needed in the ordered phaseto calculate the mixed and resonant responses. As discussedabove, the mixed and resonant response functions are ex-tracted from the corresponding multitime correlation func-tions. For the mixed one, the appropriate response function isbuilt on the stress tensor and two current operators, as fol-lows:

,f ,i1,2,3 = T1jf2ji3 . 16

The symbol T is a time ordering operator . . . =Tre− H

¯ /Z. Here we have introduced a compact nota-tion for the contraction of the stress tensor and current op-erators Eqs. 10 and 11 for q=0, kif=0 with the polar-ization vectors, as follows:

=

ei , e

f ,

ji =

ei j,

jf =

ef j, 17

respectively. The next step is to perform the Fourier transfor-mation from imaginary time to imaginary Matsubara fre-quency, and, as a result, the mixed correlation function isrepresented as a sum over Matsubara frequencies of the gen-eralized polarizations as follows:

,f ,iii − i f,i f,− ii = Tm

m−f ,m+i−f ,mM + m+i,m+i−f ,m

M .

18

Here we introduce the shorthand notation m−f ,m+i−f ,mM

=Mim− i f , im+ ii− i f , im for the dependence on thefermionic im= iT2m+1 and bosonic il= i2Tl Matsub-ara frequencies. The corresponding Feynman diagrams forthe generalized contributions to the mixed response functionare shown in Fig. 1, where the first and third diagrams cor-respond to the first term in Eq. 18 and the other two dia-grams correspond to the second one.

For the resonant response function, we construct the four-time correlation function with four current operators as fol-lows:

i,f ,f ,i1,2,3,4 = Tji1jf2jf3ji419

and in the same way as for the mixed one, the resonantresponse function is expressed as a sum of the generalizedpolarizations over Matsubara frequencies

i,f ,f ,i− ii,i f,− i f,ii

= Tm

m,m−f ,m+i−f ,m−fR,I + m,m+f,m−i+f ,m+f

R,I

+ m,m+i,m+i−f ,m−fR,II + m,m−f ,m+i−f ,m+i

R,II . 20

The corresponding Feynman diagrams for the generalizedcontributions to the resonant response function are shown in

Fig. 2, where we introduce additional sublattice indices a tos. Each term in Eq. 20 corresponds to a separate line in Fig.2, respectively. There are also other contributions to the four-time correlation function in Eq. 19 but they do not contrib-ute to the scattering cross section see Ref. 3 for details. Forthe B1g and B2g symmetries, the generalized polarizationm,m−f ,m+i−f ,m−f

R,I is a sum of the first two diagrams in the firstline of Fig. 2 the bare loop and the vertical renormalization

FIG. 1. Color online Feynman diagrams for the generalizedpolarizations of the mixed response function. Due to the static na-ture of the irreducible charge vertex of the Falicov-Kimball model,we have im= im.

FIG. 2. Color online Feynman diagrams for the generalizedpolarizations of the resonant response function.

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and the generalized polarization m,m+i,m+i−f ,m−fR,II contains

only the first diagram in the third line the bare loop,whereas for the A1g symmetry, all diagrams in the corre-sponding lines the bare loop, the vertical renormalization,and the horizontal renormalization contribute.

The next step is to derive analytic expressions for thesemixed and resonant generalized polarizations. There are twotypes of Feynman diagrams for both the mixed and resonantcontributions as well as for the nonresonant one16: bareloops and renormalized loops see Figs. 1 and 2. First weconsider the bare loops and then the renormalized ones. Thebare term for the mixed response 1,2,3

M,b in the CDW phase isequal to

1,2,3M,b =

1

N

kjkijk

fkGk,1AAGk,2

BAGk,3BB + Gk,1

AAGk,2BBGk,3

AB

+ Gk,1ABGk,2

AAGk,3BB + Gk,1

ABGk,2ABGk,3

AB + Gk,1BAGk,2

BAGk,3BA

+ Gk,1BAGk,2

BBGk,3AA + Gk,1

BBGk,2AAGk,3

BA + Gk,1BBGk,2

ABGk,3AA

=1

2i1 + d

B − 1Bi3 + d

A − 3A + i1 + d

B − 1B

i2 + dA − 2

A + i2 + dB − 2

Bi3 + dA − 3

A

+ i2 + dA − 2

Ai3 + dB − 3

B + i1 + dA − 1

A

i2 + dB − 2

B + i1 + dA − 1

Ai3 + dB − 3

B

Z1FZ1

Z22 − Z1

2Z32 − Z1

2+

Z2FZ2

Z12 − Z2

2Z32 − Z2

2

+Z3FZ3

Z12 − Z3

2Z22 − Z3

2 + Z1

3FZ1

Z22 − Z1

2Z32 − Z1

2

+Z2

3FZ2

Z12 − Z2

2Z32 − Z2

2+

Z33FZ3

Z12 − Z3

2Z22 − Z3

2 . 21

Here, we use the shorthand notation for frequencies: i1,i2, and i3→1, 2, and 3, with jk

if=eif k

kand k

= ei 2k

kk e

f , and Z defined by

Z = + dA − A + d

B − B , 22

where

FZ = d1

Z − 23

is the Hilbert transform of the noninteracting density ofstates, which satisfies =exp−2 / t2 / t for theinfinite-dimensional hypercubic lattice.

The bare loop for the resonant response 1,2,3,4R,b Fig. 2 in

the CDW phase is equal to

1,2,3,4R,b =

1

N

k

ab,cd

A,B

hg,ln

A,B

jkijk

fjkijk

fGk,1na Gk,2

bc Gk,3dh Gk,3

gl

=1

4i1 + dA − 1

Ai2 + dB − 2

B

i3 + dA − 3

Ai4 + dB − 4

B + i1 + dB − 1

B

i2 + dA − 2

Ai3 + dB − 3

Bi4 + dA

− 4A1Z1,Z2,Z3,Z4 +

1,. . .,4

i + dA −

Ai

+ dB −

B1Z1,Z2,Z3,Z4 + 21Z1,Z2,Z3,Z4 .

24

Here we introduce three quantities 1Z1 , Z2 , Z3 , Z4,1Z1 , Z2 , Z3 , Z4, and 1Z1 , Z2 , Z3 , Z4, which are equal to

1Z1,Z2,Z3,Z4 =1

N

k

1

Z12 − k

2Z22 − k

2Z32 − k

2Z42 − k

2

=FZ1/Z1

Z22 − Z1

2Z32 − Z1

2Z42 − Z1

2

+FZ2/Z2

Z12 − Z2

2Z32 − Z2

2Z42 − Z2

2

+FZ3/Z3

Z12 − Z3

2Z22 − Z3

2Z42 − Z3

2

+FZ4/Z4

Z12 − Z4

2Z22 − Z4

2Z32 − Z4

2, 25

1Z1,Z2,Z3,Z4 =1

N

k

k2

Z12 − k

2Z22 − k

2Z32 − k

2Z42 − k

2

=Z1FZ1

Z22 − Z1

2Z32 − Z1

2Z42 − Z1

2

+Z2FZ2

Z12 − Z2

2Z32 − Z2

2Z42 − Z2

2

+Z3FZ3

Z12 − Z3

2Z22 − Z3

2Z42 − Z3

2

+Z4FZ4

Z12 − Z4

2Z22 − Z4

2Z32 − Z4

226

and

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1Z1,Z2,Z3,Z4 =1

N

k

k4

Z12 − k

2Z22 − k

2Z32 − k

2Z42 − k

2

=Z1

3FZ1

Z22 − Z1

2Z32 − Z1

2Z42 − Z1

2

+Z2

3FZ2

Z12 − Z2

2Z32 − Z2

2Z42 − Z2

2

+Z3

3FZ3

Z12 − Z3

2Z22 − Z3

2Z42 − Z3

2

+Z4

3FZ4

Z12 − Z4

2Z22 − Z4

2Z32 − Z4

2, 27

respectively.The renormalized loops in the Feynman diagrams in Figs.

1 and 2 describe the charge screening effects through thereducible charge vertex, which is defined through the irre-ducible one by a Bethe-Salpeter equation. In the DMFT ap-proach, the irreducible charge vertex a is local but is differ-ent for different sublattices in the CDW ordered phase seeRef. 16. Nevertheless, it has the same functional form whenexpressed as a functional of the Green’s function and self-energy as in the normal state27–29 and is equal to

aim,im;il = mmm,m+la ,

m,m+la =

1

T

ma − m+l

a

Gmaa − Gm+l

aa 28

for the Falicov-Kimball model an explicit formula for othermodels is unknown. This expression also follows from thepartially integrated Ward identity derived by Janiš.30 Becausein the CDW phase the irreducible charge vertex is local bothin the lattice and sublattice indices, the reducible one de-pends on two sublattice indices and is defined by the Bethe-Salpeter equation

m,m+lab = abm,m+l

a + Tm,m+la

c

m,m+lac m,m+l

cb , 29

where we introduce the bare susceptibility

m,m+lab = −

1

N

kGk,m

ab Gk,m+lba . 30

The lattice Green’s functions can be derived from the Dysonequation in Eq. 4 and are equal to

Gk,mAA =

im + − mB

Zm2 − k

2,

Gk,mBB =

im + − mA

Zm2 − k

2,

Gk,mAB = Gk,m

BA =k

Zm2 − k

2. 31

Expressions for the renormalized loops have a similar formfor all contributions nonresonant, mixed, and resonant anddiffer only in the loops attached to the left and right sides ofthe total reducible charge vertex. The renormalized loop forthe mixed response is then equal to

1,2,3M,r = j j

Ai1,i2,i3 j jBi1,i2,i3T1,3

AA 1,3AB

1,3BA 1,3

BB

Ai1,i3

Bi1,i3 , 32

where we introduce the quantities

j jAi1,i2,i3

=1

N

kjkijk

fGk,1

AA Gk,2

BA Gk,3

BA + Gk,1

AA Gk,2

BB Gk,3

AA

+ Gk,1

AB Gk,2

AA Gk,3

BA + Gk,1

AB Gk,2

AB Gk,3

AA

= i1 + 2 + 3 + 3dB − 1

B − 2B − 3

B

Z1FZ1

Z22 − Z1

2Z32 − Z1

2

+Z2FZ2

Z12 − Z2

2Z32 − Z2

2+

Z1FZ3

Z12 − Z3

2Z22 − Z3

2

+ i1 + dB − 1

Bi2 + dA − 2

Ai3 + dB − 3

B

FZ1/Z1

Z22 − Z1

2Z32 − Z1

2

+FZ2/Z2

Z12 − Z2

2Z32 − Z2

2+

FZ3/Z3

Z12 − Z3

2Z22 − Z3

2 33

and j jBi1 , i2 , i3, obtained from Eq. 33 by the replace-

ment A↔B, to the left of the charge vertex with

Ai1,i3 =

1

N

kkGk,1

AA Gk,3

BA + Gk,1

AB Gk,3

AA = i1 + 3

+ 2dB − 1

B − 3B Z1FZ1 − Z3FZ3

Z32 − Z1

2 34

and Bi1 , i3, obtained from Eq. 34 by the replacement

A↔B, to the right of the charge vertex. Now we can find theexact expression for the vertex corrections defined by Eq.32 with the following form:

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1,2,3M,r =

1

1,3 j j

Ai1,i2,i3T1,3A 1,3

ABT1,3B

Bi1,i3

+ j jAi1,i2,i31 − T1,3

B 1,3BBT1,3

A Ai1,i3

+ j jBi1,i2,i31 − T1,3

A 1,3AAT1,3

B Bi1,i3

+ j jBi1,i2,i3T1,3

B 1,3BAT1,3

A Ai1,i3 , 35

where

1,3 = 1 − T1,3A 1,3

AA1 − T1,3B 1,3

BB − T1,3A 1,3

ABT1,3B 1,3

BA .

36

For the resonant response function, the renormalized loops inFeynman diagrams are defined in the same way as the mixedone and in a compact form we have

1,2,3,4R,r =

1

1,3 j j

Ai1,i2,i3

T1,3A 1,3

ABT1,3B j j

Bi3,i4,i1 + j jAi1,i2,i3

1 − T1,3B 1,3

BBT1,3A j j

Ai3,i4,i1

+ j jBi1,i2,i3

1 − T1,3A 1,3

AAT1,3B j j

Bi3,i4,i1

+ j jBi1,i2,i3T1,3

B 1,3BAT1,3

A j jAi3,i4,i1 .

37

Now we have the same quantities j jAi1 , i2 , i3 to the left

and to the right of the charge vertex. For nonresonant scat-tering, the renormalized contributions have the same formwith j j

a i1 , i2 , i3 replaced by ai1 , i3 see Ref. 16.

The total expression for the mixed generalized polariza-tion is finally obtained as the sum of both the bare and renor-malized contributions

1,2,3M = 1,2,3

M,b + 1,2,3M,r 38

on the imaginary axis. Now we have to perform an analyticcontinuation to the real axis. First we replace the sum over

Matsubara frequencies by an integral over the real axis. Nextwe analytically continue Matsubara frequencies to the realaxis in the following order: first ii− i f = ii− i f→ i0+

followed by iif→if i0+, iif →if i0+, and finally=i−i= f− f →0 in Eq. 18. Then the mixed re-sponse function is expressed directly in terms of the gener-alized polarizations as

M =1

2i2−

+

df − f +

ReM − f + i0+, + + i0+, − i0+

− M − f + i0+, + − i0+, − i0+

+ M − f − i0+, + + i0+, − i0+

− M − f − i0+, + − i0+, − i0+

+ M + i + i0+, + + i0+, − i0+

− M + i + i0+, + − i0+, − i0+

+ M + i − i0+, + + i0+, − i0+

− M + i − i0+, + − i0+, − i0+ ,

39

where f=1 / exp +1. is the Fermi-Dirac distributionfunction. Since the imaginary-axis form of the response isexpressed as a functional of the Green’s functions and self-energies, one simply replaces the appropriate Matsubara fre-quency arguments by the real frequencies, according to thedifferent terms listed above. This is a tedious, but straight-forward exercise to yield the final formulas, which are toocumbersome to include here.

For the resonant response function, an analytical continu-ation onto the real axis is more complicated but the generalapproach remains the same and final expression is thefollowing:

Rq, =1

2i2−

+

df − f +

lim→0

R,I − i0+, − f − i0+, + + i0+, − f + i0+

− R,I + i0+, − f − i0+, + + i0+, − f + i0+

+ R,I + i0+, − f − i0+, + − i0+, − f + i0+

− R,I − i0+, − f − i0+, + − i0+, − f + i0+

+ R,I − i0+, + i − i0+, + + i0+, + i + i0+

− R,I + i0+, + i − i0+, + + i0+, + i + i0+

+ R,I + i0+, + i − i0+, + − i0+, + i + i0+

− R,I − i0+, + i − i0+, + − i0+, + i + i0+

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+ 2 ReR,II − i0+, + i + i0+, + + i0+, − f + i0+

− R,II − i0+, + i + i0+, + − i0+, − f + i0+

+ R,II − i0+, − f − i0+, + + i0+, + i − i0+

− R,II − i0+, − f − i0+, + − i0+, + i − i0+ . 40

Now we can specify the different contributions to the reso-nant response in the different symmetry channels. In the B1gand B2g channels, the generalized polarizations R,II containonly the bare loop contributions the first diagrams in the lasttwo lines of Fig. 2

B1g,1,2,3,4R,II = 1,2,3,4

R,b ,

B2g,1,2,3,4R,II = B1g,1,2,3,4

R,II . 41

On the other hand, the generalized polarization R,I containsboth the bare and vertically renormalized contributions thefirst two diagrams in the first two lines of Fig. 2 in the B1gand B2g symmetry channels

B1g,1,2,3,4R,I = 1,2,3,4

R,b + 1,2,3,4R,r ,

B2g,1,2,3,4R,I = B1g,1,2,3,4

R,I . 42

In the A1g channel, all diagrams in Fig. 2 contribute hence

A1g,1,2,3,4R,I = A1g,1,2,3,4

R,II = 31,2,3,4R,b + 1,2,3,4

R,r + 2,3,4,1R,r .

43

It should be noted that some renormalized terms in Eq. 40contain nominal divergences in the limit →0 connectedwith vanishing determinants in Eq. 36 which are found inthe denominators of Eq. 37 but the contribution of these

terms to the response is actually finite. In the case of theuniform phase of the Falicov-Kimball model, their contribu-tions were calculated analytically using l’Hopital’s rule,3 butin the case of the CDW phase, the expressions are morecumbersome, so we calculate the limit →0 numerically.

V. NUMERICAL RESULTS

Now that all of the formal developments are complete, weare ready to discuss the numerical results found by calculat-ing the total Raman response function for different symmetrychannels and different interaction strengths as functions of Twithin the ordered phase. We shall consider two cases: thecase of a weakly scattering metal in the normal state U=0.5, Tc=0.0336 and the case of a strongly correlated in-sulator in the normal state U=2.5, Tc=0.0724; both casesare insulators at zero temperature due to the CDW order. Inprevious work,15 we have calculated the temperature evolu-tion of the single particle DOS in the CDW phase of theFalicov-Kimball model. Here we present figures of the DOSfor the temperatures that we calculate the Raman responseall temperatures are below Tc: T=0.02 for the case of U=0.5 Fig. 3 and T=0.06 for the case of U=2.5 Fig. 4,respectively.

One common feature of the CDW-ordered DOS is thepresence of a sharp inverse square-rootlike feature at U /2 for

-3 -2 -1 0 1 2 3

Frequency ω/t*

0

1

2

3

4

A(ω

)t*

U/2↓↓

−U/2

↓↓

E/2

−E/2

ABtotal

FIG. 3. Color online Conduction electron DOS at T=0.02 forU=0.5. The solid black curve is the total DOS while the dashed redline is for the A sublattice and the dotted-dashed blue line is for theB sublattice. Note how there is a divergence at the band edge oneach sublattice which develops as T→0 and that the subgap statesdisappear as T→0. Finally, we have marked the locations of theband edge at U /2 and of the peak of the subgap states at E /2.

-3 -2 -1 0 1 2 3

Frequency ω/t*

0

2

4

6

8

A(ω

)t*

ABtotal

↓↓

↓↓

U/2−U/2

E/2−E/2↓↓

FIG. 4. Color online Conduction electron DOS at T=0.06 forU=2.5. The solid black curve is the total DOS while the dashed redline is for the A sublattice and the dot-dashed blue line is for the Bsublattice. Note how there is a divergence at the band edge on eachsublattice which develops as T→0, and that the subgap states dis-appear as T→0. Finally, we have marked the locations of the bandedge at U /2 and of the peak of the subgap states at E /2. TheDOS has upper and lower Mott shoulders for the strongly correlatedsystem indicated by unlabeled arrows.

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sublattice A and −U /2 for sublattice B, which frame the gapas T→0. There also are bands of subgap states with a maxi-mum DOS at E /2; we have E0.18 for U=0.5 and E1.7 for U=2.5. These subgap states originate from thermalexcitations of the CDW order and they vanish at zero tem-perature. In addition, for the case of the strongly correlatedinsulator U=2.5, one can observe in Fig. 4 additional shoul-ders at 1.82, which are resulting from the upper and lowerHubbard bands of the high temperature normal state Mottinsulator. At zero temperature the states below zero energyare filled and the states above it are empty. At finite tempera-ture, due to thermal occupation, there are some empty statesbelow the chemical potential and some occupied ones above.These thermally activated states give contributions to the op-tical conductivity and to the nonresonant Raman scattering,creating different peaks in those functions.15,16 A large mainpeak at U corresponds to single-particle transitions from thelower occupied to the upper empty bands of the CDW, whichare separated by a gap of width U. This peak will becomemore enhanced as T→0. Peaks also occur at U+E /2,which correspond to single-particle transitions from thelower occupied CDW band at −U /2 to the upper empty sub-gap states at E /2 and from the lower occupied subgap statesat −E /2 to the upper empty CDW band at U /2. In addition,we see peaks at E corresponding to transitions from thelower occupied subgap states at −E /2 to the upper emptysubgap states at E /2. The intensity of these peaks will de-crease as T is lowered since the subgap states will lose spec-tral weight and eventually vanish. Finally, there is an addi-tional peak at U−E /2, which corresponds to transitionsbetween the almost fully occupied lower CDW band at −U /2and the lower subgap states at −E /2 and between the almostempty upper subgap states at E /2 and the upper CDW bandat U /2. The intensity of this peak will also shrink as T islowered. We anticipate all of this structure will also to beseen in the total electronic Raman scattering, but the detailsof the temperature dependence, or of the resonant effects aredifficult to guess without performing the calculations. We dosee, however, that we have a wide number of different “gapedges” where one would expect large resonant effects. Thelargest should occur when the photon energy is equal to Ubut we should also see them at UE /2 and E.

Analysis of the expression in Eq. 40 gives that, in addi-tion to the nonresonant peaks at =U, U+E /2, E, andU−E /2, there can also exist peaks which originate fromtwo particle transitions, i.e., = 3E−U /2, U+E /2, E,U−E, and U−E /2, some of which coincide with singleparticle transition energies. In addition, there can be strongresonant enhancement when either i or f approach theseenergies.

In Fig. 5, we plot the total Raman response at T=0.02 forthe B1g symmetry channel with U=0.5 for different energiesof the incident photons. This case corresponds to a moder-ately correlated metal in the high-temperature phase with aCDW gap of size 0.5. The total Raman response function forthe B1g symmetry contains two contributions: the nonreso-nant contribution dashed line, which is the only contribu-tion at very high photon energies i→, and the resonantcontribution, which is also the total resonant response forthe B2g symmetry. For small values of i, we observe only a

continuous enhancement of the spectra until i is largeenough to create excitations across the smallest subgaps inthe thermally excited DOS. The i=0.3 and i=0.346 curvescorrespond to the initial transition of the electron from thelower CDW band to the upper subgap states with a furthertransition to the lower subgap states with an energy lossaround U−E /2=0.16 and from the lower subgapstates to the upper CDW band with a further transition to theupper subgap states with an energy loss around E=0.18 see panel a for details. In addition, there is a peakwhich corresponds to the two particle excitations around

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Transferred frequency Ω/t*

0.0

0.5

1.0

1.5

2.0

χ B1g

(Ω)

(arb

.uni

ts)

ωi=0.154

ωi=0.192

ωi=0.25

ωi=0.3

ωi=∞

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Transferred frequency Ω/t*

0

10

20

30

40

χ B1g

(Ω)

(arb

.uni

ts)

ωi=0.3

ωi=0.346

ωi=0.4

ωi=0.7

ωi=0.8

ωi=0.9

ωi=1

ωi=∞

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Transferred frequency Ω/t*

0

100

200

300

400

500

600

700

χ B1g

(Ω)

(arb

.uni

ts)

ωi=0.4

ωi=0.5

ωi=0.6

ωi=0.7

ωi=0.8

ωi=0.9

ωi=1

ωi=∞

(c)

FIG. 5. Color online Total Raman spectra for B1g symmetryfor different values of the incident photon frequency and for differ-ent vertical scales in the different panels with T=0.02 for U=0.5.Colors are used for the different incident frequencies, which canalso be read off by examining the location of the unphysical diver-gence when →i on the hypercubic lattice. The nonresonant re-sponse the case of i= is also shown with a dashed line.

RESONANT RAMAN SCATTERING EFFECTS IN A… PHYSICAL REVIEW B 82, 155115 2010

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3E−U /2=0.02. When the energy of the incident photonsis tuned out resonance with these subgap states e.g., i=0.4, the intensity of the peaks rapidly decreases until weapproach the next resonance at i=U, which corresponds tothe initial transitions from the lower to upper CDW bands,with further transitions to all states below. In this case, weobserve the largest resonant enhancement see panel c ofmore than a factor of 1000. Note that we also have “joint”resonance effects, as there are multiple peaks resonating withthis incident photon energy, but the resonance rapidly de-creases and becomes small again once i reaches about 0.7see panel b. Increasing the incident photon energy furtherleads to a continuous decrease of the resonant response with-out any significant change in its shape; the high-energy peaksimply moves to the higher frequencies and the responsesettles into the nonresonant one. Note that every curve showsa large peak in the limit where →i. This peak is an arti-fact of the infinite-dimensional limit and the hypercubic lat-tice, and is not expected to be seen in any real materialsystem.

Similar behavior is observed for the case of a stronglycorrelated insulator in the normal state at U=2.5 and T=0.06 in Fig. 6. The main differences with the previous caseare connected with two points. First, the gap is larger and thesubgap states are wider separated. Hence, the response isvery small in the low-energy part of the spectrum and forlow initial photon frequencies. Second, the single-particleexcitation energies are quite different. For the case of U=0.5, the energies of the single particle excitations U−E /2=0.16 and E=0.18 are close to each other and thecorresponding peaks of the response functions effectivelymerge. Now these peaks at U−E /2=0.4 and E=1.7 arewell separated and can be distinguished in the spectrum. Inaddition, as was seen for the A1g total Raman response in thenormal state of the Falicov-Kimball model,3 the mixed con-tribution becomes large and negative for large-enough valuesof the transferred frequency and can completely cancel theresonant contribution when one is in the Mott insulatorphase.17 Moreover, for some values of the sum of themixed and resonant contributions is negative and the totalRaman response for the A1g symmetry becomes smaller thanthe nonresonant one not shown here.

Another important feature to examine in the total Ramanresponse is the resonant profile of the response, which is acut through the spectra with a fixed value of the transferredenergy while varying the incident photon frequency i. InFigs. 7 and 8, we plot the total Raman response functions forA1g symmetry response is very similar for different symme-try channels on log scale at various fixed transferred fre-quencies as a function of the incident photon frequencyi.

In the case of U=0.5, for small values of the transferredfrequency =0.1, we observe a wide peak centered aroundi0.3, which correspond to the joint resonance when iU+E /2 is tuned to the single-particle transitions fromthe lower CDW band to the upper subgap states and f =i−E is tuned to transitions between the lower and uppersubgap states. Another sharp peak at i=0.6 corresponds totransitions with the scattered frequency f =U. For largervalues of the transferred frequency , the resonant profiles

become more complicated and dramatically change as thetransferred frequency is increased. This complicated behav-ior is caused by the requirement to satisfy the resonanceconditions when the frequencies i, f, and =i− f mustbe tuned to the available single-particle transitions. Due tothis constraint not all of the main resonances are seen, suchas the one at i=U. But for the large values of the trans-ferred frequency U, when only transitions between thelower and upper CDW bands are involved, the shape of theresonant profiles changes smoothly and slowly.

0 1 2 3 4

Transferred frequency Ω/t*

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

χ B1g

(Ω)

(arb

.uni

ts)

ωi=1.5

ωi=1.7

ωi=1.9

ωi=∞

(a)

0 1 2 3 4

Transferred frequency Ω/t*

0.0

0.5

1.0

1.5

2.0

2.5

3.0

χ B1g

(Ω)

(arb

.uni

ts)

ωi=1.7

ωi=1.9

ωi=2.1

ωi=2.3

ωi=2.8

ωi=3

ωi=3.2

ωi=3.4

ωi=3.6

ωi=3.8

ωi=4

ωi=∞

(b)

0 1 2 3 4

Transferred frequency Ω/t*

0

20

40

60

80

100

χ B1g

(Ω)

(arb

.uni

ts)

ωi=1.9

ωi=2.1

ωi=2.5

ωi=2.6

ωi=2.8

ωi=3

ωi=3.2

ωi=∞

(c)

FIG. 6. Color online Total Raman spectra for B1g symmetryfor different values of the incident photon frequency and for differ-ent vertical scales in the different panels with T=0.06 for U=2.5.Colors are used for the different incident frequencies, which canalso be read off by examining the location of the unphysical diver-gence when →i on the hypercubic lattice. The nonresonant re-sponse the case of i= is also shown with a dashed line.

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For large values of U=2.5, when the peaks of the singleparticle DOS as well as the energies of the single particletransitions are well separated, the resonant profiles displaymuch more complicated behavior see Fig. 8, where we showjust one symmetry channel since all channels are very similaron the log scale. The overall profiles are significantly en-hanced when the transferred frequency is larger than about0.8. The profiles also illustrate peaks which change shapedramatically as is changed. Such behavior is similar towhat was seen for the resonant profiles in the normal state.5

VI. CONCLUSIONS

In this work, we have shown how one can solve for theexact total electronic Raman response of a CDW insulatorthat is formed via a nesting instability. Since the DOS recon-structs significantly below Tc, we also see a significantchange in the Raman response as a function of T. The exactsolution is made possible for the Falicov-Kimball model inthe infinite-dimensional limit, where DMFT is exact. We usethe Falicov-Kimball model because the charge vertex isknown exactly for this model.

Our main results are that there are a large number ofstrong resonances associated with all of the different peaks inthe ordered-phase DOS, which has significant subgap statesat low T. The strongest resonance occurs between the statesseparated by U corresponding to the T=0 gap. Since mostCDW systems have gaps less than an electron volt, this reso-

nance would not normally be able to be seen with opticallight. If the incident photon frequency is larger than the gap,we can, nevertheless, see some joint resonances, where lowerenergy peaks resonate, similar to what was seen in previousnormal-state calculations. In any case, we feel these resultsindicate that there should be very interesting Raman scatter-ing structures seen in experiment when one examines reso-nant effects in materials where the ordering yields a diver-gence in the single-particle DOS at T=0, such as the CDWcase we examined here. Hopefully, these kinds of experi-ments will be undertaken soon.

More interesting is the case when the incident photon en-ergy can be on the order of the CDW gap. To do this, weneed to find materials with larger gaps than most currentlyknown CDW systems. Perhaps these kinds of materials canbe found in the future and the experiments we envision car-ried out on them as well. In any case, what is clear is thatresonant effects to electronic Raman scattering in orderedsystems can yield a wide range of interesting results, even ifa microscopic description of the physical behavior is chal-lenging.

ACKNOWLEDGMENTS

J.K.F. acknowledges support from the Department of En-ergy, Office of Basic Energy Science, under Grant No. DE-FG02-08ER46542. The collaboration was supported by theDepartment of Energy, Office of Basic Energy Science, un-der Grant No. DE-FG02-08ER46540.

1 T. P. Devereaux and R. Hackl, Rev. Mod. Phys. 79, 175 2007.2 A. M. Shvaika, O. Vorobyov, J. K. Freericks, and T. P. De-

vereaux, Phys. Rev. Lett. 93, 137402 2004.3 A. M. Shvaika, O. Vorobyov, J. K. Freericks, and T. P. De-

vereaux, Phys. Rev. B 71, 045120 2005.4 A. M. Shvaika, O. Vorobyov, J. K. Freericks, and T. P. De-

vereaux, Physica B 359-361, 705 2005.5 A. M. Shvaika, O. Vorobyov, J. K. Freericks, and T. P. De-

vereaux, J. Phys. Chem. Solids 67, 336 2006.6 A. Perucchi, L. Degiorgi, and R. E. Thorne, Phys. Rev. B 69,

195114 2004.7 S. V. Dordevic, D. N. Basov, R. C. Dynes, B. Ruzicka, V.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Incident photon energy ωi/t

*

10-4

10-3

10-2

10-1

100

101

102

103

χ A1g

(ωi)

(ar b

.uni

ts)

Ω=0.1Ω=0.192Ω=0.25Ω=0.346Ω=0.5Ω=0.6

FIG. 7. Color online Resonant profiles for A1g symmetry in asemilog plot for different values of the transferred photon frequencyat T=0.02 for U=0.5. Different colors denote different transferredfrequency which can also be found from the unphysical diver-gence at i→.

0 1 2 3 4

Incident photon energy ωi/t

*

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

χ A1g

(ωi)

(arb

.uni

ts)

Ω=0.1Ω=0.5Ω=0.9Ω=1.3Ω=1.7Ω=2.5Ω=2.9

FIG. 8. Color online Resonant profiles for A1g symmetry in asemilog plot for different values of the transferred photon frequencyat T=0.06 for U=2.5. Different colors denote different transferredfrequency which can also be found from the unphysical diver-gence at i→.

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Vescoli, L. Degiorgi, H. Berger, R. Gaál, L. Forró, and E.Bucher, Eur. Phys. J. B 33, 15 2003.

8 N. Ru, C. L. Condron, G. Y. Margulis, K. Y. Shin, J. Laverock, S.B. Dugdale, M. F. Toney, and I. R. Fisher, Phys. Rev. B 77,035114 2008.

9 S. Tajima, S. Uchida, A. Masaki, H. Takagi, K. Kitazawa, S.Tanaka, and A. Katsui, Phys. Rev. B 32, 6302 1985; S. H.Blanton, R. T. Collins, K. H. Kelleher, L. D. Rotter, Z.Schlesinger, D. G. Hinks, and Y. Zheng, ibid. 47, 996 1993;M. A. Karlow, S. L. Cooper, A. L. Kotz, M. V. Klein, P. D. Han,and D. A. Payne, ibid. 48, 6499 1993; R. P. S. M. Lobo and F.Gervais, ibid. 52, 13294 1995; H.-T. Kim, ibid. 54, 90 1996;S. M. Hasanuzzaman, K. Iwano, and K. Nasu, J. Phys. Soc. Jpn.68, 1376 1999.

10 C. S. Snow, J. F. Karpus, S. L. Cooper, T. E. Kidd, and T.-C.Chiang, Phys. Rev. Lett. 91, 136402 2003; H. Barath, M. Kim,J. F. Karpus, S. L. Cooper, P. Abbamonte, E. Fradkin, E. Moro-san, and R. J. Cava, ibid. 100, 106402 2008.

11 L. M. Falicov and J. C. Kimball, Phys. Rev. Lett. 22, 9971969.

12 U. Brandt and C. Mielsch, Z. Phys. B: Condens. Matter 75, 3651989.

13 J. K. Freericks and V. Zlatić, Rev. Mod. Phys. 75, 1333 2003.14 S. R. Hassan and H. R. Krishnamurthy, Phys. Rev. B 76, 205109

2007.15 O. P. Matveev, A. M. Shvaika, and J. K. Freericks, Phys. Rev. B

77, 035102 2008.16 O. P. Matveev, A. M. Shvaika, and J. K. Freericks, Phys. Rev. B

79, 115130 2009.17 O. P. Matveev, A. M. Shvaika, and J. K. Freericks, J. Phys. Stud.

13, 4703 2009.18 U. Brandt and C. Mielsch, Z. Phys. B: Condens. Matter 79, 295

1990.19 W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 1989.20 P. G. J. van Dongen, Phys. Rev. B 45, 2267 1992.21 L. Chen, J. K. Freericks, and B. A. Jones, Phys. Rev. B 68,

153102 2003.22 C. Gruber, N. Macris, P. Royer, and J. K. Freericks, Phys. Rev. B

63, 165111 2001.23 B. S. Shastry and B. I. Shraiman, Phys. Rev. Lett. 65, 1068

1990.24 B. S. Shastry and B. I. Shraiman, Int. J. Mod. Phys. B 5, 365

1991.25 J. K. Freericks and T. P. Devereaux, Condens. Matter Phys. 4,

149 2001.26 J. K. Freericks and T. P. Devereaux, Phys. Rev. B 64, 125110

2001.27 A. M. Shvaika, Physica C 341-348, 177 2000.28 J. K. Freericks and P. Miller, Phys. Rev. B 62, 10022 2000.29 A. M. Shvaika, J. Phys. Stud. 5, 349 2001.30 V. Janiš, J. Phys.: Condens. Matter 15, L311 2003.

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