Modeling bond-order wave instabilities in doped frustrated antiferromagnets:Valence bond solids at fractional filling

M. Indergand,1 A. Läuchli,2 S. Capponi,3 and M. Sigrist11Theoretische Physik, ETH-Hönggerberg, CH-8093 Zürich

2Institut Romand de Recherche Numérique en Physique des Matériaux (IRRMA), EPFL, CH-1015 Lausanne3Laboratoire de Physique Théorique, CNRS UMR 5152, Université Paul Sabatier, F-31062 Toulouse

�Received 22 March 2006; revised manuscript received 19 June 2006; published 31 August 2006�

We explore both analytically and numerically the properties of doped t-J models on a class of highlyfrustrated lattices, such as the kagomé and the pyrochlore lattices. Focusing on a particular sign of the hoppingintegral and antiferromagnetic exchange, we find a generic symmetry-breaking instability toward a twofold-degenerate ground state at a fractional filling below half filling. These states show modulated bond strengthsand only break lattice symmetries. They can be seen as a generalization of the well-known valence bond solidstates to fractional filling.

DOI: 10.1103/PhysRevB.74.064429 PACS number�s�: 71.27.�a, 73.22.Gk, 71.30.�h, 71.35.�y

I. INTRODUCTION

Highly frustrated quantum magnets are fascinating andcomplex systems where the macroscopic ground-state degen-eracy at the classical level leads to many intriguing phenom-ena at the quantum level. The ground-state properties of spinS=1/2 Heisenberg antiferromagnets on the kagomé and py-rochlore lattices remain still puzzling and controversial inmany aspects. While the magnetic properties of the Heisen-berg and extended models have indeed been studied for quitesome time, the investigation of highly frustrated magnetsupon doping with mobile charge carriers has startedrecently.1–6 Such interest has been motivated for example bythe observation that in some strongly correlated materials,such as the spinel compound LiV2O4, itinerant charge carri-ers and frustrated magnetic fluctuations interact strongly.7

Furthermore, the possibility of creating optical kagomé lat-tices in the context of cold atomic gases has been pointedout,8 making it possible to “simulate” interacting fermionicor bosonic models in an artificial setting.9

At this point we should stress that the behavior in asimple single-band model at weak and at strong correlationsis not expected to be related in a trivial way. The weak-coupling limit allows us to discuss the electronic propertieswithin the picture of itinerant electrons in momentum spacebased on the notions of a Fermi surface and Fermi surfaceinstabilities �see, e.g., Refs. 1 and 4�. Considering, for ex-ample, the Fermi surfaces of a triangular or a kagomé latticeat half filling we do not find any obvious signature of themagnetic frustration present at large U. Although at weakcoupling these systems do not seem to be particularly spe-cial, at intermediate to strong coupling the high density oflow-energy fluctuations of the highly frustrated systems dis-play characteristic features from which the physics of thefrustrated system of localized degrees of freedom willemerge.2,3

In the following we study a class of highly frustrated lat-tices, the so-called bisimplex lattices,10 which are composedof corner-sharing simplices residing on a bipartite underlyinglattice. We restrict ourselves to the triangle and the tetrahe-dron as the basic building blocks in the following. This class

hosts lattices such as the kagomé and pyrochlore lattices.Our main result is the spontaneous symmetry breaking

between simplices that are located on different sublattices ofthe underlying bipartite lattice. This instability takes place atthe electron density n=2/3 for triangle-based lattices like thekagomé lattice and at the density n=1/2 for tetrahedron-based lattices like the pyrochlore lattice and leads to a bond-order wave �BOW� phase where the kinetic energy is stag-gered for neighboring simplices. This instability is driven bya cooperative effect of the kinetic energy and the exchangeinteractions.

The outline of the paper is the following. In Sec. II weintroduce the lattices and the model and provide general ar-guments for occurrence of spontaneous symmetry breaking.These arguments are based on an analysis of the spectrum ofisolated simplices and on a doped quantum dimer model. Wethen report in Sec. III mean-field calculations for the kagoméand pyrochlore lattices which underline the symmetry-breaking tendency. In Sec. IV we present numerical resultsfor the kagomé lattice that were obtained by exact diagonal-ization �ED� and by the contractor renormalization �CORE�algorithm. In Sec. V the one-dimensional �1D� analog of thekagomé lattice, the so-called kagomé strip, is treated withtwo powerful methods that are available for 1D systems: thedensity matrix renormalization group �DMRG� and a fermi-onic renormalization group �RG� and bosonization analysis.Finally we summarize and conclude in Sec. VI.

II. GENERAL MOTIVATION

The bisimplex lattices shown in Fig. 1 consist of corner-sharing simplices �triangles or tetrahedra� that are located onan underlying bipartite lattice. Therefore, we can separate thetriangles and the tetrahedra into two different classes, whichis visualized in Fig. 1 by a different line style �light and boldbonds�. To refer to the simplices of a given class we callthem “up” and “down” simplices, and we use the same ter-minology to distinguish the triangles, the tetrahedra, or thelattice bonds.

In this section we provide two different arguments whyfor correlated electrons on such lattices at fractional filling

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with two electrons per up simplex �one electron per simplex�a spontaneous symmetry breaking can be expected, resultingin a phase where the up simplices differ from the down sim-plices. For the kagomé and pyrochlore lattices the inversionsymmetry is broken in this phase, whereas for the lower-dimensional analogs, the kagomé strip and the checkerboardlattice, translational symmetry is broken. The following twoarguments apply generally to all bisimplex lattices shown inFig. 1. They are simple and illustrate the basic underlyingphysics. In the remaining sections we will provide detailednumerical and analytical evidence for several of these bisim-plex lattices, which show that these arguments provide thecorrect picture. In the following we use �unless otherwisespecified� the t-J model to describe the correlated electronson these lattices. The t-J Hamiltonian is given by

Ht-J = − t��ij�

��

P�ci�† cj� + H.c.�P + J�

�ij��Si · S j −

1

4ninj� ,

�1�

where the restriction to the subspace of singly occupied sitesis enforced by the projection operator P=�i�1−ni↑ni↓�. Thehopping amplitude t is always chosen to be positive. A nega-tive sign of t will most likely induce ferromagnetic tenden-cies at the fillings we are considering.11

A. The limit of decoupled simplices

To get a basic understanding of the effect of doping inhighly frustrated lattices we first consider the limit of decou-pled simplices by turning off the couplings within the downsimplices. To connect this limit with the uniform lattice weuse the parameter � �0���1� that tunes the coupling

strength of the down bonds as ��t ,�J�, while the couplingstrength on the up bonds is kept constant at �t ,J�. For��1 the inversion symmetry between the up and the downsimplices is explicitly broken by the Hamiltonian. The eigen-

values of Ht-J and their degeneracies are listed in Table I fora single triangle and a single tetrahedron. For t�0 andJ�0, there is a nondegenerate state with Ne=2 that has thelowest energy of all states and is separated from the remain-ing spectrum by a finite gap. This state has at the same timethe lowest kinetic energy �−2t or −4t, respectively� of allstates and gains the maximal exchange energy �−J� for twospins. This state is no longer frustrated because it minimizesthe kinetic and the exchange energy at the same time. Afterhaving revealed this particularly stable state with two elec-trons on a single unit, we know that for �=0 the system withtwo electrons per up simplex has a nondegenerate groundstate and a finite gap to all excitations. In the tight-bindingmodel the gap decreases with increasing � and vanishes ex-actly at �=1. For interacting electrons, however, the strongcorrelations that are induced by the constraint in the kineticterm of the t-J model, and the frustration of the exchangeterm that arises from the strong frustration of the lattice, areaccompanied by a tendency to occupy rather local states �inour case, they still keep a substantial part of their kineticenergy� and to pair up the electrons in nearest-neighbor sin-glets. Therefore, it is a reasonable possibility that the systemeven for �=1 has a finite gap and breaks the symmetry be-tween the up and the down simplices spontaneously in orderto profit from the bipartite and nonfrustrated structure of the

FIG. 1. �Color online� Four different bisimplex lattices. Thekagomé lattice �a� and the pyrochlore lattice �b� together with theirlower-dimensional analogs the kagomé strip �c� and the checker-board lattice �d�. The two types of corner-sharing units �up vsdown� are distinguished by the linewidth. They correspond to thebond-order wave symmetry-breaking pattern occurring at n=2/3 onthe triangle-based lattices and at n=1/2 on the tetrahedron-basedlattices.

TABLE I. Classification of the eigenstates of the t-J model on atriangle and on a tetrahedron. The degeneracy is given in the formr �2S+1�, where r is the dimension of the irreducible representa-tion of S3, �S4�, and S is the total spin of the state. The asteriskdenotes the states retained in the CORE calculations for the kagomésystem �see text�.

Ne

Triangle Tetrahedron

Energy Degeneracy Energy Degeneracy

0 0 11 0 11

1 −2t 12* −3t 12

t 22 t 32

2 −2t−J 11* −4t−J 11

t−J 21 −J 31

−t 23* 2t−J 21

2t 13 −2t 33

2t 33

3 −3J /2 22* −2t−3J /2 32

0 14 −3J /2 22

2t−3J /2 32

−t 34

3t 14

4 −3J 21

−2J 33

0 15

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underlying lattice. Such an instability has the character of abond-order wave—i.e., modulated expectation values of thebond energies—and yields an insulating state which breaksinversion or translational symmetry.

B. Doped quantum dimer model

In the previous sections we considered the case �=0,where the Hamiltonian itself is not invariant under inversionsymmetry. In order to get some insight into the mechanismof spontaneous symmetry breaking, it is desirable to treat upand down simplices on equal footing. In the following, wepresent a simple but illustrative model with a symmetricHamiltonian that breaks spontaneously the symmetry be-tween the up and down simplices in the ground state.

A close inspection of the wave function of the lowest-energy eigenstate of two electrons on either a triangle or atetrahedron reveals that it consists of the equal-amplitudesuperposition of all possible positions of the singlet formedby the two electrons:

GS� =1

N�i�j

�ci,↑† cj,↓

† − ci,↓† cj,↑

† �0� , �2�

where the normalization N=3 for the triangle and 6 forthe tetrahedron. This wave function motivates us to design asimple quantum dimer model which on each triangle prefersthe exact wave function described above. Such a Hamil-tonian reads, for example, for the kagomé lattice

�3�

where the Hilbert space consists of all coverings of thekagomé lattice with Nc nearest-neighbor dimers and Ncmonomers, Nc counting the number of unit cells. This corre-sponds to the situation at n=2/3 in the t–J model. The in-terpretation is simple: the antiferromagnetic exchange termfavors all the electrons pairing up into singlets, while thekinetic energy term delocalizes the singlets on a triangle. Thequantum dimer models for the tetrahedron-based lattices aredefined by letting a single singlet resonate on a tetrahedron.This simple model allows us to find the exact ground state onthese lattices. The ground state is twofold degenerate andeach state is the direct product of equal-amplitude resonanceson the same type of triangles or tetrahedra, either all up or alldown. In such a situation each resonating dimer can indepen-dently fully optimize its kinetic energy. The argument hasmuch in common with the reasoning for the close-packeddimer model on the pyrochlore lattice discussed in Ref. 12.

Although this model is only a cartoon version of the realelectronic system, it illustrates how the tendency of the elec-trons to form nearest-neighbor singlets obstructs the motionof the singlets between corner-sharing simplices, but within agiven simplex an individual singlet can hop without ob-stacles and optimize its kinetic energy. The bipartite natureof the underlying lattice allows for the localization of thesinglets on simplices without interference and triggers in thisway the spontaneous symmetry breaking.

III. MEAN-FIELD DISCUSSION

In this section we present a mean-field calculation for thekagomé and the pyrochlore lattices. The mean-field discus-sion is particularly valuable for the pyrochlore lattice, as dueto its higher dimension it is less affected by fluctuations andis presently not treatable with numerical methods. We canshow that in the mean-field analysis the spontaneous inver-sion symmetry breaking, discussed in the previous sections,is also the natural and leading instability.

We start by discussing the properties of the nearest-neighbor tight-binding model on the kagomé and pyrochlorelattices, given by

H0 = − �N − t�r,�

�m�n

��=±1

cr+�am,�† cr+�an,�, �4�

where �� �↑ , ↓ � is the spin index and the indices m ,n runfrom zero to the dimension of the lattice, d. Furthermore, r isan elementary lattice vector connecting unit cells and thevectors a0 , . . . ,ad point to the vertices of an elementary tri-angle �tetrahedron� in the kagomé �pyrochlore� lattice, a0

=0. N is the total electron number operator and � is thechemical potential. In the following we will use units wheret=1 and we will always choose �=−1 for the kagomé and�=−2 for the pyrochlore lattice, which corresponds to two

electrons per unit cell. H0 can be diagonalized in reciprocalspace and can be written as

H0 = �k,m,�

km�km�† �km�, �5�

with

− k0 = k1 k, km = d + 1 for m � 1. �6�

For the kagomé lattice we have

k = 1 + 8 cos�k1/2�cos�k2/2�cos��k1 − k2�/2� �7�

with km=k ·am and we refer to Ref. 13 for an explicit formulafor k for the pyrochlore lattice.

The three bands of the kagomé lattice consist of one flat-band and two dispersing bands. The dispersing bands areidentical to the bands of a honeycomb lattice. They areshown together with the density of states per unit cell andspin in Fig. 2. Note that around the points K and −K thedispersion shows a Dirac spectrum, i.e., the bands k0 and k1 touch at these points with linear dispersion. For the given

FIG. 2. The kagomé bands and the density of states per unit celland spin. The energy is measured in units of t.

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chemical potential the Fermi surface reduces to points at Kand −K and the density of states vanishes linearly with , i.e.,we have D� �� for small .

The four bands of the pyrochlore lattice consist of twoflatbands and two dispersing bands. The dispersing bands areidentical to the bands of a diamond lattice. They are showntogether with the density of states per unit cell and spin inFig. 3. Note that k vanishes along the lines connecting Xand W. The density of states also vanishes linearly at zero upto logarithmic corrections, i.e., we have D� �� log� � forsmall .

Systems with this form of the density of states at theFermi level are neither band insulators nor normal metals;therefore, they are sometimes called semimetals or zero-gapsemiconductors. Although they have an even number of elec-trons per unit cell and no fractionally filled bands, they haveno energy gap at the Fermi surface. Fermi surface instabili-ties are suppressed in this situation. There is no Cooper in-stability that leads to an obvious breakdown of perturbationtheory for arbitrarily small attractive interactions, as theparticle-particle polarization function involves the conver-gent integral �d D� � /2 at zero temperature. For the half-filled honeycomb lattice it has been shown that the Coulombinteractions lead to non-Fermi-liquid behavior and thatstrong enough Coulomb interactions lead to antiferromag-netic order and to the opening of a charge gap.14–17 Thesituation in the kagomé and pyrochlore lattices at the fillingconsidered here is different. Because the lattices are not athalf filling, it is not obvious that even arbitrarily large Uwould enforce a charge gap �Mott insulator� and antiferro-magnetic order would be hampered by the frustrated topol-ogy of the lattice. However, if we consider the triangles �tet-rahedra� as the fundamental units of our lattice we obtain thehoneycomb �diamond� lattice and the properties of this un-derlying bipartite lattice will be reflected in the ground stateand provide a way to circumvent the frustration effects.

Our goal is to study the electron-electron interactions of

the Ht-J-V Hamiltonian given by

Ht-J-V = PH0P + ��ij�

JSi · S j + Vninj . �8�

We will show that both the exchange and the repulsion termsfavor the bond-order wave instability. As the projection op-erator P is difficult to handle in analytic calculations, theprojection is often approximated by a purely statistical renor-malization of the Hamiltonian with Gutzwiller factors.18 We

obtain a renormalized Hamiltonian without constraints givenby

Hr = gtH0 + ��ij�

JgJSi · S j + Vninj . �9�

The renormalization is given by the Gutzwiller factors gt=2� / �1+�� and gJ=4/ �1+��2 and � is the hole doping mea-sured from half filling. Note that the nearest-neighbor repul-sion is not renormalized by a statistical factor.

In the following we will determine the critical J and V forspontaneous symmetry breaking in this model within mean-field theory. Superconductivity is a possible way of sponta-neous symmetry breaking. As it is an instability in theparticle-particle channel, the relation k= −k, which is en-sured by inversion and time reversal symmetry, plays an es-sential role. Concerning the symmetry of the order param-eter, we can restrict ourselves to singlet pairing in the spinsector, because the nearest-neighbor interaction is antiferro-magnetic, and to s-wave pairing in the orbital sector, becausein this way we obtain a nodeless, even gap function.

Another possibility of spontaneous symmetry breaking isan instability in the particle-hole channel. Such instabilitiestend to occur if a nesting condition of the form k=− k+q issatisfied. In general, this condition is not ensured by basicsymmetries and therefore instabilities in the particle-holechannel are much more special than superconducting insta-bilities. In our case, the relation k0=− k1 can be consideredas perfect nesting with q=0. Therefore, the relevant questionis which one of the two considered instabilities is dominantin our system. In order to answer this question we considerthe following single-particle Hamiltonian:

Htrial = H0 + �phHph + �ppHpp �10�

where we have introduced the two quadratic Hamiltonians

Hph = �r,�

�m�n

��=±1

�cr+�am,�† cr+�an,�,

Hpp = �r

�m�n

��=±1

�cr+�am,↓cr+�an,↑ + H.c.� . �11�

The idea is to calculate the expectation value of Hr for theground state of Htrial and to choose the variational parameters�pp and �ph such that this expectation value is minimized. In

terms of the operators that diagonalize H0 we can express thepairing operators as

Hph = �k,�

ixk�k0�† �k1� + H.c.,

Hpp = �k,m

�km�−km↑�km↓ + H.c. �12�

with the relations

km = �km − �, xk2 = 0

2 − k2 . �13�

For small values of �ph and �pp we can expand the ground-state expectation value of Hr in terms of �pp

2 and �ph2 . Using

the Wick theorem, we obtain up to higher-order terms

FIG. 3. The pyrochlore bands and the density of states per unitcell and spin. The energy is measured in units of t.

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�E

bN= �ph

2 Iph� t −3J

8�Iph − �� −

V

2�Iph − ���

+ �pp2 Ipp� t −

3J

8�Ipp − �� +

V

2�Ipp + ��� , �14�

where �E is the deviation from the ground-state expectationvalue with �ph=�pp=0. N is the number of unit cells, t= tgt,

J=JgJ, b is the number of bonds in the unit cell, and

� =1

b�

�0�− − ��D� �d ,

Iph =1

b�

�0

02 − 2

D� �d ,

Ipp =1

b� � + ��2

2 D� �d . �15�

Note that only the density of states enters these formulasbecause we are restricting ourselves to q=0 instabilities. Thesystem spontaneously breaks inversion �U�1�� symmetry, ifthe coefficient of �ph

2 ��pp2 � in Eq. �14� changes sign. If we

assume that only one of the parameters V and J is nonzero,we obtain the following expressions for the critical values:

Jcph =

8gtt

3gJ�Iph − ��, Vc

ph =2gtt

�Iph − ��, �16�

Jcpp =

8gtt

3gJ�Ipp − ��, Vc

pp =− 2gtt

�Ipp + ��. �17�

The numerical values for Jc and Vc are given in Table II. Onecan see that the tendency for inversion symmetry breaking ismuch stronger than the tendency for superconductivity inboth lattices and that both the antiferromagnetic J and therepulsive V support the inversion symmetry breaking. Theintegral Iph is large because the factor 0

2− 2 takes its maxi-mum at =0 whereas the factor � +��2 in the integral Ipp ismuch smaller for small values of . In other words, super-conductivity has the handicap that the potential is propor-tional to the dispersion �k, therefore it is small at the Fermisurface and is finite only due to the finite value of �. Thenearest-neighbor repulsion is harmful for Cooper �particle-particle� pairing, as can be seen from Table II. In the particle-hole instability, however, two particles tend to form a singleton every second triangle �tetrahedron� on the kagomé �pyro-chlore� lattice. In this way the singlet is still mobile and

keeps-dt of its kinetic energy and at the same time reducesthe nearest-neighbor repulsion energy from 4V /3 �3V /2� toV on every second triangle �tetrahedron�. On the triangles�tetrahedra� without a singlet, the expectation value of thenearest-neighbor repulsion is, however, still 4V /3 �3V /2�. Inthe limit where the kinetic energy is negligible �t�V ,J�other phases may also appear. It is therefore important toemphasize that a finite kinetic energy is necessary to stabilizethe bond- order wave, because this phase arises due to theinterplay between the kinetic and interaction energy.

The limit of large V was recently discussed in the contextof LiV2O4 by Yushankhai et al. in Ref. 19 for the pyrochlorelattice with n=1/2. The possibility of inversion symmetrybreaking was not considered in that study. But if V is of theorder of t, the optimization of the kinetic and the repulsionenergy can lead to a compromise which breaks the inversionsymmetry.

In the bond-order wave phase that we found in this sec-tion for the kagomé and the pyrochlore lattices, the up tri-angles �tetrahedra� have a higher expectation value of thekinetic energy than the down triangles �tetrahedra�. Further-more, a gap proportional to �ph opens at the Fermi surface.Therefore, the system made a transition from a semimetal toan insulator. This transition is similar to the Peierls metal-insulator transition, where a half-filled system spontaneouslylowers its crystal symmetry in order to open a gap at theFermi surface. Phonons or the elasticity of the crystal play acrucial role in the Peierls transition. In our case, as weshowed, the transition can be driven by a purely electronicmechanism in an infinitely rigid lattice. In reality, the crystalstructure will always relax and in this way additionally en-hance the transition.

IV. NUMERICAL RESULTS FOR THE KAGOMÉ LATTICE

In this section we compare the predictions obtained in thepreceding sections to numerical results for the kagomé lat-tice. We will first discuss some exact diagonalization resultsfor the kagomé lattice at n=2/3, which confirm the BOWclearly and then we will study the dependence of the excita-tion gaps on the parameter � using the contractor renormal-ization method. In essence the numerical results corroboratethe analytical predictions on the presence of a bond-orderwave instability for the kagomé lattice.

A. Exact diagonalization results

The analytical arguments presented in Secs. II and III pre-dict a bond-order wave instability at filling n=2/3. In finite,

TABLE II. The parameters for the kagomé �K� and pyrochlore �P� lattices. The critical values are givenin units of t. The coefficient of �ph �bond-order wave� in Eq. �14� is negative for J�Jc

ph �V=0� or forrepulsive V�Vc

ph �J=0�. The coefficient of �pp �superconductivity� is negative for J�Jcpp �V=0� or for

attractive V�Vcpp �J=0�.

d � b 0 � gt gJ � Iph Ipp Jcph Jc

pp Vcph Vc

pp

K 2 −1 6 3 1/3 1/2 9/4 0.43 1.08 0.59 0.91 3.58 1.53 −0.98

P 3 −2 12 4 1/2 2/3 16/9 0.32 1.05 0.62 1.36 3.33 1.81 −1.43

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periodic systems this instability can be detected with a cor-relation function of the bond strength, of either the kineticterm or the exchange term. Here we chose to work with thekinetic term, but the exchange term gives similar results. Thecorrelation function is defined as

CK��i, j�,�k,l�� = �K�i, j�K�k,l�� − �K�i, j���K�k,l�� ,

�18�

where

K�i, j� = − ��

ci,�† cj,� + H.c., �19�

and �i , j� and �k , l� denote two different nearest-neighborbonds of the kagomé lattice that have no common site. Thiscorrelation function has been calculated for all distances inthe ground state of a finite kagomé sample with 21 sites,containing seven holes, at J / t=0.4. The result is plotted inFig. 4. The reference bond uniquely belongs to a certain classof triangles �up triangle in our case�. Based on the theoreticalpicture one expects the correlation function to be positive forall bonds on the same type of triangles and negative on theothers. This is indeed what is seen in Fig. 4. We have alsocalculated the same quantity for J / t=1 and 2 and the bond-order wave correlations �not shown� were becoming evenstronger for larger J / t. In this respect the ED calculationsconfirm the qualitative picture developed above, that the ho-mogeneous t-J model on the kagomé lattice at n=2/3 has anintrinsic instability toward a spontaneous breaking of the in-version symmetry.

B. CORE results

We know from Sec. II A, where we studied the limit ofdecoupled units, that the system has a finite gap in all sectorsfor �=0. In this section we show how the gaps of the differ-ent sectors of excitations depend on the parameter �.

The CORE algorithm,20–22 which is based on the exis-tence of strong subunits �triangles� that are only weaklycoupled, provides a suitable method to perform such a study.This is certainly true for small values of � where the systemis naturally divided into weakly coupled subunits. For �=1the CORE algorithm imposes a certain bias, as up and down

triangles are not treated on equal footing; however, we havegood evidence from ED and from the mean-field analysisthat the system itself has a strong tendency to break inver-sion symmetry spontaneously and therefore we can expectthe CORE results to be reliable even for �=1. In fact, wecompare the CORE results to the ED results for smaller clus-ters, and the good agreement shows that the finite excitationgaps at �=1 are not an artifact of the method.

The CORE method extends the range of tractable sizes offinite clusters, based on a careful selection of relevant low-energy degrees of freedom. In order to apply this algorithm,the lattice has to be divided into blocks; here, we naturallychoose the up triangles. A reduced Hilbert space is definedby retaining a certain number of low-lying states on eachblock. The choice of the states to keep depends also on thequantities to be obtained. While for a ground-state calcula-tion fewer states already provide good results, one has toretain usually more states to calculate the excited states. Herewe choose to keep the four lowest states in the three-electronsector, the seven lowest states with two electrons and the twolowest states in the one-electron sector. These states are de-noted with an asterisk in Table I. This choice leads to areduction of the local triangle basis from 27 down to 13states, thus allowing one indeed to perform simulations onlarger lattices than would be possible by conventional ED.

Then, by computing the exact low-lying eigenstates oftwo coupled triangles, we calculate the effective interactionsat interaction range 2 for each value of � and we neglectlonger-range terms. Comparison to ED data on the smallerclusters shows that this approach gives very good results.

The basic excitation gaps of interest in the present prob-lem are the spin gap, the single-particle gap, and the two-particle gap. These are defined as follows:

�S=1 = E�Ne,1� − E�Ne,0� , �20�

�1p =1

2�E�Ne + 1,1/2� + E�Ne − 1,1/2�� − E�Ne,0� ,

�21�

�2p =1

2�E�Ne + 2,0� + E�Ne − 2,0�� − E�Ne,0� , �22�

where E�Ne ,Sz� denotes the ground-state energy in the sectorwith Ne electrons and spin polarization Sz.

We have determined these gaps on kagomé finite-sizesamples at n=2/3 and J / t=1 containing 18 to 27 sites. Twodifferent versions of samples with 18 and 24 sites have beentreated �v1 and v2�. The results are displayed in Fig. 5. Thereare two main observations: �1� the gaps do not close for any�� �0,1�, giving additional evidence for the proposed sym-metry breaking; �2� there is a strong dependence of the gapcurves on the specific sample. Note that there is no discrep-ancy between ED and CORE results. The second phenom-enon can be understood from the discretization of the finite-size Brillouin zones: indeed, the measured gaps directlydepend on the distance between the closest point in the Bril-louin zone to the corner of the zone, the K point, which is thepoint where the gap opens in the mean-field picture �Fig. 2�.

FIG. 4. �Color online� Correlation function of the kinetic energy�Eq. �18�� of a 21- site kagomé sample at n=2/3 and J / t=0.4. Theblack, empty bonds denote the same reference bond, the red, fullbonds negative and the blue, dashed bonds positive correlations.The line strength is proportional to the magnitude of thecorrelations.

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The �18, v2�- and 27-site samples both contain this specificpoint and differ only slightly in the values of the gap. Thus,our numerical data support the claim of a finite gap for all�� �0,1�. The strong dependence is at the same time also ahint toward a sizable dispersion of the excitations in thissystem.

V. DMRG AND RG RESULTS FOR THE KAGOMÉ STRIP

In this section we study with two different methods thekagomé strip, shown in Fig. 1�c�, as the one-dimensional�1D� analog of the kagomé lattice. This lattice has been in-troduced in Ref. 23, where it was shown to share some of thepeculiar magnetic properties of the 2D kagomé lattice. Wereport extensive density matrix renormalization groupcalculations,24 for both the t-J and the Hubbard model. Fur-thermore, we apply the renormalization group and bosoniza-tion techniques to the weak-coupling Hubbard model. Our

results show that for both the Hubbard and t-J models, andirrespective of the coupling strength, the BOW phase is re-alized in the kagomé strip at n=2/3.

A. DMRG results

The kagomé strip—being 1D—offers the opportunity toperform DMRG simulations, thus allowing a rather detailednumerical study of large systems. We first discuss the prop-erties of the t-J model at n=2/3 on this lattice, and thenmake a connection by investigating the Hubbard model atdifferent values of U to the analytical weak-coupling resultsof the following section. For both models we report numeri-cal evidence for the presence of the bond-order wave insta-bility for a large range of interaction strengths.

In contrast to the periodic systems considered abovewithin ED, the DMRG works most efficiently for openboundary conditions. In the present context this has the ad-ditional advantage that for even length L of the strip only oneof the two degenerate ground states is favored, and we candirectly measure the local bond strength. For the purpose ofillustration we show the local bond strengths for a system ofL=24 in Fig. 6. The upper panel shows the difference of thelocal kinetic energy with respect to the average, while thelower panel shows the local expectation value of the spinexchange term, using the same convention.

The calculated pattern resembles the schematic picturedrawn in Fig. 1�c�. In order to address the behavior in thethermodynamic limit we measure the bond strength alterna-tion, i.e., the difference between the expectation values of theoperators K�i , j� and Si ·S j in the middle of the system fordifferent lengths L and values of J / t. The scaling of thesequantities is shown in Fig. 7. The finite-size corrections arerather small and all the order parameters extrapolate to finitevalues, irrespective of the value of J / t. Note that even for thecase J / t=0 there is a finite alternation of both the kineticenergy and the magnetic exchange term. The alternation ofthe magnetic exchange energy is roughly the same for allvalues of J / t. The alternation of the kinetic energy, however,is increased with growing J / t ratio.

FIG. 6. �Color online� DMRG results for a L=24 kagomé ladder at J / t=0.4 and n=2/3. �a� Local bond strength deviation of the kineticterm. Red, full bonds are stronger �lower in energy� than the average kinetic energy per bond. Blue, dashed bonds are weaker than averagebonds. �b� Local bond strength deviation of the exchange term. The color pattern is the same as in the upper panel. The thickness of the bonddenotes the deviation from the average value per bond. Note that the patterns of the kinetic and exchange terms are in phase.

FIG. 5. �Color online� Excitation gaps of the t-J model on thekagomé lattice at J / t=1 as a function of the parameter �, whichdenotes the ratio of the intertriangle to the intratriangle couplings.The gaps are obtained by the CORE method for different samplesizes �and geometries�. For the smaller samples �18, v1; 18, v2; 21�ED data are shown for comparison at �=1.

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Next we address the question of the excitation gaps in thesymmetry-broken phase. The theoretical picture predicts aninsulating state with a finite gap to all excitations above thetwofold-degenerate ground state. We calculate the single-particle charge gap and the spin gap defined in Eqs. �21� and�20�, respectively. The calculated gaps are shown in Fig. 8.The finite-size gaps are extrapolated to L=� with a simplequadratic fit. All gaps extrapolate to a finite value, in agree-ment with the predictions. The charge gap more or less fol-lows the increase of the alternation of the kinetic energyshown in Fig. 7, i.e., the gap is roughly multiplied by a factorof 3 in going from J / t=0 to 2. The behavior of the spin gapis mainly driven by the fact that it scales with J / t. Note thateven in the case J / t=0 the spin gaps seem to remain finite. Itwill be an interesting question to characterize the precisenature of the charge and spin excitations. This will be left fora future study.

The weak-coupling RG calculations in the following sec-tion are performed for Hubbard on-site interactions. Al-

though we expect the behavior of the t-J model and theHubbard model at large U to be similar, we have explicitlycalculated the alternation of the kinetic energy for the Hub-bard model as a function of U / t. The results displayed in Fig.9 show that this quantity has a maximum around U / t�10–15, and interpolates between the exponentially smallorder parameter at weak U / t and the result for the t-J modelat J=0, which corresponds to U=�. These results thereforesuggest that for the particular case of the kagomé strip theweak-coupling phase is adiabatically connected to thestrong-coupling limit.

B. Weak-coupling discussion

We consider in this section the weak-coupling Hubbardmodel with the interaction

Hint = U�r

cr,↑† cr,↓

† cr,↓cr,↑ �23�

on the 1D kagomé strip. We map this weak local Coulombrepulsion on the original kagomé strip to an effective inter-action for the underlying half-filled two-leg ladder. We willshow in the following that for the one-dimensional kagoméstrip, applying the RG and bosonization techniques that weredeveloped for the half-filled two-leg ladder, we find in fact acharge density wave �CDW� instability in the effective

FIG. 7. �Color online� DMRG results for the alternation of thebond strength of the kinetic term and the spin exchange term as afunction of inverse system size 1/L, for different values of J / t.

FIG. 8. �Color online� DMRG results for the spin gap andsingle-particle gap for J / t=0,0.4,1 ,2, as a function of inverse sys-tem size 1/L.

FIG. 10. The kagomé strip bands and the density of states perunit cell and spin. The energy is measured in units of t.

FIG. 9. �Color online� DMRG results for the kinetic energyalternation for Hubbard kagomé strips at n=2/3 of lengths L=32and 48. The modulation is nonmonotonic as a function of U / t andshows a maximum around U / t�10–15.

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model for the two-leg ladder which corresponds to a BOWinstability on the kagomé strip.

The tight-binding bands of the kagomé strip with �=−tare shown in Fig. 10. The dispersing bands are the same asthe bands of a two-leg ladder. The flatband originates fromstates that are trapped within one rhombus. The density ofstates has square-root singularities at ±t, ±3t and a � peak at3t. The Fermi surface is given by the four points ±kF1 and±kF2, where kF1=� /3 and kF2=2� /3. There is a finite den-sity of states at the Fermi surface. The kagomé strip can beviewed as a kagomé lattice tube, i.e., a kagomé lattice withfinite width and periodic boundary conditions. In order to seethat the bands in Fig. 10 are in fact a cut through the kagomédispersion shown in Fig. 2, one has to shift one of the dis-persing bands by �. This difference arises because our nota-tion is chosen to emphasize the similarities of the kagoméstrip to the two-leg ladder.

In contrast to the kagomé and pyrochlore lattices, the den-sity of states at the Fermi surface is finite for the kagoméstrip and we therefore expect qualitative changes in this 1Dsystem even for weak interactions. We perform a weak-coupling RG and bosonization analysis for the kagomé strip,and we show that the bond-order wave instability is alreadypresent for arbitrary weak coupling. In this section we willonly present the results of this analysis and refer to the Ap-pendix for further details.

We derive an effective interaction for the two-leg ladder,which corresponds in weak coupling to the local Coulombrepulsion on the kagomé strip �23�. In this derivation we candrop terms that involve the high-energy states of the flatbandand focus on the states in the dispersing bands. We denotethe annihilation operator of these states by �k,�=�k,i,� wherek is the momentum along the strip and i� �1,2� is the bandindex. If we rewrite the Hamiltonian Hint in terms of thesenew operators we obtain the interaction,

Hint →U

L�

k1,. . .,k4

�gk1,. . .,k4

�k1,↑† �k2,↓

† �k3,↓�k4,↑† , �24�

where the prime over the sum restricts the sum tomomentum-conserving k values. For weak interaction wecan replace kl in gk1,. . .,k4

by �kFil, il� and we obtain the simple

expression

gk1,. . .,k4= e−iq/2��i1i2

�i3i4+ �i1i3

�i2i4+ �i1i4

�i2i3�/6, �25�

where q=k1+k2−k3−k4.The effective interaction �24� can now be expressed in

terms of left- and right-moving currents and in this way wefind the initial values for the RG equations of the two-legladder. The integration of the RG equations with these initialvalues converges to an analytic solution that was identifiedby bosonization techniques as a charge-density-wavesolution.25 This means that the operator

OCDW =1

L�k,�

�k,1,�† �k+�,2,� + �k+�,2,�

† �k,1,� �26�

acquires a finite value. The bond-order wave order parameteron the kagomé strip is given by the expectation value of anoperator OBOW.

The operators OCDW and OBOW transform identically un-der all symmetries of the system and, therefore, they describethe same phase.

In addition, OCDW is the effective operator on the two-legladder for OBOW, i.e., if one does the same substitutions aswe did for deriving Eq. �24� one sees that OBOW→OCDW, ifone chooses the right prefactor in the definition of OBOW.

We have shown that the bond-order wave instability thatis expected to occur at rather strong interactions according tothe arguments of the preceding sections is in fact alreadypresent in weak coupling for the one-dimensional kagoméstrip. Together with the DMRG results of the preceding sec-tion we provide convincing evidence that the BOW phase onthe kagomé strip is present both in the Hubbard model for allvalues of U and in the t-J model.

VI. CONCLUSIONS

In summary we have studied the occurrence of a bond-order wave instability in several bisimplex lattices. We pro-vided evidence that this instability occurs quite generally inthese lattices at the fractional filling of one electron per sim-plex �two electrons per up simplex�, if the correlations—i.e.,antiferromagnetic nearest-neighbor exchange and/or nearest-neighbor repulsion—are strong enough.

In weak coupling the physical properties of the system aredominated by the dimensionality of the lattice, by its Fermi-ology, and by the density of states at the Fermi energy. Weshow that in the intermediate-coupling regime, where thekinetic and the interaction energies are comparable, at thefilling with two electrons per up simplex, the physical prop-erties of these highly frustrated lattices are dominated bylocal states on the simplex. The bipartite and corner-sharingarrangement of the simplices allows the creation of isolatedor only weakly interacting simplices with low energy byspontaneously breaking inversion or translational symmetry.This knowledge provides a good starting point for series ex-pansions or further CORE calculations.

The magnetic interaction and the chosen sign of the dis-persion lead to a tendency to form nearest-neighbor singlets,and nearest-neighbor repulsion leads to a tendency to avoidconfigurations with more than two electrons per simplex. Ifthe underlying lattice is bipartite the system finds a way tosatisfy both tendencies simultaneously by localizing singletson every second simplex. This localization leads only to apartial loss of the kinetic energy, because the singlets canstill delocalize within the simplex. It is the cooperation be-tween the kinetic and the interaction energy which stabilizesthe bond-order wave state. Note that the bond-order waveinstability does not lead to an inhomogeneous charge distri-bution on the lattice.

The bond-order wave states, which we find on the differ-ent lattices, provide a natural generalization of the well-

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known valence bond solid states �e.g., dimerized phases,plaquette phases� found in many frustrated spin models tosituations away from half filling where a description in termsof spin variables only breaks down. The density is still arational fraction, but n=2/3 in the kagomé and kagomé stripcases while n=1/2 in the pyrochlore and checkerboardcases. These states are approximately direct products of sin-glets on triangles or tetrahedra, similar to the conventionalpicture of a dimerized phase. In contrast to the phases at n=1, however, the present instability involves a cooperativeeffect of both magnetic exchange and kinetic energy.

An interesting task is to study the properties of a lightlydoped bond-order wave phase. It can be assumed that thebond-order wave order parameter decreases rather quicklywith increased or decreased doping. However, it is conceiv-able that away from the commensurate filling the bond-orderwave order parameter coexists with a small superconductingorder parameter. This phase would at the same time breaklattice symmetries and the U�1� gauge symmetry and wouldbe therefore similar to a supersolid. A closer investigation ofthese issues has, however, be left for further studies.

In general, we conclude that the bond-order wave insta-bility occurs quite generally in bisimplex lattices and forphysically reasonable models and interaction parameters. Weshow that doping frustrated spin models can lead to newphases. Our study may also be a step toward the understand-ing of the interplay of frustrated magnetic fluctuations anditinerant charge carriers, which could play a role for examplein the unconventional heavy fermion material LiV2O4 �Ref.7� or in NaxCoO2.

ACKNOWLEDGMENTS

We thank T. M. Rice, K. Wakabayashi, and D. Poilblancfor stimulating discussions. We acknowledge support by theSwiss National Fund and NCCR MaNEP. Computationswere performed on IBM Regatta machines of CSCS Mannoand IDRIS Orsay.

APPENDIX: RG ANALYSIS: KAGOMÉ STRIP

The tight-binding Hamiltonian for the kagomé strip withperiodic boundary conditions is given by

H0 = − t�r=1

L

��=±1

��

�cr,�,�† cr+�,�,�

+ cr,0,�† �cr,�,� + cr+�,�u,�� + H.c.� − �N , �A1�

where �� �↑ , ↓ � are the spin indices and N is the numberoperator. The chemical potential � will be fixed to −t=−1 inthe following. It is convenient to introduce Fourier-transformed operators

cr,x =1L

�k

eik�r−x/2�ck,x, x � �±1,0� , �A2�

where the k sum runs over the L k values in �−� ,��. We canwrite this Hamiltonian in a diagonal form

H0 = �k

�l=1

3

��

k,l�k,l,�† �k,l,�,

and obtain the energies

k,1 = 1 − 2 cos�k�, k,2 = − 1 − 2 cos�k�, k,3 = 3.

�A3�

and the operators

��k,1,�

�k,2,�

�k,3,�� =

12�k�

�k 0 − �k

�k 2 �k

1 − 2�k 1��ck,−1

ck,0

ck,+1� �A4�

with �k=2cos �k /2� and �k=1+�k2 and k� �−� ,��.

The local Coulomb interaction introduced in Eq. �23� canbe written as

Hint =U

L�

k1,. . .,k4

��

x=−1

1

e−ixq/2ck1,x,↑† ck2,x,↓

† ck3,x,↓ck4,x,↑. �A5�

The sum over the momenta k1 , . . . ,k4 is restricted such thatq=k1+k2−k3−k4 is a multiple of 2�. Note that the appearingphase factor is important to determine the sign of the um-klapp scattering processes correctly. We obtain the effectivelow-energy Hamiltonian �24� from Eq. �A5� by doing thesubstitutions

ck,±,� → �12

�k,1,� +16

�k,2,�,

ck,0,� →2

3�k,2,�. �A6�

These substitutions rules are obtained from Eq. �A4� if weset k=kF1 in the first row and k=kF2 in the second row anddrop the third row in the matrix of the transformation.

In this way we can map the weak-coupling Hubbardmodel on the kagomé strip to an effective weak-couplingmodel on the two-leg ladder. The problem of a weak-coupling two-leg ladder has been extensively studied byrenormalization group and bosonization techniques.25–27 Wewill adopt here the notation of Lin et al. in Ref. 25. A generalweak interaction can be conveniently expressed in terms ofleft and right moving currents. Dropping purely chiral termsthe momentum-conserving four fermion interactions can bewritten as

H�1� = bij� JRijJLij − bij

�JRij · JLij + f ij� JRiiJLjj − f ij

�JRii · JLjj .

�A7�

To avoid double counting we set f ii=0. Furthermore, thesymmetry relations f12= f21 �parity�, b12=b21 �Hermiticity�,and b11=b22 �only at half filling� hold. We have therefore sixindependent coefficients. For our interaction we find the val-ues

4b11� = b11

� = U, 4b12� = b12

� =U

3, 4f12

� = f12� =

U

3.

In addition we have umklapp terms given by

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H�2� = uij� IRij

† ILi j − uij�IRij

† · ILi j + H.c.

with uii�=0, u11

� =u22� , u12

� =u21� and u12

� =u21� . Here we have the

values

u12� = 0, u12

� = −U

12, u11

� = −U

24.

Integrating the RG equations with these initial values showsthat the solution converges to the analytic solution of the RG

equation where all coupling constants except for b11� and b11

�

diverge with fixed ratios given by

f12� = −

1

4f12

� = b12� = −

1

4b12

� =1

2u12

� = − 2u12� = − 2u11

� = g � 0.

�A8�

This solution was identified by bosonization techniques as acharge-density-wave solution.

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