Sandro Sorella et al- Chiral Spin Liquid Wave Function and the Lieb-Schultz-Mattis Theorem

8/3/2019 Sandro Sorella et al- Chiral Spin Liquid Wave Function and the Lieb-Schultz-Mattis Theorem

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Chiral Spin Liquid Wave Function and the Lieb-Schultz-Mattis Theorem

Sandro Sorella,1 Luca Capriotti,2 Federico Becca,1 and Alberto Parola3

1 INFM-Democritos, National Simulation Centre, and SISSA, I-34014 Trieste, Italy2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030, USA

3 Istituto Nazionale per la Fisica della Materia and Dipartimento di Scienze, Universita dellInsubria, I-22100 Como, Italy(Received 30 June 2003; published 17 December 2003)

We study a chiral spin liquid wave function defined as a Gutzwiller projected BCS state with acomplex pairing function. After projection, spontaneous dimerization is found for any odd but finitenumber of chains, thus satisfying the Lieb-Schultz-Mattis theorem, whereas for an even number ofchains there is no dimerization. The two-di mensional thermodynamic lim it is consistently reached for alarge number of chains since the dimer order parameter vanishes in this limit. This property clearlysupports the possibility of a spin liquid ground state in two dimensions with a gap to all physicalexcitations and with no broken translation symmetry.

DOI: 10.1103/PhysRevLett.91.257005 PACS numbers: 74.20.Mn, 71.10.Fd, 71.10.Pm, 71.27.+a

A long time a fter its first proposal [1], the existence of aspin liquid ground state (GS) in two-dimensional (2D)quantum spin one-half models is still a very controversial

issue. This is main ly because all one-di mensional (1D) orquasi-1D spin models that can be solved exactly, eithernumerically or analytically [24], display a gap to thespin excitations only when a broken translation symmetr yis found in GS (e.g., i n spin-Peierls systems [2,3]) or whenthe unit cell contains an even number of spin 1=2 elec-trons (e.g., in the two-chain Heisenberg model [4]).Hence, in these models the electronic correlations donot play a crucial role since their GS can be adiabaticallyconnected to a band insulator without any transition. Th isimportant property of insulators, which clearly holds in1D systems, has been speculated to be generally valideven in higher dimensions, as it appears to follow from a

general result, the Lieb-Schultz-Mattis (LSM) theorem[5], whose range of validity has been extended to moreinteresting 2D cases [6,7].

Recently, there has been an intense theoretical andnumerical investigation of nonmagnetic wave functionsobtained after Gutzwiller projection of the GS of a BCSHamiltonian [811]. On a rectangular Lx Ly lattice,this state can be written in the following general form:

jp-BCSi PPG

"Xk

fkccyk;"cc

yk;#

#N=2

j0i; (1)

where N is the number of electrons (equal to the numberof sites, i.e., N Lx Ly), PPG is the Gutzwiller projec-tion onto the subspace of no doubly occupied sites, and ccyk;"and ccyk;# are creation operators of a spin-up or a spin-downelectron, respectively. These are defined in a plane-wavestate with momentum k allowed by the chosen boundaryconditions: periodic (PBC) or antiperiodic (APBC) ineach direction. It is worth noting that, after the projec-tion, the wave function (1) corresponds to PBC on the spinHamiltonian, regardless of the choice of boundary con-

ditions on the electronic states. The pairing function fkcan be easily related to the gap function k and the baredispersion k 2coskx cosky of the BCS Hamil-

tonian (BCSH) [9], by means of the simple relation fk k=k Ek, where the gap function k can be in generalany complex function even under inversion (k k),as required here for a singlet wave function [9], and Ek

jkj2 2k

qrepresents the spin-half excitation energies

of the BCSH.As clearly pointed out by Wen [12], in the presence of a

finite gap BCS in the thermodynamic limit, such thatEk BCS > 0, the corresponding BCS finite correlationlength is expected to be robust under Gutzwiller projec-tion. Here, we restrict to this class of nonmagnetic states,considering the projected BCS (p-BCS) state which isobtained by a d id gap function of the form

k x2y2coskx cosky ixy sinkx sinky: (2)

In the following, we will consider the case ofx2y2 2and xy 1. This wave function breaks the time reversalsymmetry TT and the parity symmetry (x $ y in 2D) PP,whereas TT PP is instead a well-defined symmetry.Hence, this spin liquid wave function may have a non-vanishing value of the chiral order parameter [13]:

OO C 1

N

Xi

SSi SSidx SSidy; (3)

with dx

1; 0 and dy

0; 1. Ch iral spin liquids wereintroduced a long time ago [13,14]; however, to ourknowledge, this is the first attempt to represent this classof states in the p-BCS framework. In finite-size systems,it is convenient to define a wave function which satisfiesall the lattice symmetries, including parity. This can bedone by suitably choosing the overall phase of the com-plex wave function (1) and taking its real part. In thiscase, a spontaneously broken lattice symmetry can occuronly in the thermodynamic limit.

P H Y S I C A L R E V I E W L E T T E R S week ending19 DECEMBER 2003VOLUME 91, NUMBER 25

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In the following, by using a variational Monte Carloapproach, we consider in more detail the relation of thep-BCS wave functions with the LSM theorem. Given ashort-range spin Hamiltonian HH, on a Lx Ly rectangle(PBC on the x direction are assumed) and a variationalstate j0i with given momentum, we can define anothervariational state, j00i OOLSMj0i, by means of the LSMoperator:

OO LSM exp

"iXr

2x

LxSSzr

#; (4)

where r x;y indicates the position of each site on thelattice. The new variational state j00i has the followingproperties: (i) its energy expectation value differs at mostbyOLy=Lx from the variational energy ofj0i and (ii) ifLy is odd, regardless of the boundary conditions on the ydirection, the momenta parallel to x corresponding toj0i and j

00i differ by . Hence, h

00j0i 0 [6].

For a 1D or quasi-1D system with an odd number ofchains Ly and vanishing aspect ratio (Ly=Lx ! 0 for

Lx ! 1), by applying the LSM operator to the actualGS, it is possible to construct an excitation of the systemwith momentum ; 0 which becomes degenerate withthe GS in t he thermodynamic limit. This implies either agapless spectrum or, in the presence of a finite gap, atwofold degenerate GS with a doubling of the unit cell anda spontaneously broken t ranslation symmetry. For in-stance, the presence of a singlet zero-energy excitationwith momentum ; 0 is just a characteristic of sponta-neous spin-Peierls dimerization, as it appears, for ex-ample, in the Majumdar-Ghosh chain [2]. This result,holding rigorously i n the l imit of vanishing aspect ratio,has been argued to apply in general for 2D systems [6,7].

In the following, we will show, with an explicit example,that this result in 2D does not necessarily imply sponta-neous dimerization, but topological degeneracy of the GS.

It is simple to show that OOLSMjp-BCSi jp-BCS0i,

namely, the same type of wave function of Eq. (1) isobtained, with the changes below:

k ! kk; (5)

fk ! ffkk fkk=Lx;0 fk; (6)

where the new quantized momenta kk k =Lx; 0 areobtained by interchanging PBC with APBC in the xdirection only and Eq. (6) means that the pairing functionis calculated with the old momenta k: fkcc

ykk;"ccy

kk;#. By

definition, the wave function jp-BCS0i has therefore thesame quantum numbers predicted by the LSM theorem,the change of momentum being implied by Eq. (5). Thereason why the momentum of the wave function (1) can benonzero for an odd number of chains is indeed rathersubtle but easy to verify. Indeed the x-translation operatorwith APBC translates all creation operators, but the onesbelonging to the boundary are also multiplied by a phase

factor 1. This translation operator always leaves in-variant the jp-BCS0i wave function. However, for a spinstate with one electron per site, each configuration hasalways Ly electrons at the boundary, so that the physicalspin translation operation (defined with PBC), differsfrom the APBC one for an overall phase 1Ly , namely,a momentum ; 0 for an odd number of chains.Analogously, the excitations obtained by modifying

only the boundary conditions in the BCSH (in the xand/or y direction), namely, using Eq. (5) (and/or itsequivalent for the y direction) and fk ! fkk, may displayin 2D the topological degeneracy of this spin liquid wavefunction [10,11].

We now assume that the p-BCS wave function (1) withk given by Eq. (2) represents the GS of some short-rangeHamiltonian. Certainly, explicit Hamiltonians withshort-range off-diagonal matrix elements can be con-structed by a simple inversion scheme [15]. This can beobtained considering a given basis of configurations fjxig,where the positions and the spins of the electrons aredefined. First, we expand the wave function (1) in this

local basis as jp-BCSi Pxxjxi. Then, starting from a

local and short-range Hamiltonian HH with matrix ele-ments Hx;x0 (e.g., a frustrated Heisenberg model), it ispossible to define an effective Hamiltonian HHeff by mod-ifying the matrix elements as follows:

Heffx0;x

8>:Veffx for x0 xHx0;x if sx0;x < 00 if sx0;x > 0;

(7)

where sx0;x x0Hx0;xx for x0 x and sx;x 0,

and the diagonal potential is given by Veffx Px0;sx0;x


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1xx0O2SP=36 const, being OSP the di mer order pa-

rameter [3]. As shown in Fig. 1 for the th ree-chain case,spontaneous dimerization is obtained in the thermody-namic system for odd-leg ladders. Here, in strong analogy

with the one-dimensional Heisenberg model in thegapped phase [2,3], the broken translation symmetryallows the system to satisfy t he LSM theorem. In contrast,on any even-leg ladder, the p-BCS state does not breaktranslational invariance, as shown in Fig. 1 for the four-chain system. Despite the dichotomy between the odd-and even-leg cases, the 2D thermodynamic limit can bestill consistently defined. In fact, as it is clearly shown inFig. 2, though a finite dimer order para meter is obtainedfor any odd-leg ladder, the order parameter is exponen-tially decreasing with the number of chains. This impliesthat the broken symmetry, which is correctly obtained forodd but finite number of chains, represents an irrelevant

effect in the 2D thermodynamic system. Nonetheless, inthe 2D system, the GS can possess degenerate topologicalexcitations [17]. Indeed, no simple (two-spin) local op-erator connects the degenerate stat es in 2D. Moreover, thematrix element of t he most relevant dimer operator withmomentum ; 0 between the two degenerate singletstates, which is finite on any finite number of chains,decreases exponentially with increasing Ly, as it isbounded by the order parameter (see Fig. 2). No otherorder is detected by studying dimer correlations.Remarkably, by studying the chiral-chiral correlationsat the maximum distance along the leg direction, it ispossible to show that the chiral order parameter O

Cremains a genuine feature of this variational wave func-tion even in the 2D limit [see Fig. 3(a)].

We have given here a clear example that a spin liquidGS can be stable in 2D, yet satisfying all the knownconstraints given by the LSM theorem. Indeed, sponta-neous broken translation symmetry is obtained for anyodd number of chains, a remarkable feature since beforeprojection the wave function is translationally invariant.As the number of chains increases, the chiral spin liquid

appears only in 2D systems, where the spin-Peierls dimerorder parameter converges to zero, and no broken trans-lation symmetry is implied in the thermodynamic limit.

The chiral spin liquid described by the p-BCS wavefunction is a lso consistent with a recent extension of theLSM to 2D systems with finite aspect ratio Ly=Lx [7]. Aspointed out by Oshikawa, the state j00i OOLSMj0i isnot necessarily degenerate with the starting wave func-tion j0i, as in the usual LSM construction. However,through a well-defined adiabatic evolutionin analogywith Laughlins treatment of the quantum Hall effectone can obtain a different state j ~00i, with the same spatialquantum numbers of j00i and degenerate with j0i. If

j0i is described by a p-BCS state (1), by a small change

FIG. 1 (color online). Dimer-dimer correlation functions as afunction of t he distance for three a nd four chains with PBC onboth directions.

FIG. 2 (color online). (left panel) Finite-size scaling of thedimer order parameter in the rectangular geometries with anodd number of chains. ( right panel) Dimer order parameter as afunction of the number of chains. The full circles in the rightpanel refer to the short-range resonating valence bond state,defined as the superposition of all nearest-neighbor dimercoverings of the lattice with the same weights, having ! 1in 2D.

FIG. 3 (color online). (a) Chiral order parameter as a functionof the number of chains. (b) Difference of the nearest-neighbortotal energy Enn NhSSi SSjie hSSi SSjio between twoorthogonal states jei and joi predicted by the LSM theorem.(c) Finite-size scaling of the dimer order parameter inrectangular lattices with finite aspect ratio. For (b) and (c),Ly Lx 1.


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of the pair ing function of the state j00i, i.e.,ffkk ! fkk, it is

easy to define a state j ~00i with the same momentumimplied by the LSM theorem but expected to be degen-erate with j0i. In fact, the wave function j ~

00i can be

obtained with the same BCSH with APBC in the xdirection. Then, in the presence of a finite gap in theexcitation spectrum BCS > 0, the wave functions j0iand j ~00i have t he same value of any physical, i.e., local,

operator, and, in particular, the Hamiltonian, so that theconsidered states are degenerate in the thermodynamiclimit. In particular, the difference of the nearest-neighbortotal energy of the two states is vanishing exponentiallywith the number of chains [see Fig. 3(b)].

The state j ~00i, obtained by the adiabatic evolution ofthe LSM excitation, j00i OOLSMj0i, is no longer con-nected to j0i by any physical operator, and, therefore, nospontaneous dimerization is implied in the thermody-namic limit. Indeed, as shown in Fig. 3(c), also in ge-ometries with nonzero aspect ratio the p-BCS wavefunction has a finite dimer correlation length. We havetherefore shown a clear counterexample to the so-called

Oshikawa conjecture [7] that no spin liquid is possible fora spin one-half model in two or higher dimensions.

In general, projected wave functions of the type (1)with a finite gap BCS > 0 do not necessarily break timereversal. For instance, a gapped BCS spectrum can beobtained with a chemical potential outside the band [11],with a gap function with d is symmetry [12], or simplywith s symmetry. For all these wave functions, a finitedimer order parameter, in some cases very small butalways nonzero, is found in lattices with infinite aspectratio, in agreement with the LSM theorem. Therefore, webelieve that the broken time reversal is not a necessarycondition to satisfy the LSM theorem.

The finite dimer correlation length represents a re-markable property of the chiral p-BCS wave function.For instance, the conventional short-range resonatingvalence bond (RVB) [18] displays instead power-lawdimer correlations in 2D [19], and therefore gapless fea-tures which may describe a singular point rather than a2D spin liquid phase. The fundamental difference be-tween our chiral RVB and the short-range one is due tothe violation of the Marshall sign rule in the former case.This property is observed, for instance, in the GS of theJ1 J2 model in the strongly frustrated region J2=J1 0:5, where this complex wave function has a lower varia-tional energy compared with the real ones [9]. Thus, weexpect t hat a true spin liquid phase cannot be stabilized inthe spin models where the wave function signs are notallowed to change as a function of the parameters of theHamiltonian [20]. The complex wave function withd-wave symmetry proposed here appears to be a reason-able way to open a gap close to a gapless antiferromag-netic phase in a translationally invariant spin modelviolating the Marshall sign rule, i.e., a generic frustrated

model on t he square lattice. In th is case, the chiral orderparameter can be very small, vanishing for xy ! 0 whenthe correlation length ! 1.

In conclusion, we have presented a clear example t hat aspin liquid GS with a gap to all physical excitations,though being with one electron per unit cell, can berealized without violating the LSM t heorem and its gen-eralizations in 2D. Our results provide a clear support to

the possibility of a true Mott insulator at zero tempera-ture, showing that the effect of correlation may be highlynontrivial in 2D systems.

One of us (S. S.) than ks R. Laughlin for many usefuldiscussions. We also thank D. J. Scalapino for useful com-ments, a nd for his kind hospitality at UCSB (S. S.). Thiswork was substantially supported by INFM-PRA-MALODI, and part ially by MIUR -COFIN 01. L. C. wassupported by NSF under Grant No. DMR02-11166.

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2095 (1987); Phys. Rev. B 39, 11879 (1988).[15] S. Sorella, cond-mat/0201388.[16] S. Sachdev and K. Park , An n. Phys. (N.Y.) 298, 58 (2002),

and references therein.[17] Two states jei and joi are topologically distinct if

hej ÂAjoi 0 and hej ÂAjei hoj ÂAjoi for any local andextensive operator ÂA.

[18] N. Read and B. Chakraborty, Phys. Rev. B 40, 7133(1989); N. Read and S. Sachdev, Phys. Rev. Lett. 66,1773 (1991); N. E. Bonesteel, Phys. Rev. B 40, 8954(1989).

[19] M. E. Fisher and J. Stephenson, Phys. Rev. 132, 1411(1963).

[20] A.W. Sandvik et al., Phys. Rev. Lett. 89, 247201 (2002).


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Sandro Sorella et al- Chiral Spin Liquid Wave Function and the Lieb-Schultz-Mattis Theorem

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