zheng weihong et al- phase diagram for a class of spin-1/2 heisenberg models interpolating between...

8/3/2019 Zheng Weihong et al- Phase diagram for a class of spin-1/2 Heisenberg models interpolating between the square-la

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Phase diagram for a class of spin- 12 Heisenberg models interpolating between the square-lattice,the triangular-lattice, and the linear-chain limits

Zheng Weihong * and Ross H. McKenzie School of Physics, The University of New South Wales, Sydney, NSW 2052, Australia

Rajiv R. P. Singh Department of Physics, University of California, Davis, California 95616

Received 16 December 1998

We study the spin- 12 Heisenberg models on an anisotropic two-dimensional lattice which interpolates be-tween the square lattice at one end, a set of decoupled spin chains on the other end, and the triangular-latticeHeisenberg model in between. By series expansions around two different dimer ground states and aroundvarious commensurate and incommensurate magnetically ordered states, we establish the phase diagram forthis model of a frustrated antiferromagnet. We nd a particularly rich phase diagram due to the interplay of magnetic frustration, quantum uctuations, and varying dimensionality. There is a large region of the usualtwo-sublattice Neel phase, a three-sublattice phase for the triangular-lattice model, a region of incommensuratemagnetic order around the triangular-lattice model, and regions in parameter space where there is no magneticorder. We nd that the incommensurate ordering wave vector is in general altered from its classical value byquantum uctuations. The regime of weakly coupled chains is particularly interesting and appears to be nearlycritical. S0163-1829 99 10421-1

I. INTRODUCTION

Quantum phases and phase transitions in two-dimensionalHeisenberg models are of great current interest. 1 Followingmuch experimental, theoretical, and numerical work, a fairlycomprehensive and consistent picture has emerged for theordered phases of unfrustrated Heisenberg models at zeroand nite temperatures as well as for the quantum criticalpoints separating them from the quantum disorderedphases. 2,3 The case of frustrated Heisenberg models poten-tially includes much richer phenomena and is relatively lessunderstood. Here, the lattice models that have been studiedthe most are the square-lattice nearest- and second-neighborinteraction or J nn - J nn n ) model, where as a function of J nn n / J nn , the two- and four-sublattice collinear Ne el phasesare separated by a region of quantum-disordered groundstates. 4 Theoretical arguments and various numerical resultssuggest that this intermediate phase is spontaneously dimer-ized. Other models that have received considerable attentionare the triangular-lattice and Kagome-lattice Heisenbergmodels. 57 The former appears to be magnetically ordered inthe three-sublattice noncollinear pattern, whereas the latteralmost surely has a nonmagnetic ground state. The nature of the magnetically disordered phases and low-energy excita-tions in these frustrated Heisenberg models is a subject of considerable interest.

Here we study a class of Heisenberg models that interpo-late between the square-lattice, the triangular-lattice, and thelinear-chain Heisenberg models. It can be dened in terms of a square lattice of spin- 12 operators, with a nearest neighborHeisenberg interaction J 2 and a second-neighbor interaction J 1 along only one of the diagonals of the lattice. In the limit J 1 0, we recover the square-lattice Heisenberg model. Inthe limit J 2 0, the model reduces to decoupled spin chains

running along the diagonals of the square lattice. For small J 2 / J 1 the model consists of weakly coupled chains withfrustrated interchain coupling. For J 2 J 1, the model is ex-actly equivalent to the triangular-lattice Heisenberg model.

This model is of particular interest for at least two rea-sons. First, within a single Hamiltonian, it provides a way of interpolating between several well-known one- and two-dimensional models. One can study the role of frustration ingoing from one to two dimensions, 8 as well as the stability of commensurate and incommensurate magnetic order with re-spect to quantum uctuations. 9 Second, the model is of directrelevance to the magnetic phases of various quasi-two-dimensional organic superconductors. It has been argued thatthis model should describe the spin degrees of freedom of the insulating phase of the layered molecular crystals,

-(BEDT-TTF) 2 X .10 These materials probably have J 1 / J 2

0.31. This ratio varies with the anion X and should varywith uniaxial stress applied within the layer along thediagonal. 11 A very slight variant on this model with J 1 / J 2

4 has been proposed to describe the insulating phase of -(BEDT-TTF) 2RbZn(SCN) 4 which has a spin gap. Recent

studies of the associated Hubbard model on an anisotropictriangular lattice at half-lling suggest that the wave vectorassociated with the spin excitations determines the type of superconducting order. 12 Consequently, it appears that agood understanding of this Heisenberg model may be a pre-requisite to understanding organic superconductivity.

In this paper we study the zero-temperature phase dia-gram of this model by series expansions around variousdimerized and magnetically ordered phases. Because of frus-tration and the large coordination number of the lattice, itwould be difcult to get a comparable treatment of thismodel by any other numerical method. We will only con-sider J 1 0, J 2 0; that is, both couplings are antiferromag-

PHYSICAL REVIEW B 1 JUNE 1999-IIVOLUME 59, NUMBER 22

PRB 590163-1829/99/59 22 /14367 9 /$15.00 14 367 1999 The American Physical Society


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netic, which introduces frustration into the model. We ndthat the Ne el order of the square-lattice Heisenberg modelpersists up to J 1 / J 2 0.7, where it goes continuously to zero.It is interesting to note that Neel order exists beyond it re-gime of classical stability, providing an example of quantumorder by disorder. 9 In the region 0.7 J 1 / J 2 0.9, there is nomagnetic order and the ground state is dimerized in the co-lumnar pattern. For larger values of J 1 / J 2 , there is incom-mensurate or spiral long-range order. Except for thetriangular-lattice point ( J 1 / J 2 1), we nd that the orderingwave vector is different from the classical Heisenberg model.At the triangular-lattice point, the dimer expansions indicatea magnetic instability at an ordering wave vector which co-incides with the classical value. Furthermore, our Ising ex-pansion series in this case reduce to previously derived seriesexpansions directly for the triangular-lattice model. 5

The region of large J 1 / J 2 is very interesting and is notfully resolved by our series expansions. In the limit of small J 2 , we have Heisenberg spin chains coupled by frustratingzigzag interactions. For just two such chains, this problemwas studied by White and Afeck. 13 This zigzag ladder has

incommensurate spin correlations, a small gap, and a spon-taneous dimerization, which breaks the degeneracy of theweak zigzag bonds but leaves the strong chain bonds equiva-lent. Our numerical results suggest that such a state is de-nitely not favored for the two-dimensional system. This isnot hard to understand as such a broken symmetry phase canonly have the strong bonds between every alternate pair of chains. We suggest three possible scenarios for this phase: iIt has weak incommensurate spiral order, which vanishesrapidly with a power greater than unity as J 2 0. ii Thisphase is dimerized along the strong J 1 bonds and has a spingap. iii The phase remains critical without any long-rangeorder. The ground state energies obtained from the Ising ex-

pansions around a spiral phase and around a dimer statealong the strong bonds remain nearly equal through much of this phase. In all three cases the system is close to beingcritical.

The elementary excitations of this phase and how they areconnected to the excitations in isolated spin chains should bevery interesting to investigate. 14 The low-energy propertiesof models with noncollinear order are described by the SO 3nonlinear model. 15,16 In the quantum-disordered phase of this model the elementary excitations are deconned spin- 12objects, or spinons, 17,3 in contrast to the magnons or spin-1triplets that are the elementary excitations in most two-dimensional antiferromagnets. Thus, this model, with spiralclassical order, provides us with candidate systems to look for spinon excitations.

The plan of the paper is as follows. In Sec. II we discussthe model and the various series expansions that are carriedout. In Sec. III we discuss the results for the phase diagram,the ground-state energy, the excitation spectra, and the cor-relation functions of the model. Finally, in the last section wepresent our conclusions and discuss some future directions.

II. SERIES EXPANSIONS

The Heisenberg antiferromagnet on an anisotropic trian-gular lattice is equivalent to the Heisenberg antiferromagneton a square lattice with nearest-neighbor interactions and one

of the diagonal next-nearest-neighbor interactions as shownin Fig. 1. The Hamiltonian is

H J 1in

Si Sn J 2i j

Si S j , 1

where i j means pairs of nearest neighbors and in meansone of the diagonal next-nearest-neighbor pairs, as illustratedin Fig. 1. We denote the ratio of the couplings as y, that is, y J 1 / J 2 . In the present paper, we study only the case of antiferromagnetic coupling, where both J 1 and J 2 are posi-tive. In the limit J 2 0, the model is equivalent to decoupledone-dimensional Heisenberg chains. In the limit J 1 0, themodel is equivalent to the two-dimensional square-latticeHeisenberg model. For J 1 J 2 , the model is equivalent to theHeisenberg model on a triangular lattice.

Let us discuss the classical ground state of this system: If we assume Si lies in the x- z plane, we can rotate all thespins related to a reference spin; so we have a ferromagneticground state. The classical ground state of this system hasnearest-neighbor spins that differ by an angle q and spins joined by a diagonal bond that differ by 2 q , where q is

q , J 1 J 2 /2,

arccos J 2 /2 J 1 , J 1 J 2 /2.2

The angle q as a function of J 1 /( J 1 J 2) is shown as thesolid curve in Fig. 2.

After this rotation, the transformed Hamiltonian is

H H 1 J 1 H 2 J 2 H 3 , 3

where

H 1 J 1cos 2qin

S i zSn

z J 2cos qi j

S i zS j

z ,

H 2in

S i ySn

y cos 2q S i xSn

x sin 2q S i zSn

x S i xSn

z ,

H 3i j

S i yS j

y cos q S i xS j

x sin q S i zS j

x S i xS j

z . 4

We have studied this system by using linked-cluster ex-pansion methods at zero temperature including Ising expan-sions, and dimer expansions about two different dimerizationpatterns. The linked-cluster expansion method has been pre-viously reviewed in several articles 1820 and will not be re-peated here. Here we will only summarize the expansion

FIG. 1. The anisotropic triangular lattice with coupling J 1 and J 2 .

14 368 PRB 59ZHENG WEIHONG, ROSS H. McKENZIE, AND RAJIV R. P. SINGH


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methods used, and the results derived from them will bepresented. The series coefcients are not given here, but areavailable upon request. Previous studies using these methodshave agreed quantitatively with other numerical studies in-cluding quantum Monte Carlo and exact diagonalization

studies for unfrustrated spin models. For frustrated two-dimensional models, both the Monte Carlo and the exactdiagonalization methods face problems due to the minussigns in case of the former and tremendous variations withsize and shapes of the clusters in case of the latter . Thus, webelieve that it would be difcult to get a comparable numeri-cal treatment of this strongly frustrated model by other nu-merical methods.

A. Ising expansions

To construct a T 0 expansion about the classical groundstate for this system, one has to introduce an anisotropy pa-rameter , and write the Hamiltonian for the Heisenberg-Ising model as

H H 0 xV , 5

where

H 0 H 1 t i

S i z,

V J 1 H 2 J 2 H 3 t i

S i z . 6

The last term of strength t in both H 0 and V is a local eldterm, which can be included to improve convergence. The

limits x 0 and x 1 correspond to the Ising model and theisotropic Heisenberg model, respectively. The operator H 0 istaken as the unperturbed Hamiltonian, with the unperturbedground state being the usual ferromagnetically ordered state.The operator V is treated as a perturbation.

To determine whether the system is in the ordered phase,we need to compute the order parameter M dened by

M 1 N i

S i z , 7

where the angular brackets denote ground-state expectationvalues. Actually, in this expansion, we can choose q differentfrom that given in Eq. 2 ; the favored q should give a lowerenergy. If q , we are expanding about the usual Neelphase.

Ising series have been calculated for the ground-state en-ergy per site, E 0 / N , the order parameter M , and the tripletexcitation spectrum for the case of q only for severalratios of couplings x and simultaneously for several valuesof t up to order x10. This calculation involves 116 137 linked

clusters of up to 10 sites. The series for the case of y 1have been computed previously, 5 and our results agree withthese previous results. This is a highly nontrivial check onthe correctness of the expansion coefcients, as the aniso-tropic square-lattice graphs considered here are completelydifferent from the triangular-lattice graphs considered inRef. 5 and the representation of the Hamiltonian as aHeisenberg-Ising model is also supercially different.

At the next stage of the analysis, we try to extrapolate theseries to the isotropic point ( x 1) for those values of theexchange coupling parameters which lie within the orderedphase at x 1. For this purpose, we rst transform the seriesto a new variable

1 1 x 1/2, 8

to remove the singularity at x 1 predicted by spin-wavetheory. This was rst proposed by Huse 21 and was also usedin earlier work on the square lattice case. 22 We then use bothintegrated rst-order inhomogeneous differentialapproximants 23 and Pade approximants to extrapolate the se-ries to the isotropic point 1( x 1). The results of theIsing expansions will be presented in later sections.

B. Dimer expansions

For this system, we have carried out two types of dimer

expansions corresponding to two types of dimer coverings.One is the columnar covering displayed in Fig. 3, and theother is the diagonal covering displayed in Fig. 4 a .

1. Columnar dimer expansions

In this expansion, we take the columnar bonds D de-noted by the thick solid bonds in Fig. 3 as the unperturbedHamiltonian H 0 and the rest of the bonds as perturbation;that is, we dene the following dimerized J 1- J 2 model:

H J 2( i , j) D

Si S j x J 2( i , j) D

Si S j J 1in

Si S j .

9

FIG. 2. Phase diagram of the model. The wave vector q is plot-ted as a function of J 1 /( J 1 J 2). The solid curve is the classicalordering wave vector, while the points are estimates of the locationof the minimum triplet gap from both columnar dimer expansionssolid circles and diagonal dimer expansions open circles . The

long dashed lines are the t given in Eq. 16 . The vertical shortdashed lines indicate the region of spontaneously dimerized phase,and vertical dotted lines indicate another possible transition point.

PRB 59 14 369PHASE DIAGRAM FOR A CLASS OF SPIN- 12 . . .


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We take the rst term in above H as the unperturbed Hamil-tonian and the second term as perturbation. The unperturbedground state is a product state of nearest-neighbor singlet

dimers and the perturbation couples these among themselvesand with the pair-triplet states. At x 1, one recovers theoriginal Hamiltonian 1 .

Two types of series can be obtained in this way. By xing y, we can compute series in the single variable x up to order L,

E / J 2i 0

L

c i y xi. 10

In the other approach we keep both x and y as expansionparameters and obtain double series of the form

E / J 2i 0

L

j 0

i

d i j yi x j, 11

where the coefcients d i j are computed up to order i L.The series has been computed up to order L 9 for the

ground-state energy E 0 and up to order L 8 for the tripletexcitation spectrum (k x ,k y). This calculation involves49 684 linked clusters of up to 9 sites.

Note that if the centers of the dimers are treated as latticesites, one obtains a rectangular lattice with spacing 2 a in the x direction and a in the y direction, where a is the latticespacing of original lattice. In characterizing the excitationspectrum presented in the later sections we set the latticespacing in both the x and y directions to be 1. In this nota-tion, the wave vector corresponding to the classical orderingis (q ,q ), which varies from (0, ) to ( /2, /2) as J 1 / J 2 increases from 0 to .

The leading order series for the triplet excitation spectrum

is

k x ,k y / J 2 1 x12 cos k x 1 J 1 /2 J 2 cos k y

J 1 /2 J 2 cos k x k y O x2 . 12

Note that the minimum of the above spectrum may not co-incide with the ordering wave vector ( q ,q ) of the clas-sical system.

2. Diagonal dimer expansions

In this expansion, we take the diagonal bonds D denotedby the thick dashed bonds in Fig. 4 a as unperturbed Hamil-tonian H 0 and rest of the bonds as perturbation. That is, wedene the following dimerized J 1- J 2 model:

H J 1( i ,n ) D

Si S j x J 1( i ,n ) D

Si S j J 2i j

Si S j .

13

Again at x 1, one recovers the original Hamiltonian 1 .For convenience, we have transformed the lattice shown

in Fig. 4 a to the equivalent one shown in Fig. 4 b . Againthe dimer centers in Fig. 4 b lie on a rectangular lattice withspacing 2 a in the x direction and a in y the direction. Incharacterizing the excitation spectrum presented in the latersections we also set the lattice spacing in both the x and y

directions to be 1. In this representation the wave vectorcorresponding to the classical ordering is ( 2q ,q ), whichvaries from ( , ) to (0, /2) as J 1 / J 2 increases from 0 to

.As in the case of columnar dimer expansions, two types

of series have been computed up to order x9 for the ground-state energy E 0 and up to order x

8 for the triplet excitationspectrum (k x ,k y). This calculation involves the samelinked clusters as the columnar dimer expansions.

The leading order series for triplet excitation spectrum is

k x ,k y 1 x12 cos k x J 2 /2 J 1 cos k y

J 2 /2 J 1 cos k x k y O x2 . 14

FIG. 3. Columnar dimerization pattern. The thick solid linesrepresent strongest bonds J 2 , and thin solid lines and dashed linesrepresent weaker bonds xJ 2 and xJ 1 , respectively. Note that thedimer centers lie on a rectangle lattice with spacing 2 a in the xdirection and a in the y direction. In characterizing the excitationspectrum we set the lattice spacing in both the x and y directions tobe 1.

FIG. 4. a Diagonal dimerization pattern. The thick dashedlines represent strongest bonds J 1, and thin dashed lines and solidlines represent weaker bonds xJ 1 and xJ 2 , respectively. b Theequivalent lattice to a used in series expansions. Again note thatthe dimer centers lie on a rectangle lattice with spacing 2 a in the xdirection and a in the y direction. In characterizing the excitationspectrum we set the lattice spacing in both the x and y directions tobe 1.



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Note that the minimum of is at ( 2q ,q ), with q denedin Eq. 2 , which is just the ordering wave vector of theclassical system.

III. RESULTS

Having obtained the series for the various expansionsabove we present in this section the results of series analysis.We use integrated rst-order inhomogeneous differentialapproximants 23 and Pade approximants to extrapolate the se-ries.

A. Ground-state energy

Figure 5 shows the ground-state energy per site, E 0 / N ,for the Hamiltonian in Eq. 1 obtained from various expan-sions, We see that the Ising expansions about the Ne el ordergive the lowest energy and also show the best convergencein the region of J 1 / J 2 0.7. The Ising expansions about theclassical ground state i.e., q dened by Eq. 2 give thelowest energy in the region of 0.9 J 1 / J 2 1.2, the colum-nar dimer expansions give the lowest energy for 0.65

J 1 / J 2 0.9, and the diagonal dimer expansions give thelowest energy for J 1 / J 2 1.2. Whenever the dimer expan-sions indicate an ordering wave vector different from theclassical one, we have also computed the series expansionaround the spiral state with this modied wave vector. Thesealways give lower ground-state energies than those for theclassical wave vector. Also, for J 1 / J 2 1.2, these modiedIsing expansions lead to ground-state energies that are al-most identical with the diagonal dimer expansions.

B. Magnetization

The order parameter M obtained from the Ising expan-sions about the classical ground state is shown in Fig. 6. Thedependence on the amount of magnetic frustration is verysimilar to that obtained from linear spin-wave theory. 24 Thestaggered magnetization associated with Ne el order exists upto J 1 / J 2 0.7, rather than up to J 1 / J 2 0.5, as in the classi-cal system. We can also see that the magnetization M van-ishes over the region 0.7 J 1 / J 2 0.9, suggesting that thismay belong to a quantum-disordered phase. For J 1 / J 2 0.9,the system may have spiral order.

The fact that Ne el order survives beyond the classicallystable region is an example of promotion of collinear orderby quantum uctuations. A similar stabilization was found inthe 1/4 depleted square-lattice Heisenberg model relevant tothe material CaV 3O7 by Kontani et al .

25 This can be viewedas a part of the more general phenomena of quantum orderby disorder, 9 where quantum uctuations help select and sta-bilize an appropriate type of order, typically collinear, in face

of near classical degeneracy.

C. Triplet excitation spectrum

Figure 7 shows the triplet excitation spectra (k x ,k y) for J 1 / J 2 0,0.3,0.6 obtained from the Ising expansions aboutNeel order, where we can see the energy gap vanishes at( , ) and its equivalent point.

We now present the spectra calculated from the columnardimer expansions. We have previously noted that the leadingorder columnar dimer expansions give the minimum of thetriplet excitation spectrum close to the classical orderingwave vector. Figure 8 shows the triplet excitation spectra

(k x ,k y) for the columnar dimerized J 1- J 2 model at x

FIG. 5. The ground-state energy per site, E 0 / N ( J 1 J 2), asfunction of J 1 /( J 1 J 2), obtained from the Ising expansions aboutthe Neel order i.e., q ) A , the Ising expansions about theclassical spiral order i.e., q dened by Eq. 2 B , the Isingexpansions using the q obtained by dimer expansions C , the co-lumnar dimer expansions D , and the diagonal dimer expansionE . The result at J 2 0 is the exact results for the ground-state

energy of a antiferromagnetic spin chain Ref. 28 .

FIG. 6. The order parameter M vs J 1 / J 2 . The solid circles witherror bars are the estimates from the Ising expansions about Ne elorder i.e., q ) , while the open circles are the estimates fromIsing expansion about the classical spiral order i.e., q dened byEq. 2 .



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0.5 and coupling ratios J 1 / J 2 0.6,0.8,0.9,1,1.1,1.25. Itcan be seen that the minimum gap is located at (0, ) for J 1 / J 2 0.7, or (0,0) for J 1 / J 2 1.25. For J 1 / J 2 over theregion 0.81.1, the minimum is close to the classical order-ing wave vector.

Extrapolating the columnar dimer series to x 1 by theintegrated differential approximants, one can get the tripletexcitation spectra for the fairly narrow parameter range 0.7

J 1 / J 2 0.9, and the results for J 1 / J 2 0.75 and 0.85 areshown in Fig. 9. We nd that the minimum gap is nonzerofor these couplings. The minimum triplet gap as a function of J 1 / J 2 is given in Fig. 10. We nd that up to J 1 / J 2 0.75, theminimum gap is located at (0, ), and for J 1 / J 2 0.75, the

system is in an incommensurate phase, with the location of minimum gap given in Fig. 2.

The excitation spectra for the regime of small J 2 / J 1 willbe discussed in a later section.

D. Phase transitions and critical exponents

In this section we describe various estimations of thephase boundaries and the calculations of critical exponents

associated with the transitions. We note that the estimates of critical exponents are not particularly accurate. The uncer-tainties shown are a measure of the spread among the Pade

FIG. 7. Plot of triplet excitation spectra (k x ,k y) derived fromthe Ising expansions about Neel order along high-symmetry cutsthrough the Brillouin zone for coupling ratios J 1 / J 2 0,0.3,0.6.

FIG. 8. Plot of triplet excitation spectrum (k x ,k y) for the co-lumnar dimerized J 1- J 2 model at x 0.5 for coupling ratios J 1 / J 2

0.6,0.8,0.9,1,1.1,1.25. The large circles indicate the locations of classical ordering wave vectors.

FIG. 9. Plot of triplet excitation spectrum (k x ,k y) derivedfrom the columnar dimer expansions for the coupling ratios J 1 / J 2

0.75,0.85 and x 1. The large solid circles indicate the locationsof classical ordering wave vector.

FIG. 10. The minimum triplet excitation gap vs J 1 / J 2 derivedfrom the columnar dimer expansions. The solid circles indicate theminimum gap is located at (0, ) .



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approximants. Given the short length of the series, they arebest regarded as effective exponents.

First, combining the results of the Ising and the columnardimer expansions, we estimate that Ne el order disappears at J 1 / J 2 0.68(3). The phase adjacent to that appears to bespontaneously dimerized in the columnar pattern. This phaseis analogous to the middle spontaneously dimerized phase in J nn - J nn n square-lattice model.

4 Our results are consistentwith the transition from the Neel to the dimerized phase be-ing a second-order transition, since, near the critical point,the ground-state energy from the columnar dimer expansionmatches very smoothly those from the Ising expansions. Fora rst-order transition, there should be a discontinuity in theslope at the transition. Similarly, we estimate the transitionpoint between spontaneously dimerized phase and the spiralphase to be at J 1 / J 2 0.91(3).

Phase boundaries for the columnar dimerized Hamiltoniancan be more fully studied by the columnar dimer expansions.By applying the D log Pade approximants to the triplet seriesat the wave vector of the minimum gap, we can locate thecritical points at which the gap closes. The results are pre-sented in Fig. 11, where in our analysis we have assumedthat the minimum gap is located at the classical ordering

wave vector for J 1 / J 2 in the range 0.81.2. The associatedcritical exponent is about 0.8(1), with not much variationwith J 1 / J 2 . It would be interesting to calculate the spectralweight for the triplet excitations to see if they vanish, sug-gesting the appearance of spinon excitations. 14 These resultsare not accurate enough to decide if the transitions belong tothe O 3 nonlinear model universality class ( 0.70)Ref. 26 or the O 4 exponents associated with the SO 3

nonlinear sigma model ( 0.74). 15,27 For J 1 / J 2 in the re-gion 0.70.9, we nd critical points xc larger than unity.This is evidence that this phase is spontaneously dimerized.

For diagonal dimer expansions, we can also get the phaseboundary by using the D log Pade approximants to locate thecritical points where the gap vanishes. The results are shown

in Fig. 12. The associated critical exponent is found to beapproximately 0.75 10 , and also does not vary much with J 2 / J 1 . For J 2 / J 1 1.75 the minimum of the energy gap isclearly ( , ), which is the classical ordering wave vector.

E. Weakly coupled zig-zag chains

In this section we discuss the excitation spectra for small J 2 / J 1 , when the system can be thought of as weakly coupledspin chains. The coupling between the chains is frustrated,

and has been called zigzag coupling in the literature.For small values of J 2 / J 1 , the critical points in the diag-onal dimer expansions lie very close to unity, and it is verydifcult to locate the wave vector of the gap minimum at x

1. Figure 13 gives the dispersion at x 1 for the case J 20 i.e., decoupled spin chains . Here we get a small but

nonzero minimum gap, due to the fact that we have not takeninto account the singular behavior of the spin chain. This canbe done by extrapolating the series in a new variable

1 1 x 2/3, 15

in which case the gap does vanish. For J 2 nonzero, we do notknow if x 1 is gapped or critical, and hence do not knowwhich extrapolation variable to use. For simplicity, we dis-cuss primarily an extrapolation in the original variable,which assumes that there is no singularity at x 1. Note,however, that even with this extrapolation we do get a atdispersion between (0,0) and (0, /2), as expected. As weincrease J 2 / J 1 up to J 2 / J 1 0.2), we still get a very atdispersion over the region (0,0) and (0, /2). The gap overthis region is also very small similar to that for J 2 0), andin fact it is zero within error bars if we extrapolate the seriesusing the variable . As an example, the dispersion is shownin Fig. 14 for J 2 / J 1 0.2 extrapolated assuming no criticalpoints for x 1). At larger values of J 2 / J 1 , for example for J 2 / J 1 0.5, the dispersion is more pronounced and is shownin Fig. 15. We can see that the position of the minimum gap

FIG. 11. The critical point xc as a function of J 1 /( J 1 J 2) asobtained from D log Pade approximants to the minimum energy gapof columnar dimer expansions. The locations of minimum gap are(0, ) for J 1 / J 2 0.6, or near classical ordering wave vector(q ,q ) for 0.95 J 1 / J 2 1.2, or 0,0 for J 1 / J 2 1.25.

FIG. 12. The critical point xc as a function of J 1 /( J 1 J 2) asobtained from D log Pade approximants to the minimum energy gapof diagonal dimer expansions. The locations of minimum gap are(0,0) for J 2 / J 1 0.75, or near classical ordering wave vector( 2q ,q ) for 0.75 J 2 / J 1 1.75, or ( , ) for J 2 / J 1 1.75.



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is not located exactly at the classical ordering wave vector,but deviates from it along the direction connecting (0, /2)and ( , ). This location of the minimum gap is plotted inFig. 2.

The wave vector q obtained from both columnar dimerexpansions and diagonal dimer expansions can be tted verywell by the simple form

qarccos 3 J 2 / J 1 2 /2 , 0.75 J 1 / J 2 1,

arccos 2 3 J 1 / J 2 21

, J 1 / J 2 1, 16

which is shown by a dashed lined in Fig. 2.Using the above results for q, we can obtain more reliable

results for the ground-state energy by the Ising expansion,and the results are shown in Fig. 5. The results for the mag-netization M obtained using this wave vector agree withinerror bars with the results obtained using the classical wave

vector, shown in Fig. 6 and, hence, are not shown.

IV. CONCLUSIONS

In this paper we have studied a class of Heisenberg mod-els, which interpolate between the square-lattice, triangular-lattice, and linear-chain models. As discussed in the Intro-duction, these models should describe the magneticproperties of the insulating phase of a range of layered or-ganic superconductors. Furthermore, using uniaxial stress itmay be possible to vary the parameter J 1 / J 2 and inducesome of the quantum phase transitions considered in this

paper. We nd that the model has a particularly rich phasediagram: Ne el, spiral, dimerized, and possible critical phasesoccur. Except, perhaps, for the region of weakly coupledchains, the phase diagram is rather well determined by ourseries expansions, and it would be difcult for any othernu-merical method to have comparable accuracy. Some of themost notable features are that the two-sublattice Ne el phaseappears stable beyond its classical regime of stability, and inthe spiral phase, the ordering wave vector is different fromthe classical ordering wave vector except for the case of thetriangular-lattice Heisenberg model. The region of weaklycoupled chains is nearly critical. In studying the instability of the dimerized phase to the spiral phase, the critical exponentsfor various dimer expansions give values of in the range

FIG. 14. Plot of triplet excitation spectrum (k x ,k y) along se-lected contours in the Brillouin zone derived from the diagonaldimer expansions for coupling ratios J 2 / J 1 0.2. The large solidcircle indicates the position of the classical ordering wave vectorand q x ( 2q ) / q 0.12 for J 2 / J 1 0.2.

FIG. 15. Plot of triplet excitation spectrum (k x ,k y) along se-lected contours in the Brillouin zone derived from the diagonal

dimer expansions for coupling ratios J 2 / J 1 0.5. The large solidpoint indicates the position of classical ordering wave vector, whilethe large open square indicates the position of minimum gap. Thesolid open points in the rightmost windows are spectra along con-tours connecting (0,0) and the classical ordering wave vector theposition of minimum gap by a straight line up to k y .

FIG. 13. Plot of triplet excitation spectrum (k x ,k y) along se-lected contours the in Brillouin zone derived from the diagonal

dimer expansions for coupling ratios J 2 / J 1 0. The large solidcircle indicates the position of the classical ordering wave vectorand q x ( 2q ) / q 0 for J 2 / J 1 0.



9/9

0.65 0.9 which is consistent with either O (4) or O (3 )universality class. This, as well as the possibility of spinonexcitations in the quantum disordered phases of this model,deserves further attention.

ACKNOWLEDGMENTS

This work is supported in part by a grant from the Na-

tional Science Foundation DMR-9616574 R.R.P.S. , the

Gordon Godfrey Bequest for Theoretical Physics at the Uni-versity of NSW, and by the Australian Research CouncilZ.W. and R.M. . The computation has been performed on

Silicon Graphics Power Challenge and Convex machines.We thank the New South Wales Center for Parallel Comput-ing for facilities and assistance with the calculations. Wethank J. Merino, J. Oitmaa, C.J. Hamer, and S. Sachdev forhelpful discussions. We thank C.J. Hamer and J. Oitmaa for

providing some of the computer codes we used.

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