Effect of anisotropy on the ground-state magnetic ordering of the spin-half quantum $J_1^{X

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

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8E?ect of anisotropy on the ground-state magnetic ordering of the

spin-half quantum J XXZ 1–J XXZ 2model on the square lattice

R.F.Bishop,1,2P.H.Y.Li,1,2R.Darradi,3J.Schulenburg,4and J.Richter 3

1School of Physics and Astronomy,Schuster Building,The University of Manchester,Manchester,M139PL,UK 2School of Physics and Astronomy,University of Minnesota,116Church Street SE,Minneapolis,Minnesota 55455,USA 3Institut f¨u r Theoretische Physik,Universit¨a t Magdeburg,P.O.Box 4120,39016Magdeburg,Germany 4Universit¨a tsrechenzentrum,Universit¨a t Magdeburg,P.O.Box 4120,39016Magdeburg,Germany

1

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

Abstract

We study the zero-temperature phase diagram of the2D quantum J XXZ

1–J XXZ

2

spin-1/2

anisotropic Heisenberg model on the square lattice.In particular,the e?ects of the anisotropy?on the z-aligned N´e el and(collinear)stripe states,as well as on the xy-planar-aligned N´e el and collinear stripe states,are examined.All four of these quasiclassical states are chosen in turn as model states on top of which we systematically include the quantum correlations using a coupled cluster method analysis carried out to very high orders.We?nd strong evidence for two quantum triple points (QTP’s)at(?c=?0.10±0.15,J c2/J1=0.505±0.015)and(?c=2.05±0.15,J c2/J1=0.530±0.015), between which an intermediate magnetically-disordered phase emerges to separate the quasiclassi-cal N´e el and stripe collinear phases.Above the upper QTP(? 2.0)we?nd a direct?rst-order phase transition between the N´e el and stripe phases,exactly as for the classical case.The z-aligned and xy-planar-aligned phases meet precisely at?=1,also as for the classical case.For all values of the anisotropy parameter between those of the two QTP’s there exists a narrow range of values of J2/J1,αc1(?)<J2/J1<αc2(?),centered near the point of maximum classical frustration, J2/J1=1

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

I.INTRODUCTION

The exchange interactions that lead to collective magnetic behavior are clearly of purely quantum-mechanical origin.Nevertheless,the underlying quantum nature has often safely been ignored in describing,at least at the qualitative level,many magnetic phenomena of interest in the past.On the other hand,the investigation of magnetic systems and magnetic phenomena where the intrinsically quantal e?ects play a dominant role,and hence have to be accounted for in detail,has evolved in recent years to become a burgeoning area at the forefront of condensed matter theory.Thus,the investigation of quantum magnets and their phase transitions,both quantum and thermal,has developed into an extremely active area of research.

From the experimental viewpoint major impetus has come both from the discovery of high-temperature superconductors and,since then,from the ever-increasing ability of mate-rials scientists to fabricate a by now bewildering array of novel magnetic systems of reduced dimensionality,which display interesting quantum phenomena.1While high-temperature superconductivity has raised the question of the link between the mechanism of supercon-ductivity in the cuprates,for example,and spin?uctuations and magnetic order in one-dimensional(1D)and two-dimensional(2D)spin-half antiferromagnets,the new magnetic materials exhibit a wealth of new quantum phenomena of enormous interest in their own right.

For example,in1D systems,the universal paradigm of Tomonaga-Luttinger liquid2,3 behavior has occupied a key position of interest,since Fermi liquid theory breaks down in1D.More generally,in all restricted geometries the interplay between reduced dimen-sionality,competing interactions and strong quantum?uctuations,generates a plethora of new states of condensed matter beyond the usual states of quasiclassical long-range order (LRO).Thus,for high-temperature superconductivity,for example,it is suggested4that quantum spin?uctuation and frustration due to doping could lead to the collapse of the 2D N´e el-ordered antiferromagnetic phase present at zero doping,and that this could be the clue for the superconducting behavior.This,and many similar experimental observations for other magnetic materials of reduced dimensionality,has intensi?ed the study of order-disorder quantum phase transitions.Thus,low-dimensional quantum antiferromagnets have attracted much recent attention as model systems in which strong quantum?uctuations

3

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

might be able to destroy magnetic LRO in the ground state(GS).In the present paper we consider a system of N→∞spin-1/2particles on a spatially isotropic2D square lattice.

The spin-1/2Heisenberg antiferromagnet with only nearest-neighbor(NN)bonds,all of equal strength,exhibits magnetic LRO at zero temperature on such bipartite lattices as the square lattice considered here.A key mechanism that can then destroy the LRO for such systems,with a given lattice and spins of a given spin quantum number s,is the introduction of competing or frustrating bonds on top of the NN bonds.The interested reader is referred to Refs.[1,5]for a more detailed discussion of2D spin systems in general.

An archetypal model of the above type that has attracted much theoretical attention in recent years(see,e.g.,Refs.[6,7,8,9,10,11,12,13,14,15,16,17,18,19])is the2D spin-1/2J1–J2 model on a square lattice with both NN and next-nearest-neighbor(NNN)antiferromagnetic interactions,with strength J1>0and J2>0respectively.The NN bonds J1>0promote N´e el antiferromagnetic order,while the NNN bonds J2>0act to frustrate or compete with this order.All such frustrated quantum magnets continue to be of great theoretical interest because of the possible spin-liquid and other such novel magnetically disordered phases that they can exhibit(and see,e.g.,Ref.[20]).The recent syntheses of magnetic materials that can be well described by the spin-1/2J1–J2model on the2D square lattice,such as the undoped precursors to the high-temperature superconducting cuprates for small J2/J1 values,VOMoO4for intermediate J2/J1values,21and Li2VOSiO4for large J2/J1values,22,23 has fuelled further theoretical interest in the model.

The properties of the spin-1/2J1–J2model on the2D square lattice are well understood in the limits when J2=0or J1=0.For the case when J2=0,and the classical GS is perfectly N´e el-ordered,the quantum?uctuations are not su?ciently strong enough to destroy the N´e el LRO,although the staggered magnetization is reduced to about61%of its classical value.Indeed,the best estimates for this order parameter are61.4±0.1% from quantum Monte Carlo studies,2463.5%from exact diagonalizations of small clusters,25 61.4±0.2%from series expansions,2661.5±0.5%from the coupled cluster method(CCM) employed here,27,28,29and61.4%from third-order spin-wave theory.30Clearly,they all agree remarkably well in this J2=0limit.The opposite limit of large J2is a classic example8of the phenomenon of order by disorder.31,32Thus,in the case where J1→0with J2=0and?xed, the two sublattices each order antiferromagnetically at the classical level,but in directions which are independent of each other.This degeneracy is lifted by quantum?uctuations and

4

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

the GS becomes magnetically ordered collinearly as a stripe phase consisting of successive alternating rows(or columns)of parallel spins.

For intermediate values of J2/J1it is now widely accepted that the quantum spin-1/2 J1–J2model on the2D square lattice has a ground-state(gs)phase diagram showing the above two phases with quasiclassical LRO(viz.,a N´e el-ordered(π,π)phase at smaller values of J2/J1,and a collinear stripe-ordered phase of the columnar(π,0)or row(0,π) type at larger values of J2/J1),separated by an intermediate quantum paramagnetic phase without magnetic LRO in the parameter regimeαc1<J2/J1<αc2,whereαc1≈0.4and αc2≈0.6.The precise nature of the intermediate magnetically-disordered phase is still not fully resolved.Suggested candidates include a homogeneous spin-liquid state of various types with no broken symmetry(see,e.g.,Ref.[19]),or a valence-bond solid(VBS)phase with some broken symmetry.Possible spin-liquid states include a resonating-valence-bond (RVB)state proposed by Anderson,4which has been supported more recently by variational quantum Monte Carlo studies.14Other studies7,33,34,35,36have supported a spontaneously dimerized state for the intermediate phase with both translational and rotational symmetry broken,and thus representing a columnar VBS phase.Yet other studies13,37have supported instead a plaquette VBS state for the intermediate phase,with translational symmetry broken but with rotational symmetry preserved.

There has also been considerable discussion in recent years as to whether the quan-tum phase transition between the quasiclassical N´e el phase and the magnetically disordered (intermediate paramagnetic)phase in the spin-1/2J1–J2model on the2D square lattice is?rst-order or of continuous second-order type.A particularly intriguing suggestion by Senthil et al.38is that there is a second-order phase transition in the model between the N´e el state and the intermediate disordered state(which these authors argue is a VBS state), which is not described by a Ginzburg-Landau-type critical theory,but is rather described in terms of a decon?ned quantum critical point.Such direct second-order quantum phase transitions between two states with di?erent broken symmetries,and which are hence char-acterized by two seemingly independent order parameters,are di?cult to understand within the standard critical theory approach of Ginzburg and Landau,as we indicate below.

Thus,the competition between two such distinct kinds of quantum order associated with di?erent broken symmetries would lead generically in the Ginzburg-Landau scenario to one of only three possibilities:(i)a?rst-order transition between the two states,(ii)an intermediate

5

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

region of co-existence between both phases with both kinds of order present,or(iii)a region of intermediate phase with neither of the orders of these two phases present.A direct second-order transition between states of di?erent broken symmetries is only permissible within the standard Ginzburg-Landau critical theory if it arises by an accidental?ne-tuning of the disparate order parameters to a multicritical point.Thus,for the spin-1/2J1–J2 model on the2D square lattice and its quantum phase transition suggested by Senthil et al.,38it would require the completely accidental coincidence(or near coincidence)of the point where the magnetic order parameter(i.e.,the staggered magnetization)vanishes for the N´e el phase with the point where the dimer order parameter vanishes for the VBS phase. Since each of these phases has a di?erent broken symmetry(viz.,spin-rotation symmetry for the N´e el phase and the lattice symmetry for the VBS phase),one would naively expect that each transition is described by its own independent order parameter(i.e.,the staggered magnetization for the N´e el phase and the dimer order parameter for the VBS phase)and that the two transitions should hence be mutually independent.

By contrast,the“decon?ned”type of quantum phase transition postulated by Senthil et al.38permits direct second-order quantum phase transitions between such states with di?erent forms of broken symmetry.In their scenario the quantum critical points still sep-arate phases characterized by order parameters of the conventional(i.e.,in their language,“con?ning”)kind,but their proposed new critical theory involves fractional degrees of free-dom(viz.,spinons for the spin-1/2J1–J2model on the2D square lattice)that interact via an emergent gauge?eld.For our speci?c example the order parameters of both the N´e el and VBS phases discussed above are represented in terms of the spinons,which themselves become“decon?ned”exactly at the critical point.The postulate that the spinons are the fundamental constituents of both order parameters then a?ords a natural explanation for the direct second-order phase transition between two states of the system that otherwise seem very di?erent on the basis of their broken symmetries.

We note,however,that the decon?ned phase transition theory of Senthil et al.38is still the subject of controversy.Other authors believe that the phase transition in the spin-1/2 J1–J2model on the2D square lattice from the Ne´e l phase to the intermediate magnetically-disordered phase need not be due to a decon?nement of spinons.For example,Sirker et al.36have argued on the basis of both spin-wave theory and numerical results from series expansion analyses,that this transition is more likely to be a(weakly)?rst-order transition

6

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

between the Ne´e l phase and a VBS phase with columnar dimerization.Other authors have also proposed other,perhaps less radical,mechanisms to explain such second-order phase transitions(if they exist)and their seeming disagreement(except by accidental?ne tuning) with Ginzburg-Landau theory.What seems clearly to be a minimal requirement is that the order parameters of the two phases with di?erent broken symmetry should be related in some way.Thus,a Ginzburg-Landau-type theory can only be preserved if it contains additional terms in the e?ective theory that represent interactions between the two order parameters.For example,just such an e?ective theory has been proposed for the2D spin-1/2J1–J2model on the square lattice by Sushkov et al.,39and further discussed by Sirker et al.36

From the classical viewpoint frustrated models often exhibit“accidental”degeneracy,and the degree of such degeneracy,which can vary enormously,has become widely viewed as a measure of the frustration.Among the e?ects that can act to lift any such degeneracy are thermal?uctuations,quantum?uctuations,and such“perturbations”as spin-orbit interac-tions,spin-lattice couplings,further neglected exchange terms,and impurities,all of which might be present in actual materials.In the present paper we focus particular attention on the role of quantum?uctuations.From the quantum viewpoint such frustrated quantum magnets as the spin-1/2J1–J2model on the2D square lattice often have ground states that are macroscopically degenerate.This feature leads naturally to an increased sensitivity of the underlying Hamiltonian to the presence of small perturbations.In particular,the pres-ence in real systems that are well characterised by the J1–J2model,of anisotropies,either in spin space or in real space,naturally raises the issue of how robust are the properties of the model against any such perturbations.

Combining the above two viewpoints,it is clear that it is of particular interest in the study of frustrated quantum magnets to focus special attention on the mechanisms or parameters that are available to us to“tune”or vary the quantum?uctuations that play such a key role in determining their gs phase structures.Apart from changing the spin quantum number or the dimensionality and lattice type of the system,or tuning the relative strengths of the competing exchange interactions,another key mechanism is the introduction of anisotropy into the existing exchange bonds.Such anisotropy can be either in real space40,41,42,43,44,45 or in spin space.46,47,48,49

In order to investigate the e?ect in real space an interesting generalization of the pure

7

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

J1–J2model has been introduced recently by Nersesyan and Tsvelik40and further studied by other groups including ourselves.41,42,43,44,45This generalization,the so-called J1–J′1–J2 model,introduces a spatial anisotropy into the2D J1–J2model on the square lattice by allowing the NN bonds to have di?erent strengths J1and J′1in the two orthogonal spatial lattice dimensions,while keeping all of the NNN bonds across the diagonals to have the same strength J2.In previous work of our own44,45on this J1–J′1–J2model we studied the e?ect of the coupling J′1on the quasiclassical N´e el-ordered and stripe-ordered phases for both the spin-1/2and spin-1cases.For the spin-1/2case,44we found the surprising and novel result that there exists a quantum triple point below which there is a second-order phase transition between the quasiclassical N´e el and columnar stripe-ordered phases with magnetic LRO,whereas only above this point are these two phases separated by the intermediate magnetically disordered phase seen in the pure spin-1/2J1–J2model on the2D square lattice(i.e.,at J′1=J1).We found that the quantum critical points for both of the quasiclassical phases with magnetic LRO increase as the coupling ratio J′1/J1is increased, and an intermediate phase with no magnetic LRO emerges only when J′1/J1 0.6,with strong indications of a quantum triple point at J′1/J1=0.60±0.03,J2/J1=0.33±0.02.For J′1/J1=1,the results agree with the previously known results of the J1–J2model described above.

In the present paper we generalize the spin-1/2J1–J2model on the2D square lattice in a di?erent direction by allowing the bonds to become anisotropic in spin space rather than in real space.Such spin anisotropy is relevant experimentally as well as theoretically,since it is likely to be present,if only weakly,in any real material.Furthermore,the intermediate magnetically-disordered phase is likely to be particularly sensitive to any tuning of the quantum?uctuations,as we have seen above in the case of spatial anisotropy.Indeed,other evidence indicates that the intermediate phase might even disappear altogether in certain situations,such as increasing the dimensionality or the spin quantum number.

Thus,for example,the in?uence of frustration and quantum?uctuations on the magnetic ordering in the GS of the spin-1/2J1–J2model on the body-centered cubic(bcc)lattice has been studied using exact diagonalization of small lattices and linear spin-wave theory,50and also by using linked-cluster series expansions.51Contrary to the results for the corresponding model on the square lattice,it was found for the bcc lattice that frustration and quantum ?uctuations do not lead to a quantum disordered phase for strong frustration.Rather,the

8

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

results of all approaches suggest a?rst-order quantum phase transition at a value J2/J1≈0.70from the quasiclassical N´e el phase at low J2to a quasiclassical collinear phase at large J2.Similarly,the intermediate phase can also disappear when the spin quantum number s is increased for the J1–J2model on the2D square lattice.Thus,we45found no evidence for a magnetically disordered state(for larger values of J2/J1)for the s=1case,by contrast with the s=1/2)case.44Instead,we found a quantum tricritical point in the s=1case of the J1–J′1–J2model on the2D square lattice at J′1/J1=0.66±0.03,J2/J1=0.35±0.02, where a line of second-order phase transitions between the quasiclassical N´e el and columar stripe-ordered phases(for J′1/J1 0.66)meets a line of?rst-order phase transitions between the same two phases(for J′1/J1 0.66).

As in our previous work44,45involving the e?ect of spatial anisotropy on the spin-1/2 and spin-1J1–J2models on the2D square lattice,we again employ the coupled cluster method(CCM)to investigate now the e?ect on the same model of spin anisotropy.The CCM is one of the most powerful techniques in microscopic quantum many-body theory.52,53 It has been applied successfully to many quantum magnets.27,54,55,56,57,58,59It is capable of calculating with high accuracy the ground-and excited-state properties of spin systems.In particular,it is an e?ective tool for studying highly frustrated quantum magnets,where such other numerical methods as the quantum Monte Carlo method and the exact diagonalization method are often severely limited in practice,e.g.,by the“minus-sign problem”and the very small sizes of the spin systems that can be handled in practice with available computing resources,respectively.

II.THE MODEL

The usual2D spin-1/2J1–J2model is an isotropic Heisenberg model on a square lattice with two kinds of exchange bonds,with strength J1for the NN bonds along both the row and the column directions,and with strength J2for the NNN bonds along the diagonals,as shown in Fig.1(a).Here we generalize the model by including an anisotropy in spin space in both the NN and NNN bonds.We are aware of only a very few earlier investigations with a similar goal.46,47,48The two most detailed have studied the extreme limits where either the frustrating NNN interaction becomes anisotropic but the NN interaction remains isotropic46(viz.,the J1–J XXZ

model)and the opposite case where the NN interaction be-

2

9

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

(a)(b)(c)(d) (e)

FIG.1:(a)The J XXZ

1–J XXZ

2

Heisenberg model;—J1;---J2;(b)and(c)z-aligned states for

the N´e el and stripe columnar phases respectively;(d)and(e)planar x-aligned states for the N´e el and stripe columnar phases respectively.Arrows in(b),(c),(d)and(e)represent spins situated on the sites of the square lattice[symbolized by?in(a)].

comes anisotropic but the NNN interaction remains isotropic47(viz.,the J XXZ

1

–J2model). In real materials one might expect both exchange interactions to become anisotropic.To

our knowledge the only study of this case48(viz.,the J XXZ

1–J XXZ

2

model)has been done

using the rather crude tool of linear spin-wave theory(LSWT),from which it is notori-ously di?cult to draw any?rm quantitative conclusions about the positions of the gs phase boundaries of a system.It is equally di?cult to use LSWT to predict with con?dence either the number of phases present in the gs phase diagram or the nature of the quantum phase transitions between them.We comment further on the application of spin-wave theory to the J1–J2model and its generalizations in Sec.V.The aim of the present paper is to use the

CCM,as a much more accurate many-body tool,to investigate the spin-1/2J XXZ

1–J XXZ

2

model on the2D square lattice.

In order to keep the size of the parameter space manageable the anisotropy parameter?

is assumed to be the same in both exchange terms,thus yielding the so-called J XXZ

1–J XXZ

2

model,whose Hamiltonian is described by

H=J1 i,j (s x i s x j+s y i s y j+?s z i s z j)

+J2 i,k (s x i s x k+s y i s y k+?s z i s z k),(1) where the sums over i,j and i,k run over all NN and NNN pairs respectively,counting each bond once and once only.We are interested only in the case of competing antiferro-magnetic bonds,J1>0and J2>0,and henceforth,for all of the results shown in Sec.

10

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

IV,we set J1=1.Similarly,we shall be interested essentially only in the region?>0 (although for reasons discussed below in Sec.IV we shall show results also for small negative values of?).

This model has two types of classical antiferromagnetic ground states,namely a z-aligned state for?>1and an xy-planar-aligned state for0<?<1.Since all directions in the xy-plane in spin space are equivalent,we may choose the direction arbitrarily to be the x-direction,say.Both of these z-aligned and x-aligned states further divide into a N´e el(π,π) state and stripe states(columnar stripe(π,0)and row stripe(0,π)),the spin orientations of which are shown in Figs.1(b,c,d,e)accordingly.There is clearly a symmetry under the interchange of rows and columns,which implies that we need only consider the columnar stripe states.The(?rst-order)classical phase transition occurs at J c2=1

J1,and the columnar stripe states being the classical

2

GS for J2>1

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

In terms of the set{|Φ ;C+

},the CCM employs the exponential parametrization

I

|Ψ =e S|Φ ,S= I=0S I C+I(5a) for the exact gs ket energy eigenstate.Its counterpart for the exact gs bra energy eigenstate is chosen as

?Ψ|= Φ|?S e S,?S=1+ I=0?S I C?I.(5b)

It is important to note that while the parametrizations of Eqs.(5a)and(5b)are not man-ifestly Hermitian-conjugate,they do preserve the important Hellmann-Feynman theorem at all levels of approximation(viz.,when the complete set of many-particle con?gurations{I} is truncated).53Furthermore the amplitudes(S I,?S I)form canonically conjugate pairs in a time-dependent version of the CCM,by contrast with the pairs(S I,S?I),coming from a manifestly Hermitian-conjugate representation for ?Ψ|=( Φ|e S?e S|Φ)?1 Φ|e S?,that are not canonically conjugate to one another.53

The static gs CCM correlation operators,S and?S,contain the real c-number correlation coe?cients,S I and?S I,that need to be calculated.Clearly,once the coe?cients{S I,?S I} are known,all other gs properties of the many-body system can be derived from them.To ?nd the gs correlation coe?cients we simply insert the parametrizations of Eqs.(5a,b)into

and the Schr¨o dinger equations(4a,b)and project onto the complete sets of states Φ|C?

I

C+

|Φ ,http://www.77cn.com.cnpletely equivalently,we may simply demand that the gs energy I

expectation value,¯H≡ ?Ψ|H|Ψ ,is minimized with respect to the entire set{S I,?S I}.In either case we are easily led to the equations

e?S H e S|Φ =0;?I=0,(6a)

Φ|C?

I

]e S|Φ =0;?I=0,(6b)

Φ|?S e?S[H,C+

I

which we then solve for the set{S I,?S I}.Equation(6a)also shows that the gs energy at the stationary point has the simple form

E=E({S I})= Φ|e?S H e S|Φ .(7) It is important to realize that this(bi-)variational formulation does not necessarily lead to an upper bound for E when the summations for S and?S in Eqs.(5a,b)are truncated,

12

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

due to the lack of manifest Hermiticity when such approximations are made.Nonetheless, one can prove53that the important Hellmann-Feynman theorem is preserved in all such approximations.

We note that Eq.(6a)represents a coupled set of non-linear multinomial equations for the c-number correlation coe?cients{S I}.The nested commutator expansion of the similarity-transformed Hamiltonian,

1

e?S H e S=H+[H,S]+

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

generalized multi-con?gurational creation operators C+

I

are simple products of single spin-

raising operators,C+

I →s+

k1

s+

k2

···s+

k m

,where s±

k

≡s x k±is y

k

,and(s x k,s y

k

,s z k)are the usual

SU(2)spin operators on lattice site k.For the quasiclassical magnetically-ordered states that we calculate here,the order parameter is the sublattice magnetization,M,which is given within our local spin coordinates de?ned above as

M≡?

1

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

TABLE I:Numbers of fundamental con?gurations(?f.c.)retained in the CCM LSUB n approxi-

mation for the z-aligned states and the planar x-aligned states of the s=1/2J XXZ

1–J XXZ

2

model.

z-aligned states planar x-aligned states

Scheme?f.c.?f.c.

N´e el stripe N´e el stripe

LSUB21112

LSUB4791018

LSUB675106131252

LSUB81287192227935532

LSUB10296054582574206148127

The?nal step in any CCM calculation is then to extrapolate the approximate LSUB n

results to the exact,n→∞,limit.Although no fundamental theory is known on how

the LSUB n data for such physical quantities as the gs energy per spin,E/N,and the gs

staggered magnetization,M,scale with n in the n→∞,limit,we have a great deal of

experience in doing so from previous calculations.27,28,44,45,54,55,58,59,61Thus,we employ here

the same well-tested LSUB n scaling laws as we have used,for example,for the J1–J′1–J2

model,44,45namely

E/N=a0+a1n?2+a2n?4(10) for the gs energy per spin,and

M=b0+n?0.5 b1+b2n?1 (11) for the gs staggered magnetization,both of which have been successfully used previously

for systems showing an order-disorder quantum phase transition.An alternative leading

power-law extrapolation scheme for the order parameter,

M=c0+c1n?c2,(12)

15

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

has also been successfully used previously to determine the phase transition points.For most systems with order-disorder transitions the two extrapolation schemes of Eqs.(11) and(12)give remarkably similar results almost everywhere,as demonstrated explicitly,for example,for the case of quasi-one-dimensional quantum Heisenberg antiferromagnets with a weak interchain coupling.61However,in regions very near quantum triple points the form of Eq.(11)is more robust than that of Eq.(12)due to the addition of the next-to-leading correction term,as has been explained in detail elsewhere.44Hence,in this work we use the extrapolation schemes of Eqs.(10)and(11).

Obviously,better results are obtained from the LSUB n extrapolation schemes if the data with the lowest n values are not used in the?ts.However,a robust and stable?t to any?tting formula with m unknown parameters is generally only obtained by using at least(m+1) data points.In particular,a?t to only m data points should be avoided whenever possible. In our case both?tting schemes in Eqs.(10)and(11)have m=3unknown parameters to be determined.For all four model states we have LSUB n data with n={2,4,6,8,10}, and it is clear that the optimal?ts should be obtained using the sets n={4,6,8,10}. All the extrapolated results that we present below in Sec.IV are obtained in precisely this way.However,we have also extrapolated E/N and M using the sets n={2,4,6,8,10}and n={4,6,8}.In almost all cases they lead to very similar results,which adds credence to the stability of our numerical results and to the validity of our conclusions presented below.

IV.RESULTS

Figure2shows the extrapolated results for the gs energy per spin as a function of J2(with J1=1)for various values of?,for the z-aligned and planar x-aligned model states.For each model state,two sets of curves are shown,one(for smaller values of J2)using the N´e el state, and the other(for larger values of J2)using the stripe state.As we have discussed in detail elsewhere,53,54,57the coupled sets of LSUB n equations(6a)have natural termination points (at least for values n>2)for some critical value of a control parameter(here the anisotropy,?),beyond which no real solutions to the equations exist.The extrapolation of such LSUB n termination points for?xed values of?to the n→∞limit can sometimes be used as a method to calculate the physical phase boundary for the phase with ordering described by the CCM model state being used.However,since other methods exist to de?ne the phase

16

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

?1

?0.9?0.8?0.7?0.6

?0.5 0.35

0.4

0.45

0.5

0.55 0.6

0.65

0.7

E /N

J 2

N éel stripe

?=1.0?=1.4?=1.8?=2.2?=2.5

E max point : N éel

stripe (a)z -aligned states ?0.6

?0.55?0.5?0.45?0.4

?0.35?0.3 0.35

0.4

0.45

0.5

0.55 0.6

0.65

0.7

E /N

J 2

N éel stripe

?=?0.2?=0.2?=0.6?=1.0

E max point : N éel

stripe (b)planar x -aligned states

FIG.2:Extrapolated CCM LSUB n results using the z -aligned and planar x -aligned states for the

gs energy per spin,E/N ,for the N´e el and stripe phases of the s =1/2J XXZ 1–J XXZ 2

model.The LSUB n results are extrapolated in the limit n →∞using the sets n ={4,6,8,10}for both the z -aligned states and the planar x -aligned states.The NN exchange coupling J 1=1.The meaning of the E max points shown is described in the text.

transition points,which are usually more precise and more robust for extrapolation (as we discuss below),we have not attempted such an analysis here.

Instead,in Fig.2,the E max points shown,for each set of calculations based on one of the four CCM model states used,are either those natural termination points described above for the highest (LSUB10)level of approximation we have implemented,or the points where the gs energy becomes a maximum should the latter occur ?rst (i.e.,as one approaches the termination point).The advantage of this usage of the E max points is that we do not then display gs energy data in any appreciable regimes where LSUB n calculations with very large values of n (higher than can feasibly be implemented)would not have solutions,by dint of having terminated already.

Curves such as those shown in Fig.2(a)illustrate very clearly that the corresponding pairs of gs energy curves for the z -aligned N´e el and stripe phases cross one another for all values of ?above some critical value,? 2.1.The crossings occur with a clear discontinuity in slope,as is completely characteristic of a ?rst-order phase transition,exactly as observed in the classical (i.e.,s →∞)case.Furthermore,the direct ?rst-order phase transition between the z -aligned N´e el and stripe phases that is thereby indicated for all values of ? 2.1,occurs (for all such values of ?)very close to the classical phase boundary J 2=

1

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0

0.2

0.4

0.6

0.8

1

M

J 2

N éel

stripe

?=1.0?=1.2?=1.5?=1.8?=2.0?=2.2

(a)z -aligned states 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0

0.2

0.4

0.6

0.8

1

M

J 2

N éel stripe

?=?0.2?=0.0?=0.2?=0.4?=0.8?=1.0

(b)planar x -aligned states

FIG.3:Extrapolated CCM LSUB n results using the z -aligned and planar x -aligned states for the

gs staggered magnetization,M ,for the N´e el and stripe phases of the s =1/2J XXZ 1–J XXZ 2

model.The LSUB n results are extrapolated in the limit n →∞using the sets n ={4,6,8,10}for both the z -aligned states and the planar x -aligned states.The NN exchange coupling J 1=1.

of maximum (classical)frustration.Conversely,curves such as those shown in Fig.2(a)for values of ?in the range 1<? 2.1also illustrate clearly that the corresponding pairs of gs energy curves for the z -aligned N´e el and stripe phases do not intersect one another.In this regime we thus have clear preliminary evidence for the opening up of an intermediate phase between the N´e el and stripe phases.The corresponding curves in Fig.2(b)for values of ?<1tell a similar story,with an intermediate phase similarly indicated to exist between the xy -planar-aligned N´e el and stripe phases for values of ?in the range ?0.1 ?<1.We show in Fig.3corresponding indicative sets of CCM results,based on the same four model states,for the gs order parameter (viz.,the staggered magnetization),to those shown in Fig.2for the gs energy.The staggered magnetization data completely reinforce the phase structure of the model as deduced above from the gs energy data.Thus,let us now denote by M c the quantum phase transition point deduced from curves such as those shown in Fig.3,where M c is de?ned to be either (a)the point where corresponding pairs of CCM staggered magnetization curves (for the same value of ?),based on the N´e el and stripe model states,intersect one another if they do so at a physical value M ≥0;or (b)if they do not so intersect at a value M ≥0,the two points where the corresponding values of the staggered magnetization go to zero.

Clearly,case (a)here corresponds to a direct phase transition between the N´e el and stripe

18

We study the zero-temperature phase diagram of the 2D quantum $J_{1}^{XXZ}$--$J_{2}^{XXZ}$ spin-1/2 anisotropic Heisenberg model on the square lattice. In particular, the effects of the anisotropy $\Delta$ on the $z$-aligned N\'{e}el and (collinear) stripe

phases,which will generally be?rst-order if the intersection point has a value M=0(and, exceptionally,second-order,if the crossing occurs exactly at M=0).On the other hand, case(b)corresponds to the situation where the points where the LRO vanishes for both quasiclassical(i.e.,N´e el-ordered and stripe-ordered)phases are,at least naively,indicative of a second-order phase transition from each of these phases to some unknown intermedi-ate magnetically-disordered phase.We return to a discussion of the actual order of such transitions in Sec.V.In summary,we hence de?ne the staggered magnetization criterion for a quantum critical point as the point where there is an indication of a phase transi-tion between the two states by their order parameters becoming equal,or where the order parameter vanishes,whichever occurs?rst.A detailed discussion of this order parameter criterion and its relation to the stricter energy crossing criterion may be found elsewhere.59 From curves such as those shown in Fig.3(a)we see that for? 1.95for the z-aligned states,there exists an intermediate region between the critical points at which M→0for the N´e el and stripe phases.Conversely,for? 1.95the two curves for the order parameters M of the quantum N´e el and stripe phases for the same value of?meet at a?nite value, M>0,as is typical of a?rst-order transition.Similarly,Fig.3(b)shows that for the planar x-aligned states,there exists an intermediate region between the critical points at which M→0for the N´e el and stripe phases for all values of?in the range?0.15 ?<1. Again,the two curves for the order parameters M of the N´e el and stripe phases for the same value of?intersect at a value M>0for? ?0.15.In order to show more explicitly how the quantum phase transitions are driven by anisotropy,?,we display the same data for the extrapolated results for the order parameter,M,somewhat di?erently in Fig.4,where we plot M as a function of?for various values of J2around the value J2=0.5,corresponding to the point of maximum(classical)frustration.

By putting together data of the sort shown in Figs.2,3,and4we are able to deduce

the gs phase diagram of our2D spin-1/2J XXZ

1–J XXZ

2

model on the square lattice,from our

CCM calculations based on the four model states with quasiclassical antiferromagnetic LRO (viz.,the N´e el and stripe states for both the z-aligned and planar xy-aligned cases).We show in Fig.5the zero-temperature gs phase diagram,as deduced from the order parameter criterion,and using our extrapolated LSUB n data sets with n={4,6,8,10},shown as the critical value J c2for the NNN exchange coupling J2as a function of anisotropy?(with NN exchange coupling strength J1=1).Very similar results are obtained from using the energy

19

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