the coupled cluster method applied to quantum magnetism

7 The Coupled Cluster MethodApplied to Quantum Magnetism

Damian J.J. Farnell! and Raymond F. Bishop2

1 Unit of Ophthalmology, Department of Medicine, University ClinicalDepartments, Daulby Street, University of LiverpooL Liverpool 1,69 3GA,United Kingdom, d. farnelHlliverpool. ac. uk

2 Department of Physics, University of Manchester Institute of Science andTechnology (UlvlIST), P.O. Box 88, l\lanchester M60 lQD. United Kingdom,[email protected]

Abstract. The Coupled Cluster Method (CC\I) is one of the most powerful anduniversally applied techniques of quanturn many-body theory. In particular, it hasbeen used extensively in order to investigate many types of lattice quantum spin system at zero temperature. The ground- and excited-state properties of these systernsmay now be determined routinely to great accuracy. In this Chapter we present anoverview of the CCM formalism and we describe how the CC1\1 is applied in detaiL \Ve illustrate the power and versatility of the method by presenting results forfour diH'erent spin models. These are, namely, the XXZ model, a Heisenberg modelwith bonds of differing strengths on the square lattice. a model which interpolates between the Kagome- and triangular-lattice antiferromagnets. and a frustratedferrimagnetic spin system on the square lattice. vVe consider the ground-state properties of all of these systems and we present accurate results for the excitationenergies of the spin-half square-lattice XXZ model. vVe utilise an "extcnded" SUB2approximation scheme. and we demonstrate how this approximation Illay be solvedexactly by using Fourier transform methods or, alternatively, by determining andsolving the SUB2-m problem. \Ve also present the rcsults of "localised" approximation schemes called the LSUBm or SUBm-m schemes. \Ve note t hat we mustutilise computational techniques in order to solvc these localised approximationschemes to "high order" vVe show that we are able to determine the positions ofquantum phase transitions with much accuracy, and we demonstrate that we areable to determine their quantum criticality by using the CClvi in conjunction withthe coherent anomaly method (CAM). Also. we illustrate that the CCM lnay beused in order to determine the "nodal surfaces" of lattice quantum spin systems,Finally. we show how connections to cumulant series expansions ma:-' be made bydetermining the perturbation series of a spin-half triangular-lattice antiferromagnetusing the CCM at various levels of LSUBm approximation,

7.1 Introduction

Key experimental observations in fields such as supcrfluidity, superconductivity, nuclear structure, quantum chemistry, quantum magnetism and strOllgly correlated electronic systems have often implied that the strong quantumcorrelations inherent in these systems should be fully included, at least conceptually, in any theoretical calculations that aim fully to describe their basic

D.J.J. Farnc~ll and fl.P. T1ishop, The Coupled Cluster I\1f't.hud Applied to Quant HIll I\:1agrlPtisIll.

Lee!. Noll" PLys, 645. :l07 :l'l~ (20()4)http://www.springerlink.com/© Springer-Verlag BC'rlill H~idelbprg 2001

308 D.J.J. Farnell and H..F. Bishop

properties. Until fairly recently a COlllmon problem in many of these fields hasbeen that the "conceptual school" of quantum many-body theory (Ql\IBT)has been rat.her divorced from the "quantitative school" of fully microscopicQl\JBT. In this context the cOllceptual school typically simplifies the original.fundamental theoretical model to a more tractable one. This is done eitherby replacing the original Hamiltonian with a simpler or effective one thatstill implies or includes the most important of the observed f~;atures, and/orby postulating that these key features can be captured via an (approximate)wave function with specific inbuilt correlations. The BCS state for supercondm:tors is a typical exarnp!e of the latter. By contrast. the quantitative schoolattempts to solve' t he original quantum many-body problem as accurately aspossible. Nowadays the boulldaries between the two schools are becoming increasingly blurred for several import ant reasons. Thus. on one hand. many ofthe most interesting problerns, such as high- temperature superconductivity.are so difficult that neither school can present convincing solutions. On theother hand. the techniques now available in the field of ab initio Ql\IBT havebecome increasingly refined over the last 15 years or so. and have also becomemore acces::;iblc to a wider group of rc::;earchcrs who can utilise t.he experienceand expertise built up in other fit'ld::; of application.

.4 b initio techniques of microscopic QJ\lBT are. at their bc::;t. designedto include the import.ant t'ffects of quantum correlations in an unbiased and::;ystematic manner. In particular. over the last decade or so. some of theQl\IBT too]::; that have proveu to ])(' versatile in describing H'ry accuratelya wide range of both finite and ext ended systems of interest in physics andchemistry. and which are defined in continuous space. have begun to be applied to quantum lattice sy.stem.s. They arc now beginning to provide unifiedtreatmenh of such systems. which can not only compete, for example, withother much more computationally inten::;ive :otochastic simulations, but canalso provide an almo:ot unique Illeans to ::;tudy in a sy::;tematic and unbia.sedmanner the phy::;ically interesting (zero-temperature) quantum phast' transition::; that many .such quantum lattice systems display in abundant varie(y.At the same time the conceptual school of QI\lBT can often provide a goodstarting point for the quantitative school. as we shall see in mon' detail below. in t he form of "Illodel" reference .states that become the .starting orzeroth-order approximations OlJ top of which further many-body corrdatiOll::;can be sy::;tematically included within well-defined hierarchical approximationschemes.

}'oremost among the most versatile technique:o in the modern arsenal ofQMBT arc tho::;e such as quantum Monte Carlo (Q"lC) methods [1-4] thecorrelated basis function (CBF) method [5 15] and the coupled clu::;(er method (CCM) [16-24], on the la::;t of which we concentrate in this Chapter.The latter two method::; are undoubtedly the most powerful and most univer::;ally applicable of all fully microscopic techniques presently available forob initio calculations in QMBT. Each of the above methods ha::; its own particular strengths and weaknes:oes, as we discuss in more detail below. Before

7 The Coupled Cluster ?vlethod Applied to Quantum Magnetism 309

doing so, hmvever, we first give a short overview of tl1(~ CBF method since, forreasons discussed more fully below, we shall concentrate our main attentionhereafter on the CCM.

The most common, and perhaps the simplest, of the variational methodsin QJVIBT are based on trial wave functions of the (Bijl-Dingle-)Jastrowform [25]. Early calculations of this sort relied on various cluster expansions of the ensuing approximate matrix elements [25--27]. It was realisedlater that these variational approaches may also be formulated diagrammatically [28]. This feature has been of considerable help in the construction ofsuch powerful approximations as the Pen~us-Yevick and hypernetted chain(HNC) summations and their variants, which have their origins in the classical theory of liquids and which have been adapted for both bosonic andfermionic systems [7.29]. The review article by Clark [8] gives a good overview of the variational theory sketched above as applied to f'xtended nuclearmatter. The interested reader is also referred to [30].

Two basic flaws mar the above variational approaches. Firstly, the particular partial summations of the graphs considered by such approximations asthe HNC approach destroy one of the most attractive features of variationaltechniques, namely that they yield upper bounds to the exact ground-stateenergy. Secondly. even a complete summation of graphs (or a variationalI\Ionte Carlo evaluation of the corresponding expectation values) for a giventrial wave function (of Jastrow type, for example) gives only the exact variational result and not the true ground state. This latter deficiency may beremedied by the inclusion of more general state-dependent correlations andhigher-order correlation functions of the Feenberg type. Alternatively, andmore generally, one may extend the Jastrow wave function to a complete setof correlated basis functions, which is the CBF approach.

The CBF method was introduced some 45 years ago by Feenberg andhis collaborators [5-7], and was later developed largely by Clark and hiscollaborators [8-11]. Introductory surveys of the method are given in [12 -15].vVe sirnply note here that the CBF method has as its central ingredient thedirect incorporation of the most important interparticle correlations into tbeapproximate wave functions on which the microscopic description is based. Atits simplest level the method only involves a single configuration, and hencereduces to ordinary variational theory. This further reduces to Jastrow theoryif the simplest reasonable choice of correlation operator is made in terms ofthe usual symmetric product over all pairs in the system of state-independenttwo-body correlation functions.

Since we shall be dealing extensively with applications of the Cc:rvI inthis Chapter, we postpone a comparable introduction of it until Sec. 1.2, andbefore doing so we return to a review of the relative merits and weaknessesof the QI\IC, CBF and CCI\I approaches to QMBT. We first note that QJ\JCtechniques are severely restricted in the choice of problems to which theycan readily be applied by the infamous "sign problem" [31,32], which ariseswhenever we have a lack of prior knowledge of the nodal surface of the many-

:no D..LJ. Farnell and R.F. Bishop

body wave function under discussion. For spin systems on a regular lattice itis often rdat.ed to t.he occurrence of (strong) frustration. Conversely, it canonly readily be circumvented wllPn we have such prior knowledge via. forexample. the 1\larshall-Peierls sign rulp [::\:3]. or some such analogous relation.Nevert.heless. Ql\IC numerical results for spin-latticp systems often provide1he benchmark for other methods for the cases in which the technique canbe applied, especially for lattices in two or more spatial dimensions. \Ve not.ein passing thnt for the special case of one spatial dimension (i.e .. chains) thenwt hods of choin> nsnally indndC' exact solntions whC'n availablC' [34 37], thedensity matrix H'llonlwlisation gronp (Di\IRG) method [:}8]. and tC'chniquC'sfrom quant llm fiC'ld !JlC'ory [;j9].

By contrast. the CBF method is not limited in the range of systemsto which it can lw applied by t hC' prespnce of stl'Ong (geometric or dyrwmic) frustration. However. it s applications up till now have been rC'stricted inpractice to a very limited nnmhcr of spin-latticC' systems (namely, thC' transverse Ising model [40/U] amI the Xl' model [44]. \Ve note that part of thereason for this limited usage of the method for problems in quantum magnetism lies in thC' faet that. in practical calcnlations. it. is often difficnlt toinclnde correlations beyond thC' two-body level in the .Jastrow-Feenlwrg trialstates. Snch higher-order correlations are often ilnportant for very accnratecalculat ions.

By further cont rast. the CC'\1 is limited neither by the presence of frnstriltion in the system nor to the inclusion of only two-body correlations. As weshall see latC'r. t he inclusion of many-body correlations bet,veen spins up toabout the 8-body level or so is nowadays quite routine. It is important to notethat thC' Goldstone linkpd-cluster tllPorpm is pxplicitly obeyed by the CCl\[at any level of approximate impiPmentaticm. and hence result;; may alway;;be deU'rmined directly from t he outset in the illfinit e-latticC' limit. IV ---+ x(w]1('re IV is the 1l111nber of ;;pins in thC' ;;ystem). This i;; in sharp contrastto the Ql\IC H'sults that are always obtained for finite-sized lat! ices. fromwhich the re;;ults for the illfinite latticC' need to 1)(' C'xtrapolatpd using finitesize scaling argulllC'nt;;. Furthermore. the very importallt Hellmann-FC'ynlllanthC'orem is also obeyed by the C(';\1 at all lpvels of approximation. On tlle'other hanel. WC' note that in order to rC'tain all of these llseful and importantfeat urC's. it t urns out to be ll('cC';;;;arv to relax the conditioll tbat the corresponding bra- and ket-st ates are lllauifpstly Hermitian conjugates of oneanother. At a given level of tnmcatiOlI. this HC'nniticity property may bC'only approximately obC'yed. although it i;; certainly restored in thC' exact limit. As we shall see. a consequence is that we lose t he property in the CCi\1that the re;;ults for the ground-state energy form au upper bound to thetrue results. In practice this lack of manifest. Hermiticity poses few actualproblems. Indeed. it can oft.ell be used as all interllal quality check on t hC'accuracy of the met hod. Finally. we note that the CC1\I lends it;;elf extremelywell for applicat ions 011 the lattice to t he use of com]mter-algebraic tedmiques both to derive and to solve the fundamental sets of coupled nonlinear

7 The Coupled Cluster Method Applied to Quantum Magnetism 311

equations that lie at its heart in practical implementations, via well-definedhierarchies of approximations.

In the rest of this Chapter we will focus attention only on spin-latticeapplications of the CCM, for reasons already cited. Nevertheless, we believethat the CBF method still has a worthy future in this field. We hope thatothers will still develop it further, since it certainly shares many highly desirable features with the CCM. Before concentrating in the rest of this Chaptersolely on the CCM, we take a final opportunity to list some of the moreimportant of these features below:

• Both methods are extremely versatile, and they have been extensivelytested. There is by now a large amount of experience in using them.

• An impressively wide range of applications to systems of physical interest has been made of one and/or the other method. These include finitenuclei; nuclear matter; quantum field theory (including systems of anharmonic oscillators, ¢4 field theory, and pion-nucleon field theory); atomsand molecules of interest in quantum chemistry; the electron gas; quantum hydrodynamics; and the liquids helium (including bulk :IHe and 4Heand their mixtures, and films).

• Both methods are capable of very high accuracy at attainable levels ofimplementation. In most applications the CBF and/or CCM results areeither the best or among the best from all available microscopic techniques. They arc now often at the point of being fully competitive with thelarge-scale QMC simulations in the cases where the latter can be perforrned.

• Neither method is restricted in principle to particular forms of the Hamiltonian. Both are easily capable of handling very complicated interactions.

• Both the CBF method and the CC]'vI are intrinsically nonperturbativein nature. Some correlations are retained to infinite order, even at thelowest levels of implementation. The CC]'vI, in particular, can often be usedto derive (or reconstruct) perturbation theory (PT) series, by a suitablechoice of truncation hierarchy for the subsets of terms retained in themulticonfigurational expansions of the ground- or excited-state correlationoperators, as described more fully below. In such cases, the CCM providesa natural analytic continuation of the PT series, which in practice isusually found to be valid far outside the radius of convergence of the PTseries, and also t.o be quantitatively superior t.o such alternative schemesas (generalised) Pade resummations.

• Although nonperturbative in principle, the CC]\I can be easily relatedto the Goldstone diagram expansions of time-independent perturbationtheory. This feature facilitates comparisons with other methods.

• Similarly, at the optimised Jastrow level implemented via the HNC approximation, the CBF method has been shown [45] to be equivalent tot.wo-body localised parquet theory, and hence to a sum of planar Feynmandiagrams of time-dependent perturbation theory.

:)12 D.,1.,J. Farnell and R.F. Bishop

• In both methods Olle may work from the outset in the bulk thermodynamic limit. N --+x. thereby avoiding problems connected with finite-sizeeffects. This is always done in the CC?\L although, for technical reasons,it is not always practicable in the CBF' case.

• Both methods have the virtue of great flexibility. One may choose "Ullcorrelated" or ;'model" ground-state reference states. for example, in manyways. In part iculal'. this presents em opportunity for the "conceptualschool" of lllall~'-body tlwory to provide a good starting-point for the"quantitative school.·· Similarly. many different approximation hierarchiesfor j he correlation operators of t he CBF and CC::\ r schemes can be envisaged. and there is again room for external experience or physical intuitionto be utilised in their choice.

• Both methods arc capable of handling phase transitions. Even when tlw"uncorrelated" or "lllodel" n{erence state is a poor dlOi("(~, both the CBFand CC\J sdwnws have been shown in particular cases to be able topredict phase changes. In t he case of the CC1\1 we discuss t his in lllOl"('detail below.

• Both methods, bnt particularly the CC\,l, often have the practical capability of implementation to high orders of approximation. The CCl\1 hasespecially be,'n shown to be very amenable to the use of computer algebra to derive the high-order basic coupled sets of nonlinear equations thatunderpin it. This feature is particularly marked for lattice systems. and itis a key reason why the CC::\l is now proving to be fully competitive' withlarge-scale Q?\1C stochastic simulations at a fraction of the computingcost. in those cases where t Ill' latter can he performed.

For further details of tlw eBF method and some of its applicatioIls tovarious quantum lattice systems. the interested reader is referred to the overview in [46], where comparisons arc also made with the eCl\!.

Henceforth we confine our attention to the eCl\!. whose applications overthe last ten or so years to quantum magnetic systems at zero temperature [47 -68] have proven to be extremely successfnl. In particular. the useof computer-algebraic implementations of the CC}\,I f()r quantum systems oflarge or infinite numbers of particles has largely been pioneered with respect to t hes{' spin-lattice problems. \Ve note too in this context that therehave 1wen subsequent applications of these highly accurate computationaleG!\l techniques to other types oflattice quantum systems, such as U(l) andSU(N) lattice gauge field theory [69-71], and the latticised O(N) nonlinearsigma model of relevance to chiral meson field theory [72].

In the remainder of this Chapter we firstly give a brief description of theeC\1 formalism. \Ve then describe four specific applications of the methodto various spin-lattice systems at zero temlwraturp. The first applicatioIl isto the unfrustrated spin-half XXZ model (or anisotropic Heisenberg model)OIl the liIlear chain and on the bipartite square lattice. This simple llloddserves both to illustrate how the method may be applied in practice aIld


to indicate the quality of the result,; attainable at practical levels of implementation. By contrast to this simple model, frustrated systems generallyare both more difficult to deal with and have richer phase diagrams, whichcontain phases of novel forms of order. Three such strongly frustrated systems are then considered. The first of these, the so-called J-J' model, is aspin-half Heisenberg model on the square lattice with two diff(~rent, competing nearest- neighbour couplings with different bond strengths arranged ina regular zigzag pattern. For the case where the bond strengths have different signs the square plaquettes are thus dynamically frustrated, wlwn~as

when the bond strengths have the same sign the rnode! exhibits competition(between magnetic order and dimerisation) without frustratioll. The thirdmodel exhibits geometric frustration, and is again a spin-half Heisenberg model that interpolates smoothly between a triangular lattice and a Kagomelattice. The last model considered is another model that includes the possibility of dynamical frustration, in which we have both nearest-neighbour andnext- nearest-neighbour Heisenberg interactions with unequal strengths. Furthermore, the rnodel is taken to represent a spin-halfjspin-one ferrimagnet inwhich one sublattice of the bipartite square lattice is populated entirely withspin-one spins, while the other sublattice is populated entirely with spin-halfspins. The Chapter is concluded with a discussion of the implications of theseillustrative results for further work, and with some ideas for future extensionsand applications of the CCl\l.

7.2 The CCl'vI Formalism

A brief description of the normal coupled cluster method (NCCM) formalismis now provided, although the interested reader is referred to [16-24,117-68].for further details. The exact ket and bra ground-state energy eigenvectors,1\[/) and (PI, of a general many-body system described by a Hamiltonian H,

are parametrised within the single-reference CCM as follows:

(7.1)

,,- ~S·(~+k_ - ~ 1 ---'I ~.

ITO

S' = 1 + LSIC,lico

(7.2)

The single model or reference state I<P) is required to have the property ofbeing a cyclic vector with respect to two well-defined Abelian subalgebrasof multi-configurational creation operators {CJ} and their Hermitian-adjointdestruction counterparts {C;- == (C;) t}. Thus, I<P) plays the role of a vacuumstate with respect to a suitable set of (mutually commuting) many-body creation operators {CJ}. Note that C I-I4» = 0, V I oj O. and that CiJ- == 1, the

314 D.J.J. Farnell and RF. Bishop

identity operator. These operators are furthermore complete in the manybody Hilbert (or Fock) space. Also, the corTe/a.tion operatoT 5 is decomposedcntirely in terms of these creation operators {ct}, which. when acting onthe model state ({ ctl<P)}). create excitations from it. We note that althoughthe manifest Hermiticity. ((Pi! .= il[t) / (WIW)). is lost. the normalisation conditions (Plw) = ((piw) = (<PI<p) == 1 are explicitly imposed. The corTela.t.ioncoefficients {Sf' Sf} are regarded as being independent variables. and the fullset {S], 151 } t Ims provides a complete description of the ground state. For instance. an arbitrary operator it will have a ground-state expectation valuegiven as.

\Ne note that the exponentiated form of the ground-state CC.!\I parametrisation of (7.2) ensures the correct counting of the independent and excitedcorrelated many-body clusters with respect to i<P) which are present in theexact ground state IW). It also eusures t he exact incorporation of the Goldstone linked-cluster theorem. which itself guarantees the size-extensivity ofall relevant extensive physical quantities. \Ne also note that any operator ina similarity transform lnay be \vritten as

(701)

The determination of the correlation coefficients {Sf, SI } is achieved bytaking appropriate projections onto the ground-state Schrbdinger equationsof (7.1). Equivalently. they may be determined variationally by requiringthe ground-state energy expectation functional H({S],Sf}), defirwd as in(7.:3). to be stationary with respect to variations in each of the (independent)variables of the full set. \Ve thereby easily derive the following coupled set ofequations.

5H /615] = 0 =? (<PiCie-sHesi<P) = O. V J f 0 : (7.G)

5lf /651 = D =? \<PISe-s[H. CiJesl<P) = D. V [ Ie 0 . (Hi)

Equation (7.5) also shows that the ground-state energy at the stationarypoint has the simple form

(7.7)

It is important to realize that this (bi- )variational formulation does not leadto an upper bound for E q when the summations for 5 and ,5' in (7.2) aretruncated, due to the lack of exact Herrniticity when such approximationsare made. However, one ca.n prove that the important Hellmann-Feynmantheorem i8 preserved in all such approximations.

7 The Coupled Cluster l\lethod Applied to Quantum Magnetism :315

vVe note that (7.5) represents a coupled set of non-linear multinomialequations for the c-number correlatioIl coefficients {Sr}. The nested commutator expansion of the similarity-transformed Hamiltonian

(7.8)

and the fact that all of the individual components of S in the sum in (7.2)commute with one another, together imply that each element of S' in (7.2) islinked directly to the Hamiltonian in each of the terms in (7.8). Thus, eachof the coupled equations (7.5) is of linked cluster type. Furthermore, each ofthese equations is of finite length when expanded. since the otherwise infinite series of (7.8) will always terminate at a finite order, provided only (asis usually the case) that each term in the second-quantised form of the Hamiltonian. H, contains a finite number of single-body destruction operators.defined with respect to the reference (vacuum) state 1<1». Hence. the CC1\1parametrisation naturally leads to a workable scheme which can lw efficientlyimplemented computationally. It is important to note that at the heart of theCC1\1 lies a similarity transformation, in contrast with the unitary transformation in a standard variational formulation in which the bra state (tP 1 issimply taken as the explicit Hermitian conjugate of Ip).

In the case of spin-lattice problems of the type considered here, the operators ct become products of spin-raising operators 8: over a set of sites{k:}, with respect to a model state 1<1» in which all spins points "downward"in some suitably chosen local spin axes. The CCl\1 formalism is exact in theIlmit of inclusion of all possible such multi-spin cluster correlations for SandS'. although in any real application this is usually impossible to achieve. It istherefore necessary to utilise various approximation schemes within Sand S.The three most commonly employed schemes previously utilised have been:(1) the SUBn scheme. in which all correlations involving only n or fewer spinsare retained, but no further restriction is made concerning their spatial separatioIl on the lattice; (2) the SUBn-m sub-approximation, in vvhich all SUBncorrelatiolls spanning a range of llO more than In adjacent lattice sites areretained; and (3) the localised LSUBm scheme. in which all multi-spin COITelations over all distinct locales on the lattice defined by m or fewer contiguoussites are retained.

An excited-state wave function, Ipc), is determined by linearly applyingall excitation operator X' to the kct-state wave function of (7.2). such that

Ip,) = Xc (;81<1» . (7.9)

This equation may now be used to determine the low-lying excitation energies, such that the Schrodinger equation, Hlp,) = E,lp,), may be combinedwith its ground-state counterpart of (7.1) to give the result,

(7.10)

:n6 D..J ..J. Farnell and R.F. Bishop

where f( == E" - E q is the excitation energy. By analogy with the ground-stateformalism, the excited-state correlation operator is written as.

X' =c LX;C;fT'0

(7.11 )

where the set {C;} of multi-spill creation operators may differ from thoseused in the ground-state parametrisation in (7.2) if the excited state hasdifferent quantum numbers than the ground state. We note that (7.11) impliesthe overlap relation (<PIlJ.fc ) = O. By applyillg (<PiC{ to (7.10) we find that.

(7.12)

which is a gellcralised set of eigenvalue equations with eigenvalues f, andcorresponding eigcnvectors Xl. for each of the excited states which satisfy(<PllJf,) = o.

\Ve note that lower orders of approximation may be determined analytically and an example of applying the LSUB2 and SUB2 approximations to thespin-half linear chain XXZ model is given later ill order to show clearly howthis is performed. However, it rapidly becomes clear that analytical detennination of the CCI\1 equations for higher orders of approximation is impractical. \Ve therefore employ computer algebraic techniques in order efficientlyto determine and solve the CC1\[ ket- and bra-state equations. A full exposition of this topic is beyond the scope of this chapter, although we notc thatthe problem esselltially becomes one of pattern matchillg in order to determine the CC1\1 ground-state ket equations. The bra-state equatiolls may bedetermined easily thereafter alld the ket - and bra-state equations are readily solved using standard techniques for the solution of coupled polynomialequatiolls (e.g., the Newton-Raphson method). The excited-state eigenvalueequations may be abo determined ill an analogous manner, and. althoughthis is not strictly necessary. we restrict the level of approximation to thesame for the excited state as for t he ground state ill calculations presentedhere. A full exposition of t he details in applying t he CC~I to high ordersof approximation is given for the ground state in [Ei4. 59. 67] and for excitedstates in [62].

)Jo[(' that thc results of SCI3m-m a!HI LSCBm approximation schemesmay be extrapolated to the exact limit. In --+ x. using various "heuristic'approaches. How to do this is not discussed herc. although the interestedreader is referred to [59.67] for more details.

7.3 The XXZ Model

\Ve wish to apply the ceM to the spin-half XXZ model on the linear chainand the square lattice in order to illustrate how one applies the CC]\1 to apractical problem and also to demonstrate the accuracy and power of the


method. vVe note that these systems are unfrustrated and, in global spincoordinates, the XXZ Hamiltonian is specified as follows,

H = ~[Q:rs:r ...L "Y, sY + LisZs Z]L..-t U 1 J I < 1 ] I ]

(i.j)

(7.13)

where the sum on (i, j) counts all nearest-neighbour pairs once. The Neel stateis the ground state in the trivial Ising limit Li --+ 00, and a phase transitionoccurs at (or near to) Li = 1. Indeed, the ground state demonstrates Neellike order in the z-direction for Li > 1 and a similar x-y planar phase for-1 < Li < 1. The system is ferromagnetic for Li < -1.

7.3.1 The CCM Applied to the XXZ ModelUsing a z-Aligned Neel Model State

We turn now to the choice of Ip) and the operators {ct} for the case ofspin-half quantum antiferromagnets on bipartite lattices, in regimes where thecorresponding classical limit is described by a Ned-like order in which all spinson each sublattice are separately aligned in some global spin axes. It is thenconvenient to introduce a different local quantisation axis and different spincoordinates on each sublattice, by a suitable rotation in spin space, so that thecorresponding reference state becomes a fully aligned ("ferromagnetic") state,with all spins pointing along, say, the negative z-axis in the correspondinglocal axes. Such rotations are cannonical tranforrnations that leave the spincommutation relations unchanged. In the same local axes, the configurationindices I --+ {k 1 , k2 ,' . " k M }. a set of site indices, such that ct --+ st

1st .

..stH' where st == sf ± iSk are the usual spin-raising and spin-loweringoperators at site k.

For the Hamiltonian of (7.1:3) we first choose the z-aligned Neel stateas our reference state (which is the exact ground state for Li --+ ex:: , and isexpected to be a good starting point for all Li > 1, down to the expectedphase transition at Li = 1). vVe then perform a rotation of the up-pointingspins by 1800 about the y-axis, such that s:I: --+ _8:r , sy --+ sy, SZ --+ -sz onthis sublattice. The Hamiltonian of (7.13) may thus be written in these localcoordinates as.,

H --~~r,,+,+, '-'-+2A,2Z]- 2 ~ [Li Sj T Si sJ DS, sJ .<u>

(7.14)

The results presented below are based on the SUB2 approximation schemeand the localised LSUBTn scheme, in which we include all rnultispin correlations over all possible distinct locales (or "lattice animals") on the latticedefined by Tn or fewer contiguous sites. \Ve include all fundamental configurations, I --+ {k1 , k2 , ...kn }, with n ::; Tn, which are distinct under the pointand space group symmetries of both the lattice and the Hamiltonian. The

:)18 D..!.J. Farnell and B.F. Bishop

numbers. N p and N[.; . of such fundamental configurations for the groundand excited states. respectively, may he further restricted by the use of additional conservation laws. f(x example. the Hamiltonian of (7. U) COlmnute,.;with the total uniform magnE'tisation. 8 7 = :Lk .0:( where the SUlll on k nmsover alllatticr sites. The ground ,.;tate j,.; known to lie in the = () su!Jsp<H'e.and hence we exclude configurations with an odd nmnber of spins or wit hunequal llllm!Jers of spins on the two equivalent ,.;ublattices. Similarl~! for theexcited state,.;. since we an' only interested in the lowest-]ying excitation. werestrict t he choice of configurations to those with 8 7 = ± 1.

7.3.2 The LSUB2 Approximationfor the Spin-Half, Linear-Chain XXZ J\lodel

\Ve ,.;tart the LSUB2 calculation !J~' speci(ying the commutation relation,.;[8t. st] = ~8~6!./, and . 8~] = 281.6u. We again note that the similaritytransform may be expanded as a series of Iw,.;ted cornrnutators in (7.4). \Vewrite the LSUB2 ket-,.;tate operator in the following simple form for the ,.;pinhalf linear chain model.

.y

Sc:cIJILs,t I . (7.15)

where i nms over all sites on t he linear chain and bi is the sole ket-statecorrelation coefficient. In this approximation we may therefore determinesimilarity transformed version::; of the ::;pin operators. given by

.s; = s;8/ = s? + bdsis~1 + st-Isi)

8/ = 8/'- 2bds?8i~1 + si lSi) - 2bfst'ls;s~1

(7.16)

\Ve note that the otherwi::;e infinite-::;eries of operator::; in the expan::;iolJ of the::;imilarity transform terminates to finite order. \Ve also note that (sn2Ip) =() for any lattice ,.;ite (which is true only for spin-half ::;ystems), and thisis implicitly assumed in the ]a,.;t of (7.16). Clearly we may also vvrite thesimilarity transformed version of t he Hamiltonian as

- 1 LIi = -- [.sT2 I

(i.·.n

(7.17)

\Ve may now substitute the expressions for the spin operators in (7.16) intothe above expression. The ground-state energy is given by

1---{Li + 21J 1 1.4 . J (7.18)


\Ve note that our expression for the ground-state energy is size-extensive(i.e., it scales lim~arly with N). as required by the Goldstone theorem whichis obeyed by the NCCM. Furthermore, this expression terminates to finiteorder, as for the similarity transformed versions of spin operators. Finally, wenote that any other non-trivial choice for S will always yield this expressionfor the ground-state energy. The task is now to find b] and we note that if wecould include all possible spin correlations in S then we" would obtain an exactresult for the ground-state em"rgy. However, this is found to be impossible toachieve for most cases in practice. and we make an approximation (such asthe LSUB2 approximation presented here). The LSUB2 ket-state equation isgiven by

~{b~ + 2L1b1 -- 1 = 0 . (7.19)

which therefore implies that the LSUB2 ground-state energy may be writtenexplicitly in terms of L1 as,

(7.20)

vVe note that this expression gives the correct result in the Ising limit L1 --+ x.\Ve again note that the bra state does not manifestly have to be the Hermitianconjugate of the ket statt~, and we note that the bra-state correlation operatorfor the LSUB2 approximation is given by.

N

5 = 1 + b] L 8j 8j+lJ

(7.21)

where the index j runs over all sites on the linear chain and b] is the solebra-state correlation coefficient in the LSUB2 approximation. In order todetermine the bra-state equation, we now explicitly determine fI I({5/, Sf }).

- N - (1 32)H=--(L1+2b])+Nb] --+L1b1 +-b] .4 2 2

such that LSUB2 bra-state equation is given from DfIUb] = 0 as

1 - -- 2" + L1b] + 3b] b] = 0 .

(7.22)

(7.23)

which gives b] = ~(L12 +3)-]/2. Finally, we note that once the values for thebra- and ket-state correlation coefficients have been determined (at a givenlevel of approxirnation) then we may also obtain the values for expectationvalues, such as the sublattice magnetisation given by

1<1» . (7.24)


The sublattice Inagnetisation is written here in terms of the "rotated" spincoordinates. \Ve note that this is given by

J\hSlll2 = 1- 4h] h]1 2L1,

= ;~ [1 -+ -J:.12+3 J

for the LSlJB2 approximation.

7.3.3 The SUB2 Approximationfor the Spin-Half, Linear-Chain XXZ Model

(7.25)

The SCB2 approximation allows us to include all possible two-spin correlatiellIS in our wave function. \Ve note that the SCB2 ket-state operator is givenby

iIi

S' = 1/2 L L h,S/8i'+,_ (7.26)

and that the index i runs over all sites on the linear chain. Furthermore.the index r runs over all lattice vedors which connect one sublattice to theother and br- is its corresponding SUB2 ket-state correlation coefficient forthis vector. \Ve again determine a similarity transformed version of the spinoperators and we are able to determine the SlJB2 equations, given by

L {(1 + 2L1b[ -+ 2bi)6fJI - 2(L1 -+ 2bJ)b, -+ L br+-scfJb,} = 0 .f> ~

(7.27)

where p runs over all (1D) nearest-m>ighbour lattice vectors. Equation (7.27)may now be solved by employing a sublattice Fourier transform. given by

1'( -) ,ir-Iflq = L(: II' (7.28)

where r again is a lattice vector (i.e .. an odd integer number in ID) whichconnects the different sublattices. This expression has an inverse given by

j 'T[ dqb, = --..;:cos(rq)F(q).

. II "

(7.29)

The SUB2 equations (7.27) and (7.28) therefore lead to an expression forr(q) given by

r(q) = ~rl ± Jl .- k2cm,2(q)] .cos(q) -

where f{ = L1 -+ 2b] and k2 = (I -+ 2L1b[ -+ '2bO /1\-2. (Note that we choose thenegative solution in (7.30) in order to reproduce results in the trivial limit

7 The Coupled Cluster Method Applied to Quantum Magnetism :521

L1 -+ 00.) These equations now yield a self-consistency requirement on thevariable b1 and they may be solved iteratively at a given value of L1. Indeed,we know that all correlation coefficients must tend to zero (namely, for SUB2:br -+ 0, V r) as L1 -+ 00 and we track this solution for large L1 by reducingL1 in small successive steps. 'rYe find that the discriminant in (7.30) becomesnegative at a critical points, L1" ~ 0.3728. Furthermore, the behaviour ofb, changes from exponential to algebraic decay with respect to r at L1".These are strong indications that the CCl'vI critical point is detecting theknown quantum phase transition in the system at L1 = 1. Furthermore, theSUB2 approximation for the ground state may be used in conjunction witha SUBI approximation for the excited state operator X" in (7.11) in orderto determine the excitation spectrum. 'rYe note that the excitation spectrumbecomes soft at the critical point, L1". This is further evidence for a phasetransition and the interested reader is referred to [48] for more details.

\Ye may also solve the SUB2-m equations directly using computationaltechniques. Indeed, we study the limit points of these approximations by usingsolution-tracking software (PITCON), which allows one to solve coupled nonlinear equations. 'rYe again track our solution from the limit L1 -+ JO downto and beyond the limit point and Fig. 7.1 shows our results. In particular,we note that we have two distinct branches, although only the upper branchis a "physical" solution. We again note that the CCM does not necessarilyalways provide an upper bound on the ground-state energy although thisis often the case for the physical solution! By tracking from a point at whichwe are sure of, the solution we guarantee that our solution is valid, and thisapproach is also used for LSUBm approximations.

VVe find that the two branches collapse onto the same line, namely, thatof the full SUB2 solution, as we increase the level of SUB2-m approximationwith respect to m. Indeed, we may plot the positions of the SUB2-m limitpoints against 1/771,2 and we note that these data points are found to be bothhighly linear and they tend to the critical value, L1", for the full SUB2 equations in the limit m -+ JO. Again, we note that the LSUBm and SUBm-mapproximations also show similar branches (namely, one "physical" and one"unphysical" branch) which appear to converge as one increases the magnitude of the truncation index, m. This is a strong indication that our LSUBmand SUBm-m critical points are also reflections of phase transitions in thereal system and that our extrapolated LSCBm and SUBm-m results shouldtend to the exact solution.

7.3.4 CCM Results for the Spin-Half Square-Lattice XXZ ModelUsing a z-Aligned Model State

'rVe shall now illustrate the power and accuracy of the CCJ\I by presentingresults in Figs. 7.27.4 respectively for the energy per spin (E,JiN) and thesublattice magnetisat.ion (JU) for the ground state. and the energy gap (I: c )

;)22 D ..3..1. Farnell and RF. Bishop

2.01.51.0

Ll

"" ."""-

"-,,- """"""""""""-

"0.5

I,I,,

\\,I\,\I\I\\\\

........

---"---r-------·t·--...........,--··-T--,---~---r---·~

SUB2-2 j II H1-- SUB2-8 ! i! -- SUB2-50 111'-' SUB2-100 i II...... SUB2-2000 U10-0 FULL SUB2111- Exact 1i

1

00

·0.3

·05 .

·0.4

·06 -

E INa'"

Fig. 7.1. CC1\l SUB2-rn and full Sl.;B2 results for the ground-state energy of thespin-half linear-chain XXZ model compared to exact results of the Bethe Ansatz[34- 37]. The CC\I model state is the z-aligned !'\eel state. SUB2-m limit pointsconverge to the SUB2 limit at de = 0.3728, at which point the solution to thefull SUB2 equations ],pcOInes complex. as m ~ x. and these are reflections of theinfinite-order phase transition at d = 1 in the ·true· system. Note that the upperbranch of the SUB2-m results are physical and the lower branch is unphysical

of the lowest-lying excited state for the spin-half XXZ model Oll the squarelattice.

vVe find that for all LSliBm approximations with m > 2 111(' physicalbranch of ground-state solutions (i.e .. the one which becomes exact in theLi ---+ x limit) terminates at a critical value Li,., such that for Li < Lie no rcalsolution exists. These LS'CBrn "critical points" are analogous to the S'C132-mlimit points of the previous subsectioll, and they are again taken to he a signalof the phase transition at (or Ileal' to) Li = 1. (Note that the "ullphysical"LSUBm branches. as seen for the SC132-111 approximations above. are notplotted here in order to present a clear illustration of our results. althoughthey certainly exist.)

The SCB2 and LSUBm results usillg the .:-aligned state as Ulodel stateare compared ill Table 7.1 for the isotropic (Li =-co 1) case with results fromlinear spin-wave theory (LSvVT) [74]. series expallsioll techniques [75]. alldquantum J\Ionte Carlo (QlVIC) simulations [76]. Figures 7.2 and 7.4 show thecorresponding results for the ground-state energy and lowest-lying excitat ionenergy f, as functions of Li. Our n~sults for the ground- and excited-stateproperties of the XXZ systems are seen to be in excellent agreeuw]]t withthose results of the best of other approaches. \lVe also not e that values for thespin stiffness of the Heisenberg model (see [77] and later on in this Chapter for

7 The Coupled Clu~ter JVlethod Applied to Quantum JVlagnetism 323

further details) are also found to be very accurate. Furthermore, calculationshave been carried out for this model usin~ an extended version of the CCM inwhich the bra-state correlation operator S is also written in an exponentiatedform, analogous to that of the ket-state operator. The intere~ted reader isreferred to [61] for more details.

7.3.5 CCM Results for the Spin-Half Square-Lattice XXZ ModelUsing a Planar Model State

There is never a unique choice of model state I<p). Ind(,ed, our choice shouldbe guided by any physical insight available to us concerning the system Of

lnore specifically, that particular phase of it which is under consideration. Inthe absence of any other insight into the quantuIll many-body system, we Illaysometimes be guided by the behaviour of the corresponding classical system.The XXZ model under consideration provides just such an illustrative example. Thus, for L1 > 1 the classical Hamiltonian of (7.13) on the 2D squarelattice (and, indeed, on any bipartite lattice) is lninimized by a perfectly antiferromagnetically Neel-ordered st.ate in the z-direction, and we have alreadyutilised this information in the preceding subsections. However, t.he classicalground-state energy is minimized by a Neel-ordered state wit.h spins point.ing

Planar Model State LSUB4Planar Model State LSUB6z-Aligned Model State LSUB4

0-0 z-Aligned Model State LSUB6X Monte Carlo

-0.65

-0.75

-0.60 =r--.---......--r----.---.------,----.---,---------,---,-------,----,--=

0.7 0.8 0.9 1.0

~

1.1 1.2 1.3

Fig. 7.2. CCM LSUBm results using the z-aligned and planar Neel model statesfor the ground-~tateenergy of the spin-half square-lattice XXZ model compared toquantum Monte Carlo re~ults of [73]. Results for the LSUB6 approximation usingboth model states end at their respective critical points

324 D.J ..J. Farnell and RF. Bishop

\\

0.60 \\

M\II

0040 Planar Model State LSUB4Planar Model State LSUB6z-Aligned Model State LSUB4

0200-0 z-Aligned Model State LSUB6

III

0.00 I10 2.0 3.0

~

Fig. 7.3. CCl\1 LSCBrn results using the z-aligned and planar Neel model statesfor the sublattice magnetisation of the spin-half square-lattice XXZ model. Resultsfor the LSUB4 and LSUI36 approximation using both model states end at theirrespective critical points

21.5

"""""""""'" i

............

. ... LSUB4 I- - SUB2 1_ .. LSUB6 l0--0 LSUB8~ Extrapolated GGM

Linear Spin-Wave Theory

/

II

oI-----~'___T_,.(---...;---_.

-1

Fig. 7.4. CClVl LSUBm results using the z-aligned NeeI state as model state forthe lowest-lying excitation energies of the spin-half square-lattice XXZ model


Table 7.1. CCM results [59,62] for the isotropic (LI = 1) spin-half square-latticeHeisenberg antiferromagnet compared to results of other methods. The numbers offundamental configurations in the ground-state and excited-state CeM wave functions for the z-aligned Neel model state are given by Nj and N r" respectively,and the number of fundamental configurations in the ground-state Ce]'vI wave function for the planar Neel model state is given by Nf. Results for the critical pointsof the z-aligned Neel model state are indicated by LI:' and results for the criticalpoints of the planar Neel model state are indicated by L1~. (Note that results for theground-state expectation values for both model states are identical for the isotropicHeisenberg model at L1 = 1)

IMethod IVl

LSUB2 -0.64833 0.841 1.407 1 1 1 - -

SUB2 -0.65083 0.827 1.178 - 0.799 1.204LSUB4 -0.66366 0.765 0.852 7 6 10 0.577 1.648LSUB6 -0.66700 0.727 0.610 75 91 131 0763 1.286LSUB8 -0.66817 0.705 0.473 1273 2011 2793 0.843Extrapolated CCM -0.6697 0.62 0.00 1.03LSWT [74] -0.658 0.606 0.0 - -- 1.0

Series Expansions [75] -0.6693(1) 0.614(2) - - -

QJ\lC [76] -0.669437(5) 0.6140(6)

along any direction in the J:;Y plane, say along the x-axis for -1 < Li < 1.Thus, in order to provide CClvI results in the region ~ 1 < Li < 1, we nowtake this state to be our model state and we shall refer to it as the "planar"model state.

In order to produce another "ferromagnetic" model state for the planarmodel state in the local frames, we rotate the axes of the left-pointing spins(i.e., those pointing in the negative :y-direction) in the planar state by 900

about the ;y-axis, and the axes of the corresponding right-pointing spins by--900 about the y-axis. (Note that the positive z-axis is defined here to pointupwards and the positive x-axis is defined to point right wards. ) Thus, thetransformations of the local axes are described by

(7.:31 )

for the left-pointing spins, and by

(7.32)

for the right-pointing spins. The transformed Hamiltonian of (7.13) may nowbe written in these local axes as

(7.33)

:)26 D..1..1. Fame!! and R.F. Bi;;hop

In thi;; ca;;(' we track the CC]\1 ;;olution for the planar model ;;tate from thepoint L1 = ~ 1. \Ve note that all of the CCI\1 correlation coeflicients are zeroat L1 = -1 b('GHl;;e the model state is an exact ground eigenstate of theHmniltoniclll of (7.:n) at this point. The results for the ground-stat(' energyusing the planar model stat(' arc plotted in Fig. 7.2. and the correspondingre;;ults for the sublattice magnet isation (U. again ddined with respect tothe rotated local spin axe;;) are shown in Fig. 7.:L FurtlJ('rmore. wc not('that the Hamiltonian for the planar model state of (7.:3:3) is idcntical to theHamiltonian for the .:-aliglJed model state of (7.17) at L1 = 1. Indeed, weobtain identical results for the ground-state expectation vahws at L1 = 1.and t hi;; is an excellent test of the validity of our results.

7.3.6 Quantum Criticality of the Antiferromagnetic PhaseThansition for the Spin-Half Square-Lattice XXZ Model

\Ve wish to investigate the qmmtllJn criticality of the pha;;e at (or neat" to)L1 = 1 for the case of t 11(' square lattice. The critical index for the ;;ingular(non-analyt ic) term in E,,) N near an LSljBm critical point L1('( In) can first beobtaincd, for example. by direct cxcunination of the anisotropy susceptibility.Yo == --iP(E'I/N)/i:JL1L. For III > 2 we find.

(7.:34)

Direct calculation for the LSlJI3m approxilnation;; using bot h the z-alignedand planar Neel model ;;tatc;; shows that for m > 2 we have no ~ 1.500 ±0.005. However, the prefactors X::' in (7.34) are themselve;; strongly dependenton the truncation index Til. \Ve may now use a variant of the so-called coherentanomaly method (CAI\I) of Suzuki [7~] to extract further information. Thus.we attempt to fit,,::' with the coherent anomaly form,

x::' ---'t K!L1,(x) -- L1,(m)I" : L1-+ L1,(x) (7.:35 )

where K is a constant. Thus, as explained by Suzuki [7~]. one lllay intuit orprove that the exact ,,"11 (L1) has the critical form,

(7.:~G)

where f" is a cOIl;;lant.A CAl\.[ analysis along thest' lines of the LSUBm re;;ults based on the

z-aligned Neel state gives 1/ ~ 1.25 using the L1~1(4) and L1~1(6) data. and1/ ~ 0.97 u;;ing the L1~1 (6) and L1~1 (8) data. We tIm;; obtain Ii ;;ingular terlllin Eq/N near L1~1 with a critical exponent 2 - 00 + l/ ~ 1.50 --1.75. Thi;; mayb(~ compared with the corresponding value of 3/2 for both the mean-fieldlike CCl'vI SUB2 approximation (in \'I'hich all 2-spin-flip correlation termsare retained, however far apart on the lattice) and linear spin-wavE' theory(LS\VT). A similar treatment for the plallar model state yields a criticalexponent of 2 - 00 + 1/ ~ 1.77. which i;; in good agreement with the resultfor the z-aligned Ned model state.

7 The Coupled Cluster Method Applied to Quantum J'vlagnetism 327

0.015x

• 0 LSUB4

• LSUB60

LSUB8x

0.010 0 SUB20

\fiR x•

0.005 0

0x

• x0 •

I0~0.000

2.0 3.0 4.0 5.0

RFig. 7.5. Re,;ults for the Ising-expansion coefficients, plotted as a function of thelattice distance R, corre;,;pondillg to two-body excitations with respect to the modelstate !()r the spin-half, square lattice Heisenberg model at L1 = 1 obtained via theLSUBrn approximatioIl scheme (with III = {4, G, 8}) and the SUB2 approximatioIl.(Figure taken from [:)2])

7.3.7 CCM Prediction of the Nodal Surface of the Spin-HalfSquare-Lattice Heisenberg Model

\Ve consider an expansion of the ground-state wave function in a completeIsing basis {!I)} (in terms of the local coordinates after rotation). This maybe again written as, liP) = Lt ![rtll), where the sums over 1 goes over all 2N

Ising states, and we find that this expression naturally leads from (7.2) (alsosee [32,60]) to an exact mapping of the CCl\f correlation coefficients {Sr } tothe Ising-expansion coefficients {tJ!{}, which is given by

(7.:37)

It is possible to match the terms in the exponential to the 'target' configuration of Cj in (7.37), and so obtain a numerical value for the {![rr}coefIicients once the CCM bot-state equations have been derived and solvedfor a given value of the anisotropy. Note that we may plot the Ising-expansioncoefIicients as a function of the lattice distance R, corresponding to two-bodyexcitations with respect to the model state, and results are shown in Fig. 7.5.

\Ve observe that all of the coefficients are found to be positive. and thisshows that the exact l\Iarshall-Peierls sign rule is being obeyed for our abinitio calculation. \Ve note that no snch condition is imposed in our CCJ\Itreatment of this model. Indeed, it is also the case that all other four- or

:328 D.Ll. Farnell and H.F. Bishop

higher-body terms have corresponding Ising expansion terms ,vllich are positive. \Ve also note that tlw Ising expansion coefficients appear to cOIlvergerapidly with increasing levels of approximation, and that a strength of theCCl\! is that it may be applied to even very strong!y frustrated system:o whereno ana!ogues of the \Iarshall-Peierb sign rule arc usually known.

\Ve note that it might be possible to nse the CC1\I in order to simnlateaccurately the nodal surface of quantulll problem and this information mightbe fed into a fixed-node QJ\IC calcnlation in order to simulate very accnratclythe propertie:o of this system. Indeed. general rnles might be inferred fromthe CCJ\I data amI, if :00. an exact :oolution. to wit hin QMC :otati:oticallimits.might be determined. The intprested reader is referred to [:n.:)2.GO] for moreinformation.

7.4 The J_JI Model: A Square-Lattice Modelwith Competing Nearest-Neighbour Bonds

\Ve now wish to show that the CCJ\I can treat frustrated system:o as ea:oily a:ounfru:otrated :oystems, and we begin by noting that the JJ' model is a spinhalf Hei:oenberg model on a square lattice with two kinds of nearest-neighbourbonds ,I and J'. as shown in Fig. 7.6.

II = J L Si . 8; + ,I' L Si . 8;.

(I), (ij)2

(7.:38)

The sums over (ij) 1. and (ijh represent sum" over the nearest-neighbourbonds "hown in Fig. 7.6 by da"hed and "olid lines re"pectively. Each square-

~-----.-e:("-l-------11f :2 : l) : 6I I I I, "

~ ~ ¥- ----H,, ",

:t +~---- e(' cb~.--------.1: :2 :J : G

I I I I

/; \ : : : J, :;:t it t

, ~ j 0:3-------~,

.\8

80

Fig. 7.6. IlluHtration of the J J' model of (7.:)8). with two kinds of wgularlycliHtributed nearest-Iwighbour exchange bonds. J (dashed lines) and oF (solid lines)and its classical spiral state (1) > 0) shown f(JI' the ferromagndic case (J < O.J' > jJi/:{). (Fignrc taken fmlll [nJ)


lattice plaquette consists of three J bonds and one J' bonei. A model withsuch a zigzag pattern has been treated by various methods [64,77,79-82]. Forthe cases in which J and J' have different signs (i.e., one bond is ferromagneticwhile the other is antiferromagnetic) the plaquettes are frustrated, whereascompetition without frustration is realized for antiferrornagnetic bonds (J' >o and J > 0).

Using this model we discuss the influence of quantum fluctuations onthe ground-state phase diagram and in particular on the nature of the zerotemperature phase transitions from phases with collinear magnetic order atsmall frustration to phases with noncollinear spiral order at large frustration.The role of quantum fluctuations is examined by comparing ferromagneticspiral and antiferrornagnetic-spiral transitions within the same model. \\Thereas for the classical version of the J J' model both situations can be mappedonto each other, the quantum model behaves differently in the two cases andthis is because of the different nature of the collinear state. The quantumNeel state on two-dimensional lattices exhibits strong quantum fluctuations.For example, as we saw in the previous section, the sublattice magnetisationof the Heisenberg antiferromagnet (HAF) on the square lattice is only about60% of its classical value. By contrast, the ferromagnetic state is the same forthe quantum and the classical model and there are no quantum fiuctuationsin this state.

The classical ground state of this J -J' model is collinear (i.e.. ferromagnetic or antiferromagnetic depending on the sign of J) for the unfrustratedcases. For IJ'I > PI/:) (and J and J' having different signs) the frustrationis large enough in order to force the ground state to be a noncollinear stateof spiral nature with a characteristic pitch angle P = ±Ipcrl given by

{o

IpeI! = 1 1I arccos (-2 i 1 + 1.1' i )

IJ'I < i:~1

\,f'l > U.l- :1(7.39)

Figure 7.6 shows the classical spiral state for the ferromagnetic case (J <0, J' > IJ!/:3) where the spin orientations at A and B lattice sites as numberedon the figure are defined the angle Bn = nPcI. For the anti ferromagnetic case(J > 0, J' < -J/3) all of the spins on one sublattice are reversed. We notethat P = 0 corresponds to the collinear state. The classical transition betweenthe collinear and the noncollinear state is of second order and takes place atthe critical point J;' = -J/3. Figure 7.7 gives an illustration of the completeclassical ground-state phase diagram.

\Ne choose the spiral state with the characteristic angle ep (illustrated inFig. 7.6) as our CCM model state. Further details concerning the treatmentof the J-J' model via the CCM are given in [64,77,81]. \\Tc calculate theground state and the low-lying excitations of the Hamiltonian of (7.:38). Weuse the CCJ\I for high orders of approximation up to LSUB8 which contains4986 fundamental configurations for the Neel model state with <P = 0 and42160 fundamental configurations for a helical model state with 1/ i" O. (\Nc

330 D ..LJ. Farnell and R.F. Bishop

noneoll.spiral

«p>O)

-.+i---·-1r,---r\-.-\+-/3-----+-",. J'

r

\ noneol!.eollll1e r\ferro \ spiral(<p=o) \ ((p>O)

-I \\

Fig. 7.7. Classical ground-state phasc diagram for the J J' model on the squarelattice with competing nearest-neighbour bomb, indicating the collinear N6el andferromagnetic phases and the noncolliuear spiral phases for various values of J andJ'. (Figure taken from [77])

note that such a large number of configurations as the latter case may beconsidered only by using parallel processing techniques. although this is notperformed here. The interested reader is referred to [1:5:3] for more details ofa parallelised implementation of the CC\L) By way of comparison we alsoexact ly diagonalise finitc' sized lattices of up to N = :32 spins with periodicboundary conditions. \Ve extrapolate to the infinite-lattice limit using standard finite-size scaling laws.

For sufficiently strong antiferromagnel ic .J' bonds the J ./' model is characterised by a tendency to singlet pairing of the tvvo spins coupled by a ./'bond. and hence the long-range magnetic (collinear or noncollinear) order isdestroyed. \Ve observe clear indications of a second-order phase transition to aquantmn paramagnetic dimerised phase at a certain erit ical valuE' of J' = J:.However the only case examined in detail here is the antifelTomagnetic case(./ = +1). Evidence of a phase transition to a dimerised phase is indicaU~d

by the sublattice magnetisation (sec Fig. 7.1:5). The results of the CCl\I andexact diagonalisations (ED) Clgree well with each other and with the resultJ: ~ 2.56 from cmnulant series expansions [79]. whereas by contrast renormalised spin wave theories (HS\VT) clearly overestimate the order. \Ve alsouote that another indication of a dimerised phase is t he appearance of a gap.:1 between the singlet ground state and the first triplet excitation. The gapappears to open in the range 2.5 -:::J: .::;:).0 for both the ED and CCl\I calculations (see Fig. 7.1:5). This result is in good agreement with the correspondingestimates for the critical point using tlw magnetisation.

The phase transition to the dimerised phase is also indicated by tlw spinstiffness p-,. The ground-state stiffness is a variable which indicates the distance of the ground state from criticality. and the breakdown of the Nc~c1

long-range order is thus accompanied by (J8 going to zero. The spin stiffness

7 The Coupled Cluster l\Iethod Applied to Quantum Magnetism 331

mO.2

4.03.0Orof;l0--!-il-i'l.0-~t;I2:t::.0~

J'

0.4 ~-r-r-r-'~--.-.--,T', ~CCM~·,.---r~l~ED

-- LSWT------- RSWT O.Ord.

'r~~:~:: ~u,,~J~:< ,',i

·0.0 1.0 2.0 3.0 4.0 5.0 6.0J'

Fig. 7.8. Indications of a phase transition to the dimerised phase for the J-J' modelon the square lattice with competing nearest-neighbour bonds, with J = +1. Leftgraph: sublattice magnetisation versus J' using the CCM, exact diagonalisation,and spin wave theories. Right graph: spin gap versus JI using the CCl'vI comparedto results of exact diagonalisation. (Figure taken from [77])

measures the amount of energy used in introducing a twist () to the directionof spin between every pair of neighbouring rows, such that

(7.40)) _ (P Eo(()) Ir H - d()2 N 0=0'

and this quantity may be calculated directly using the CCM.We note that the magnetic order parameters may only tell us whether

certain types of long-range order are present, whereas the spin stiffness hasthe advantage of being unbiased with respect to the nature of the ordering.The spin stiffness constitutes, together with the spin wave velocity, the fundamental parameter that determines the low-energy dynamics of magneticsystems [84]. The CCl\I LSUBn results are given in Fig. 7.9. vVe calculate thestiffness using two different directions of in-plane rows, i.c., rows parallel tothe J' bonds and rows perpendicular to the J' bonds. \Ve note that, althoughthe results of the stiffness for the two directions are different in general (seeFig. 7.9), the phase transition points (i.e., the values of J' where PH becomeszero) agree well with each other for the various LSUBn approximations although the extrapolated CC:l\1 results are expected to be even more accurate.Our calculations predict that J~ ~ 2.8 which is again in good agn~ement

with the results of the other methods. We note that this phase transitionto the dimerised phase is expected to belong to the three-dimensional 0(3)universality class as indicated by the value of the correlation length criticalexponent [82].

\tVe now consider the frustrated region of the J _.]' model for J and J'with different signs. We note that classically there is a second-order phasetransition from collinear order to noncollinear order at J' = -J/3 for both


,1=1

o. 4 ["""~",,~~n-rc~"TTT~n-rc-rrn~~""'''3~ -- LSUBn '

~ extrapolated 1,1=1

2

-_.- LSUBn~--...., ~ extrapolated

0.3 ~/;;~\/; \ \

Psx ~/\\ \\

0.2r \\\t \ \~\4od \ 6 \

\ \oq ~. '\" "'0 2.0 3.0 4.0 5.0

J'

Fig. 7.9. The spin stiffness versus J' for two different rows within the plane of thelattice are good indicators of a phase transition to the dimerised phase for the JJ'model on the square lattice with competing nearest-neighbour bonds. with J = +1.Left graph: rows along the :r direction (parallel to the J' bonds). R.ight graph: rowsalong the y direction (perpendicular to the J' bonds). (Figure taken from [77])

Fig. 7.10. Pitch angle <P versus IJ' i for the quantum and the classical case of the JJ' model on the square lattice with competing nearest-neighbour bonds, Although<P is classically the same for the ferromagnetic case (J = ~ 1, J' > 0) and for theantiferromagnetic case (J = +1, J' < 0) we note that the quantum pitch angle isdifferent for both cases, The curves to the left of the classical (dashed) curve belongto J = -1 and those to the right of it belong to J = +1, (Figure taken from [77])

antiferromagnetic and ferromagnetic nearest-neighbour J-bonds. By contrast,the behaviour of the quantum model for the two cases is diflerent concerningthe phase transition. In particular. we find that the critical point is shifted to.1' ~ -1.35 (see Fig. 7.10) for the antiferromagnetic case (.1 = +1), although110 such shift is observed for the ferromagnetic case (J = --1).

The exact diagonalisation (ED) data of the structure factor S(k) (seeFig. 7.11) also agree with these findings. For .1 = +1 the collinear Neel order

7 The Coupled Clustcr!'v1ethod Applied to Quantum Magnetism 33:)

-4.0 -3.0 -2.0 - 1.0 -00J' J'

,,I i...~...., .

5.0

>.< 4.0ot.;'230

QJ

~t 2 .0;:J>.<

-<-'[fJ

1.0

ED (N=8x4):J=+l

QJ>.<2.0;:J

-<-'()

;:J>.<01.0

4.0 ~)== ~ ~J4,0)~ S n/2,0)

>.< ~ S 3n/4,0)..s 3.0 classical()

<1J'+-<

Fig. 7.11. Ground-state structure factor 8(k) ex: Li.JEA ei(Rj R i )k(S" sJ (note

that the indices i and j run over one sublattice) for a 8 x 4 lattice of the J -J' modelon the square lattice with cornpeting nearest-neighbour bonds for the quantum andthe classical case for various spiral vectors k for antiferromagnetic J = +1 (leftgraph) and ferromagnetic J = -1 (right graph). (Figure taken from [77])

[k = (0,0)] becomes unstable in comparison to the noncollinear spiral order[k = (7T / 4, 0)] in the classical model for Y ~ - 0.36. We note tha t this occursonly for Y.:::: - 0.95 in the quantum case. The situation for the ferromagnetic case (.I = -1) is again different, and the results of the structure factorshow that the transition from k = (0,0) (collinear ferromagnetic order) tok = (7T /4,0) (spiral order) takes place at nearly the same value of Y :":;; 0.36 forboth the classical and the quantum cases. VVe may also use the difference between the amount of the on-sit.e magnetic moment (Si) and its classical value(S,)d = 1/2 as a measure of quantum fluctuations. \Ve compare the strengthof quantum fluctuations near the collinear-noncollinear transitions for boththe anti ferromagnetic and the ferromagnetic cases. Although the quantumfluctuations are particularly strong for .I = +1 near the antiferromagneticspiral transition (leading to an on-site magnetic moment less then 20% ofits classical value [64]), there are virtually no quantum fluctuations at theferromagnetic-spiral transition for .I = ·-1 because the on-site magnetic mornent takes its classical value 1/2 up to .I' :":;; 0.36 (cf. [81]). Hence the shift. ofthe critical .I: in the anti ferromagnetic case can clearly be attributed to thestrong quantum fluctuations.

\Ve rnay summarise by saying that our findings are generally consistentwith the statement that quantum fluctuations (which we have in the antiferromagnetic case only) prefer a collinear ordering. \Ve note that the quantumcollinear state can survive for the quantum model studied here into a classicaJly frustrated region in which classical theory indicates that the collinearstate is already unstable. In addition, our results indicate that there is asecond-order phase transition for the ferromagnetic case (.I = -1) which is

:)34 D.LI. Farnell and ftF. 13ishop

.1.1

spiral

ferro i

II~

spiral

-I

(7.11 )

Fig. 7.12. Phase diagrams of the.1 f model on the sqnare lattice with competingnearest-neighbour bonds for the classical case (left graph) and for the quantHIIlcase (right graph). The dashed jiwc indicates a first-order phase transition. whilethe ot her transitions awol' second order. (Figure taken from [77])

in agreement with classical theory. The collinear-lJoncolliliPar tran::;ition in theantiferromagnetic ca::;e (J = +1) is probably of first order for the quantummodel (d. Fig. 7.10 amI discussion in [64]) which comjJ<lJ"('S to a sc'cond-ordertransition for the classical case. Figure 7.12 compares the phase diagram::; oft hi::; J J' modd for tire classical and the quantum cases.

7.5 An Interpolating KagomejTriangle Model

\Ve also wish to study another ::;trongly frustrated ::;pin-half Heisenberg model. namdy one which interpolates smoothly between the triangular-latticeantiferromagnet (TAF) [59.8587] and tIl(' Kagomc-lattice [65.88,89] antift'rromagnet (KAF). \Ve shall refer to this as the interpolating Kagomejtrianglemodel (illustrated in Fig. 7.1:)). and the Hamiltonian i::; given by

II = J L S; . 5j + J' L S; . S,

(I.j) {,,}

where (i.j) nllls over all nearest-neighbour (n.n.) bonds on the Kagomc latt ice. and {t. Ii-} nm::; over all n.n. bomb which connect the Kagomc latticesites to those other sites on an underlying triangular lattice. Note that eachbond is cOllllted once and once only. \Ve explicitly set J = 1 throughout thispaper. and we note tlwt M J' = 1 we thus have the TAF and at .r = 0 wehave the KAF.

7.5.1 CCM Treatment of the Interpolating KagomejTriangleModel

For the interpolating Kagomcjtriangle model described by (7.41). we choosea model state leT/) in which the lattice is divided into three sublattices. denoted {AJ3.C}. The spin::; on sublattic:e A are oriented along the negative

7 The Coupled Clu~ter Method Applied to Quantum Magnetism 335

(a) (b)

Fig. 7.13. The interpolating Kagome/triangle model i~ illustrated in diagram (a),where the bonds of strength J between Kagomc lattice sites are indicated by thethick solid lines and the non-Kagorne bonds of strength .r on the underlying triangular lattice sites are indicated by the "broken" lines. The triangular latticeHeisenberg antiferromagnet (TAF) is illustrated in diagram (b), and we note thatthe two models are equivalent when J = J'. The quadrilateral unit cells for bothcases are also illustrated. The interpolating Kagomc/triangle model contains foursites per unit cell, whereas the TAF has only one site per unit cell. (Figure takenfrom [65])

z-aXls, and spins on sublattices 13 and C are oriented at +120° and -120°,respectively. with respect to the spins on sublattice A. Our local axes arechosen by rotating about the ;ii-axis the spin axes on sublattices 13 and C by-120° and + 120° respectively, and by leaving the spin axes on sublattice Aunchanged. Under these canonical transformations.

•e'Br; 1 r; V3 z, -+ -28U - 2 8B

8j] -+ shV3 r 1 z8n --+ 28B - 28R

r 1 r V3,8(, -+ -28c' + 2 8 (.

82: -+ 827 'z V3 .:r; 1,2

8e -+ -'28e - 25C . (7.42)

The model state I<p) now appears mathematically to consist purely ofSpillS pointing downwards along the z-axis, and the Hamiltonian (for J = 1)is given in terms of these rotated local spin axes as,

II V3A( Z + ,2.- e+s Z S-SZ)+ -4- 8 i 5 j + 8 i oS,) - "i 'j - , i 'j.

(7.43)

:336 D.J.J. Farnell and RF. Bishop

I\ote that i aud j run only over the NJ{ sites on the Kagomc lattice.whercas k: runs over those non-Kagomc sites on the (underlying) triangularlat tice. N indicates the total number of triangular-lattice sites, and eachbond is counted once and once only. \Ve also note that we have multipliedall of the off-diagonal terms in the new Hamiltonian by a factor of A. \Neshall use this factor in order to det ermilH' the perturbation series arouud theIsing limit (/\ = 0) for the ground-state energy and sublattice magnetisation.The case A = 1 corresjJonds to our isotropic Heisenberg case of (7.41). Thesymbol -+ indicates an explicit bond din:ctionality in the Hamiltonian gi venby (7.43). namely. the tl/Tce directed nearest-neighbour bonds included in(7.43) point fr01n sublattice sites A to B. B to C. and C to A for both typesof bonc!. \Ne now perform high-order LSUBm calculations for this model viaa computational procedure for the Hamiltonian of (7.4:3).

7.5.2 CCM Results for the Ground-State Properties

\Ve note that for the CC1\[ treatment of the interpolating Kagomc/trianglemode! presented here (and see [65] for further details) the unit cell contain"four lattice sites (see Fig. 7.1:3). By contra"L previous calculations [59] forthe TAF used a unit cell containing only a single site per unit cell. Hence. theinterpolating Kagomc/triangle model has many more "fundamental" configurations than the TAF model at equivalent levels of approximation. However.we find that those configuration" which are not equivalent for the interpolating Kagomc/triangle mc)(le! but aTe equivalent for the TAF have CCIvI correlation coefficients {Sf, Sf} which become equal at the TAF point. ,F-c= l.Hence. the CCM naturally and without bias refleds the extra amount ofsymmetry of the interpolating KagOl1H"/triangle model at this one particularpoiut. This is an excellent indicator of the validity of the CCM treatmentof this model. The results for t he interpolating Kagomc/triangk model at.1' = 1 tIm" also exactly agn'(' with those of a previous CC~1 treatnwnt ofthe TAF [59].

\Ve now set A = 1 for the rernainder of thi" subsection and again we"track" the "trivial" solution for large .1' for decreasing values of J' untilwe reach a critical value of .1;, at which the solution to the CCJVI equationsbreak,; down. Results for .1;, for this model are presented in Table 7.2. A simple"heuri"tic" extrapolation of these results gives a value of J; = O.O±O.] for theposition of this phase tran"ition point. Thi" result indicates that the classicalthree-"ublattice "\eel-like order. of which about 50% remains for the TAF.completely di"appears at a point very near to the KAF point (.7' '-c~ 0).

The results for the ground-"tate energy arP shown in Fig. 7.14 and inTable 7.2. These results are seen to be highly converged with respect to eachother over the whole of the regioll 0 ::-: .1' -S 1. The results for the groundstate energies of the KAF and TAF model in Table 7.2 agree well wit h resultsof other technique". Indeed, 'eve helieve that the extrapolated CCJ\I re"ults


-0.20 '----'

-0.25

-0.30 &r---0.35 CJ

EiN -0.40

LSUB2-0.45 LSUB3

-0.50 LSUB4LSUB5

-0.55 lSUB6

-0.60 i ~.~-1.0 -0.5 0.0 0.5 1.0

J'

Fig. 7.14. CCM results for the ground-state energy per spin of the interpolating Kagome/triangle model (with .I = 1) using the LSUBm approximation withm = {2. 3. 4. 5, 6}. The boxes indicatc the CCM critical points, .I;, and a simpleextrapolation in thc limit m --+ oc implics that .I,; = 0.0 ± 0.1. (Figurc takenfrom [65])

Table 7.2. CCM results [65] for the ground-state energy per spin and sublatticemagnetisation of the TAF and KAF models using the LSUBm approximation withm. = {2, 3, 4, 5. 6}. CCM critical values• .I;. of the iuterpolating Kagon1(;/trianglemodel (with .I = 1), which are themselvcs indicators of a phase transition point inthe true system. are also given. Comparison is made in the last row with thc resultsof other calculation~

i I KAF! TAF I .I .I' 1c-Ir-n-ji-'---:E=-g-/""'N--'-·l-,-~i----'E=-.q-/""'N-'-· --'-1--.c-Mc-Jcr,--+i---'-.Ic7"",--11

2 --0.37796 0.8065 -0.50290 0.8578:3 -0..'39470 0.7338 -0.51911 0.8045 -0.6834 -0.10871 0.6415 -0.5:3427 O.72n -0.217.5 -0.41392 0.5860 -0.53869 0.6958 -0.2446 -0.41767 0.5504 -0.54290 0.6561 -0.088oc -0.4252 0.366 -0.5505 0 ..516 O.O±O.Jc.f. -0.43 ( [89]) 0.0 -0.551 ( [85]) 0.5 ( [86.87])

are unquestionably arnong the most accurate results available for the groundstate energies of the TAF and KAF.

vVe now wish to describe how much of the original classical ordering of themodel state remains for the quantum system. If one considers llol1-Kagome

::\:38 D.LI. Fanwll and R.F. Bishop

1.00

----------10.75

MK0.50 , /

II

I

0.25

/ I -------=~~--:-=~--=~/y --_:~- -------~

I'. /!f:='/, I I /I 'j ;'. \ 'I . - ------,I ; I '1/1

; i LSUB2'I I, I LSUB311'1il! i I LSUB4

, :i i; I ~~~:~0.00 ~..~_.........J,-------,: I

-0.8 -0.4 0.0 0.4 0.8J'

Fig. 7.15. CC:\I results for the sublatticc lllagnctisation of the interpolatingKagome/triangle model (with J = 1) using the LSUBm approximation withm = p. :{, 4. 5. 6}. (Figure taken from [65])

lattice sites then t he SpillS on these sites are effectively "frozen" into theiroriginal directions (of the model state) at .I' = O. Hence. we believe that therelevant quantity to be considered for this model is the average value of 81:(again after rotation of thc' local spin axes) where k runs only over the Nf(Kagome lattice sites, given by

(7.44)

The results for l\l f( are presented in Fig. 7. Hi and in Table 7.'2. The puzzling"upturn" of "UK for the LSUB5 data is an artifact- and typically such behaviour only ever occurs when one enters a phase in which the model statebecomes an increasingly bad starting point. Although the extrapolated valuefor 1\1 K specifically at the KAF point relllains nOll-zero. the LSUB6 resultgoes to zero very close to the KAF point. Cel\! results are thns fully consistent with the hypothesis that. unlike the TAF, the ground state of the KAFdoes not contain any Neel ordering.

7.5.3 Evaluation of the Perturbation Series Using CCM

Finally, it is instructive to make contact with the cUlllulant series expansionsfor the anisotropic TAF (i.e., .I' = J = 1) with respect to the parameter /\.

7 The Coupled Cluster l\Iethod Applied to Quantum Magnetism 3:39

Table 7.3. Expansion coefficients in powers of ),. up to the L5th order for theground-state energy per spin, E g / N and the sublattice magnetisatioIl, iII, for theanisotropic spin- ~ triangular-lattice Heisenberg antiferromagnet obtained from theCCrvI equations in the LSUB6 approximation. The exact series expansions up tothe lIth order obtained by Singh and Huse [85] are also included for comparison.(Table taken from [59])

IC)n1erILSUB6: Eg/NIExact: Eq/NI LSUB6: j\ll Exact: M I

0 -0.3750000 -0.:3750000 1 11 0.0000000 0.0000000 0 02 -0.1687500 -0.1687500 -0.27 -0.27

3 0.0:337500 0.0:337500 0.108 0.1084 -0.0443371 -0.044:B71 -0.2726916 -0.2726916

5 0.02042.59 0.0204259 0.17179.51 0.1717951

6 -0.0283291 -0.0283291 -0.3315263 - 0.:3::Il52();j

7 0.0311703 0.0315349 0.4060277 0.41107378 -0.0357291 -0.0476598 -0.5:3:31858 -0.738220:39 0.0541263 0.0685087 0.8894023 1.178130310 --0'()771681 -0.102,5446 -1.:3927:395 -2.0109889~11 0.1294.578 O.L565.522 2.4179612 3.4012839 ,12 -0.1848858 ? -4.0426184 ?13 0.28.57225 ? 6.80865:38 ?

14 -0.446:3496 ? -11.488761 ?15 0.7021061 ? 19.38805:3 i "

\Ve have computed the perturbative CCM solutions of Eg/N and the sublattice magnetisation 1\1. as defined in (7.24) with respect to the local spinaxes, in terms of the anisotropy parameter A. In Table 7.3 we tabulate theexpansion coefficients from the LSUB6 approxilnation, together with the corresponding results from exact series expansions [85]. We note that the LSUB6approximation reproduces the exact series expansion up to the c;ixth order.\Ve conjecture that the LS{JBm approximation reproduces the e.Tact seriesexpansion to the same mth order. J\Ioreover, the fact that the corresponding values of several of the higher-order expansion coefficients from boththe CCI\I LSUB6 perturbative solution and the exact series expansion remain close to each other shows that the exponential parametrisation of theCCJ\I with the inclusion of multi-spin correlations up to a certain order alsocaptures the dominant contribntions to correlations of a few higher orders inthe series expansions.

7.6 The J 1-J2 Ferrimagnet

\Ve now briefly present results for another frustrated model in which we haveboth llearest- and next-nearest-lleighbour antiferromagnetic bonds. (The in-

340 D..J..J. Farnell and R..F. Bishop

terested reader is referred to [68.77J for more details.) The Hamiltonian forthe square-lattice spin-half/spin-one.h .h ferrimagnet is given by

II = J! LSi' SJ + ·h L Si .SI,

(i.j) {i.k}

(7.45)

\vhere j) runs over all nearest-neighbour bonds on the square lattice. {i. k}nlllS over all next-nearest-neighbour bonds. vVe assume that .it = 1 andh > 0 throughout this article and so this model is fru,stmted. Note thatOllC sublat tice (A) of the square lattice is populated entirelv by spin-Olwspins (SA = 1) and the other sublattice (B) is populated entirely by spin-halfspins (Sf} = 1/2). This model is an extension of the well-known spin-halfJ! h model on the squan- lattice (see e.g. [90-96J and references therein)which serves as a canonical model for the discussion of an order-disorderquantum phase transition driven by the interplay of quantum fluctuationsand frust ration.

it feature of such ferrimagrwt ic spin systems is that the Lieb-l\Iattis tlworem may be obeyed (if frustration is excluded) such that the ground state hasa magnetic moment per spin of strength (.'q - S n) = 1/2. !'\ote in particularthat this property is obeyed for tll(' ferrimagnet of (7.15) at .h = n. and thusa macroscopic lattice magnetisatioll exists for this case.

vVe note that many ferrimagnetic materials haw' recently been fabricated and various examples are the "ladder" systems: \InCu(pba Off) (H2 0):1(where pbaOH=2-hydroxy-L 3-propylenebisoxamato) and .\InCu(pba) (H20h. H2 0 (where pba = L3-propylenebisoxmnato) [97 99]. These materials contain magnetic atoms .\In (SA = 5/2) and eu (sn ccc 1/2).

The classical behaviour of the square-lattice spin-half/spin-OllP ferrimagnet of (7.4;")) is also interesting and three distinct phases are predicted. Thefirst such phase for .h S 0.25 is one in which the ground-state is the collinearferrimagnetic !\pe! st ate (shown in Fig. 7.16). A second-order phase transition theu occurs within this classical pictnre to a phase in which the spin-onespins may cant at an angle e. although the spin-half spins do not changetheir direction (also shown in Fig. 7.16). This state is henceforth referred toas the "spin-flop" state. A first-orcler phase transition to a collinear statein which next-nean>st-neighbonr spins are antiparallel (again. see Fig. 7.lG)then occurs classically, and this s1 ate is referred t () hen' as the "collinearstriped" state. Notp that this state in the classicallllodel is degenerate withstates canting at an arbitrary angle between spins on sublattice A and spinson sublattice B. However. this degeneracy is lifted by quantum fillCt uatioJ]swhich select the collinear state [92. IOO.WI]. \Ve notp that the spin-flop andcollinear striped states are "incolllnwnsnrate" in thp SPllse that no value ofthe angle 0 may be chosen such t hat the two states are equivalpnt. vVe notphowever t hat the l\6d and spin-flop states arp eqnivalent wlwn e= it.

A furt her motivation for studying this model is that exact calculations offinite-sized lattices indicate that the behaviour of the quantum ferrirnagnet of

7 The Coupled Cluster IVlethod Applied to Quantum Magnetism 341

a b cFig. 7.16. The classical ground states used to treat the square-lattice spinhalf/spin-one .h-J2 ferrirnagnet via the CCrvI. These states are namely: a) thecollinear ferrirnagnetic Neel state: b) the collinear "striped" state: and c) the "spinflop" state. Note that long arrows indicate spin-one spins (and their orientation)and short arrows indicate the spin-half spins. The primitive unit cells are shownby the ovals for the Neel and striped states and by the rectangles for the spin-flopstate. Full lines indicate the nearest-neighbour bonds and dashed lines indicate thenext-nearest-neighbour bonds. (Figure taken from [77])

(7.45) may be much different from the behaviour of its linear chain counterpart and from the square-lattice spin-half J]-.h antiferromagnet. These exactcalculations of finite-sized lattices suggest that the behaviour of the squarelattice ferrimagnet is analogous to that of the classical behaviour outlinedabove.

Three model states are used to treat the square-lattice spin-half/spinone J 1-.h ferrimagnet in order to provide results across various regimes ofdiffering quantum order. TIlt) first such model state is the collinear Neel state,and the primitive unit cell in this case contains a spin-half site and a spin-onesite (shown in Fig. 7.16). The underlying Bravais lattice is formed from twovectors (/2, /2) and (/2, -/2). There is also an 8-point symn1(~try group,namely: rotations of 0°, 90°. 180°, and 270°; and reflections about the lines.r = 0, :y = 0, :y = x, and y = -x. The collinear striped state (also shownin Fig. 7.16) is also used as a model state, in which spins for even values of.1: along the x-axis point "downwards" and spins for odd values of :r point·'upwards." The primitive unit cell again contains only two spins, althoughthis time only four of the eight point group symmetry operations are allowed,namely: rotations of 0° and 180°; and reflections about the lines :r = 0 andy = O.\Ve note that rotations of the local axes of 180° about the y-axis ofevery spin is carried out such that each spin now appears (matlwmatically)to point "downwards". Each spin may now be treated equivalently.

It is noted that (in the original unrotated spin coordinates) 8j. == 2..:; si =o is preserved for all CCM approximations for these two models states in orderto reduce the number of fundamental configurations at a given LSUBrn orSUBm-m approximation level.

342 D.J.J. Farnell and R.F. Bishop

";fIJl>0

>;,/./.

..~

0.60.4~.__~. __ -'------- J ; ~' _

- Exact Diagonalisations (N=20)X GeM Extrapolated• Spin-Wave Theoryo Classical

0.2

o~-ooo~-~OC- ·----1---

00 <000 0

o 0o 0

o 0o 0

o 0o 0

o 0o 0

o 0o 0

oo

oo

oo

oo

-1 <iT

-0.6

-0.8 ~

-1.2o

J2Fig. 7.17. Ground-state encrgy per spin of the square-Iatticc spin-half/spin-one ,hJ2 ferrirnagnet versus .h for the CC:\l method using threc model states comparedto results of exact diagonalisations of finite-sized lattices. (]\;ote that wc have setJ I = 1)

The third model state is the spin-flop state. \Ve note that there is noequivalent conserved quantity to 8~ for the spin-flop model state. althoughsingle-body correlations are explicitly excluded from this calculation as theyare (in some sense) already included by the rotation of the local axes ofspins. It should however be noted that this is an explicit assumption of thecalculation for the spin-flop rnodel state.

The amount of ordering on each sublatt icc is represented by.

(7.4G)

where iiI nms over all spin-one lattice sites and in nms Over all spin-half lattice sitt's. Note that. as usual. all of the spins for all of the models states havebeen rotated such that all spins appear mathematically to point downwards.The quantities ml and m2 arc the expectation values of the magnetic moment in the z-direction on the A and B sublattice. respectively, with respectto a given model state and represent order parameters for this model.

The ground-state energies predicted by the CCr\I using the three modelstates are shown in Fig. 7.17 and once again CCr\I results are in good agreement with results of exact diagonalisations (ED) of finite-sized lattices andspin-wave theory. Results for the sublattice magnetisations shown in Fig. 7.18

7 The Coupled Cluster Method Applied to Quantum Magnetism :H:J

1I Tn]

o<!lo~ ..

08 •

•0.6 (S,X\ ••

r,~

•.....1n2

0.4OdJo~

=" "C'X<Xoo

0.2 ~

TIll

. . . .

Ocl . 0*

o L..-~~......-I-\--,,-:~--~o 0.2 04

Fig. 7.18. Sublattice magnetisations, 1n] and 1n2 of (7.46), of the square~latticc

spin~half/spiIl~one .h-h ferrimagnet versus -h for the CCM method using threemodel states compared to results of exact: diagonalisations (ED) of finite-sized lattices. Note that full lines are those results for the sublattice magnetisation usinglinear spin-wave theory, open circles arc those results of exact diagonalisations, andCeM results arc indicated by the filled circles. (Figure taken from [68])

also show good correspondence with results of exact diagonalisations (ED)of finite-sized lattices and spin-wave theory. Hence, the CCM yield.s excellentquantitative accuracy for the ground-state properties of the spin-half/spinone .11 -.h ferrimagnet across a wide range of the next-ncarest-neighbour bondstrength .h by the use of three different model states. The CCM thus provides a comprehensive picture of the ordering in the ground state, an accurateprediction of the phase diagram, and even evidence regarding the order ofthe phase transitions.

The results for the ground-state properties of the spin-half/spin-one .11

.12 ferrimagnet at .11 = 1 and .12 = 0 based on the Ned model state arepresented in Table 7.4 and we see that our raw SDBrn-rn results appear toconverge rapidly. An ext.rapolation in the lirnit rn -+ (X) is also performed forthe .11 h ferrimagnet at .1] = 1 and h = 0 in order to provide even betteraccuracy.

The positions of quantum phase transition points as a function of (l =.h/.h are also shown in Fig. 7.18. A second-order phase transition is observed at (le, and CCJ\I results place this at Gel = 0.27, which is slightly abovethe classical value. By contrast, CCM results predict a first-order phase transition at etC2 ~ 0.5 and this result is in good agreement with those resultsof both spin-wave theory and exact diagonalisations. Another possible phaseis also indicated in Fig. 7.18, namely, one is which we have finite and nonzero sublattice magnetisation on the spin-one lattice sites and zero sublattice

344 D.LI. Farnell and RF. Bishop

Table 7.4. Results for the ground-state energy and amounts of sublattice lllH

gnetisations m I and m2. on the spin-one and spin-half sites respectively. of thesquare-lattice spin-half'jspin-ouc .II .h ferrimagnet at .II = 1 and h = 0 basedon the Neel model state [77]. 'iote that llh· indicates the number of fundamentalconfigurations at a given level of LSUBm or SliBm-m approximation. CCJ\I results are compared to exact diagonalisations (ED) of finite-sized lattices. Heuristicextrapolations in t h,' limit m ---tX are performed

SUB2-2 I -1.1925820.928480.42848SUB4-4 1:3 -1.204922 0.90781 0.40781SUB6-6 268-1.2062710.903330.40;);:\4LSlJI36 279 -1.206281 0.903290.40::\30

Extrapolated CC\J -1.20fj9 0.898 0.398ED (N = Hi) 1.2181340.875lf)iO.:>7515,ED (N = 20) :-1.2120500.88482iO.384831

magnetisation OIl the spin-half lattice sites. The onset of this phase withincreasing .h is indicated by the symbol. n*.

7.7 Conclusion

\Ve have seen in this chapter that the CC\I may be applied to various quantum spin systems at zero temperatnre. In particular, suggestive results for thepositions of CC:t\I critical points were observed. and these points were seento correspond closely to the occurrence of quantum phase transitions in the"rear' system. Furthermore. quantitatively accurate results for expectationvalues with respect to the ground and excited state;.; were determined. The;.;eresults were scen to bc competitive with the best results of other approximatemethods.

A possible use of high-ordcr CCl\1 techniques might bc to predict excitation spectra of quantum magnets to great accuracy. Fnrthermore, this wouldmcan that a direct connectioIl might be made to results of neutron scattering experiments. Also. the application of the CC:t\1 to quantum spin systemswhich exhibit novd ordering. such as those with ground states which demonstrate dimer- or plaquette-tiolid ordering, iti another pOtitiible future goal.Furthermore. the extension of high-order techniques to bosonic and fermionicsystems is potisiblc in future.

High-order CervI techniques might abo be applied at even greater levels ofapproximation with the aid of parallel processing techniques [8~j]. Indeed. theCCl\I is well-suited to such an implementation a.nd recent CGi'vI calculationsusing parallel procetising t echniqucs have been carried out for approximatelylO4 fundamental configurations. \Ve believe that an increase of at leatit ano-


ther order of magnitude in the number of fundamental configurations shouldeasily be possible in the near future by using such techniques.

The extension of the CervI to quantum spin systems at non-zero temperatures might also be accomplished by using thermo-field theory. The application of the CCM at both zero and non-zero temperatures might then help toexplain the subtle interplay of quantum and thermal fluctuations in drivingphase transitions over a wide range of physical parameters.

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the coupled cluster method applied to quantum magnetism

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