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PHYSICAL REVIEW B, VOLUME 64, 020406~R!

Extended cluster algorithm in quantum simulations

Hiromi Otsuka*Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan

~Received 3 April 2001; published 20 June 2001!

We propose an extension of the quantum cluster algorithm by a combined use of the Fortuin-Kasteleynmapping and the Hubbard-Stratonovich transformation. To describe the idea, we consider theS5

12 XXZ chain

model and express the partition function as the sum in an extended configuration space of spins, graphs, andfields. Then it is clarified that this algorithm possesses a computationally tractable continuous-time limit andmaintains virtues of the quantum cluster algorithm. Numerical simulations are performed and its applicabilityis demonstrated. Further, we discuss potential gains in our algorithm.

DOI: 10.1103/PhysRevB.64.020406 PACS number~s!: 75.10.Jm, 02.70.Rr, 05.30.2d, 05.50.1q

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Recently, substantial developments have been achievethe methodological aspect of world-line quantum MonCarlo ~QMC! simulations, where applicabilities of thmethod as well as their efficiencies have been drasticimproved.1–4 Obviously, these innovations have bebrought by natures of clusters5 constructed by the FortuinKasteleyn ~FK! mapping,6 and thus simulations workingwith such geometrical objects open a window of opportunfor expanding the scope of numerical approaches.7,8 We will,in this paper, propose one extension of the cluster algoritwhere a part of the interactions in a system is managedthe FK clusters and the rest via the so-called HubbaStratonovich~HS! fields.9,10 These two kinds of auxiliaryvariables have been so far independently used in eachtext, while their aims are both in expanding interacting stems by some more preferable/tractable ones. Thereforegive a formulation of a MC algorithm in the extended cofiguration space including these variables, and then clathe nature of our approach.

As the simplest example to make the idea concrete, leconsider theS5 1

2 XXZ chain:

H5J(i

~SixSi 11

x 1SiySi 11

y 1gSizSi 11

z !, ~1!

(J.0). As mentioned, we separate the Hamiltonian into tparts: Specifically, here we setH15J(Si•Si 11 and H2

5Jl(SizSi 11

z with lªg21. Then the path-integral representation of the partition function is given by the standaargument.11,12 In the Sz-diagonal basis, the result is

Z5Tr e2bH5Tr~e2eH!Lt.(S

W1~S!W2~S!, ~2!

where the spin configuration,S, represents a set of spin varables $Sk% on (111)-dimensional space-time siteskP$( i , j )u0< i ,L,0< j ,2Lt% with lattice spacingsa ande/2, and expresses a set of conformations of so-called wlines. Hereafter, we takea51 (J51) as an unit of length inthe space~unit of energy!. In Eq. ~2!, a partial Boltzmannweight Wn(S) stands for a contribution from the HamitonianHn ; in the present case, they are expressed by a puct of the local Boltzmann weights on interacting plaquetor on spacelike bonds:

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W1~S!5 )p51

L3Lt

w1~Sp!, W2~S!5 )b51

L32Lt

w2~Sb!. ~3!

The indexp ~b! specifies a plaquette~spacelike bond! and thevariableSp (Sb) denotes a set of spins at the corners ofpth plaquette~at the ends of thebth bond!, p5$( i , j ),(i11,j ),(i , j 11),(i 11,j 11)% @b5$( i , j ),(i 11,j )%#. Whilethe four-spin plaquette interactionw1(Sp) is expressed bythe matrix element of exp(2e Si•Si 11), the two-spin space-like bond interaction w2(Sb) is given as exp@2(e/2)lS( i , j )S( i 11,j )#.

First, with respect toW1(S), we shall proceed with theFK mapping by whichw1(Sp) is expanded to a linear combination of normalized tensorsd(Sp ,Gp) with appropriateweightsv(Gp)>0:

w1~Sp!5(Gp

d~Sp ,Gp!v~Gp!. ~4!

Performing the expansion of the local Boltzmann weigon each interacting plaquette gives the expression

W1~S!5(G

W1~S,G!5(G

D~S,G!V~G!, ~5!

where the graph configuration,G, is defined by a set of locagraph variables$Gp%, and represents a set of conformatioof loops in space-time due to the grouping conditionsspins,D(S,G)Þ0.

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HIROMI OTSUKA PHYSICAL REVIEW B 64 020406~R!

Second, we considerW2(S). To decouple the two-spininteraction, we introduce the auxiliary fieldUbPR on eachspacelike bond, i.e., using the HS transformation:9

w2~Sb!}E dUbe2(e/2)$12

Ub22Aulu[S( i , j )6S( i 11,j )]Ub%.

~6!

Hereafter the easy-plain~easy-axis! anisotropic caseugu,1(g.1) refers to the upper~lower! sign in such expressionsThis equation shows that by virtue of the introduction ofUb ,S( i , j ) andS( i 11,j ) do not directly interact with each otheso that, as we shall see below, loop flips can be indepdently performed in MC simulations under a fixed field codition. Here we comment on a property of the HS transfmation. Actually, there exist some possible typesdecoupling fields other than the real variable. For examHirsch found that the Ising variable can be used for the saaim.10 However, as was also pointed out in the reference,could not take the continuous-time limit if working withfixed-length field like the Ising variable. For this reason,have employed the real variable. One may think that tchoice brings about some disadvantages in numerical slations, but it will turn out that a compatibility between thHS transformation using real fields and the loop algoritbecomes excellent in the continuous-time limit.

Performing the HS transformation on each spacebond,W2(S) is expressed as

W2~S!5(U

W2~S,U !5(U

M ~S,U !X~U !, ~7!

where the field configuration,U, is defined by a set of fieldvariables$Ub% and the multiple integral is denoted by(U .While X(U) is the spin-configuration independent Gaussdistribution term,M (S,U) shows the ‘‘Zeeman coupling’with auxiliary introduced fields:

M ~S,U !5expS e

2 (k

AuluSkHkD . ~8!

HereHk is given as a linear combination ofUb whose bondincludes thekth site.

Consequently, we obtain an expression of the partitfunction described by a sum in the extended configuraspace of spins, graphs and fields as

Z. (S,G,U

W1~S,G!W2~S,U !. ~9!

For a givenS, the ways to generateG and U are indepen-dently defined byW1(S,G) and W2(S,U). And now, it isclear that the nature of clusters in random fields is the msubject in our algorithm. In the following, we shall explathe method to perform numerical simulations in theS,G,Uspace.

According to the Kandel-Domany formalism,13 we sto-chastically generate a graph configurationG0 for a givenS0

in the way thatW1(S0,G0)5W1(S,G0) for all S compatiblewith G0. Suppose thatG0 partitionsL32Lt space-time sites

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into N0 loopsLn (n51, . . . ,N0). Then, since Eq.~8! can bedivided into a product of the loop contribution asM (S0,U)

5)n51N0 An with

An5expS e

2 (kPLn

AuluSkHkD , ~10!

the flipping probability of thenth loop is given by, for ex-ample,Pn51/(11A n

2) in the heat-bath method under thfixed field configuration. Consequently, our new algorithmaintains an independence of loop-flip probabilities forvalue ofl, and it reduces to the ordinary one, i.e.,Pn5 1

2 inthe l→0 limit. Especially, the ‘‘freezing graph’’ whichshould be introduced in the loop-cluster algorithm withugu.1 is absent here. Instead, we should take an ensembleerage of the loop system over fields, which may reviveteractions between loops.

Next, let us consider how to treat the fields under a givS0. Equation~6! shows thatUb obeys Gaussian probabilitdistribution function P(Ub)5(1/A2ps)exp@2(Ub

2m)2/2s2# with (m,s2)5(Aulu@S( i , j )6S( i 11,j )#,2/e).Therefore, generating a field configurationU0 according tothe probability and summing up the Zeeman energy Eq.~10!,we can evaluate a flipping probability of spins on each loand thus generate the next spin configurationS1. Schematicrepresentation of our MC algorithm is given in Fig. 1.

At this stage, we discuss the continuous-time limite→0with (e/2) j→tP@0,b). As Beard and Wiese pointed ouconformations of world lines are economically stored incomputer’s memory as a sequence of times where the ‘‘trsition’’ of a world line to one of the neighboring siteoccurs.4 Then, conformations of loops are shaped accordto the ‘‘forced transition’’ and ‘‘optional decay’’ spacelikelinks as well as to the ‘‘forced continuation’’ timelike linksa part of the loops is schematically shown in Fig. 2. Withthis framework of the continuous-time algorithm, Eq.~10! isexpressed as a contour integral along the loop:

An5expSAulu RLn

dkSkHkD . ~11!

Here we redefine space-time pointkª( i ,t); dk is a lineelement of the loop andSk5const. (Hk50) on the timelike~spacelike! links. For example, let us evaluate a part of tZeeman coupling on a segment of the loop~i.e., from A to Bin Fig. 2!. SinceSk is constant on the segment, we only neto calculate the integral *t

t4dt8Ub8 (b85$( i ,t8),(i

FIG. 1. Schematic representation of the present algorithm.

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EXTENDED CLUSTER ALGORITHM IN QUANTUM SIMULATIONS PHYSICAL REVIEW B64 020406~R!

11,t8)%). Further, sinceUb is the white noise obeyingP(Ub), a probability distribution function of a stochastvariable,

j~Dt,Sb!ªEt

t1Dt

dt8Ub8 , ~12!

is also given byP(j) with (m,s2)5(DtAulu@S( i ,t)6S( i11,t)#,Dt), if two neighboring spins travel together forDtin time direction without changing their spin values. For tpresent example, we can obtain the integral as a sum of tGaussian random numbers representing contributions fR1, R2, and R3 regions in the figure.

Consequently, as illustrated, generating Gaussian rannumbers@Eq. ~12!# and summing up the Zeeman eneralong the loop, we can evaluatePn also in the continuoustime limit. An important observation here is that we do nneed to store infinite numbers of real variablesUb , but finitenumbers of their integrated values on the loop segmewhich makes it possible to perform simulations in the lim

Here we mention the method to measure physical quaties in our algorithm. Our simulations can also receivebenefits of so-called improved estimators,14 where wepromptly sum up an observable with respect to 2N configu-rations ofS compatible with a givenG. Actually, there existsome varieties to construct them due to, for example,possibility of a reassignment of loop-flip probabilitiewhereas we give a simplest expression onSz-diagonal spin-spin correlation (Si

zSi 8z ) IE . Under a fixedG andU, contribu-

tions come from two spins on the same and different loop(21)i 2 i 8/4 and (122Pn)(122Pn8)SkSk8 , respectively~seeFig. 2!. For the latter case, indexk85( i 8,t)PLn8 . There-fore, we only needPn for the calculation of (Si

zSi 8z ) IE .

Now, we shall show some numerical simulation resultsthe S5 1

2 XXZ chain using the algorithm explained abovWe here restricted ourselves to its ground state properAccording to the exact solution provided by the Bethe-ansmethod,15 they are summarized as follows. For21,g<1, it

FIG. 2. An example of loop conformations. Solid lines~dottedlines! show up-spin~down-spin! segments and horizontal brokelines denote ‘‘decay points’’ to a neighboring site. A part of tZeeman energy on the segment from A to B is given as the sumcontributions from R1, R2, and R3 regions. The pointsk1 , k4

PLn andk2 , k3PLn8 .

02040

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is in the spin-liquid phase where ‘‘the liquid parameter’’K isgiven as16 1/K5 1

2 1(1/p)arcsing and the spin correlationasymptotically behaves as ^Si

zS0z&.Ku i u221const.

(21)i u i u2K without the logarithmic correction atg51.17 Af-ter the Berezinskii-Kosterlitz-Thouless transition atg51,18

the Neel phase is realized in 1,g. We treated systems up tL566 (L[2 mod 4) and up tob580. A least-square-fitting curve for even sites’ data of^Si

zS0z& was used to evalu-

ate the staggered component at the largest distancei maxªL/2, i.e., a difference between interpolated value and Mdata ati max, C( i max), was calculated. In Fig. 3, we ploti maxversus i maxC(imax) in a log-log scale~the parameter20.2<g<1.4). Asymptotically expected behaviors are also givby dotted lines with slopes 12K (20.2<g<0.8), and 1(g51.4) which indicates an existence of the long-range N´elorder. From this figure, we find that our numerical simulatidata well agree with the exact result and further the increat g51 can be regarded as the logarithmic correction duethe marginally irrelevant coupling.17 Therefore, we cancheck the validity of our algorithm. Although the samplinefficiency is in a practical level throughout our simulationwe can recognize its lowering with the increase of the cpling constantAulu.

Last, to illustrate one of the potential gains in our algrithm, we discuss its significance in fermion simulations.recently proposed by Chandrasekharan and Wiese,‘‘meron-cluster’’ idea can solve the so-called sign problefor a class of systems, e.g., the spinless fermions~SF! onbipartite lattices@see Eq.~5! in Ref. 3#. The Boltzmannweight is then expressed by that of theS5 1

2 XXZ modelwith the sign factor Sign@S# which stems from Fermi statistics and has a topological means of world lines. In this cawithin the framework of the normal cluster algorithm, thcorrespondingXXZ model cannot be in the easy-plain aisotropy region to receive benefits of the idea. On the othand, since our algorithm can control types of graphs usesimulations while keeping the independence of the loops,can treat, for example, noninteracting SF without using ‘‘agonal breakup’’~as demonstrated in our simulations atg

ofFIG. 3. i max versus i maxC(imax) in a log-log scale. Slopes o

dotted lines show exact values of 12K for 20.2<g<0.8 and 1 forg51.4 which indicates an existence of the long-range Ne´el order.

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HIROMI OTSUKA PHYSICAL REVIEW B 64 020406~R!

50). This fact enables us to use simple expressions ofproved estimators for fermionic observables, e.g., Sign@S#.Suppose thatM is a set of merons whose flip changes tsign, then (Sign@S#) IE5Sign@S#)nPM(122Pn). WhenMis an empty set, (Sign@S#) IE51 due to the existence of ‘‘thereference configuration’’ with Sign51.3 Consequently, al-though the exact cancellation in nonzero-meron secwhich was explained in Ref. 3 does not occur due toZeeman coupling on loops~hence, the sign problem still remains!, one can get considerable improvements of sampefficiency also for SF in the easy-plain anisotropy regionemploying improved estimators.

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Our algorithm is also applicable for Hubbard-type modwhich are relevant to the high-Tc superconductivity and fur-ther, since the idea to use both FK and HS transformationquite general, it can be also applied to the classical syste

The author has benefited from discussions about the locluster algorithm with N. Kawashima. He is also gratefulY. Tomita and Y. Okabe for helpful discussions. Main computations were performed using the facilities of Tokyo Mropolitan University and the Supercomputer Center, Institfor Solid State Physics, University of Tokyo.

.

*Email address: otsuka@phys.metro-u.ac.jp1For example, H.G. Evertz, inNumerical Methods for Lattice

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~1969! ~suppl.!; C.M. Fortuin and P.W. Kasteleyn, Physica~Utrecht! 57, 536 ~1972!.

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~1993!.

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Prog. Theor. Phys.56, 1454~1976!.12M. Barma and B.S. Shastry, Phys. Rev. B18, 3351~1978!.13D. Kandel and E. Domany, Phys. Rev. B43, 8539~1991!.14U. Wolff, Phys. Rev. Lett.60, 1461~1988!.15J. des Cloizeaux and M. Gaudin, J. Math. Phys.7, 1384~1966!;

C.N. Yang and C.P. Yang, Phys. Rev.150, 321~1966!; 150, 327~1966!.

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