IL NUOVO CIMENTO VOL. 105 B, N. 1 Gennaio 1990

Employing the Ising Representation to Implement Nonlocal Monte Carlo Updating in O(N) Models.

A. PATRASCIOIU

Physics Department and Center for the Study of Complex Systems University of Arizona - Tucson, AZ 85721

(ricevuto il 26 Maggio 1989)

Summary. - - O(N) ferromagnets, anisotropic or not, with standard nearest- neighbour interaction are expressed as N Ising models coupled with certain auxiliary fields. A hybrid Monte Carlo algorithm is then developed. It consists of global updates of the Ising variables using the Fortuin-Kasteleyn representation, combined with ordinary heat-bath or Metropolis updates of the remaining variables. The method generalizes to Z(2N) models. A numerical test for Z(8) at ~ = 1.1 is reported. It reveals a dramatic reduction in critical slowing-down.

PACS 02.50 - Probability theory, stochastic processes and statistics. PACS 05.50 - Lattice theory and statistics; Ising problems. PACS l l . 1 5 . H a - Lattice gauge theory.

Consider an O(N) spin. I t can be specified as

(1) S = (S1, S2,..., SN), N

where - 1 < Si < 1 and ~ S ~ = 1. An a l ternat ive specification of the same spin is possible as 1

(2) S = (zlAl, z~A2... r

where now zj = + 1 and 0 < A j< 1. Thus a set of N Ising-like var iables can natural ly be identified. I t is instruct ive to express the Gibbs factor in t e r m s of

the new variables in eq. (2). F o r ordinary neares t -ne ighbour interaction, the

91

92 A. PATRASCIOIU

partition function on a cubic lattice A reads

(3) Z = ~ dAx,~ A~,~- 1 exp ~ ~A~,~Au,~,~% ~ . {ax,i=• {oc,y) 1

As can be seen from eq. (3), the original O(N) model can be expressed as a system consisting of N Ising magnets interacting with the field A. However, because of the range of A, the N Ising systems are always ferromagnetic, hence amenable to the Fortuin-Kasteleyn (FK)[1] treatment, which relates the Potts model to a well-defined percolation problem. This representation was used by Sweeny [2] to study critical behaviour in the Potts model. One of his remarkable finds was that this percolation approach was not affected by critical slowing- down. This observation was quantified ulteriorly by Swendsen and Wang [3] who found that this updating procedure leads to a correlation time growing as L ~ rather than the usual L 2.

The strategy to be adopted becomes therefore clear: in a Monte Carlo study of the O(N) spin system with nearest-neighbour interaction, the Ising-like variables will be updated in a global fashion using the FK prescription, while the remaining variables will be updated in a local fashion, using the heat-bath or Metropolis algorithm. The FK prescription, though nonlocal, is very simple and fast to implement. At given At, to update ~j one identifies all connected sets of occupied links: a link is occupied only if the two ~j's at its ends are equal and then, only with probability 1-exp[-2flAx.jAy,j]. Each set of Ising variables connected by occupied links is now flipped with probability 1/2. The procedure is repeated sequentially for all the N Ising-like variables, after which, leaving them untouched, one updates in a local fashion the Aj's. In applying these ideas to the study of the model Z(8) in two dimensions (2D), to be reported below, I found it convenient to perform several (30) heat-bath updates of the Aj's for each update of the Ising-like variables. This may generally be the case, since as recalled above, the FK procedure has much shorter correlation time than ordinary local updating. It is obvious that the proposed Monte Carlo updating scheme obeys detailed balance and is ergodic. As my numerical results for Z(8) illustrate, the algorithm produces very fast convergence of typical observables (susceptibility, disorder parameter) and reduces tremendously critical slowing-down.

The idea described above generalizes immediately to anisotropic O(N) models (in eq. (3) ~--->fii). Also to certain discrete models such as Z(2N) or Z(4N), N = 1, 2 ... In the first case one introduces one set of Ising variables and restricts 0 < r < =, while in the second one there are two sets of Ising variables being used and 0 < r < =/2. This latter is the scheme I adopted to investigate Z(4) and Z(8) in 2D. For Z(4), which with this standard action is nothing but two decoupled Ising models, one knows rigorously that tim = - log (~r~ _ 1) and that v = 0.25. Both statements were verified first by varying ~ around 0.88 and monitoring

EMPLOYING T H E ISING R E P R E S E N T A T I O N ETC. 93

(S>, then by measuring the dependence of the susceptibility Z on the linear size of the lattice L and seeing that it obeys the behavior z = cL175 a t ~crt.

For Z(8), based on recent estimates of Tc for 0(2) [4, 5], one expects ~ ~ 1.1. While I made no attempt to determine precisely T~, the observed behavior of the disorder parameter is consistent with ~crt ~ 1.1. Following the ideas expressed in a previous paper [6], I define the disorder parameter as

(4) < [ LJ~ 7\71/2

D S - exp[-2ZEoSi,oS~,lJ)J �9

This observable plays the role of (S> for the dual system. It should vanish in the KT phase and become nonzero in the massive high-temperature phase. It is a notoriously impossible quantity to measure with ordinary Monte Carlo updating techniques, especially around To. In fig. 1 I present the values measured with

8.C

6 .C L1 0 186

c~ 4.0 c

B

2.0

I l I I ~, 0 10 20 30 40 L

Fig. 1. - The dependence of the disorder variable eq. (4) upon L for Z(8) at ~ = 1.1.

the present algorithm, as a function of L, at ~ = 1.1; they seem to fall on a perfect exponential. Figure 2 displays the behavior of the susceptibility Z:

with L at ~ = 1.1 for Z(8). Defining the critical exponent v as

(6) Z = c L 2 - ~ ' ,

the numerical results verify this relation with ~ = 0.246 + 0.002. In the same fig. 2, one sees the behaviour of the correlation time ~ for the susceptibility z (eq. (5)), function of L at ~ = 1.1. The data suggest

(7) ~ = c L z , z = 0.75.

94 A. PATRASCIOIU

6.(:

r-

5.0

4..0

\ \

\ \

j /

/ /

I I, nlO

J / X~ L 1.75

.Q /

/

~ v ~L 0.75

\

bn20 I,,n 30 bn40 bn

1.0

I

D.5

Fig. 2. - The dependence of the susceptibility z and its correlation time ~ upon L for Z(8) at ~= 1.1.

The numerical value of z is roughly twice the value measured by Swendsen and Wang [3] for the Ising model. It is difficult to obtain an accurate value for z because the correlation times are very short (even for L = 40, much smaller than 1).

Based on the above reported numerical experiments, I conclude that the proposed algorithm could successfully be employed to investigate critical behavior in O(N) and Z(N) ferromagnets with or without anisotropy (in any dimension). Another nonlocal updating algorithm for studying isotropic O(N) models, whose connection to the FK representation is less clear, has been proposed [7] and utilised recently by Wolff to study the 0(2) model [5]. That algorithm proved clearly superior to recent variations of Adler's overrelaxation method [4], which, being a locally updating scheme, is in my opinion of limited usefulness in investigating critical behaviour. I do not know how the algorithm proposed in this paper compares to Wolffs. The latter was reported displaying slower convergence for local quantities, such as the energy density. The present algorithm, with its mixture of local (heat-bath) and nonlocal updates, has no such problems. It also seems to display a faster convergence rate per spin update. It remains to be seen if the same is true when one compares the CPU time needed to achieve a certain accuracy.

I am grateful for beneficial discussions with J.-L. Richard and E. Seiler and to C. Newman for emphasizing the usefulness of the Fortuin-Kasteleyn representation. The numerical work was performed at the John von Neumann Supercomputer Center through a grant from NSF.

EMPLOYING THE ISING REPRESENTATION ETC. 95

R E F E R E N C E S

[1] C. M. FORTUIN and P. W. KASTELEYN: J. Phys. Soc. Jpn. Suppl., 26, 86 (1987). [2] M. SWEENY: Phys. Rev. B, 27, 4445 (1983). [3] R. H. SWENDSEN and J.-S. WANG: Phys. Rev. Lett., 58, 86 (1987). [4] R. GUPTA, J. DELAPP, G. BATROUNI, G. C. Fox, C. F. BAILLIE and J. APOSTO-

LAKIS: Phys. Rev. Lett., 61, 1996 (1988). [5] U. WOLFF: Collective Monte Carlo updating in a high precision study of the X-Y

model, preprint University of Kiel (1988). [6] A. PATRASCIOIU: Phys. Rev. Lett., 58, 2285 (1987). [7] V. WOLFF: Phys. Rev. Lett., 62, 361 (1989).

�9 R I A S S U N T O (*)

Si esprimono ferromagneti O(N), anisotropici o no, con interazione standard dei vicini prossimi come modelli di Ising N accoppiati a certi campi ausiliari. Si sviluppa poi un algoritmo ibrido di Monte Carlo. Esso consiste di aggiornamenti globali delle variabili di Ising usando la rappresentazione di Fortuin-Kasteleyn combinati con aggiornamenti di Metropolis o del bagno di calore ordinari. I1 metodo si generalizza ai modelli Z(2N). Si riporta un test numerico per Z(8) con ~ = 1.1. Esso rivela una riduzione drammatica nel rallentamento critico.

(*) Traduzione a cura della Redazione.

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(*) Flepe~eikeuo peOaKt~uegt.

7 - Il Nuovo Cimento B.