On generalized cluster algorithms for frustrated spin models

P.D. Coddington and L. Han

Physics Department, Syracuse University, Syracuse, NY 13244, USA (Received 1 January 1992)

Cluster update algorithms have proved to be very effective for many different spin models, however they fail for frustrated spin systems. Recently a generalized cluster algorithm was introduced which works extremely well for the fully frustrated Ising model on a square lattice. However we have applied this method to the triangular and cubic lattice fully frustrated Ising model and found that it works poorly, producing clusters which are too large. PACS numbers 05.50.+q, 75.40.Mg

I. INTRODUCTION

The cluster update algorithm of Swendsen and Wang [?] provides a great improvement in the efficiency of Monte Carlo simulations over the Metropolis algorithm [?] and other local update algorithms by dramatically reducing critical slowing down [?,?] in certain ferromagnetic and anti-ferromagnetic spin models. However the Swendsen-Wang algorithm fails for frustrated spin models, which are the systems for which critical slowing down is the most extreme. The reason for the failure of the algorithm is that the clusters which are created are much too large at the critical point, with the algorithm often producing a cluster which encompasses a large fraction (if not all) of the sites in the lattice, so that flipping the spins in this cluster is virtually the same as a trivial global spin flip.

Recently Kandel, Ben-Av and Domany (KBD) introduced a generalized cluster algorithm [?,?] which works very well for the two dimensional fully frustrated Ising model [?].

In section 2 we present the method we used for implementing the KBD general cluster method for a particular spin model. We apply this to the fully frustrated Ising model on a triangular lattice (section 3) and a cubic lattice (section 4).

II. FORMULATING A GENERAL CLUSTER

ALGORITHM

III. THE TRIANGULAR LATTICE ISING

ANTIFERROMAGNET

We first attempted to generalize the KBD cluster algorithm to a very simple fully frustrated model, the Ising anti-ferromagnet on a triangular lattice. This model is very similar to the fully frustrated Ising model on a square lattice, being in the same universality class and also having a critical point at zero temperature [?,?].

IV. THE 3-D FULLY FRUSTRATED ISING MODEL

We next attempted to apply the generalized cluster Monte Carlo idea to the three dimensional fully frustrated Ising model [?,?,?]. This is

V. CONCLUSIONS

ACKNOWLEDGMENTS

We would like to thank Ofer Biham, Danny Kandel and Enzo Marinari for valuable discussions. This work was sponsored in part by Department of Energy grants DE-FG03-85ER25009 and DE-AC03-81ER40050, and by a grant from the IBM corporation.

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TABLE I. The Poisson ratio defined as the ratio of lateral contraction to longitudinal expansion for uniaxial stress. Experimental values are given for comparison.

σ  σ

Predicted  Observed^{a}  Predicted  Observed^{a}
Cu  0.48  0.36  Al  0.47  0.33
Ag  0.48  0.37  Tl  0.47  0.35
Au  0.48  0.36  Sn  0.46  0.33
Mg  0.47  0.35  Pb  0.46  0.40–0.45
Zn  0.47  0.25  Pb  0.49  0.43
K  0.49  0.44

aHere we also cite a reference