VOLUME 65 20 AUGUST 1990 NUMBER 8 Cluster Dynamics for Fully Frustrated Systems Daniel Kandel, Radel Ben-Av, and Eytan Domany Electronics and Physics Departments, Weizmann Institute of ScienceRe, hovot 76100, Israel (Received 8 June 1990) We present a novel cluster algorithm for Monte Carlo simulations of the fully frustrated Ising model on the square lattice. The new method does not suffer from problems of metastability, and is extremely e%cient even at T 0. Our algorithm is a special case of a more general Monte Carlo simulation scheme. The general scheme unifies many cluster algorithms that were developed recently in order to accelerate Monte Carlo simulations. PACS numbers: 05.50.+q, 75.40. Mg The work of Swendsen and Wang' (SW) on accelera- tion of simulations of ferromagnetic Potts models opened a new field of interest in computational physics. The im- proved efficiency of their cluster algorithm gave hope that similar methods may be used to accelerate simula- tions of other systems, for which standard techniques are very inefficient. Indeed, generalizations appeared soon after the work of SW was published. Clearly, one needs different cluster algorithms to ac- celerate simulations of different models and physical sys- tems. A most important unsolved problem is that of simulating models with competing interactions and frus- tration. All known algorithms ' become inefficient when competing interactions are introduced. They fail to identify the "correct" clusters, and in most cases al- most all of the lattice ends up in the same cluster, lead- ing to a trivial move. Thus, it is still very difficult, if not impossible, to perform simulations of spin glasses at low temperatures. Many optimization problems (e. g. , the wiring problem in computer design, the problem of finding the location of atoms in the unit cell of a crystal from x-ray scattering information, ' etc. ) fall into this class of models with frustration. Simulated annealing, which is the most efficient method for such problems, also suffers from severe slowing down. The "replica" Monte Carlo algorithm" of SW improves simulations of the two-dimensional Ising spin glass, but is not as effec- tive in other cases. For example, it is much less efficient than the algorithm we present here for the fully frustrat- ed Ising model on the square lattice. This Letter makes a first step towards solving some of the problems in the simulation of frustrated systems. We propose a novel cluster algorithm for simulating the fully frustrated Ising model on the square lattice. We show that it is extremely efficient even at T=O, and does not have metastable states; hence we move between ground states of the model without simulated annealing. The paper is organized as follows: First, we describe the algorithm in detail, and compare its performance at T=O to that of Metropolis et al. '2 We find that while typical time scales of the Metropolis algorithm diverge very strongly as a function of system size, our algorithin does not suffer from significant slowing down. Conse- quently, we can easily measure the dependence of the magnetic susceptibility on system size, and confirm that at T =0 the system behaves as a ferromagnet at criticali- ty. As we increase L, the linear size of the system, the susceptibility g diverges as X-L ", where ri= —, ', in agreement with exact results. ' We also give an intuitive explanation for the extraordinary performance of the al- gorithm. Last, we demonstrate that our new method is a special case of a more general cluster Monte Carlo tech- nique. The general scheme includes as special cases oth- er previously developed cluster algorithms. ' We show that this general method satisfies the detailed bal- ance condition, and conclude that our algorithm is a legi- timate Monte Carlo procedure. Description of the model and the algorithm We. — 1990 The American Physical Society 941