We propose an efficient Monte Carlo algorithm for simulating a ``hardly-relaxing" system, in which many replicas with different temperatures are simultaneously simulated and a virtual process exchanging configurations of these replica is introduced. This exchange process is expected to let the system at low temperatures escape from a local minimum. By using this algorithm the three-dimensional $\pm J$ Ising spin glass model is studied. The ergodicity time in this method is found much smaller than that of the multi-canonical method. In particular the time correlation function almost follows an exponential decay whose relaxation time is comparable to the ergodicity time at low temperatures. It suggests that the system relaxes very rapidly through the exchange process even in the low temperature phase.Comment: 10 pages + uuencoded 5 Postscript figures, REVTe

We analyze models of spin glasses on the two-dimensional square lattice by exploiting symmetry arguments. The replicated partition functions of the Ising and related spin glasses are shown to have many remarkable symmetry properties as functions of the edge Boltzmann factors. It is shown that the applications of homogeneous and Hadamard inverses to the edge Boltzmann matrix indicate reduced complexities when the elements of the matrix satisfy certain conditions, suggesting that the system has special simplicities under such conditions. Using these duality and symmetry arguments we present a conjecture on the exact location of the multicritical point in the phase diagram. Let us start by considering the Hamiltonian of the ±J Ising modelwhere J ij is ferromagnetic J(> 0) with probability p and antiferromagnetic −J with 1 − p. The sum extends over nearest neighbours on the square lattice. The randomness in J ij will be treated by the replica method. We will mainly consider the case of positive integer n, the number of replicas. Some results for the quenched limit n → 0 will be discussed in subsequent sections. The constraints of Ising spins and the ±J distribution of the interactions will be relaxed later in the present section.

Determination of the precise location of the multicritical point and phase boundary is a target of active current research in the theory of spin glasses. [1][2][3][4][5][6][7][8][9][10][11] In this short note we develop a duality argument to predict the location of the multicritical point and the shape of the phase boundary in models of spin glasses on the square lattice.The first system we treat is a random Z q model with gauge symmetry which includes the AEJ Ising model and the Potts gauge glass. Following the notation of Wu and Wang, 12) the partition function isis the quenched randomness, and VðÁÞ is a periodic function with period q. The sum in the exponent runs over neighbouring sites on the square lattice. It is straightforward to generalize the duality transformation in the Wu-Wang formalism 12) to the random model, and the result iswhere ij denotes i À j and ij is the bond variable on the dual lattice corresponding to ij and eṼ V is the Fourier transform of e V . The prime in the sum indicates that the bond variables ij and ij are constrained such that their sums over a closed loop vanish.12) Thus the duality transformation consists in replacing the exponential in (2) with that in (3):Average over quenched randomness is dealt with by the replica method. A generalization of the existing argument 13) suggests that it is convenient to consider the following quantity

An efficient Monte Carlo algorithm for simulating hardly-relaxing systems is proposed. By using this algorithm the three-dimensional ± J Ising spin glass model is studied. The result shows that reasonable values of the critical temperature and of the critical exponents can be obtained within Monte Carlo steps much shorter than the observation time a conventional simulation usually requires.

We study bond percolations on hierarchical scale-free networks with the open bond probability of the shortcutsp and that of the ordinary bonds p. The system has a critical phase in which the percolating probability P takes an intermediate value 0 < P < 1. Using generating function approach, we calculate the fractal exponent ψ of the root clusters to show that ψ varies continuously withp in the critical phase. We confirm numerically that the distribution ns of cluster size s in the critical phase obeys a power law ns ∝ s −τ , where τ satisfies the scaling relation τ = 1 + ψ −1 . In addition the critical exponent β(p) of the order parameter varies asp, from β ≃ 0.164694 atp = 0 to infinity atp =pc = 5/32.