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arXiv:cond-mat/0610822 · Physical Review Letters

Efficient Algorithm for Random-Bond Ising Models in 2D

Yen Lee Loh, E. W. Carlson

Published Nov 30, 2006• 31 citations
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Abstract

We present an efficient algorithm for calculating the properties of Ising models in two dimensions, directly in the spin basis, without the need for mapping to fermion or dimer models. The algorithm computes the partition function and correlation functions at a single temperature on any planar network of N Ising spins in O(N;{3/2}) time or less. The method can handle continuous or discrete bond disorder and is especially efficient in the case of bond or site dilution, where it executes in O(NlnN) time near the percolation threshold. We demonstrate its feasibility on the ferromagnetic Ising model and the +/-J random-bond Ising model and discuss the regime of applicability in cases of full frustration such as the Ising antiferromagnet on a triangular lattice.

DOI
10.1103/physrevlett.97.227205

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We present an efficient algorithm for calculating the properties of Ising models in two dimensions, directly in the spin basis, without the need for mapping to fermion or dimer models. The algorithm computes the partition function and correlation functions at a single temperature on any planar network of N Ising spins in O(N;{3/2}) time or less. The method can handle continuous or discrete bond disorder and is especially efficient in the case of bond or site dilution, where it executes in O(NlnN) time near the percolation threshold. We demonstrate its feasibility on the ferromagnetic Ising model and the +/-J random-bond Ising model and discuss the regime of applicability in cases of full frustration such as the Ising antiferromagnet on a triangular lattice.
Recently, we developed and implemented the bond-propagation algorithm for calculating the partition function and correlation functions of random-bond Ising models in two dimensions [Y. L. Loh and E. W. Carlson, Phys. Rev. Lett. 97, 227205 (2006)]. The algorithm is the fastest available for calculating these quantities near the percolation threshold. In this paper, we show how to extend the bond-propagation algorithm to directly calculate thermodynamic functions by applying the algorithm to derivatives of the partition function and we derive explicit expressions for this transformation. We also discuss variations of the original bond-propagation procedure within the larger context of $Y\ensuremath{\nabla}Y$ reducibility and discuss the relation of this class of algorithm to other algorithms developed for Ising systems. We conclude with a discussion on the outlook for applying similar algorithms to other models.
We develop a recently proposed mapping of the two-dimensional Ising model with random exchange (RBIM) via the transfer matrix, to a network model for a disordered system of noninteracting fermions. The RBIM transforms in this way to a localization problem belonging to one of a set of nonstandard symmetry classes, known as class D; the transition between paramagnet and ferromagnet is equivalent to a delocalization transition between an insulator and a quantum Hall conductor. We establish the mapping as an exact and efficient tool for numerical analysis: using it, the computational effort required to study a system of width M is proportional to M3,{M}^{3},M3, and not exponential in M as with conventional algorithms. We show how the approach may be used to calculate for the RBIM the free energy, typical correlation lengths in quasi-one dimension for both the spin and the disorder operators, and the even powers of spin-spin correlation functions and their disorder averages. We examine in detail the square-lattice, nearest-neighbor $\ifmmode\pm\else\textpm\fi{}J$ RBIM, in which bonds are independently antiferromagnetic with probability p, and ferromagnetic with probability $1\ensuremath{-}p.$ Studying temperatures T> 0.4J,T>~0.4J,T> 0.4J, we obtain precise coordinates in the $p\ensuremath{-}T$ plane for points on the phase boundary between ferromagnet and paramagnet, and for the multicritical (Nishimori) point. We demonstrate scaling flow towards the pure Ising fixed point at small p, and determine critical exponents at the multicritical point.
Extending the recent work of the bond-propagation algorithm [Y. L. Loh and E. W. Carlson, Phys. Rev. Lett. 97, 227205 (2006)], we present an efficient algorithm for calculating the correlation functions between any two (or more) spins of Ising models in two dimensions. With this algorithm, the spontaneous magnetization, correlation length, and susceptibility can be calculated. The results for the usual Ising model with different size lattices are compared with the exact results. We also present the correlation functions for the $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model.
We study the percolation behavior of in two dimensional Ising model with spontaneous magnetization for pure and diluted lattices containing spins. Expression for partition function is calculated for pure lattice and is applied to find out the same for diluted lattices. Specific example is taken for nearest neighbor interaction in two dimensional square lattices and validity range of the calculation is checked. Behavior of partition function is studied at different temperatures and at different number of spins.
We address the issue of universality in two-dimensional disordered Ising systems, by considering long, finite-width strips of ferromagnetic Ising spins with randomly distributed couplings. We calculate the free energy and spin-spin correlation functions (from which averaged correlation lengths, ${\ensuremath{\xi}}^{\mathrm{av}}$, are computed) by transfer-matrix methods. An ansatz for the size dependence of logarithmic corrections to ${\ensuremath{\xi}}^{\mathrm{av}}$ is proposed. Data for both random-bond and site-diluted systems show that pure system behavior (with $\ensuremath{\nu}=1$) is recovered if these corrections are incorporated, discarding the weak-universality scenario.