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Physical Review B

Efficient algorithm for computing correlation functions of the two-dimensional random-bond Ising model

Xintian Wu, Juan Zhao

Published Sep 3, 2009• 6 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.80.104402

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•Dec 1, 2009•0 citations
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±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ random-bond Ising model.
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 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.
The self-consistent approach to the two-dimensional Ising model with quenched random bonds is extended to the full lattice theory described by four real fermions. A calculation of the averaged spin-spin correlation function for large separation of the spins in the disorder-dominated phase indicates an exponential decay of this quantity and therefore a vanishing spontaneous magnetization. The corresponding correlation length is proportional to 1/ eta 2 , where eta denotes the order parameter of the new phase introduced by Ziegler (1990).
The energy-energy correlation function of the two-dimensional Ising model with weakly fluctuating random bonds is evaluated in the large scale limit. Two correlation lengths exist in contrast to one correlation length in the pure 2D Ising model: one is finite whereas the other is divergent at the critical points. The corresponding exponent of the divergent correlation length is νe=1/2ν_e=1/2νe​=1/2 in contrast to the pure system where νe=1ν_e=1νe​=1. The calculation is based on a previously developed effective field theory for the energy density fluctuations.
We calculate the two-point correlation function and magnetic susceptibility in the anisotropic 2D Ising model on a lattice with one infinite and the other finite dimension, along which periodic boundary conditions are imposed. Using exact expressions for a part of lattice form factors, we propose the formulas for arbitrary spin matrix elements, thus providing a possibility to compute all multipoint correlation functions in the anisotropic Ising model on cylindrical and toroidal lattices. The scaling limit of the corresponding expressions is also analyzed.
Abstract This chapter summarizes the many results of the exactly solved Ising model in two dimensions at H = 0 including: the partition function, free energy, spontaneous magnetization, the Toeplitz determinant representations of the correlation functions, the Painlevé VI differential equation for the diagonal two spin correlations, the form factor expansion of the magnetic susceptibility, the scaling limit, the boundary properties of the half plane lattice, and the random layered lattice. For H ≠ 0 the Lee-Yang circle theorem, the free energy and magnetization for H/kBT = iπ/2 expansions for small H, the results of Zamolodchikov at T = Tc and a discussion of extended analyticity are given.