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arXiv:cond-mat/0703043 · Physical Review B

Bond-propagation algorithm for thermodynamic functions in general two-dimensional Ising models

Yen Lee Loh, E. W. Carlson, M. Y. J. Tan

Published Jul 5, 2007• 20 citations
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Abstract

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.

DOI
10.1103/physrevb.76.014404

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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∇YY\ensuremath{\nabla}YY\ensuremath∇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.
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 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.
It was recently shown [Phys. Rev. Lett. 110, 227201 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.227201] that the critical behavior of the random-field Ising model in three dimensions is ruled by a single universality class. This conclusion was reached only after a proper taming of the large scaling corrections of the model by applying a combined approach of various techniques, coming from the zero- and positive-temperature toolboxes of statistical physics. In the present contribution we provide a detailed description of this combined scheme, explaining in detail the zero-temperature numerical scheme and developing the generalized fluctuation-dissipation formula that allowed us to compute connected and disconnected correlation functions of the model. We discuss the error evolution of our method and we illustrate the infinite limit-size extrapolation of several observables within phenomenological renormalization. We present an extension of the quotients method that allows us to obtain estimates of the critical exponent α of the specific heat of the model via the scaling of the bond energy and we discuss the self-averaging properties of the system and the algorithmic aspects of the maximum-flow algorithm used.
A recently introduced cluster acceleration algorithm for spin models [J. Machta et al., Phys. Rev. Lett. 75, 2792 (1995)] based on invasion percolation is generalized to temperatures away from the critical point using concepts of random bond percolation. The generalized algorithm is equivalent to an "optimized" version of the Swendsen-Wang cluster algorithm [R. H. Swendsen and J. S. Wang, Phys. Rev. Lett. 58, 86 (1987)], and we demonstrate its success and speed at all temperatures for the d=2d=2d=2 Ising model. We argue that the previously discussed connection to self-organized criticality may be viewed as one limit of the fixed-cluster algorithm.
This is a continuation of an effort to reduce to the solution of a characteristic value problem the rigorous calculation of thermodynamic properties of systems in which the intermolecular forces are sufficiently short ranged so that practically the entire potential energy of the system results from interactions between nearest neighbors. The partition function of such a system can be expressed in terms of the largest characteristic value of a linear operator equation and finally as the ratio of partition functions of systems with relatively few particles whose potential energy functions differ somewhat from those in the original system. A method of evaluating grand partition functions and one of introducing interactions between more distant neighbors are discussed. The general theory is applied to the calculation of magnetization, internal energy, and specific heat of two-dimensional ferromagnets on the basis of the Ising model. There seems to exist a λ-point phase transition in the change from ferromagnetic to nonferromagnetic states.
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.