Diagonalization of replicated transfer matrices for disordered Ising spin systems

Abstract

We present an alternative procedure for solving the eigenvalue problem of replicated transfer matrices describing disordered spin systems with (random) 1D nearest neighbor bonds and/or random fields, possibly in combination with (random) long range bonds. Our method is based on transforming the original eigenvalue problem for a $2^n\times 2^n$ matrix (where $n\to 0$) into an eigenvalue problem for integral operators. We first develop our formalism for the Ising chain with random bonds and fields, where we recover known results. We then apply our methods to models of spins which interact simultaneously via a one-dimensional ring and via more complex long-range connectivity structures, e.g. $1+\infty$ dimensional neural networks and `small world' magnets. Numerical simulations confirm our predictions satisfactorily.

24 pages, LaTex, IOP macros

References in corpus (6)

• Replicated Transfer Matrix Analysis of Ising Spin Models on `Small World' Lattices
• Coupled dynamics of fast spins and slow exchange interactions in the XY spin-glass
• Replica structure of one--dimensional Ising models
• Hierarchical Self-Programming in Recurrent Neural Networks
• Attractor Modulation and Proliferation in 1+$\infty$ Dimensional Neural Networks
• A replica approach to products of random matrices

Cited by in corpus (10)

• Entropies of complex networks with hierarchically constrained topologies
• The ordered phase of the one-dimensional Ising spin glass with long-range interactions
• Replica symmetry breaking in the `small world' spin glass
• Spin models on random graphs with controlled topologies beyond degree constraints
• A solvable model of the genesis of amino-acid sequences via coupled dynamics of folding and slow genetic variation
• Dynamic rewiring in small world networks
• On metastable configurations of small-world networks
• One-dimensional disordered Ising models by replica and cavity methods
• Transfer operator analysis of the parallel dynamics of disordered Ising chains
• Statistical mechanics of LDPC codes on channels with memory