Diagonalization of replicated transfer matrices for disordered Ising spin systems

Details Diagonalization of replicated transfer matrices for disordered Ising spin systems Diagonalization of replicated transfer matrices for disordered Ising spin systems

2004-05-12

arxiv.org/abs/cond-mat/0405269v2

Physics

Condensed Matter

Disordered Systems and Neural Networks

24 pages, LaTex, IOP macros

Scientific paper

10.1088/0305-4470/37/35/003

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.

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