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 neighbour 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 × 2 n matrix (where n → 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 onedimensional ring and via more complex long-range connectivity structures, e.g., (1 + ∞)-dimensional neural networks and 'small-world' magnets. Numerical simulations confirm our predictions satisfactorily.

Ordered phase of the one-dimensional Ising spin glass with long-range interactions

“…The 2 n × 2 n matrix K ab is readily diagonalized 18,19 . In general there are n + 1 distinct eigenvalues, whose degeneracies are 1, n, n(n − 1)/2, · · · , n p , · · · , 1.…”

Transfer operator analysis of the parallel dynamics of disordered Ising chains

“…Expression (8) is for N → ∞ dominated by the largest eigenvalue λ max of (9), provided its spectrum is discrete at λ max . In an equilibrium replica analysis [9,10] the relevant kernel would have replicated spins as arguments; here the arguments are spin 'paths', field 'paths' and conjugate field 'paths' through time. The fields θ i and ψ i were only introduced for generating perturbations, so we may expand Z[ψ] in powers of these fields.…”

Metastable configurations of small-world networks

“…The Kronecker deltas on the right-hand side of the above expressions impose the transformation (14). Similarly to equation (13) we now havê…”