Efficient generation of series expansions for ±J Ising spin glasses in a classical or a quantum field

Abstract

We discuss generation of series expansions for Ising spin glasses with a symmetric $\ifmmode\pm\else\textpm\fi{}J$ (i.e., bimodal) distribution on $d$-dimensional hypercubic lattices using linked-cluster methods. Simplifications for the bimodal distribution allow us to go to higher order than for a general distribution. We discuss two types of problems, one classical and one quantum. The classical problem is that of the Ising spin glass in a longitudinal magnetic field $h$, for which we obtain high temperature series expansions in variables $tanh(J/T)$ and $tanh(h/T)$. The quantum problem is a $T=0$ study of the Ising spin glass in a transverse magnetic field ${h}_{T}$ for which we obtain a perturbation theory in powers of $J/{h}_{T}$. These methods require (i) enumeration and counting of all connected clusters that can be embedded in the lattice up to some order $n$, and (ii) an evaluation of the contribution of each cluster for the quantity being calculated, known as the weight. We discuss a general method that takes the much smaller list (and count) of all no free-end (NFE) clusters on a lattice up to some order $n$ and automatically generates all other clusters and their counts up to the same order. The weights for finite clusters in both cases have a simple graphical interpretation that allows us to proceed efficiently for a general configuration of the $\ifmmode\pm\else\textpm\fi{}J$ bonds and at the end perform suitable disorder averaging. The order of our computations is limited by the weight calculations for the high-temperature expansions of the classical model, while they are limited by graph counting for the $T=0$ quantum system. Details of the calculational methods are presented.