Efficient generation of series expansions for $\pm J$ Ising spin-glasses in a classical or a quantum (transverse) field

TL;DR

This paper develops efficient linked-cluster methods to generate high-order series expansions for $\

Contribution

It introduces a simplified approach for generating series expansions of bimodal $"+ J\

Findings

Higher order series expansions achieved for bimodal distributions.

A general method for enumerating clusters up to a certain order.

Efficient calculation of weights using graphical interpretation.

Abstract

We discuss generation of series expansions for Ising spin-glasses with a symmetric $\pm J$ (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 problem, 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 \textit{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 $\pm J$ 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.