The Thermodynamic Limit of Spin Systems on Random Graphs

TL;DR

This paper develops a continuous framework using graphons to analyze quantum and classical spin systems on dense graphs, deriving integral equations that describe their thermodynamic properties and validating results with numerical methods.

Contribution

It introduces a novel graphon-based approach to describe spin systems in the thermodynamic limit, providing exact integral equations and solutions for complex models.

Findings

Derived coupled non-linear Fredholm integral equations for spin systems.

Obtained analytical solutions for various quantum and classical models.

Numerical results converge to analytical predictions with increasing system size.

Abstract

We utilise the graphon--a continuous mathematical object which represents the limit of convergent sequences of dense graphs--to formulate a general, continuous description of quantum spin systems in thermal equilibrium when the average co-ordination number grows extensively in the system size. Specifically, we derive a closed set of coupled non-linear Fredholm integral equations which govern the properties of the system. The graphon forms the kernel of these equations and their solution yields exact expressions for the macroscopic observables in the system in the thermodynamic limit. We analyse these equations for both quantum and classical spin systems, recovering known results and providing novel analytical solutions for a range of more complex cases. We supplement this with controlled, finite-size numerical calculations using Monte-Carlo and Tensor Network methods, showing their convergence towards our analytical results with increasing system size.