Computing spectral bounds of the Heisenberg ferromagnet from geometric considerations

TL;DR

This paper presents a polynomial-time algorithm to compute upper bounds on low-energy eigenvalues of the Heisenberg ferromagnet using graph geometric properties, and explores methods for estimating lower bounds through isoperimetric inequalities.

Contribution

It introduces a novel connection between Heisenberg models and graph symmetric products, enabling efficient eigenvalue bounds computation via combinatorial optimization.

Findings

Upper bounds on energy eigenvalues can be computed using generalized diameters.

Lower bounds relate to isoperimetric inequalities of symmetric graph products.

Potential polynomial-time algorithms depend on solving the edge-isoperimetric problem.

Abstract

We give a polynomial-time algorithm for computing upper bounds on some of the smaller energy eigenvalues in a spin-1/2 ferromagnetic Heisenberg model with any graph $G$ for the underlying interactions. An important ingredient is the connection between Heisenberg models and the symmetric products of $G$. Our algorithms for computing upper bounds are based on generalized diameters of graphs. Computing the upper bounds amounts to solving the minimum assignment problem on $G$, which has well-known polynomial-time algorithms from the field of combinatorial optimization. We also study the possibility of computing the lower bounds on some of the smaller energy eigenvalues of Heisenberg models. This amounts to estimating the isoperimetric inequalities of the symmetric product of graphs. By using connections with discrete Sobolev inequalities, we show that this can be performed by considering just the vertex-induced subgraphs of $G$. If our conjecture for a polynomial time approximation algorithm to solve the edge-isoperimetric problem holds, then our proposed method of estimating the energy eigenvalues via approximating the edge-isoperimetric properties of vertex-induced subgraphs will yield a polynomial time algorithm for estimating the smaller energy eigenvalues of the Heisenberg ferromagnet.