Converting Lattices into Networks: The Heisenberg Model and Its Generalizations with Long-Range Interactions

TL;DR

This paper transforms lattice models into network representations, solves the Heisenberg model using group theory, generalizes it to new exactly solvable models, and analyzes eigenstate degeneracies related to particle interactions.

Contribution

It introduces a novel network-based approach to lattice models, linking group theory to solve and generalize the Heisenberg model with long-range interactions.

Findings

Higher eigenstate degeneracy with increased particle-pair interactions

Exact solutions for generalized Heisenberg models

Eigenvalue analysis of networks with varying links

Abstract

In this paper, we convert the lattice configurations into networks with different modes of links and consider models on networks with arbitrary numbers of interacting particle-pairs. We solve the Heisenberg model by revealing the relation between the Casimir operator of the unitary group and the conjugacy-class operator of the permutation group. We generalize the Heisenberg model by this relation and give a series of exactly solvable models. Moreover, by numerically calculating the eigenvalue of Heisenberg models and random walks on network with different numbers of links, we show that a system on lattice configurations with interactions between more particle-pairs have higher degeneracy of eigenstates. The highest degeneracy of eigenstates of a lattice model is discussed.