Semi-classical Analysis of Spin Systems near Critical Energies

TL;DR

This paper develops a semi-classical method to analyze the spectral properties of $su(2)$ Hamiltonians near critical energies where classical dynamics exhibit hyperbolic points, providing algebraic relations for eigenvalues and observable matrix elements.

Contribution

The paper introduces a novel semi-classical approach to determine eigenvalues and matrix elements near critical energies in spin systems, validated by numerical comparisons.

Findings

Eigenvalue relations near critical energies derived analytically.

Agreement between semi-classical predictions and numerical results.

Observable matrix elements computed and validated against numerics.

Abstract

The spectral properties of $su(2)$ Hamiltonians are studied for energies near the critical classical energy $\epsilon_c$ for which the corresponding classical dynamics presents hyperbolic points (HP). A general method leading to an algebraic relation for eigenvalues in the vicinity of $\epsilon_c$ is obtained in the thermodynamic limit, when the semi-classical parameter $n^{-1}=(2s)^{-1}$ goes to zero (where $s$ is the total spin of the system). Two applications of this method are given and compared with numerics. Matrix elements of observables, computed between states with energy near $\epsilon_c$, are also computed and shown to be in agreement with the numerical results.