Quantum Ising model in transverse and longitudinal fields: chaotic wave functions

TL;DR

This paper develops a statistical model for eigenfunctions of the chaotic quantum Ising model in transverse and longitudinal fields, accounting for corrections from higher moments and symmetry effects, and validates it against numerical data.

Contribution

It introduces a detailed statistical model for chaotic eigenfunctions of the quantum Ising model, including symmetry and higher-moment corrections, with validation against numerical results.

Findings

Wave function coefficients follow Gaussian distribution in the large spin limit.

Corrections from higher moments and symmetry significantly improve model accuracy.

Model agrees well with numerical calculations of wave function moments.

Abstract

The construction of a statistical model for eigenfunctions of the Ising model in transverse and longitudinal fields is discussed in detail for the chaotic case. When the number of spins is large, each wave function coefficient has the Gaussian distribution with zero mean and the variance calculated from the first two moments of the Hamiltonian. The main part of the paper is devoted to the discussion of different corrections to the asymptotic result. One type of corrections is related with higher order moments of the Hamiltonian and can be taken into account by Gibbs-like formulae. Another corrections are due to symmetry contributions which manifest as different numbers of non-zero real and complex coefficients. Statistical model with these corrections included agrees well with numerical calculations of wave function moments.