The Loschmidt Spectral Form Factor

TL;DR

This paper introduces the Loschmidt Spectral Form Factor, a time-domain correlation measure for Hamiltonian ensembles, revealing exponential sensitivity of spectral details to perturbations and providing analytical calculations across various models.

Contribution

It generalizes the spectral form factor to a Loschmidt version, linking spectral fluctuations to covariance of correlated Hamiltonian ensembles and analyzing its exponential sensitivity.

Findings

Averaged Loschmidt SFF is proportional to e^{iλT}T with complex λ.

Spectral details are exponentially sensitive to perturbations.

Analytical calculations of λ in random matrix, defect, and hydrodynamic models.

Abstract

The Spectral Form Factor (SFF) measures the fluctuations in the density of states of a Hamiltonian. We consider a generalization of the SFF called the Loschmidt Spectral Form Factor, $\textrm{tr}[e^{iH_1T}]\textrm{tr} [e^{-iH_2T}]$, for $H_1-H_2$ small. If the ensemble average of the SFF is the variance of the density fluctuations for a single Hamiltonian drawn from the ensemble, the averaged Loschmidt SFF is the covariance for two Hamiltonians drawn from a correlated ensemble. This object is a time-domain version of the parametric correlations studied in the quantum chaos and random matrix literatures. We show analytically that the averaged Loschmidt SFF is proportional to $e^{i\lambda T}T$ for a complex rate $\lambda$ with a positive imaginary part, showing in a quantitative way that the long-time details of the spectrum are exponentially more sensitive to perturbations than the short-time properties. We calculate $\lambda$ in a number of cases, including random matrix theory, theories with a single localized defect, and hydrodynamic theories.