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.