Thermalisation as Diffusion in Hilbert Space

Abstract

We develop a microscopic theory of thermalisation for a thermometer coupled to a many-body bath beyond standard Markovian and Fermi-golden-rule assumptions. By modeling interaction matrix elements in the non-interacting basis as independent random variables, we derive a diffusion-propagator expression for the reduced dynamics and show that relaxation is controlled by the distribution of interaction-induced level broadenings. The theory predicts a thermalisation timescale set by the inverse typical broadening and yields a non-Markovian generalization of global balance. Exact-diagonalization tests for heavy-tailed L{é}vy couplings, an all-to-all transverse-field Ising model, and the one-dimensional Imbrie model show good agreement with these predictions.

Figures (5)

• Figure 1: Diagrammatic representation of the cavity equation for the self-energy. Note the similarity to the standard non-crossing approximation, where the dashed line denotes a pairing according to Wick's theorem. Here, the crossed line indicates that we only consider terms where two perturbation-matrix elements are complex conjugates.
• Figure 2: Diagrammatic representation of the leading contribution to the propagator from Eq. (\ref{['eq: propagator definition']}). The argument of the top Green function is $\epsilon + i \varkappa/2$, whereas the argument of the bottom Green function is $\epsilon - i \varkappa/2$.
• Figure 3: The Laplace transform of the $zz$ autocorrelation function for the Lévy model is plotted for different bath sizes $N \equiv 2^L$. The solid lines represent the direct evaluation of Eq. (\ref{['eq: averaged propagator']}). The dashed lines represent the theoretical evaluation of Eq. (\ref{['eq: main spin']}). We use the parameters $\mu = 1.5$, $\Delta = 0.9$, and $g = 0.4$. All data are averaged over at least 30 realizations.
• Figure 4: The Laplace transform of the $zz$ autocorrelation function for the TFIM is plotted for different bath sizes $L$. The solid lines represent the direct evaluation of Eq. (\ref{['eq: averaged propagator']}). The dashed lines represent the theoretical evaluation of Eq. (\ref{['eq: main spin']}). We use the parameters $\Delta = 0.9$ and $g = 0.4$. All data are averaged over at least 30 realizations.
• Figure 5: The Laplace transform of the $zz$ autocorrelation function for the Imbrie model is plotted for different bath sizes $L$. The solid lines represent the direct evaluation of Eq. (\ref{['eq: averaged propagator']}). The dashed lines represent the theoretical evaluation of Eq. (\ref{['eq: main spin']}). We use the parameters $\Delta = 0.9$, $g = 0.4$, and $W = 2$. All data are averaged over at least 30 realizations.