Thermalisation as Diffusion in Hilbert Space

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.

💡 Research Summary

The paper presents a microscopic theory of thermalisation for a small quantum “thermometer” coupled to a many‑body bath that goes beyond the usual Markovian master‑equation and Fermi‑golden‑rule (FGR) approximations. The authors model the interaction matrix elements (V_{\text{int}}) in the non‑interacting basis as independent random variables, an assumption that can be viewed as an extension of the eigenstate thermalisation hypothesis (ETH) to the interaction operator itself. This allows them to treat situations where the distribution of matrix elements is heavy‑tailed and may lack a finite second moment, a regime where standard FGR fails.

Starting from the full Hamiltonian ( \hat H = \hat H_T + \hat H_B + \hat V_{\text{int}} ), they express the reduced diagonal density‑matrix elements of the thermometer, (p_{\alpha\beta}(t)), in terms of products of Green’s functions of the combined system. By focusing on long‑time contributions (poles in opposite half‑planes) they introduce a diffusion‑propagator \