Breakdown of Fermi's Golden Rule in 1d systems at non-zero temperature

TL;DR

This paper investigates the decay of quasiparticles in 1d quantum systems at finite temperature, revealing a logarithmic correction to the decay rate predicted by Fermi's Golden Rule, using numerical and diagrammatic methods.

Contribution

It introduces a combined numerical and analytical approach to resolve the divergence in quasiparticle decay rates in 1d systems, predicting a logarithmic enhancement.

Findings

Quasiparticle decay rate scales as Δ² log(1/Δ²) in 1d systems.

Numerical simulations confirm the logarithmic correction to FGR.

The effect applies broadly to various quantum and classical systems with quasiparticles.

Abstract

In interacting quantum systems, the single-particle Green's function is expected to decay in time due to the interaction induced decoherence of quasiparticles. In the limit of weak interaction strengths ($\Delta$), a naive application of Fermi's Golden Rule (FGR) predicts an $\mathcal{O}(\Delta^{2})$ quasiparticle decay rate. However, for 1d fermions on the lattice at $T>0$, this calculation gives a divergent result and the scaling of the quasiparticle lifetime with interaction strength remains an open question. In this work we propose a solution to this question: combining numerical simulations using the recently introduced dissipation-assisted operator evolution (DAOE) method, with non-perturbative diagrammatic re-summations, we predict a logarithmic enhancement of the quasiparticle decay rate $\tau^{-1} \sim \Delta^{2} \log \Delta^{-2}$. We argue that this effect is present in a wide variety of well-known weakly interacting quantum fermionic and bosonic systems, and even in some classical systems, provided the non-interacting limit has quasiparticles with a generic dispersion.