Non-equilibrium 1D many-body problems and asymptotic properties of Toeplitz determinants

TL;DR

This paper extends non-equilibrium bosonization to hard-core bosons, analyzing Toeplitz determinants to understand long-time asymptotics, dephasing, and power-law behavior in interacting quantum systems out of equilibrium.

Contribution

The authors generalize non-equilibrium bosonization to Tonks-Girardeau gases and connect various problems through Toeplitz determinants, providing new insights into their asymptotic behavior.

Findings

Dephasing rates depend on interaction strength and non-equilibrium state.

Power-law scaling exponents are influenced by system parameters.

Long-time asymptotics are characterized using Szegő and Fisher-Hartwig theorems.

Abstract

Non-equilibrium bosonization technique facilitates the solution of a number of important many-body problems out of equilibrium, including the Fermi-edge singularity, the tunneling spectroscopy and full counting statistics of interacting fermions forming a Luttinger liquid. We generalize the method to non-equilibrium hard-core bosons (Tonks-Girardeau gas) and establish interrelations between all these problems. The results can be expressed in terms of Fredholm determinants of Toeplitz type. We analyze the long time asymptotics of such determinants, using Szeg\H{o} and Fisher-Hartwig theorems. Our analysis yields dephasing rates as well as power-law scaling behavior, with exponents depending not only on the interaction strength but also on the non-equilibrium state of the system.