Ballistic transport in classical and quantum integrable systems

TL;DR

This paper reviews the unique ballistic transport properties of classical and quantum integrable systems, emphasizing the role of conservation laws and analyzing specific models like the Toda chain and a quantum bath.

Contribution

It provides a comprehensive discussion on the mathematical techniques and physical mechanisms underlying ballistic transport in integrable systems, highlighting new insights into their dynamic behavior.

Findings

Conservation laws underpin ballistic transport in integrable systems.

The Toda chain exhibits singular long-time current correlations.

The Drude weight characterizes the ballistic nature of transport.

Abstract

In this essay, we first sketch the development of ideas on the extraordinary dynamics of integrable classical nonlinear and quantum many body Hamiltonians. In particular, we comment on the state of mathematical techniques available for analyzing their thermodynamic and dynamic properties. Then, we discuss the unconventional finite temperature transport of integrable systems using as example the classical Toda chain and the toy model of a quantum particle interacting with a fermionic bath in one dimension; we focus on the singular long time asymptotic of current-current correlations, we introduce the notion of the Drude weight and we emphasize the role played by conservation laws in establishing the ballistic character of transport in these systems.