Lower bounds for ballistic current and noise in non-equilibrium quantum steady states

TL;DR

This paper establishes lower bounds on steady-state current and noise in non-equilibrium quantum systems, revealing fundamental limits of ballistic transport and generalized sound velocities far from equilibrium.

Contribution

It derives a Lieb-Robinson bound-based inequality for non-equilibrium steady-state currents, connecting equilibrium averages to ballistic transport in quantum many-body systems.

Findings

Lower bounds on steady-state current established

Bound on energy current noise derived

Inequality saturated at one-dimensional criticality

Abstract

Let an infinite, homogeneous, many-body quantum system be unitarily evolved for a long time from a state where two halves are independently thermalized. One says that a non-equilibrium steady state emerges if there are nonzero steady currents in the central region. In particular, their presence is a signature of ballistic transport. We analyze the consequences of the current observable being a conserved density; near equilibrium this is known to give rise to linear wave propagation and a nonzero Drude peak. Using the Lieb-Robinson bound, we derive, under a certain regularity condition, a lower bound for the non-equilibrium steady-state current determined by equilibrium averages. This shows and quantifies the presence of ballistic transport far from equilibrium. The inequality suggests the definition of "nonlinear sound velocities", which specialize to the sound velocity near equilibrium in non-integrable models, and "generalized sound velocities", which encode generalized Gibbs thermalization in integrable models. These are bounded by the Lieb-Robinson velocity. The inequality also gives rise to a bound on the energy current noise in the case of pure energy transport. We show that the inequality is satisfied in many models where exact results are available, and that it is saturated at one-dimensional criticality.