Low temperature quantum bounds on simple models

TL;DR

This paper investigates quantum bounds on transport and chaos in simple models, revealing how quantum effects, geometry, and energy scales influence these bounds, especially in low-temperature regimes.

Contribution

It provides a physical understanding of quantum bounds on chaos and viscosity by analyzing classical and quantum dynamics on curved manifolds, highlighting the role of quantum effects and geometry.

Findings

Quantum viscosity and Lyapunov exponent depend on de Broglie wavelength and geometric scales.

Quantum effects can violate classical bounds at low temperatures.

Chaos is limited by quantum curvature effects, especially at low energies.

Abstract

In the past few years, there has been considerable activity around a set of quantum bounds on transport coefficients (viscosity) and chaos (Lyapunov exponent), relevant at low temperatures. The interest comes from the fact that Black-Hole models seem to saturate all of them. The goal of this work is to gain physical intuition about the quantum mechanisms that enforce these bounds on simple models. To this aim, we consider classical and quantum free dynamics on curved manifolds. These systems exhibit chaos up to the lowest temperatures and - as we discuss - they violate the bounds in the classical limit. First of all, we show that the quantum dimensionless viscosity and the Lyapunov exponent only depend on the de Broglie length and a geometric length-scale, thus establishing the scale at which quantum effects become relevant. Then, we focus on the bound on the Lyapunov exponent and identify three different ways in which quantum effects arise in practice. We illustrate our findings on a toy model given by the surface of constant negative curvature - a paradigmatic model of quantum chaos - glued to a cylinder. By exact solution and numerical investigations, we show how the chaotic behaviour is limited by the quantum effects of the curvature itself. Interestingly, we find that at the lowest energies the bound to chaos is dominated by the longest length scales, and it is therefore a collective effect.