Low temperature quantum bounds on simple models

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.

arXiv comment22 pages, 11 figures. Revised manuscript

References in corpus (22)

• Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics
• A bound on chaos
• The approach to thermal equilibrium in quantized chaotic systems
• Saturation of electrical resistivity
• Lyapunov Exponent and Out-of-Time-Ordered Correlator's Growth Rate in a Chaotic System
• Hyperbolic Lattices in Circuit Quantum Electrodynamics
• Loschmidt Echo
• Jamming versus Glass Transitions
• Weak Quantum Chaos
• Renormalization of the Inverse Square Potential
• Chaotic Scattering Theory of Transport and Reaction-Rate Coefficients
• Quantum Simulation of Hyperbolic Space with Circuit Quantum Electrodynamics: From Graphs to Geometry
• Bridging entanglement dynamics and chaos in semiclassical systems
• On the rigidity of a hard sphere glass near random close packing
• Minimal quantum viscosity from fundamental physical constants
• Temperature dependence of butterfly effect in a classical many-body system
• Classical many-body chaos with and without quasiparticles
• Quantum echo dynamics in the Sherrington-Kirkpatrick model
• Scrambling versus relaxation in Fermi and non-Fermi liquids
• Super-exponential scrambling of Out-of-time-ordered correlators
• Quasi-integrable systems are slow to thermalize but may be good scramblers
• Quantum Kolmogorov-Sinai entropy and Pesin relation

Cited by in corpus (7)

• Quantum Dynamics in Krylov Space: Methods and Applications
• Selberg trace formula in hyperbolic band theory
• Finite-time Landauer principle beyond weak coupling
• Quantum bounds on the generalized Lyapunov exponents
• Scaling laws of the out-of-time-order correlators at the transition to the spontaneous $\mathcal{P}\mathcal{T}$-symmetry breaking in a Floquet system
• A Bound on Thermalization from Diffusive Fluctuations
• Analyticity constraints bound the decay of the spectral form factor