[Paper Review] Interaction-induced transition in the quantum chaotic dynamics of a disordered metal

Sergey Syzranov, Alexey V. Gorshkov+1 more|arXiv (Cornell University)|Sep 26, 2017

Quantum many-body systems4 citations

TL;DR

This paper demonstrates that a weakly disordered metal with short-range interactions undergoes a transition from quantum-chaotic to non-chaotic dynamics as temperature or interaction strength increases. Using the out-of-time-ordered correlator (OTOC) of momentum operators, it shows exponential growth at low temperatures/weak interactions—driven by quasiparticle trajectories near the Fermi surface—while inelastic scattering at high temperatures or strong interactions suppresses this growth, signaling a transition to non-chaotic behavior.

ABSTRACT

We demonstrate that a weakly disordered metal with short-range interactions exhibits a transition in the quantum chaotic dynamics when changing the temperature or the interaction strength. For weak interactions, the system displays exponential growth of the out-of-time-ordered correlator (OTOC) of the current operator. The Lyapunov exponent of this growth is temperature-independent in the limit of vanishing interaction. With increasing the temperature or the interaction strength, the system undergoes a transition to a non-chaotic behaviour, for which the exponential growth of the OTOC is absent. We conjecture that the transition manifests itself in the quasiparticle energy-level statistics and also discuss ways of its explicit observation in cold-atom setups.

Motivation & Objective

• To investigate the conditions under which quantum chaotic dynamics emerge in weakly disordered metals with short-range interactions.
• To determine how temperature and interaction strength affect the out-of-time-ordered correlator (OTOC) of momentum operators.
• To explore the connection between OTOC behavior and quasiparticle energy-level statistics.
• To propose experimental observability of the transition in ultracold atomic systems via momentum correlations in double layers.

Proposed method

• The study employs the OTOC of the total momentum operator $ \hat{P}_z $, defined as $ F(t) = \langle [\hat{P}_z(t), \hat{P}_z(0)]^2 \rangle $, to probe quantum chaotic dynamics.
• The time evolution of the OTOC is modeled as $ F(t) \propto \exp\left(2\lambda t - \frac{2t}{\tau(T)}\right) $, where $ \lambda $ is the Lyapunov exponent and $ \tau(T)^{-1} \propto T^2 $ is the inelastic scattering rate.
• The Lyapunov exponent $ \lambda $ is derived from quasiparticle dynamics at the Fermi surface and is found to be temperature-independent in the non-interacting limit.
• The transition is identified when the inelastic scattering rate $ \tau(T)^{-1} $ becomes comparable to the Lyapunov exponent $ \lambda $, suppressing exponential OTOC growth.
• The authors use a mapping to two-layer systems to suggest experimental observation via momentum correlation measurements in ultracold atoms.
• A conjecture is made that the transition may be accompanied by a change in quasiparticle level statistics, from Wigner-Dyson to Poisson-like.

Experimental results

Research questions

• RQ1 Under what conditions does a weakly disordered metal transition from quantum-chaotic to non-chaotic dynamics?
• RQ2 How does the temperature or interaction strength affect the exponential growth of the OTOC of momentum operators?
• RQ3 What is the role of inelastic scattering in suppressing quantum chaos in disordered metals?
• RQ4 Can the transition between chaotic and non-chaotic behavior be detected through quasiparticle energy-level statistics?
• RQ5 How can this transition be experimentally observed in ultracold atomic systems?

Key findings

• The OTOC of the total momentum operator exhibits exponential growth at low temperatures and weak interactions, with a Lyapunov exponent $ \lambda $ that is independent of temperature in the non-interacting limit.
• The exponential growth is suppressed when the inelastic scattering rate $ \tau(T)^{-1} \propto T^2 $ becomes comparable to $ \lambda $, signaling a transition to non-chaotic dynamics.
• The transition occurs at a critical temperature or interaction strength where $ \tau(T) \approx \lambda^{-1} $, marking the breakdown of classical-like trajectory divergence.
• The Lyapunov exponent $ \lambda $ is determined by quasiparticle parameters at the Fermi surface and remains finite even in the presence of weak interactions.
• The transition is conjectured to be accompanied by a change in quasiparticle level statistics, from Wigner-Dyson to Poisson-like, though this requires further analysis.
• The system's behavior is analogous to two layers of ultracold atoms in the same random potential, enabling potential experimental observation via momentum correlation measurements.