Abstract
In the past decades, it was recognized that quantum chaos, which is essential for the emergence of statistical mechanics and thermodynamics, manifests itself in the effective description of the eigenstates of chaotic Hamiltonians through random matrix ensembles and the eigenstate thermalization hypothesis. Standard measures of chaos in quantum many-body systems are level statistics and the spectral form factor. In this work, we show that the norm of the adiabatic gauge potential, the generator of adiabatic deformations between eigenstates, serves as a much more sensitive measure of quantum chaos. We are able to detect transitions from non-ergodic to ergodic behavior at perturbation strengths orders of magnitude smaller than those required for standard measures. Using this alternative probe in two generic classes of spin chains, we show that the chaotic threshold decreases exponentially with system size and that one can immediately detect integrability-breaking (chaotic) perturbations by analyzing infinitesimal perturbations even at the integrable point. In some cases, small integrability-breaking is shown to lead to anomalously slow relaxation of the system, exponentially long in system size.
Highlights
Finding signatures of chaos in the quantum world has been a long-standing puzzle [1,2,3]
It was recognized that quantum chaos, which is essential for the emergence of statistical mechanics and thermodynamics, manifests itself in the effective description of the eigenstates of chaotic Hamiltonians through random matrix ensembles and the eigenstate thermalization hypothesis
We find several, previously unexpected, results for a particular but fairly generic integrable XXZ spin chain with additional small perturbations: (i) The strength of the integrabilitybreaking perturbation scales exponentially down with the system size, much faster than in previous estimates [47,48]; (ii) integrability-breaking deformations immediately lead to an exponential scaling of the norm of the adiabatic gauge potential (AGP), showing that chaotic perturbations can be already detected in the integrable regimes; and (iii) in the presence of small integrability-breaking terms, the system can exhibit exponentially slow relaxation dynamics, which is similar to the slow dynamics observed in some classical nearly integrable systems like the Fermi-Pasta-Ulam-Tsingou (FPUT) chain [49,50,51]
Summary
Finding signatures of chaos in the quantum world has been a long-standing puzzle [1,2,3]. The sensitivity of this norm to chaotic perturbations is orders of magnitude greater than that of the aforementioned methods Using this approach, we find several, previously unexpected, results for a particular but fairly generic integrable XXZ spin chain with additional small perturbations: (i) The strength of the integrabilitybreaking perturbation scales exponentially down with the system size, much faster than in previous estimates [47,48]; (ii) integrability-breaking deformations immediately lead to an exponential scaling of the norm of the AGP, showing that chaotic perturbations can be already detected in the integrable regimes; and (iii) in the presence of small integrability-breaking terms, the system can exhibit exponentially slow relaxation dynamics, which is similar to the slow dynamics observed in some classical nearly integrable systems like the Fermi-Pasta-Ulam-Tsingou (FPUT) chain [49,50,51]. Whereas most of the previous works focus mainly on short-time effects, here we effectively focus on dynamics and operator growth at times that are exponentially long in the system size (Fig. 1)
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