Thermalization of a closed Sachdev-Ye-Kitaev system in the thermodynamic limit

Ioanna Kourkoulou

Santiago Salazar Jaramillo, Rishabh Jha, Stefan K. Kehrein

The question of thermalization of a closed quantum system is of central interest in nonequilibrium quantum many-body physics. Here we present one such study analyzing the dynamics of a closed coupled Majorana SYK system. We have a large- q SYK model prepared initially at equilibrium quenched by introducing a random hopping term, thus leading to nonequilibrium dynamics. We find that the final stationary state reaches thermal equilibrium with respect to the Green's functions and energy. Accordingly, the final state is characterized by calculating its final temperature and the thermalization rate. We provide a detailed review of analytical methods and derive the required Kadanoff-Baym equations, which are then solved using the algorithm developed in this work. Our results display rich thermalization dynamics in a closed quantum system in the thermodynamic limit.

Julian Sonner, Manuel Vielma

The eigenstate thermalization hypothesis (ETH) explains how closed unitary quantum systems can exhibit thermal behavior in pure states. In this work we examine a recently proposed microscopic model of a black hole in AdS 2 , the so-called Sachdev-Ye-Kitaev (SYK) model. We show that this model satisfies the eigenstate thermalization hypothesis by solving the system in exact diagonalization. Using these results we also study the behavior, in eigenstates, of various measures of thermalization and scrambling of information. We establish that two-point functions in finite-energy eigenstates approximate closely their thermal counterparts and that information is scrambled in individual eigenstates. We study both the eigenstates of a single random realization of the model, as well as the model obtained after averaging of the random disordered couplings. We use our results to comment on the implications for thermal states of a putative dual theory, i.e. the AdS 2 black hole.

Karamoko Soumahoro, Alison Valbuena, Roman Senkov

Valérie Bettaque, Brian Swingle

Any quantum state is fully specified by the expectation values of a complete set of Hermitian operators. For a system of Majorana fermions, such as the Sachdev-Ye-Kitaev (SYK) model, this set of observables can be taken to be all possible strings of Majorana fermion operators. The expectation values of these fermion strings in a thermal state depend erratically on the microscopic couplings that specify the SYK Hamiltonian, and we study their statistical properties directly in the thermodynamic limit using path integral techniques. When the underlying SYK Hamiltonian is chaotic, we find that these expectation values are well-modeled as real Gaussian random variables with zero mean and a variance that we compute. In contrast, for the integrable variant of SYK, we find that the expectation values are actually non-Gaussian. We then use these results to study measures of magic in the SYK thermal state, including the robustness of magic and the stabilizer Rényi entropy. We also show that our results can be quantitatively reproduced with a dual gravity calculation in the chaotic case at sufficiently low temperature. In this dual gravity model the variance of a given microscopic operator string is related to a wormhole geometry stabilized by a massive particle which is dual to the operator string. Our results thus provide a concrete and quantitative setting in which to study the relationship between randomness, wormholes and closed universes, as well as a holographic dual of quantum magic.

Julian Sonner, Manuel Vielma

Abstract The eigenstate thermalization hypothesis (ETH) explains how closed unitary quantum systems can exhibit thermal behavior in pure states. In this work we examine a recently proposed microscopic model of a black hole in AdS2, the so-called Sachdev-Ye-Kitaev (SYK) model. We show that this model satisfies the eigenstate thermalization hypothesis by solving the system in exact diagonalization. Using these results we also study the behavior, in eigenstates, of various measures of thermalization and scrambling of information. We establish that two-point functions in finite-energy eigenstates approximate closely their thermal counterparts and that information is scrambled in individual eigenstates. We study both the eigenstates of a single random realization of the model, as well as the model obtained after averaging of the random disordered couplings. We use our results to comment on the implications for thermal states of a putative dual theory, i.e. the AdS2 black hole.