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arXiv:1909.10637 · Physical review. B./Physical review. B

Evaporation dynamics of the Sachdev-Ye-Kitaev model

Pengfei Zhang

Published Dec 3, 2019• 51 citations
arXivHTMLPDF

Abstract

In this paper, we study the evaporation dynamics of the Sachdev-Ye-Kitaev model, with an initial temperature ${T}_{\ensuremath{\chi}}$, by coupling it to a thermal bath with lower temperature ${T}_{\ensuremath{\psi}}<{T}_{\ensuremath{\chi}}$ modeled by a larger SYK model. The coupling between the small system and the bath is turned on at time $t=0$. Then the system begins to evolve and finally becomes thermalized. Using the Keldysh approach, we analyze the relaxation process of the system for different temperatures and couplings. For marginal or irrelevant coupling, after a short-time energy absorption, we find a smooth thermalization of the small system where the energy relaxes before the system become thermalized. The relaxation rate of effective temperature is found to be bounded by $T$, while the energy thermalization rate increases without saturation when increasing the coupling strength. On the contrary, for the relevant coupling case, both energy and effective temperature show oscillations. We find this oscillations frequency to be coincident with the typical excitation energy of the small SYK system.

DOI
10.1103/physrevb.100.245104

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In this paper, we study the evaporation dynamics of the Sachdev-Ye-Kitaev model, with an initial temperature T\ensuremathχ{T}_{\ensuremath{\chi}}T\ensuremathχ​, by coupling it to a thermal bath with lower temperature T\ensuremathψ<T\ensuremathχ{T}_{\ensuremath{\psi}}<{T}_{\ensuremath{\chi}}T\ensuremathψ​<T\ensuremathχ​ modeled by a larger SYK model. The coupling between the small system and the bath is turned on at time t=0t=0t=0. Then the system begins to evolve and finally becomes thermalized. Using the Keldysh approach, we analyze the relaxation process of the system for different temperatures and couplings. For marginal or irrelevant coupling, after a short-time energy absorption, we find a smooth thermalization of the small system where the energy relaxes before the system become thermalized. The relaxation rate of effective temperature is found to be bounded by TTT, while the energy thermalization rate increases without saturation when increasing the coupling strength. On the contrary, for the relevant coupling case, both energy and effective temperature show oscillations. We find this oscillations frequency to be coincident with the typical excitation energy of the small SYK system.
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We investigate the non-equilibrium dynamics of a resonant level model coupled to a strongly interacting electron bath modeled by a Sachdev-Ye-Kitaev (SYK) model. Different from the well-investigated case of a structureless non-interacting Fermi gas bath leading to a temperature-independent exponential decay of the impurity orbital occupation, we find a temperature-dependent oscillatory decay. We attribute this difference to the lack of quasiparticles in the SYK model, which is reflected in its singular density of states at the Fermi level. Our results are exact and can be obtained analytically by mapping to a suitably structured Fermi gas bath as an ancillary model for the SYK bath.
The Sachdev-Ye-Kitaev (SYK) model describes Majorana fermions with random interaction, which displays many interesting properties such as non-Fermi liquid behavior, quantum chaos, emergent conformal symmetry and holographic duality. Here we consider a SYK model or a chain of SYK models with N Majorana fermion modes coupled to another SYK model with N 2 Majorana fermion modes, in which the latter has many more degrees of freedom and plays the role as a thermal bath. For a single SYK model coupled to the thermal bath, we show that although the Lyapunov exponent is still proportional to temperature, it monotonically decreases from 2π/β (β = 1/(k B T), T is temperature) to zero as the coupling strength to the thermal bath increases. For a chain of SYK models, when they are uniformly coupled to the thermal bath, we show that the butterfly velocity displays a crossover from a T \sqrt{T} T​ -dependence at relatively high temperature to a linear T-dependence at low temperature, with the crossover temperature also controlled by the coupling strength to the thermal bath. If only the end of the SYK chain is coupled to the thermal bath, the model can introduce a spatial dependence of both the Lyapunov exponent and the butterfly velocity. Our models provide canonical examples for the study of thermalization within chaotic models.
We study thermalization dynamics in a fermion-phonon variant of the Sachdev-Ye-Kitaev model coupled to an external cold thermal bath of harmonic oscillators. We find that quantum critical fermions thermalize more efficiently than phonons, in sharp contrast to the behavior in the Fermi liquid regime. In addition, after a short prethermal stage, the system acquires a quasithermal distribution given by a time-dependent effective temperature, reminiscent of ``hydrodynamic'' relaxation. All physical observables relax at the same rate which scales with the final temperature through an exponent that depends universally on the low-energy spectrum of the system and the bath. Such relaxation rate is derived using a hydrodynamic approximation in full agreement with the numerical solution of a set quantum kinetic equations derived from the Keldysh formalism for nonequilibrium Green's functions. Our results hint toward further research on the applicability of the hydrodynamic picture in the description of the late time dynamics of open quantum systems despite the absence of conserved quantities in regimes dominated by conserving collisions.
Abstract We investigate the thermalization of Sachdev–Ye–Kitaev (SYK) models coupled via random interactions following quenches from the perspective of entanglement. Previous studies have shown that when a system of two SYK models coupled by random two-body terms is quenched from the thermofield double state with sufficiently low effective temperature, the Rényi entropies do not saturate to the expected thermal values in the large- N limit. Using numerical large- N methods, we first show that the Rényi entropies in a pair SYK models coupled by two-body terms can thermalize, if quenched from a state with sufficiently high effective temperature, and hence exhibit state-dependent thermalization. In contrast, SYK models coupled by single-body terms appear to always thermalize. We provide evidence that the subthermal behavior in the former system is likely a large- N artifact by repeating the quench for finite N and finding that the saturation value of the Rényi entropy extrapolates to the expected thermal value in the N → ∞ limit. Finally, as a finer grained measure of thermalization, we compute the late-time spectral form factor of the reduced density matrix after the quench. While a single SYK dot exhibits perfect agreement with random matrix theory, both the quadratically and quartically coupled SYK models exhibit slight deviations.
The Sachdev-Ye-Kitaev (SYK4_{4}4​) model has attracted attention for its fast scrambling properties and its thermalization rate that is set only by the temperature. In this work we ask the question of whether the SYK4_{4}4​ model is also a good thermal bath, in the sense that it allows a system coupled to it to thermalize. We address this question by considering the dynamics of a system of NNN random non-interacting Majorana fermions coupled to an SYK4_{4}4​ bath with MMM Majorana fermions that we solve with Keldysh techniques in the limit of M≫N≫1M\gg N\gg 1M≫N≫1. We compare this nonequilibrium setting with a conventional bath made of free non-interacting degrees of freedom with a continous spectrum. We show that the SYK4_{4}4​ bath is more efficient in thermalising the system at weak coupling, due to its enhanced density of states at low frequency, while at strong system-bath couplings both type of environments give rise to a similar time scale for thermalisation.