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arXiv:2111.08671

Thermalization of many many-body interacting SYK models

Jan C. Louw, Stefan K. Kehrein

Published Nov 16, 2021• 0 citations
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Abstract

We investigate the non-equilibrium dynamics of complex Sachdev-Ye-Kitaev (SYK) models in the $q\rightarrow\infty$ limit, where $q/2$ denotes the order of the random Dirac fermion interaction. We extend previous results by Eberlein et al. [Phys. Rev. B 96, 205123 (2017)] to show that a single SYK $q\rightarrow\infty$ Hamiltonian for $t\geq 0$ is a perfect thermalizer in the sense that the local Green's function is instantaneously thermal. The only memories of the quantum state for $t<0$ are its charge density and its energy density at $t=0$. Our result is valid for all quantum states amenable to a~$1/q$-expansion, which are generated from an equilibrium SYK state in the asymptotic past and acted upon by an arbitrary combination of time-dependent SYK Hamiltonians for $t<0$. Importantly, this implies that a single SYK $q\rightarrow\infty$ Hamiltonian is a perfect thermalizer even for non-equilibrium states generated in this manner.

DOI
10.48550/arxiv.2111.08671

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We investigate the non-equilibrium dynamics of complex Sachdev-Ye-Kitaev (SYK) models in the q→∞q\rightarrow\inftyq→∞ limit, where q/2q/2q/2 denotes the order of the random Dirac fermion interaction. We extend previous results by Eberlein et al. [Phys. Rev. B 96, 205123 (2017)] to show that a single SYK q→∞q\rightarrow\inftyq→∞ Hamiltonian for t≥0t\geq 0t≥0 is a perfect thermalizer in the sense that the local Green's function is instantaneously thermal. The only memories of the quantum state for t<0t<0t<0 are its charge density and its energy density at t=0t=0t=0. Our result is valid for all quantum states amenable to a~1/q1/q1/q-expansion, which are generated from an equilibrium SYK state in the asymptotic past and acted upon by an arbitrary combination of time-dependent SYK Hamiltonians for t<0t<0t<0. Importantly, this implies that a single SYK q→∞q\rightarrow\inftyq→∞ Hamiltonian is a perfect thermalizer even for non-equilibrium states generated in this manner.
We investigate the nonequilibrium dynamics of complex Sachdev-Ye-Kitaev (SYK) models in the $q\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, where q/2q/2q/2 denotes the order of the random Dirac fermion interaction. We extend previous results by Eberlein et al. [Phys. Rev. B 96, 205123 (2017)] to show that a single SYK $q\ensuremath{\rightarrow}\ensuremath{\infty}$ Hamiltonian for $t\ensuremath{\ge}0$ is a perfect thermalizer in the sense that the local Green's function is instantaneously thermal. The only memories of the quantum state for t<0t<0t<0 are its charge density and its energy density at t=0t=0t=0. Our result is valid for all quantum states amenable to a 1/q1/q1/q expansion, which are generated from an equilibrium SYK state in the asymptotic past and acted upon by an arbitrary combination of time-dependent SYK Hamiltonians for t<0t<0t<0. Importantly, this implies that a single SYK $q\ensuremath{\rightarrow}\ensuremath{\infty}$ Hamiltonian is a perfect thermalizer even for nonequilibrium states generated in this manner.
Nonequilibrium dynamics of unentangled and entangled pure states in interacting quantum systems is crucial for harnessing quantum information and to understand quantum thermalization. We develop a general Schwinger-Keldysh (SK) field theory for nonequilibrium dynamics of pure states of fermions. We apply our formalism to study the time evolution of initial density inhomogeneity and multipoint correlations of pure states in the complex Sachdev-Ye-Kitaev (SYK) models. We demonstrate a remarkable universality in the dynamics of pure states in the SYK model. We show that dynamics of almost all pure states in a fixed particle number sector is solely determined by a set of universal large-NNN Kadanoff-Baym equations. Moreover, irrespective of the initial state the site- and disorder-averaged Green's function thermalizes instantaneously, whereas local and nonlocal Green's functions have finite thermalization rate. We provide understanding of our numerical and analytical large-NNN results through random-matrix theory (RMT) analysis. Furthermore, we show that the thermalization of an initial pure product state in the noninteracting SYK2{\mathrm{SYK}}_{2}SYK2​ model is independent of fermion filling and an initial density inhomogeneity decays with weak but long lived oscillations due to dephasing. In contrast, the interacting ${\mathrm{SYK}}_{q\ensuremath{\ge}4}$ model thermalizes slower than the noninteracting model and exhibits filling-dependent monotonic relaxation of initial inhomogeneity. For evolution of entangled pure states, we show that the initial entanglement is encoded in the nonlocal and/or multipoint quantum correlations that relax as the system thermalizes.
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The Sachdev-Ye-Kitaev (SYK) model is a strongly interacting fermionic system. Its dynamics are difficult to analyze with Lieb-Robinson bounds and cluster expansions due to its all-to-all disordered interactions. We prove that, with high probability over the disorder, the SYK model admits a quantum Gibbs sampler with a system-size-independent spectral gap at sufficiently high constant temperatures. This yields polynomial-time preparation of SYK Gibbs states on a quantum computer. While prior non-rigorous computations that suggested SYK thermalization were limited to studying local correlators, we emphasize that our result holds up to arbitrary inverse polynomial trace distance. Our proof introduces the pseudo-Lindbladian approach to fermions and uses an especially simple SYK analysis based on a comparison to classical dynamics.
Analytical solutions to the non-equilibrium Green's functions in large-qqq SYK models
It is known that the Sachdev-Ye-Kitaev (SYK) model is exactly solvable at leading order in 1/q1/q1/q; specifically, the instantaneously thermal time block after the quench is known in closed form. Thus far the remaining (inherently out of equilibrium) blocks have only been numerically accessible. Here we provide their analytical solutions for a quench between non-commuting q/2q/2q/2-body SYK Hamiltonians. We obtain the Green's functions in closed form for every time block. We extract a simple relation between the pre- and post-quench energies ε1∝ε0ε_1 \propto ε_0ε1​∝ε0​ leading to an exact temperature update rule. Via a Cauchy-Schwarz inequality, we show that heating is inevitable for the closed system; hence, the second law emerges geometrically. The solution is used in a companion paper to address what ``instantaneous thermalization'' means at finite qqq.
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.