Universal Equilibration Dynamics Of The Sachdev-Ye-Kitaev Model

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Soumik Bandyopadhyay¹, Philipp Uhrich¹, Alessio Paviglianiti^1,2, and Philipp Hauke¹

¹Pitaevskii BEC Center, CNR-INO and Dipartimento di Fisica, Università di Trento, Via Sommarive 14, Trento, I-38123, Italy
²International School for Advanced Studies (SISSA), via Bonomea 265, 34136 Trieste, Italy

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Abstract

Equilibrium quantum many-body systems in the vicinity of phase transitions generically manifest universality. In contrast, limited knowledge has been gained on possible universal characteristics in the non-equilibrium evolution of systems in quantum critical phases. In this context, universality is generically attributed to the insensitivity of observables to the microscopic system parameters and initial conditions. Here, we present such a universal feature in the equilibration dynamics of the Sachdev-Ye-Kitaev (SYK) Hamiltonian – a paradigmatic system of disordered, all-to-all interacting fermions that has been designed as a phenomenological description of quantum critical regions. We drive the system far away from equilibrium by performing a global quench, and track how its ensemble average relaxes to a steady state. Employing state-of-the-art numerical simulations for the exact evolution, we reveal that the disorder-averaged evolution of few-body observables, including the quantum Fisher information and low-order moments of local operators, exhibit within numerical resolution a universal equilibration process. Under a straightforward rescaling, data that correspond to different initial states collapse onto a universal curve, which can be well approximated by a Gaussian throughout large parts of the evolution. To reveal the physics behind this process, we formulate a general theoretical framework based on the Novikov–Furutsu theorem. This framework extracts the disorder-averaged dynamics of a many-body system as an effective dissipative evolution, and can have applications beyond this work. The exact non-Markovian evolution of the SYK ensemble is very well captured by Bourret–Markov approximations, which contrary to common lore become justified thanks to the extreme chaoticity of the system, and universality is revealed in a spectral analysis of the corresponding Liouvillian.

Featured image: Universal super-exponential equilibration dynamics of the QFI, $F_{mathcal{Q}}$, under the complex SYK$_4$ Hamiltonian.
(a) Illustration of the quench protocol. Left: Initial states are chosen as ground states of the Fermi–Hubbard (FH) Hamiltonian for different values of $U/mathcal{J}$ (other generic initial states yield equivalent results). The black and gray circles respectively represent occupied and empty fermionic modes. Dashed lines illustrate hopping of fermions between nearest-neighbor sites. Right: The system is evolved under the SYK$_4$ Hamiltonian, where spinless fermions (black circles) can hop to any empty fermionic mode (light gray circles). The disordered interaction strengths $left{J_{i_{1}i_{2};j_{1}j_{2}}right}$ are randomly sampled from independent Gaussian distributions.
(b) QFI averaged over $400$ disorder realizations, $mathbb{E}left[F_{mathcal{Q}}right]$, computed with respect to the operator $hat{R}$ defined in Eq. (6). Initial states from darker to lighter shade of blue are for $U/mathcal{J} = 0, 2, 4, 6, 8$, and $10$. The system equilibrates fast to the expectation value of the Gibbs infinite temperature state (dotted black line).
(c) Universality in the dynamics is revealed by rescaling to $mathcal{G}left(mathbb{E}left[F_{mathcal{Q}}right]right)$, as given in Eq. (3). Very good agreement is found to a Gaussian fit, $mathrm{exp}left[-(Jt/tau)^2right]$, with a fast decay constant of $tau = 1.52$ (dashed red curve). Data for $Q=8$ fermions occupying $N = 16$ fermionic modes.

Popular summary

The modern description of matter hinges on the concept of universality. According to this principle, a system’s microscopic details become unimportant, allowing one to describe the behavior of vastly different systems by just a few parameters. For equilibrium matter, this has a rigorous theoretical basis in the form of the minimization of the free energy. Yet, despite decade-long efforts, the situation is much less firm for quantum systems out of equilibrium. Here, we provide a piece to the puzzle of out-of-equilibrium universality. Our focus is on a paradigm model for a particularly fascinating type of quantum matter called "holographic." Such matter is currently attracting great interest because it draws deep connections to well-known theories of gravity and because it is among the most chaotic systems possible in nature.

We find numerically that the dynamics of relevant physical observables becomes fully independent of microscopic details that define the initial conditions. To explain this unexpected universal behavior, we develop a theoretical framework that describes the isolated quantum model under study through methods that are typical of open systems that interact with an environment. This framework elucidates connections between the extreme chaotic behavior of the holographic quantum model and dissipative quantum systems.

This study opens an array of follow-up questions: In which other systems can we expect similar universal behavior? Can we extend the dissipative framework to other models? And is it possible to observe these effects in a real system in Nature or in the laboratory?

► BibTeX data

@article{Bandyopadhyay2023universal, doi = {10.22331/q-2023-05-24-1022}, url = {https://doi.org/10.22331/q-2023-05-24-1022}, title = {Universal equilibration dynamics of the {S}achdev-{Y}e-{K}itaev model}, author = {Bandyopadhyay, Soumik and Uhrich, Philipp and Paviglianiti, Alessio and Hauke, Philipp}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {1022}, month = may, year = {2023}
}

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Cited by

[1] Debanjan Chowdhury, Antoine Georges, Olivier Parcollet, and Subir Sachdev, "Sachdev-Ye-Kitaev models and beyond: Window into non-Fermi liquids", Reviews of Modern Physics 94 3, 035004 (2022).

[2] Jan C. Louw and Stefan Kehrein, "Thermalization of many many-body interacting Sachdev-Ye-Kitaev models", Physical Review B 105 7, 075117 (2022).

[3] Ceren B. Dağ, Philipp Uhrich, Yidan Wang, Ian P. McCulloch, and Jad C. Halimeh, "Detecting quantum phase transitions in the quasi-stationary regime of Ising chains", arXiv:2110.02995, (2021).

[4] Alessio Paviglianiti, Soumik Bandyopadhyay, Philipp Uhrich, and Philipp Hauke, "Absence of operator growth for average equal-time observables in charge-conserved sectors of the Sachdev-Ye-Kitaev model", Journal of High Energy Physics 2023 3, 126 (2023).

[5] Philipp Uhrich, Soumik Bandyopadhyay, Nick Sauerwein, Julian Sonner, Jean-Philippe Brantut, and Philipp Hauke, "A cavity quantum electrodynamics implementation of the Sachdev--Ye--Kitaev model", arXiv:2303.11343, (2023).

[6] Ceren B. Daǧ, Philipp Uhrich, Yidan Wang, Ian P. McCulloch, and Jad C. Halimeh, "Detecting quantum phase transitions in the quasistationary regime of Ising chains", Physical Review B 107 9, 094432 (2023).

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