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arXiv:2204.06434

Are fast scramblers good thermal baths?

Ancel Larzul, S. J. Thomson, Marco Schirò

Published Apr 13, 2022• 0 citations
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Abstract

The Sachdev-Ye-Kitaev (SYK$_{4}$) 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 SYK$_{4}$ 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 $N$ random non-interacting Majorana fermions coupled to an SYK$_{4}$ bath with $M$ Majorana fermions that we solve with Keldysh techniques in the limit of $M\gg N\gg 1$. 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 SYK$_{4}$ 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.

DOI
10.48550/arxiv.2204.06434

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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.
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.
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=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.
The growth of information scrambling, captured by out-of-time-order correlation functions (OTOCs), is a central indicator of the nature of many-body quantum dynamics. Here, we compute analytically the complete time dependence of the OTOC for an integrable Sachdev-Ye-Kitaev (SYK) model, N Majoranas with random two-body interactions of infinite range, coupled to a Markovian bath at finite temperature. In the limit of no coupling to the bath, the time evolution of scrambling experiences different stages. For t≲sqrt[N], after an initial polynomial growth, the OTOC approaches saturation in a power-law fashion with oscillations superimposed. At t∼sqrt[N], the OTOC reverses trend and starts to decrease linearly in time. The reason for this linear decrease is that, despite being a subleading 1/N effect, the OTOC in this region is governed by the spectral form factor of the antisymmetric couplings of the SYK model. The linear decrease stops at t∼2N, the Heisenberg time, where saturation occurs. The effect of the environment is an overall exponential decay of the OTOC for times longer than the inverse of the coupling strength to the bath. The oscillations at t≲sqrt[N] indicate lack of thermalization-a desired feature for better performance of quantum information devices.
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.
Abstract The Sachdev-Ye-Kitaev (SYK) model is a premier toy model for quantum gravity and holography, saturating the Maldacena-Shenker-Stanford (MSS) chaos bound in its large-NNN infrared limit. This paper introduces the Fano-SYK Model, in which the random Gaussian interaction tensor of the standard SYK4\text{SYK}_4SYK4​ model is replaced by a deterministic, sparse tensor governed by the incidence geometry of the Fano plane PG(2,2)\text{PG}(2,2)PG(2,2) — the combinatorial skeleton of the G2G_2G2​ exceptional Lie group. Results Through exact diagonalisation of Majorana systems at N=7,14,21,N = 7, 14, 21,N=7,14,21, and 282828 (Hilbert space dimension up to 214=16,3842^{14} = 16,384214=16,384), we extract out-of-time-order correlators (OTOCs) to measure the pre-thermal scrambling rate. We define the suppression ratio R(N)=λRandomλFanoR(N) = \frac{\lambda_{\text{Random}}}{\lambda_{\text{Fano}}}R(N)=λFano​λRandom​​ and find R(N)∈7.0,3.7,3.4,2.85R(N) \in {7.0, 3.7, 3.4, 2.85}R(N)∈7.0,3.7,3.4,2.85 across the four system sizes. A power-law fit R(N)=R∞+aNbR(N) = R_\infty + \frac{a}{N^b}R(N)=R∞​+Nba​ yields R∞=2.723±0.489R_\infty = 2.723 \pm 0.489R∞​=2.723±0.489, with the lower 1σ1\sigma1σ bound R∞>2.23R_\infty > 2.23R∞​>2.23 rigorously excluding vanishing impedance at the thermodynamic limit. Interpretation The result provides robust finite-size evidence that the non-associative incidence structure of the Fano plane imposes a scale-invariant geometric impedance on pre-thermal operator scrambling. This positions non-associativity as a structural brake on quantum chaos, complementary to the role of ppp-adic Bruhat-Tits geometry in holographic boundary scattering. Working paper in the Adelic Simplicial Architecture (ASA) series, Portfolio A. Key Concepts Concept Definition SYK Model Sachdev-Ye-Kitaev model; a solvable large-NNN quantum mechanics exhibiting maximal chaos Fano Plane The smallest projective plane PG(2,2)\text{PG}(2,2)PG(2,2); 7 points, 7 lines, incidence-governed OTOC Out-of-time-order correlator; measures operator scrambling and information spreading MSS Bound Maldacena-Shenker-Stanford upper limit on Lyapunov exponent λL≤2πkBT/ℏ\lambda_L \leq 2\pi k_B T/\hbarλL​≤2πkB​T/ℏ Pre-thermal Early-time dynamics before equilibration; regime where scrambling dominates
We study the preparation of thermal states of the dense and sparse Sachdev-Ye-Kitaev (SYK) model using a variational quantum algorithm for 6≤N≤126 \le N \le 126≤N≤12 Majorana fermions over a wide range of temperatures. Utilizing IBM's 127-qubit quantum processor, we perform benchmark computations for the dense SYK model with N=6N = 6N=6, showing good agreement with exact results. The preparation of thermal states of a non-local random Hamiltonian with all-to-all coupling using the simulator and quantum hardware represents a significant step toward future computations of thermal out-of-time order correlators in quantum many-body systems.