Cooling the Sachdev-Ye-Kitaev model using thermofield double states

TL;DR

The paper demonstrates an efficient cooling protocol for the Sachdev–Ye–Kitaev model by adiabatically coupling two SYK copies (Maldacena–Qi) and gradually reducing their coupling μ to prepare low-temperature thermofield double states. It combines large-N Schwinger–Dyson numerics with an eigenstate thermalization framework to show a gapped adiabatic path and to predict effective temperatures and excitation gaps across μ, including a fast, semi-classical protocol whose time scales are T ∼ β log(βJ) and independent of system size N. Finite-size numerics and a detailed error analysis for both adiabatic and semi-classical ramps confirm that local observables closely follow TFD predictions even when the many-body fidelity to the true ground state is not perfect. The findings suggest broad applicability of entropy- and ETH-based cooling techniques to strongly interacting Hamiltonians and connect cooling performance to graviton-like excitations in the SYK gravity dual, with potential extensions to other holographic or non-quasiparticle systems.

Abstract

We analyze a simple and efficient experimental protocol to cool the Sachdev-Ye-Kitaev (SYK) model to low temperatures. The protocol utilizes local couplings between two copies of an SYK model to create a gapped adiabatic path, between a high temperature product state and a low temperature thermofield double state. By smoothly varying the coupling strength between these two limits, one efficiently cools the SYK model. We support these predictions-and demonstrate fast cooling to the low-temperature gravitational regime-via exact numerical solutions to the large-N equations of motion that govern the ground state and dynamical properties of the coupled system. Finally, we present a theoretical framework based upon eigenstate thermalization that provides a microscopic explanation for the efficacy of the cooling protocol; intriguingly, this suggests that the protocol may be applicable for cooling strongly-interacting quantum Hamiltonians more broadly.

Figures (6)

• Figure 1: Schematic of the experimental protocol for cooling the Sachdev-Ye-Kitaev (SYK) model kobrin2023applicationsbentsen2024approximate. Two copies of the model, with interaction strength $J$, are coupled by pairwise terms of strength $\mu$ [Eq. (\ref{['eq:H_MQ']})]. When $\mu \gg J$, the ground state is a tensor product of EPR pairs between the two copies (green). When $\mu \ll J$, the ground state is a thermofield double (TFD) state at a low temperature set by $\mu$ (purple). Our protocol prepares EPR pairs and then adiabatically interpolates to the TFD state by slowly decreasing $\mu$.
• Figure 2: Numerical comparison of the ground state of Eq. (\ref{['eq:H_MQ']}) (blue) and the TFD state (orange) in the large-$N$ limit. (a) The two-side correlator, $G_{LR} \equiv \left<i\chi_L \chi_R \right>$, vs. the single-side energy, $E \equiv \left <H_L\right>$, normalized by the magnitude of the ground-state energy of the SYK model, $E_0$. Inset: Matching the value of $G_{LR}$ yields the effective TFD temperature $\beta_{\textrm{eff}}$ as a function of $\mu$. (b) The single-side auto-correlation function, $G_{LL}(t) \equiv \left<i\chi_L(t) \chi_L(0) \right>$, for the ground state of Eq. (\ref{['eq:H_MQ']}) at $\mu = 0.1 J$ and $\mu = 0.2 J$ (inset), and the TFD state at $\beta J = 3.7$ and $\beta J = 6.1$, respectively, which correspond to $\beta \equiv \beta_{\text{eff}}(\mu)$.
• Figure 3: Dynamics of the coupled Hamiltonian in the large-$N$ limit. In all cases, the initial state is a low-temperature state of Eq. (\ref{['eq:H_MQ']}) at an initial coupling strength $\mu_i$. (a) Quench dynamics upon a small sudden change in the coupling strength. The change leads to oscillations in the single-side energy, $E \equiv \left <H_L\right>$. Inset: Example for $\mu_i = 0.4 J$ and $\mu_f = 0.38 J$. The oscillation frequency (purple circles) closely matches the graviton gap predicted by the Schwinger-Dyson equations and our thermalization-based framework (purple curve) across a range of $\mu \equiv \mu_i$supp. For comparison, the matter gap (red) of Eq. (\ref{['eq:H_MQ']}), obtained by simulating Eq. (\ref{['eq:H_MQ']}) in imaginary time garcia2019quantum, is moderately smaller. (b) Simulation of our proposed adiabatic protocol. The coupling strength is decreased adiabatically from $\mu_i = 0.5 J$ to various $\mu_f$, as $\mu(t) = (\mu_i-\mu_f) e^{-t/T_*}+\mu_f$ with $T_* = 10/J$ (inset). The single-side energy (purple) is observed to decrease alongside $\mu(t)$ and saturates a final value determined by $\mu_f$. (c) Comparison between the final value of the single-side energy (purple circles) and its value in the TFD state at $\mu_f$ (orange curve). The small offset at large $\mu_f$ arises from the small initial physical temperature used to regulate the large-$N$ numerical simulations supp. Inset: Identical comparison for the two-sided correlator, $G_{LR}$. (d) Modified adiabatic protocol with $\mu_f = 0$, i.e. $\mu(t) = \mu_i \exp(-t/T_*)$ (inset). We sweep $T_*$ to determine the minimum time $T$ required to reach a single-side energy corresponding to a desired temperature $\beta^{-1}$ (purple circles). The theory prediction $T = a \beta \log (\beta)$ is shown for $a = 0.25$ (dashed).
• Figure 4: Finite-size numerical analysis of the ground state and cooling dynamics of the coupled Hamiltonian. (a) The re-scaled coefficients, $\psi_{nn} \equiv \sqrt{D_n} c_{nn}$, of the ground-state wavefunction (circles) and the TFD state (curves) for various $\mu$ ($2N = 48$). We plot only a subset of coefficients for visual clarity. Inset: The bare coefficients $c_{nn}$ plotted as in Ref. garcia2019quantum. (b) Dynamical simulation of our adiabatic protocol ($2N = 32$). The system begins in the EPR state and the coupling decreases as $\mu(t) = \mu_i (\mu_f/\mu_i)^{(t/T_*)}$ with $\mu_i = 4J$. The infidelity of the final state with the ground state of Eq. (\ref{['eq:H_MQ']}) is shown for various $T_*$ and $\mu_f$. (c) The infidelity at $\mu_f = 0.01 J$ for various system sizes $N = 12,14,16$ (top). The single-sided energy at $\mu_f = 0.01 J$ for the same system sizes (bottom).
• Figure 5: Comparison of the adiabatic cooling protocol shown in Fig. \ref{['fig: adiabatic']}(b) of the main text under three different initial temperatures, $\beta = 6,8,$ and $10$. (a) The single-side energy $E = \langle H_L \rangle$ as a function of time, for a coupling that decreases adiabatically as $\mu(t) = (\mu_i-\mu_f) e^{-t/T_*}+\mu_f$, with $T_* = 10/J$ and various initial couplings $\mu_i$. (b) As the initial temperature decreases, the final single-side energy, evaluated at $t = 225/J$, approaches the expected value of the TFD state. (c) Similar convergence is seen in the final value of the two-sided coupling, $G_{LR}$. The simulations use a step size $\Delta t = 0.05/J$ and an initial time range $t_\textrm{init}=80/J$.