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arXiv:2609.17794

What does "instant thermalization" in large-$q$ SYK models mean?

Alexander Osterkorn, Jan C. Louw

cond-mat.str-elhep-thphysics.comp-phquant-phSubmitted Sep 15, 2026• Announced Sep 17, 2026• 0 citations
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Abstract

Motivated by the Planckian thermalization rate $Γ\sim T$ observed in the two-body interacting SYK model, we study thermalization in the $q/2$-body case. Previous studies of such systems have established the notion of instantaneous thermalization to leading order in $1/q$. It was conjectured that the thermalization may still be Planckian but with a divergent rate $Γ\sim T q$, explaining the ``instantaneous'' part. For an analytically and numerically tractable system, we calculate the effective temperatures after a quench at time $t = 0$ and indeed find a Planckian rate, albeit with the unexpected decaying behavior $Γ\sim T q^{-1} $. This is contrasted with the behavior of the causal Green's function $\mathcal{G}(t_1, t_2)$, which instantly acquires a thermal form in the two-time plane block $t_1, t_2 > 0$. The resulting picture is that instant thermalization is a meaningful concept only for this block, while the off-diagonal blocks $t_1 \cdot t_2 < 0$ are inherently non-thermal at any $q$. Since the effective temperature is obtained from correlations spanning all blocks, it inherits a finite rate set by the off-diagonal. We illustrate this directly by studying the non-thermal correlations' time dependence.

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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.

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Thermalization in different phases of charged SYK model

Tousik Samui, Nilakash Sorokhaibam

Abstract We study thermalization of charged SYK model in two different phases. We show that both the highly chaotic liquid phase and the dilute gas phase thermalize. Surprisingly the dilute gas state thermalizes instantaneously. We argue that this phenomenon arises because the system in this phase consists of only long-lived quasi-particles at very low density. The liquid state thermalizes exponentially fast. We also show that the additional introduction of random mass deformation (q = 2 SYK term) slows down thermalization but the system thermalizes exponentially fast. This is observed despite the fact that the addition of large q = 2 SYK interaction forces spectral statistics to obey Poisson statistics. An interesting new observation is that the effective temperature is non-monotonic during thermalization in the liquid state. It has a bump at relatively long time before settling down to the final value. With non-zero chemical potential, the effective temperature oscillates noticeably before settling down to the final value.

arXiv:2004.14376•Journal of High Energy Physics•Apr 1, 2021•15 citations
What does "instant thermalization" in large-qqq SYK models mean?
Motivated by the Planckian thermalization rate Γ∼TΓ\sim TΓ∼T observed in the two-body interacting SYK model, we study thermalization in the q/2q/2q/2-body case. Previous studies of such systems have established the notion of instantaneous thermalization to leading order in 1/q1/q1/q. It was conjectured that the thermalization may still be Planckian but with a divergent rate Γ∼TqΓ\sim T qΓ∼Tq, explaining the ``instantaneous'' part. For an analytically and numerically tractable system, we calculate the effective temperatures after a quench at time t=0t = 0t=0 and indeed find a Planckian rate, albeit with the unexpected decaying behavior Γ∼Tq−1Γ\sim T q^{-1} Γ∼Tq−1. This is contrasted with the behavior of the causal Green's function G(t1,t2)\mathcal{G}(t_1, t_2)G(t1​,t2​), which instantly acquires a thermal form in the two-time plane block t1,t2>0t_1, t_2 > 0t1​,t2​>0.
The resulting picture is that instant thermalization is a meaningful concept only for this block, while the off-diagonal blocks t1⋅t2<0t_1 \cdot t_2 < 0t1​⋅t2​<0 are inherently non-thermal at any qqq. Since the effective temperature is obtained from correlations spanning all blocks, it inherits a finite rate set by the off-diagonal. We illustrate this directly by studying the non-thermal correlations' time dependence.
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.
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.
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.
We show that the integrable state of charged SYK model does not thermalize even though the state has non-zero Lyapunov exponent. The chaotic state always thermalizes which is expected. So whether the system thermalizes or not depends on the initial state. We also show that the introduction of random mass deformation (q=2 SYK term) slows down thermalization but the system thermalizes exponentially fast. This is observed despite the fact that large (q=2) SYK interaction forces spectral statistics to obey Poisson statistics. This means that a precise measure of chaos and its prediction of thermalization is still lacking. In the chaotic state, the effective temperature is non-monotonic. It has a bump at relatively long time before settling down to the final value. With non-zero chemical potential, the effective temperature oscillates noticeably before settling down to the final value. The spectral asymmetry frequency does not change during quantum quenches.
A verified Kubo-Martin-Schwinger (KMS) relation, after a drive has ceased, is not sufficient evidence to prove equilibrium. We demonstrate this in a large-NNN large-qqq Sachdev-Ye-Kitaev (SYK) quench protocol. Although all post-quench fermion correlators are exactly thermal, a second quench back to the initial Hamiltonian reveals hidden memory of the initial state. It is encoded in correlators connecting to times before the first quench. This memory is an extensive nonequilibrium (NEQ) witness with its decay rate being the Lyapunov exponent λLλ_LλL​. Thus λLλ_LλL​ sets the rate at which the state becomes effectively indistinguishable from a Gibbs state. Despite being a unitary interacting many-body system, the complete NEQ real-time evolution is obtained exactly in the large-NNN, large-qqq limit, making the setup ideal for analytically studying thermalization.
Abstract We study thermalization of charged SYK model in two different phases. We show that both the highly chaotic liquid phase and the dilute gas phase thermalize. Surprisingly the dilute gas state thermalizes instantaneously. We argue that this phenomenon arises because the system in this phase consists of only long-lived quasi-particles at very low density. The liquid state thermalizes exponentially fast. We also show that the additional introduction of random mass deformation (q = 2 SYK term) slows down thermalization but the system thermalizes exponentially fast. This is observed despite the fact that the addition of large q = 2 SYK interaction forces spectral statistics to obey Poisson statistics. An interesting new observation is that the effective temperature is non-monotonic during thermalization in the liquid state. It has a bump at relatively long time before settling down to the final value. With non-zero chemical potential, the effective temperature oscillates noticeably before settling down to the final value.