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.
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.
Abstract We study chaotic-integrable transition and the nature of quantum chaos in SYK model with chemical potential. We use a novel numerical technique to calculate the partition function explicitly. We show the phase transition in the presence of large chemical potential. We also show that a mass-like term consisting of two fermion random interaction (q = 2 SYK term) does not give rise to a sharp transition. We find that turning on the chemical potential suppresses the Lyapunov exponent in the chaotic phase exponentially.
The complex Sachdev-Ye-Kitaev (SYK) model, featuring fermions with all-to-all interactions, serves as a dual paradigm for understanding non-Fermi liquid behavior and the holographic nature of charged black holes. Two defining characteristics of the standard SYK model are its maximal chaos (Lyapunov exponent
λL=2πT at temperature
T), and its finite zero-temperature residual entropy. While previous studies have largely focused on couplings drawn from a zero-mean Gaussian distribution, we investigate a generalized model with a finite mean-to-standard-deviation ratio,
g≡J0/δJ of the coupling distribution in order to get deeper insight into the evolution of chaos. We find that increasing
g yields the following effects: (i) The system remains a fast scrambler with
λL=A T, but with a suppressed coefficient
A<2π. (ii) In the limit
g→∞, out-of-time-ordered correlators (OTOCs) no longer exhibit exponential growth with
λL≃0. (iii) The spectral correlations indicative of late-time chaos maintain Wigner-Dyson level spacing statistics for all values of
g. (iv) The system preserves a finite residual entropy, albeit with reduced magnitude, for all
g values. We conclude that in this generalized SYK model, there is a chaotic to non-chaotic crossover. Moreover different measures of chaos decouple, demonstrating that the presence of finite residual entropy does not strictly imply maximal chaos.
Abstract We study non-equilibrium dynamics in SYK models using quantum quench. We consider models with two, four, and higher fermion interactions (q= 2,4, and higher) and use two different types of quench protocol, which we call step and bump quenches. We analyse evolution of fermion two-point functions without long time averaging. We observe that inq= 2 theory the two-point functions do not thermalize. We find thermalization inq= 4 and higher theories without long time averaging. We calculate two different exponents of which one is equal to the coupling and the other is proportional to the final temperature. This result is more robust than thermalization obtained from long time averaging as proposed by the eigenstate thermalization hypothesis(ETH). Thermalization achieved without long time averaging is more akin to mixing than ergodicity.
Motivated by the Planckian thermalization rate
Γ∼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∼Tq, 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
G(t1,t2), which instantly acquires a thermal form in the two-time plane block
t1,t2>0.
The resulting picture is that instant thermalization is a meaningful concept only for this block, while the off-diagonal blocks
t1⋅t2<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.