Thermalization of randomly coupled SYK models

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.

Quantum entanglement in the Sachdev—Ye—Kitaev model and its generalizations

“…For generalized SYK models that contain multiple sites, with some of the sites described by the SYK model, there is no simple approach to employ the ETH analysis, and the ED is limited to small system size N. Consequently, previous studies rely on the path-integral approach [22,[59][60][61][62][63][64][65][66][67][68][69]. We are interested in the entanglement dynamics of the system prepared in some short-range entangled state.…”

“…We are interested in the entanglement dynamics of the system prepared in some short-range entangled state. In the simplest case, we consider two SYK sites with N 1 and N 2 Majorana modes, coupled by random pair hopping [22,[59][60][61]]. In the earlytime regime, the entanglement entropy shows linear growth in time [22,[59][60][61][62][63]].…”

“…In the simplest case, we consider two SYK sites with N 1 and N 2 Majorana modes, coupled by random pair hopping [22,[59][60][61]]. In the earlytime regime, the entanglement entropy shows linear growth in time [22,[59][60][61][62][63]]. In the long-time limit, when N 1 = N 2 , it is found that in the canonical ensemble the Rényi entanglement entropy does not saturate to the thermal value due to the existence of soft modes [22,59,60].…”

“…In the earlytime regime, the entanglement entropy shows linear growth in time [22,[59][60][61][62][63]]. In the long-time limit, when N 1 = N 2 , it is found that in the canonical ensemble the Rényi entanglement entropy does not saturate to the thermal value due to the existence of soft modes [22,59,60]. On the contrary, if N 2 N 1 , the second site serves as a heat bath, and the Rényi entanglement entropy of the first site always matches the thermal entropy [60,61].…”

“…In the long-time limit, when N 1 = N 2 , it is found that in the canonical ensemble the Rényi entanglement entropy does not saturate to the thermal value due to the existence of soft modes [22,59,60]. On the contrary, if N 2 N 1 , the second site serves as a heat bath, and the Rényi entanglement entropy of the first site always matches the thermal entropy [60,61]. When the entanglement entropy approaches the thermal value, a first-order transition occurs.…”