We review our recent work [1] where we studied the chaotic property of the two coupled Sachdev-Ye-Kitaev systems exhibiting a Hawking-Page like phase transition. By computing the out-of-time-ordered correlator in the large N N limit by using the bilocal field formalism, we found that the chaos exponent of this model shows a discontinuous fall-off at the phase transition temperature. Hence in this model the Hawking-Page like transition is correlated with a transition in chaoticity, as expected from the relation between a black hole geometry and the chaotic behavior in the dual field theory.
Abstract It has been known that the large-q complex Sachdev-Ye-Kitaev (SYK) model falls under the same universality class as that of van der Waals (mean-field) and saturates the Maldacena-Shenker-Stanford (MSS) bound, both features shared by various black holes. This makes the SYK model a useful tool in probing the fundamental nature of quantum chaos and holographic duality. This work establishes the robustness of this shared universality class and chaotic properties for SYK-like models by extending to a system of coupled large-q complex SYK models of different orders. We provide a detailed derivation of thermodynamic properties, specifically the critical exponents for an observed phase transition, as well as dynamical properties, in particular the Lyapunov exponent, via the out-of-time correlator calculations. Our analysis reveals that, despite the introduction of an additional scaling parameter through interaction strength ratios, the system undergoes a continuous phase transition at low temperatures, similar to that of the single SYK model. The critical exponents align with the Landau–Ginzburg (mean-field) universality class, shared with van der Waals gases and various AdS black holes. Furthermore, we demonstrate that the coupled SYK system remains maximally chaotic in the large-q limit at low temperatures, adhering to the MSS bound, a feature consistent with the single SYK model. These findings establish robustness and open avenues for broader inquiries into the universality and chaos in complex quantum systems. We provide a detailed outlook for future work by considering the ‘very’ low-temperature regime, where we discuss relations with the Hawking–Page phase transition observed in the holographic dual black holes. We present preliminary calculations and discuss the possible follow-ups that might be taken to make the connection robust.
It has been recently proposed by Maldacena and Qi that an eternal traversable wormhole in a two-dimensional anti--de Sitter space is the gravity dual of the low temperature limit of two Sachdev-Ye-Kitaev (SYK) models coupled by a relevant interaction (which we will refer to as spin operator). We study spectral and eigenstate properties of this coupled SYK model. We find that level statistics in the tail of the spectrum, and for a sufficiently weak coupling, show substantial deviations from random matrix theory, which suggests that traversable wormholes are not quantum chaotic. By contrast, for sufficiently strong coupling, corresponding to the black hole phase, level statistics are well described by random matrix theory. This transition in level statistics coincides approximately with a previously reported Hawking-Page transition for weak coupling. We show explicitly that this thermodynamic transition turns into a sharp crossover as the coupling increases. Likewise, this critical coupling also corresponds to the one at which the overlap between the ground state and the thermofield double state (TFD) is smallest. In the range of sizes we can reach by exact diagonalization, the ground state is well approximated by the TFD only in the strong coupling limit. This is due to the fact that the ground state is close to the eigenstate of the spin operator corresponding to the lowest eigenvalue which is an exact TFD at infinite temperature. In this region, the spectral density is separated into blobs centered around the eigenvalues of the spin operator. For weaker couplings, the exponential decay of coefficients in a tensor product basis, typical of the TFD, becomes power law. Finally, we also find that the total Hamiltonian has an additional discrete symmetry which has not been reported previously.
We investigate the charged
q/2-body interacting Sachdev-Ye-Kitaev (SYK) model in the grand-canonical ensemble. By treating
q as a large parameter, we are able to analytically study its phase diagram. By varying the chemical potential or temperature, we find that the system undergoes a phase transition between low and high entropies in the maximally chaotic regime. A similar transition in entropy is seen in charged anti--de Sitter (AdS) black holes transitioning between a large and small event horizon. Approaching zero temperature, we find a first-order chaotic-to-nonchaotic quantum phase transition, where the finite extensive entropy drops to zero. This again has a gravitational analog---the Hawking-Page transition between a large black hole and thermal radiation. An analytical study of the critical phenomena associated with the continuous phase transition provides us with mean-field van der Waals critical and effective exponents. We find that all analogous power laws are shared with several charged AdS black hole phase transitions. Together, these findings indicate a connection between the charged $q\ensuremath{\rightarrow}\ensuremath{\infty}$ SYK model and black holes.
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 various aspects of the mass deformation of the SYK model which makes the black hole microstates escapable. SYK boundary states are given by a simple local boundary condition on the Majorana fermions and then evolved in Euclidean time in the SYK Hamiltonian. We study the ground state of this mass deformed SYK model in detail. We also use SYK boundary states as a variational approximation to the ground state of the mass deformed SYK model. We compare variational approximation with the exact ground state results and they showed a good agreement. We also study the time evolution of the mass deformed ground state under the SYK Hamiltonian. We give a gravity interpretation of the mass deformed ground state and its time evolutions. In gravity side, mass deformation gives a way to prepare black hole microstates that are similar to pure boundary state black holes. Escaping protocol on these ground states simply gives a global AdS2 with an IR end of the world brane. We also study the thermodynamics and quantum chaotic properties of this mass deformed SYK model. Interestingly, we do not observe the Hawking Page like phase transition in this model in spite of similarity of the Hamiltonian with eternal traversable wormhole model where we have the phase transition.