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arXiv:2306.12571 · Physical review. D/Physical review. D.

Entanglement transition and replica wormholes in the dissipative Sachdev-Ye-Kitaev model

Hanteng Wang, Chang Liu, Pengfei Zhang, Antonio M. Garcı́a-Garcı́a

Published Feb 8, 2024• 13 citations
arXivHTMLPDF

Abstract

Recent discoveries have highlighted the significance of replica wormholes in resolving the information paradox and establishing the unitarity of black hole evaporation. In this paper, we propose the dissipative Sachdev-Ye-Kitaev (SYK) model as a minimal quantum model that exhibits entanglement dynamics with features qualitatively similar to replica wormholes. As a demonstration, we investigate the entanglement growth of a pair of dissipative SYK models initialized in a thermofield double state. In the regime of large $N$ with weak dissipation, we observe a first-order entanglement transition characterized by a switch of the dominant saddle point: from replica-diagonal solutions for short times to replica wormholelike off-diagonal solutions for long times. Furthermore, we show that the signature of replica wormholes persists even at moderate $N\ensuremath{\lesssim}30$ by using the Monte Carlo quantum trajectory method. Our work may pave the way for explorations of replica wormhole physics in quantum simulators.

DOI
10.1103/physrevd.109.046005

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Recent discoveries have highlighted the significance of replica wormholes in resolving the information paradox and establishing the unitarity of black hole evaporation. In this paper, we propose the dissipative Sachdev-Ye-Kitaev (SYK) model as a minimal quantum model that exhibits entanglement dynamics with features qualitatively similar to replica wormholes. As a demonstration, we investigate the entanglement growth of a pair of dissipative SYK models initialized in a thermofield double state. In the regime of large NNN with weak dissipation, we observe a first-order entanglement transition characterized by a switch of the dominant saddle point: from replica-diagonal solutions for short times to replica wormholelike off-diagonal solutions for long times. Furthermore, we show that the signature of replica wormholes persists even at moderate N\ensuremath≲30N\ensuremath{\lesssim}30N\ensuremath≲30 by using the Monte Carlo quantum trajectory method. Our work may pave the way for explorations of replica wormhole physics in quantum simulators.
Motivated by recent studies of the information paradox in (1+1)-D anti-de Sitter spacetime with a bath described by a (1+1)-D conformal field theory, we study the dynamics of second Ŕenyi entropy of the Sachdev-Ye-Kitaev (SYK) model (χ) coupled to a Majorana chain bath (ψ). The system is prepared in the thermofield double (TFD) state and then evolved by HL + HR. For small system-bath coupling, we find that the second Rényi entropy S(2)χL,χRS^(2)_{χL,χR}S(2)χL,χR​ of the SYK model undergoes a first order transition during the evolution. In the sense of holographic duality, the long-time solution corresponds to a “replica wormhole”. The transition time corresponds to the Page time of a black hole coupled to a thermal bath. We further study the information scrambling and retrieval by introducing a classical control bit, which controls whether or not we add a perturbation in the SYK system. The mutual information between the bath and the control bit shows a positive jump at the Page time, indicating that the entanglement wedge of the bath includes an island in the holographic bulk.
Entanglement is one of the most important concepts in quantum physics. We review recent progress in understanding the quantum entanglement in many-body systems using large- N solvable models: the Sachdev—Ye—Kitaev (SYK) model and its generalizations. We present the study of entanglement entropy in the original SYK model using three different approaches: the exact diagonalization, the eigenstate thermalization hypothesis, and the path-integral representation. For coupled SYK models, the entanglement entropy shows linear growth and saturation at the thermal value. The saturation is related to replica wormholes in gravity. Finally, we consider the steady-state entanglement entropy of quantum many-body systems under repeated measurements. The traditional symmetry breaking in the enlarged replica space leads to the measurement-induced entanglement phase transition.
A bstract Motivated by recent studies of the information paradox in (1+1)-D anti-de Sitter spacetime with a bath described by a (1+1)-D conformal field theory, we study the dynamics of second Ŕenyi entropy of the Sachdev-Ye-Kitaev (SYK) model ( χ ) coupled to a Majorana chain bath ( ψ ). The system is prepared in the thermofield double (TFD) state and then evolved by H L + H R . For small system-bath coupling, we find that the second Rényi entropy $$ {S}_{\upchi L,\upchi R}^{(2)} $$ S χ L , χ R 2 of the SYK model undergoes a first order transition during the evolution. In the sense of holographic duality, the long-time solution corresponds to a “replica wormhole”. The transition time corresponds to the Page time of a black hole coupled to a thermal bath. We further study the information scrambling and retrieval by introducing a classical control bit, which controls whether or not we add a perturbation in the SYK system. The mutual information between the bath and the control bit shows a positive jump at the Page time, indicating that the entanglement wedge of the bath includes an island in the holographic bulk.
We study the open quantum dynamics of the Sachdev-Ye-Kitaev (SYK) model described by the Lindblad master equation, where the SYK model is coupled to Markovian reservoirs with jump operators that are either linear or quadratic in the Majorana fermion operators. Of particular interest for us is the time evolution of the dissipative form factor, which quantifies the average overlap between the initial and time-evolved density matrices as an open quantum generalization of the Loschmidt echo. We find that the dissipative form factor exhibits dynamical quantum phase transitions. We analytically demonstrate a discontinuous dynamical phase transition in the limit of a large number of fermion flavors, which is formally akin to the thermal phase transition in the two-coupled SYK model between the black-hole and wormhole phases. We also find continuous dynamical phase transitions that do not have counterparts in the two-coupled SYK model. While the phase transitions are sharp in the limit of a large number of fermion flavors, their qualitative signatures are present even for a finite number of fermion flavors, as we show numerically.
A pair of identical Sachdev-Ye-Kitaev (SYK) models with bilinear coupling forms a quantum dual of a traversable wormhole, with a ground state close to the so called thermofield double state. We use adiabaticity arguments and numerical simulations to show that the character of the ground state remains unchanged when interaction strengths in the two SYK models become imbalanced. Analysis of thermodynamic and dynamical quantities highlights the key signatures of the wormhole phase in the imbalanced case. Further adiabatic evolution naturally leads to the ``maximally imbalanced'' limit where fermions in a single SYK model are each coupled to a free Majorana zero mode. This limit is interesting because it could be more easily realized using various setups proposed to implement the SYK model in a laboratory. We find, based on numerical studies of the quantum mechanical model, that this limiting case is marginal in that it retains some of the characteristic signatures of the wormhole physics, such as the spectral gap and revival dynamics, but lacks others so that it likely does not represent the full-fledged wormhole dual. We discuss how this scenario could be implemented in the proposed realization of the SYK physics in quantum wires of finite length coupled to a disordered quantum dot.
We study the out-of-equilibrium dynamics of a Sachdev-Ye-Kitaev (SYK) model, NNN fermions with a qqq-body interaction of infinite range, coupled to a Markovian environment. Close to the infinite-temperature steady state, the real-time Lindbladian dynamics of this system is identical to the near-zero-temperature dynamics in Euclidean time of a two-site non-Hermitian SYK with intersite coupling whose gravity dual has been recently related to wormhole configurations. We show that the saddle-point equations in the real-time formulation are identical to those in Euclidean time. Indeed, an explicit calculation of Green's functions at low temperature, numerical for q=4q=4q=4 and analytical for q=2q=2q=2 and large qqq, illustrates this equivalence. Only for very strong coupling does the decay rate approach the linear dependence on the coupling characteristic of a dissipation-driven approach to the steady state. For q>2q>2q>2, we identify a potential gravity dual of the real-time dissipative SYK model: a double-trumpet configuration in a near-de Sitter space in two dimensions with matter. This configuration, which we term a Keldysh wormhole, is responsible for a finite decay rate even in the absence of coupling to the environment.