Replica wormhole and information retrieval in the SYK model coupled to Majorana chains

TL;DR

This work investigates information retrieval in a holography-inspired setting by coupling an SYK model to a 1+1D Majorana chain bath. Using a large-$N$ Schwinger-Dyson framework and twist-operator formalism, it identifies a first-order entropy transition at small coupling, interpreted as replica wormhole formation, with a Page-time-like transition controlling information flow between the SYK system and the bath. A classical control-bit probe reveals a jump in bath mutual information at Page time, signaling the emergence of an island and the recoverability of perturbations from the bath. The study provides a tractable quantum model that captures Page curves, replica wormholes, and information retrieval through bulk island physics, offering insights applicable to black hole information questions in a non-gravitational setting.

Abstract

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 Rényi 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 $S^{(2)}_{χ_L, χ_R}$ 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.

Figures (12)

• Figure 1: The graphical representations of (a) the $\ket{TFD(t)}$ state and (b) the reduced density operator $\rho_{\chi_L, \chi_R} (t)$.
• Figure 2: Two equivalent illustrations of the contour $C$ of path integral for computing the second Rényi entropy.
• Figure 3: (a) Numerical result for $V^2/J=0.25$, $\beta J=4$ and $\Lambda=5J$. The entorpy is a smooth function of time. (b) Numerical result for $V^2/J=0.05$, $\beta J=4$ and $\Lambda=5J$. There is a first order transition of the entropy. (c) The real part of Green's function $G(s,s')$ corresponding to the short-time saddle in (b). Here we take $t/\beta=1.5$ as an example. Orange/Blue means positive/negative while their darkness indicates magnitude. The numbers on the axes correspond to the discretization of parameter $s$ in numerics. (d) The real part of Green's function $G(s,s')$ corresponding to the long-time saddle in (b). Here we take $t/\beta=6$ as an example. Here and in latter figures we have removed tiny matrix elements $|G(s,s')|<10^{-3}$ in the plot to make the plot clearer.
• Figure 4: Comparison of the analytic formula \ref{['resshort']} with numerics for $V^2/J = 0.005$, $\beta J = 4$ and $\Lambda =5J$.
• Figure 5: (a) The long-time solution with $t=6\beta, V^2/J=0.05$. The two matrix plots are the real part and the imaginary part respectively (same in (b)). (b) The trivial solution that we do not insert any twist operators.