Spread of entanglement in a Sachdev-Ye-Kitaev chain

TL;DR

This work investigates how Rényi entropies spread in a generalized SYK chain after a global quench from a thermofield double state. Using replica methods and large-$N$ path integrals, it shows that for integer Renyi index $n>1$, the entropy saturates at a sub-thermal value, indicating prethermalization governed by slow heavy modes in contrast to rapid scrambling of reparameterization modes. The authors develop a weak-link perturbation, a Schwarzian reparameterization framework, and a two-site geometric picture to compute the entropy growth and long-time saturation, finding a linear growth at early times with a rate set by $J_1^2/(4\beta J^2)$ and a saturated value controlled by $c_v T$. The results suggest that the entanglement velocity for Rényi entropy may not reflect full thermalization in this model and motivate a detailed comparison with holographic setups involving $\mathrm{AdS}_2$ geometries, highlighting subtleties in entanglement dynamics beyond von Neumann entropy.

Abstract

We study the spread of Rényi entropy between two halves of a Sachdev-Ye-Kitaev (SYK) chain of Majorana fermions, prepared in a thermofield double (TFD) state. The SYK chain model is a model of chaotic many-body systems, which describes a one-dimensional lattice of Majorana fermions, with spatially local random quartic interaction. We find that for integer Rényi index $n>1$, the Rényi entanglement entropy saturates at a parametrically smaller value than expected. This implies that the TFD state of the SYK chain does not rapidly thermalize, despite being maximally chaotic: instead, it rapidly approaches a prethermal state. We compare our results to the signatures of thermalization observed in other quenches in the SYK model, and to intuition from nearly-$\mathrm{AdS}_2$ gravity.

Figures (11)

• Figure 1: (a) Illustration of the TFD state, which is obtained by applying imaginary time evolution $e^{-\frac{\beta}{4}H_L}$ and $e^{-\frac{\beta}{4}H_R}$ to a state $|I\rangle$ of the two-chain system. $|I\rangle$ is a direct product of local EPR pairs between the two sites in the two chains at the same spatial location. (b) The real time evolution of the TFD state by $U(t)= \exp [-i t (H_{\mathrm L}+H_{\mathrm R}) ]$ and our choice of entanglement cut. We study the Renyi entropies of the region $A=A_L\cup A_R$, with support on both chains.
• Figure 2: (a) We picture the TFD state of the SYK chain model as a half tube. The two top edges correspond to the states in left and right Hilbert space. The subregion $A=A_{\mathrm L} \cup A_{\mathrm R}$ is defined on both sides, and colored blue. The yellow shaded region corresponds to the real time evolution $U(t)$, and the circular portion of the tube represents the initial imaginary time evolution of (\ref{['eq:TFDintro']}). (b) The density matrix $\rho_A$ is then found by gluing two $|\mathrm{TFD}(t)\rangle$ states on $A^c$ and leaving $A$ (blue) 'open'. This perspective is useful in computing the partition function $Z_{A,n}$, where the blue lines play the role of branch cuts. Each replicated fermion $\chi_{\alpha}$ shifts its replica index to $\alpha+1$ when it crosses the right branch cut line (the side closer to the reader) from below and shift to $\alpha-1$ when it crosses the left branch cut line from the above. We can further deform the two horizontal blue branch cuts to a single vertical dashed branch cut (shown in red).
• Figure 3: The time contour involved in the entanglement entropy calculation. This figure is plotted in complex plane $z=\exp (i\frac{2\pi}{\beta} t_{\mathbb{C}})$, where $t_{\mathbb{C}}=\tau+it$ is the complex time variable. The red part represents the $\langle \mathrm{TFD}(t)|$ in Fig. \ref{['fig: tfd']}, and the black represents the $| \mathrm{TFD}(t) \rangle$. For later convenience, we name them as $C_1$ and $C_2$.
• Figure 4: (a) The imaginary time contour involved in the entanglement entropy calculation. (b) The calculation needs to be regularized by introducing a small separation between $C_1$ and $C_2$ by $\epsilon_{1,2}$, both of which are of order $J^{-1}$.
• Figure 5: (a) Geometric illustration of the minimization problem we need to solve. The total length of the curve is fixed to be $L$. The "string" connects $X_1$ and $X_2$ tends to pull the two points closer, while the other term of the action prefers the curve to enclose a larger area. (b) For a fixed distance $D(X_1,X_2)$ between $X_1$ and $X_2$, the joint of two arcs maximizes the area enclosed by the curve. Then the entropy is determined by the minimum of action as a function of $D(X_1,X_2)$.