Spatially Structured Entanglement from Nonequilibrium Thermal Pure States

TL;DR

This work analyzes nonequilibrium dynamics of crosscap states in (1+1)d CFTs under spatially inhomogeneous quenches generated by $ ext{SL}^{(q)}(2, ext{R})$ deformations, comparing integrable (free Dirac fermion) and holographic (large-$c$) theories. It combines twist-field techniques, the quasiparticle picture, and AdS$_3$/CFT$_2$ holography to show that certain deformations (notably $q$-Möbius, $q$-SSD, and $q$-Displacement) produce non-thermalizing dynamics with universal graph-like entanglement patterns dictated by the deformation profile, independent of microscopic details. The quasiparticle framework is extended to inhomogeneous settings and, surprisingly, yields late-time graph structures even in holographic CFTs, while holographic entanglement entropy calculations via RT/HRT reproduce the main features and reveal a geodesic interior mismatch in certain inhomogeneous cases. Overall, the paper demonstrates a deformation-driven route to control scrambling and entanglement spreading in critical systems and provides a gravity dual that corroborates the CFT results, with implications for understanding thermalization and multipartite entanglement in nonuniform quantum quenches.

Abstract

We study quantum quench dynamics in (1+1)-dimensional critical systems, starting from thermal pure states called crosscap states, and evolving them under spatially inhomogeneous Hamiltonians. The spatial inhomogeneity is introduced through a deformation of the Hamiltonian, expressed as linear combinations of the generators of the $SL^{(q)}(2,\mathbb{R})$ subalgebra of the Virasoro algebra. We analyze the free massless Dirac fermion theory and holographic conformal field theory as prototypical examples of integrable and non-integrable dynamics. Consistent with general expectations, "Möbius-type" deformations lead to thermalization in the non-integrable case, and to periodic revivals in the integrable one. In contrast, "sine-square-type" and "displacement-type" deformations prevent both thermalization and scrambling, instead producing late-time, graph-like entanglement patterns. These patterns emerge from the interplay between the deformed Hamiltonian and the crosscap initial state and appear to be universal: they are determined solely by the deformation profile while remaining largely insensitive to microscopic details. Finally, we perform a holographic calculation in three-dimensional gravity using AdS$_3$/CFT$_2$, which reproduces the main features of our (1+1)-dimensional study.

Figures (14)

• Figure 1: Schematics of EAP state (left panel) and path-integral construction of the Klein bottle (right panel). Left panel: red dots and lines denote the qubits and their EPR links, respectively; right panel: crosses represent the crosscap boundary conditions.
• Figure 2: Schematic of the quasiparticle motion. Throughout this work, the crosses and dots represent the right- and left-movers, respectively. The dashed lines indicate the EPR links between quasiparticle pairs. The coloring does not carry any physical meaning; it merely labels distinct quasiparticle pairs.
• Figure 3: Entanglement entropy (EE) and mutual information (MI) following a uniform quench in free fermionic and holographic theories. Upper panels: EE and MI for the free Dirac fermion theory. Hereafter, we set $L=10000$ for all plots. The solid blue and dashed orange curves show the EE/MI corresponding to the two crosscap states $\ket{C_{O1}}$ and $\ket{C_{O0}}$, respectively, which are defined in \ref{['eq:Compact-Boson-Crosscap-States']}. The entanglement evolution is explicitly compared with the quasiparticle prediction (dashed green), showing excellent agreement. Lower panels: EE and MI in holographic CFT. Both quantities clearly deviate from quasiparticle picture predictions; the EE saturates at the subsystem thermal entropy (dashed red) given by \ref{['eq:QP-density']}, and the MI decays to zero.
• Figure 4: EE and MI time evolution for $q$-Möbius quench. Upper panels: Evolution of EE and MI in free Dirac fermion CFT. In this setup, both the EE and the MI are well described by the quasiparticle picture and exhibit periodic revivals. In contrast to the uniform quench in Fig. \ref{['fig:Uniform-Crosscap-Quench']}, the EE plateau–decay behavior and the MI revival around $t\approx\frac{L}{2}$ are significantly altered by the inhomogeneous quench, where $L=10000$; lower panels: EE and MI in holographic CFT. Although the EE shows residual oscillations, the system nonetheless thermalizes and scrambles: both the EE and the MI no longer follow the quasiparticle predictions, and the MI decays monotonically to zero.
• Figure 5: EE and MI time evolution for $q$-SSD quench. Upper panels: EE and MI in free Dirac fermion CFT; lower panels: EE and MI in holographic CFT; left panels: conformal cooling drives $S_{A}^{(2)}(t)$ to the vacuum entanglement value (gray dashed line) given in \ref{['eq:Conformal-Cooling-Vacuum']} when $A$ contains no fixed points, in both free-fermion and holographic CFTs. The insets detail how the entanglement entropy (EE) approaches the vacuum value; the quasiparticle picture fails to capture this behavior, as its validity is restricted to order $\beta^{-1}$. Right panels: in both free-fermion and holographic CFTs, the late-time mutual information matches the graph-like pattern prediction (purple dashed line), \ref{['eq:qSSD-Asym-MI']}.