Relaxation Process During Complex Time Evolution In Two-Dimensional Integrable and Chaotic CFTs

TL;DR

The paper investigates complex time evolution in 2d CFTs, combining Lorentzian unitary dynamics with a non-unitary Euclidean evolution from post-selected measurements, to study how subsystems thermalize and how entanglement evolves. It analyzes both spatially compact and non-compact geometries, using Euclidean path integrals and holographic methods to compute energy densities and (Rényi) entanglement entropies, connecting late-time behavior to primary states with conformal weights set by the inserted operator. In holographic CFTs, the gravity dual features a heavy local operator sourcing a BTZ black hole whose horizon relaxes from an inhomogeneous to a homogeneous profile (compact case) or evolves with a δ-dependent, potentially linear growth (non-compact case), providing a geometric picture for the relaxation toward the primary-state entanglement. The results establish that complex-time evolution drives compact subsystems to primary-state entanglement structures, while non-unitary effects can imprint δ-dependent features in non-compact settings, linking boundary dynamics to horizon relaxation, and offering a framework for exploring measurement-induced non-unitary dynamics in holography.

Abstract

We investigate the complex time evolution of a vacuum state with the insertion of a local primary operator in two-dimensional conformal field theories (2d CFTs). This complex time evolution can be considered as a composite process constructed from Lorentzian time evolution and a Euclidean evolution induced by a post-selected measurement. Our main finding is that in the spatially-compact system, this complex time evolution drives the state of the subsystems to those of the primary state with the same conformal dimensions of the inserted operator. Contrary to the compact system, the subsystems of the spatially non-compact system evolve to states that depend on the non-unitary process during a certain time regime. In holographic systems with a compact spatial direction, this process induced by a heavy local operator can correspond to the relaxation from a black hole with an inhomogeneous horizon to that with a uniform one, while in the ones with a non-compact spatial direction, it can correspond to the relaxation to that with a horizon depending on the non-unitary process.

Figures (8)

• Figure 1: Chiral and anti-chiral energy densities for various $X$ and $\delta$, as a function of time. In the left panels, we show the time dependence of the chiral energy density, while in the right panels, we show that of the anti-chiral energy density. We set the parameters to be $\beta=50$, $x=0$ (insertion point), $L=20000$, $h_{\mathcal{O}}=\overline{h}_{\mathcal{O}}=1000$, $c=1000$.
• Figure 2: The growth of the second Rënyi entanglement entropy, $\Delta S_A^{(2)}(t)$, for various $\delta$ as the function of $t$. In panels, (a)–(c), we show the complex time dependence of $\Delta S_A^{(2)}(t)$ for several values of $\delta$. The parameters are set to be $\beta=10$ , $x=0$$A=[X_2,X_1]$, $X_2=2500$, $X_1=3500$, $L=20000$, and $c=1000$. Here, the blue curve is the complex time dependence of $\Delta S^{(2)}_A$, while red curve is (\ref{['eq:final']}).
• Figure 3: Entanglement entropy for various values of the deformation parameter $\delta$, as a function of $t$. Here, we set $\beta =20, x=0, L=20000 , h_\mathcal{O}= \overline{h}_\mathcal{O} =1000, c=6$. The blue curve is the complex time dependence of $S_A$ and the red curve is \ref{['eq:late-EE-prim']}.
• Figure 4: Finite-interval entanglement entropy on infinite system with various $\delta$, as a function of $t$. In this figure, we set $\beta =20, X_2=10000, X_1= 20000 ,h_\mathcal{O}= \overline{h}_\mathcal{O} =1000,$ and $c=1000$.
• Figure 5: The $\delta$-dependence of the late-time value of $\Delta S_A$ for the semi-infinite interval. Here, we set $c=1$ and $h_\mathcal{O}= \overline{h}_\mathcal{O} =1$.