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

TL;DR

This paper explores how complex time evolution in 2D conformal field theories causes subsystems to relax to primary states, revealing differences between compact and non-compact systems and their holographic duals.

Contribution

It introduces a detailed analysis of complex time evolution effects in 2D CFTs, highlighting the contrasting relaxation behaviors in compact versus non-compact systems and their holographic interpretations.

Findings

Subsystems in compact systems relax to primary states.

Non-compact systems exhibit non-unitary evolution effects.

Holographic duals show black hole horizon relaxation processes.

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.