# Emergent Spatial Structure and Entanglement Localization in Floquet Conformal Field Theory

## TL;DR

This paper analyzes energy and entanglement dynamics in Floquet-driven 1+1D conformal field theories with sine-square deformation, revealing universal features of the heating phase, including localized energy peaks and Bell pair generation, with implications for quantum entanglement.

## Contribution

The study provides an analytical description of universal features in the heating phase of Floquet CFTs, including energy localization, entanglement sharing, and a relation between energy and entanglement entropy, extending understanding beyond specific models.

## Key findings

- Energy density concentrates in two peaks, leading to exponential energy growth.
- Quantum entanglement is localized between the two peaks via Bell pairs.
- Energy and entanglement are related by a simple exponential relation involving central charge.

## Abstract

We study the energy and entanglement dynamics of $(1+1)$D conformal field theories (CFTs) under a Floquet drive with the sine-square deformed (SSD) Hamiltonian. Previous work has shown this model supports both a non-heating and a heating phase. Here we analytically establish several robust and `super-universal' features of the heating phase which rely on conformal invariance but not on the details of the CFT involved. First, we show the energy density is concentrated in two peaks in real space, a chiral and anti-chiral peak, which leads to an exponential growth in the total energy. The peak locations are set by fixed points of the M\"obius transformation. Second, all of the quantum entanglement is shared between these two peaks. In each driving period, a number of Bell pairs are generated, with one member pumped to the chiral peak, and the other member pumped to the anti-chiral peak. These Bell pairs are localized and accumulate at these two peaks, and can serve as a source of quantum entanglement. Third, in both the heating and non-heating phases we find that the total energy is related to the half system entanglement entropy by a simple relation $E(t)\propto c \exp \left( \frac{6}{c}S(t) \right)$ with $c$ being the central charge. In addition, we show that the non-heating phase, in which the energy and entanglement oscillate in time, is unstable to small fluctuations of the driving frequency in contrast to the heating phase. Finally, we point out an analogy to the periodically driven harmonic oscillator which allows us to understand global features of the phases, and introduce a quasiparticle picture to explain the spatial structure, which can be generalized to setups beyond the SSD construction.