Evolution operators in conformal field theories and conformal mappings: the entanglement Hamiltonian, the sine-square deformation, and others

TL;DR

This paper constructs and analyzes various deformed time-evolution operators in (1+1)D conformal field theories using conformal mappings, providing exact spectral properties and proposing a regularized sine-square deformation with finite-size spectrum control.

Contribution

It introduces a systematic method to construct deformed evolution operators in CFTs and proposes a regularized sine-square deformation with well-defined finite-size spectral properties.

Findings

Exact spectrum and level spacing scaling of deformed operators

Construction of a regularized sine-square deformation with finite-size spectrum

Proposal of a deformation with level spacing scaling as 1/L^2

Abstract

By making use of conformal mapping, we construct various time-evolution operators in (1+1) dimensional conformal field theories (CFTs), which take the form $\int dx\, f(x) \mathcal{H}(x)$, where $\mathcal{H}(x)$ is the Hamiltonian density of the CFT, and $f(x)$ is an envelope function. Examples of such deformed evolution operators include the entanglement Hamiltonian, and the so-called sine-square deformation of the CFT. Within our construction, the spectrum and the (finite-size) scaling of the level spacing of the deformed evolution operator are known exactly. Based on our construction, we also propose a regularized version of the sine-square deformation, which, in contrast to the original sine-square deformation, has the spectrum of the CFT defined on a spatial circle of finite circumference $L$, and for which the level spacing scales as $1/L^2$, once the circumference of the circle and the regularization parameter are suitably adjusted.