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

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 $\ensuremath{\int}dx\phantom{\rule{0.16em}{0ex}}f(x)\mathcal{H}(x)$, 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.