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

By making use of conformal mapping, we construct various time-evolution operators in (1+1)-dimensional conformal field theories (CFTs), which take the form $\int dxf\left(x\right)\mathcal{H}\left(x\right)$, where $\mathcal{H}\left(x\right)$ is the Hamiltonian density of the CFT and $f\left(x\right)$ 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.

Figure 1

Conformal map $w=lnz.$

Figure 2

Conformal map $w=ln(z+1)/(z-1).$

Figure 3

Conformal map $w=1/z$.

Figure 4

Conformal map $z=sin\left(w\right)$.

Figure 5

The finite-size spectra of the XX model (18 sites) with PBC (top panel) and OBC (bottom panel).

Figure 6

The finite-size spectra of the XX model with the SSD (18 sites) and the finite-size scaling analysis of its low-lying spectrum.

Figure 7

The finite-size spectra of the XX model with the regularized SSD (18 sites) and the finite-size scaling analysis of its low-lying spectrum. We chose $R=20.$

Figure 8

The finite-size spectra of the XX model with the square root deformation (18 sites) and the finite-size scaling analysis of its low-lying spectrum.