Universal boundary entropies in conformal field theory: A quantum Monte Carlo study

Recently, entropy corrections on nonorientable manifolds such as the Klein bottle are proposed as a universal characterization of critical systems with an emergent conformal field theory (CFT). We show that entropy correction on the Klein bottle can be interpreted as a boundary effect via transforming the Klein bottle into an orientable manifold with nonlocal boundary interactions. The interpretation reveals the conceptual connection of the Klein bottle entropy with the celebrated Affleck-Ludwig entropy in boundary CFT. We propose a generic scheme to extract these universal boundary entropies from quantum Monte Carlo calculation of partition function ratios in lattice models. Our numerical results on the Affleck-Ludwig entropy and Klein bottle entropy for the $q$-state quantum Potts chains with $q=2,3$ show excellent agreement with the CFT predictions. For the quantum Potts chain with $q=4$, the Klein bottle entropy slightly deviates from the CFT prediction, which is possibly due to marginally irrelevant terms in the low-energy effective theory.

Figure 1

(a) Path integral manifold of a 1D quantum system with periodic boundary condition in the spatial direction is an equivalent to a torus. The direction of black arrows indicate how the space or time boundaries are joined. (b) With open boundary condition in the spatial direction, the path integral manifold is a cylinder. (c) The path integral manifold of Eq. (4) corresponds to a Klein bottle due to the spatial reflection inserted in the trace.

Figure 2

(a) Transforming a Klein bottle path integral manifold into an orientable manifold via cutting, flipping, and sewing the manifolds. (b) A Klein bottle is transformed into a cylinder with nonlocal interactions along the time direction as indicated by the dashed lines on the cylinder boundaries. (c) A torus corresponds to a cylinder with two spatial boundaries joined by the nonlocal interactions indicated by the dashed lines.

Figure 3

Results of the AL entropy extracted from a linear fit of the QMC result of Eq. (3), including Ising model with (a) free and (b) fixed boundary conditions and three-state Potts model with (c) free, (d) fixed, (e) mixed, and (f) symmetric boundary conditions (abbreviated as b.c. in the figures). The intercept of the linear regression over $[{\beta }_c-1,{\beta }_c+1]$ is plotted versus the center of the fitting range ${\beta }_c$. The horizontal lines indicate the CFT predictions of the AL entropies. The size of the systems is $L=500$.

Figure 4

Quantum Monte Carlo results of the Klein bottle entropies for (a) $q=2$ (Ising), (b) $q=3$, (c) $q=4$ quantum Potts chains.