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

Abstract

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.

Klein bottle entropy of compactified boson conformal field theory

“…The slight deviation may originate from the marginally irrelevant Umklapp process at ∆ = 1. We have observed this kind of slight deviation in the q = 4 Potts model, which also has a marginally irrelevant term 14 .…”

“…As an example, in the XXZ chain where we perform our simulation, if we propose the transition update directly in the configuration space, most of such updates will be rejected, since there are usually graphs locating between the boundary sites, which will lead to the cut of closed loops in the worldline configuration after the direct transition. In other words, the direct transition update, which is originally developed for the quantum Potts model 14 , will usually break the total spin conservation which is satisfied by the XXZ model. In this case, the overlap between the two ensembles only consists of those configurations with no graph elements between the boundaries, which will quickly decay with respect to the system size L and the temperature β.…”

“…However, in practice, due to the existence of the nonuniversal term − f b β, the partition function ratio decays exponentially with β, and the error of the calculation becomes intolerable when β reaches some certain value. On the other hand, the calculation result from the finite-size lattice will converge to the universal value only when the β and L is large enough 14 .…”

“…The AL entropy emerges from the open boundary of a long cylinder, which is universal and only depends on the CFT and conformal boundary conditions. In lattice models, when L vβ, one can obtain the AL entropy by calculating the ratio between the partition functions of the systems on a long cylinder and a torus 14 ,…”

Topological and geometric universal thermodynamics in conformal field theory

“…Recently, the universal thermodynamics of 2D CFTs on nonorientable surfaces has been explored, including the Klein bottle (K 2 ) and Möbius strip [21][22][23][24]. For diagonal CFT partition functions on the Klein bottle, F K = ln Z K = πc 24vβ L + ln k, with a universal constant k = a d a /D, where d a is the quantum dimension of the ath primary field, and D = a d 2 a is the total quantum dimension.…”