Mapping topological to conformal field theories through strange correlators

TL;DR

It is argued that the different conformal boundary conditions can be obtained by applying the strange correlator concept to the different topological sectors of the string net obtained from Ocneanu's tube algebra.

Abstract

We extend the concept of strange correlators, defined for symmetry-protected phases in [You et al., Phys. Rev. Lett. 112, 247202 (2014)], to topological phases of matter by taking the inner product between string-net ground states and product states. The resulting two-dimensional partition functions are shown to be either critical or symmetry broken, as the corresponding transfer matrices inherit all matrix product operator symmetries of the string-net states. For the case of critical systems, those non-local matrix product operator symmetries are the lattice remnants of topological conformal defects in the field theory description. Following [Aasen et al., J. Phys. A 49, 354001 (2016)], we argue that the different conformal boundary conditions can be obtained by applying the strange correlator concept to the different topological sectors of the string-net obtained from Ocneanu's tube algebra. This is demonstrated by calculating the conformal field theory spectra on the lattice in the different topological sectors for the Fibonacci/hard-hexagon and Ising string-net. Additionally, we provide a complementary perspective on symmetry-preserving real-space renormalization by showing how known tensor network renormalization methods can be understood as the approximate truncation of an exactly coarse-grained strange correlator.

Figures (3)

• Figure 1: Topological sector labeling of finite-size CFT spectra (scaling dimension $\Delta$ versus momentum $p$) of twisted hard hexagon partition functions on a cylinder (extrapolated from $L_{y} = 18,21,24$). The exact topological correction to the conformal spin is denoted next to the first appearance of the respective idempotents.
• Figure 2: Topological sector labeling of finite-size CFT spectra (scaling dimension $\Delta$ versus momentum $p$) of twisted Ising partition functions on a cylinder ($L_{y} = 11$). From left to right: the $1$-twist, $\psi$-twist, and $\sigma$-twist spectra. The exact topological correction to the conformal spin is denoted next to the first appearance of the respective idempotents.
• Figure 3: Every RG step, the gates of the exact string-net coarse-graining map block the PEPS tensors in such a way that preserving the PEPS structure leads to an increased bond dimension, indicating that the coarse-grained PEPS has to be truncated in order to obtain a sustainable RG transformation.