Mapping Topological to Conformal Field Theories through strange Correlators

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, since the corresponding transfer matrices inherit all matrix product operator symmetries of the string-net states. For the case of critical systems, these nonlocal 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 on the lattice by calculating the conformal field theory spectra 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.

Figure 1

Finite-size CFT spectra (scaling dimension $\mathrm{\Delta }$ versus momentum $p$) of twisted hard-hexagon partition functions on a cylinder (extrapolated to $L_y=\infty$). Introducing an extra site in the upper- and lower-right plots induces a ${\mathbb{Z}}_3$ twist. The upper two spectra contain no twist and the lower two spectra contain a ${\mathrm{MPO}}_{\tau }$ twist. 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 $\mathrm{\Delta }$ versus momentum $p$) of twisted Ising partition functions on a cylinder ($L_y=11$) in terms of the conformal characters. The exact topological correction to the conformal spin is denoted next to the first appearance of the respective idempotents.