  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.
