Topological conformal defects with tensor networks

The critical two-dimensional classical Ising model on the square lattice has two topological conformal defects: the Z2 symmetry defect Dε and the Kramers-Wannier duality defect Dσ. These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function ZD of the critical Ising model in the presence of a topological conformal defect D is expressed in terms of the scaling dimensions Δα and conformal spins sα of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising conformal field theory. This characteristic conformal data {Δα,sα}D can be extracted from the eigenvalue spectrum of a transfer matrix MD for the partition function ZD. In this paper, we investigate the use of tensor network techniques to both represent and coarse grain the partition functions ZDε and ZDσ of the critical Ising model with either a symmetry defect Dε or a duality defect Dσ. We also explain how to coarse grain the corresponding transfer matrices MDε and MDσ, from which we can extract accurate numerical estimates of {Δα,sα}Dε and {Δα,sα}Dσ. Two key ingredients of our approach are (i) coarse graining of the defect D, which applies to any (i.e., not just topological) conformal defect and yields a set of associated scaling dimensions Δα, and (ii) construction and coarse graining of a generalized translation operator using a local unitary transformation that moves the defect, which only exist for topological conformal defects and yields the corresponding conformal spins sα.