Topological conformal defects with tensor networks

The critical two-dimensional classical Ising model on the square lattice has two topological conformal defects: the Z(2) symmetry defect D-is an element of and the Kramers-Wannier duality defect D-sigma. These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function Z(D) of the critical Ising model in the presence of a topological conformal defect D is expressed in terms of the scaling dimensions Delta(alpha) and conformal spins s(alpha) of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising conformal field theory. This characteristic conformal data {Delta(alpha), s(alpha)} D can be extracted from the eigenvalue spectrum of a transfer matrix M-D for the partition function Z(D). In this paper, we investigate the use of tensor network techniques to both represent and coarse grain the partition functions Z(D is an element of) and Z(D sigma) of the critical Ising model with either a symmetry defect D-is an element of or a duality defect D-sigma. We also explain how to coarse grain the corresponding transfer matrices M-D is an element of and M-D sigma, from which we can extract accurate numerical estimates of {Delta(alpha), s(alpha)}(D is an element of) and {Delta alpha, s(alpha)}(D sigma). 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 Delta(alpha), 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 s(alpha.)