Topological conformal defects with tensor networks

The critical 2d classical Ising model on the square lattice has two topological conformal defects: the $\mathbb{Z}_2$ symmetry defect $D_{\epsilon}$ 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 CFT. 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 $Z_{D_\epsilon}$ and $Z_{D_\sigma}$ of the critical Ising model with either a symmetry defect $D_{\epsilon}$ or a duality defect $D_{\sigma}$. We also explain how to coarse-grain the corresponding transfer matrices $M_{D_\epsilon}$ and $M_{D_\sigma}$, from which we can extract accurate numerical estimates of $\{\Delta_{\alpha}, s_{\alpha}\}_{D_{\epsilon}}$ and $\{\Delta_{\alpha}, s_{\alpha}\}_{D_{\sigma}}$. Two key new 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 $s_{\alpha}$.