Topological conformal defects with tensor networks
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abstract

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 $s_{\alpha}$.