Topological conformal defects with tensor networks

Abstract

The critical 2d classical Ising model on the square lattice has two topological conformal defects: the $\mathbb{Z}_2$ symmetry defect $D_ε$ and the Kramers-Wannier duality defect $D_σ$. 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 $Δ_α$ and conformal spins $s_α$ of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising CFT. This characteristic conformal data $\{Δ_α, s_α\}_{D}$ 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 $Z_{D_ε}$ and $Z_{D_σ}$ of the critical Ising model with either a symmetry defect $D_ε$ or a duality defect $D_σ$. We also explain how to coarse-grain the corresponding transfer matrices $M_{D_ε}$ and $M_{D_σ}$, from which we can extract accurate numerical estimates of $\{Δ_α, s_α\}_{D_ε}$ and $\{Δ_α, s_α\}_{D_σ}$. 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 $Δ_α$, 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 $s_α$.