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 $D_{\epsilon }$ 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 $s_{\alpha }$.

20 pages + 7 pages of appendices. 30 figures; v2: Added a note and Python 3 source code, plus minor fixes; v3: Typos & aesthetics

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