Topological conformal defects with tensor networks

Abstract

The critical two-dimensional classical Ising model on the square lattice has two topological conformal defects: the ${\mathbb{Z}}_{2}$ symmetry defect ${D}_{\ensuremath{\epsilon}}$ and the Kramers-Wannier duality defect ${D}_{\ensuremath{\sigma}}$. 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 ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$ and conformal spins ${s}_{\ensuremath{\alpha}}$ of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising conformal field theory. This characteristic conformal data ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{D}$ 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 ${Z}_{{D}_{\ensuremath{\epsilon}}}$ and ${Z}_{{D}_{\ensuremath{\sigma}}}$ of the critical Ising model with either a symmetry defect ${D}_{\ensuremath{\epsilon}}$ or a duality defect ${D}_{\ensuremath{\sigma}}$. We also explain how to coarse grain the corresponding transfer matrices ${M}_{{D}_{\ensuremath{\epsilon}}}$ and ${M}_{{D}_{\ensuremath{\sigma}}}$, from which we can extract accurate numerical estimates of ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\epsilon}}}$ and ${{{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}},{s}_{\ensuremath{\alpha}}}}_{{D}_{\ensuremath{\sigma}}}$. 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 ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\alpha}}$, 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 ${s}_{\ensuremath{\alpha}}$.