Topological conformal defects with tensor networks

TL;DR

<p>We present a tensor-network framework to characterize topological conformal defects in the critical 2D Ising model, focusing on the symmetry defect $D_{\epsilon}$ and the Kramers–Wannier duality defect $D_{\sigma}$. By constructing defect-augmented transfer matrices $M_{D}$ and, for topological defects, generalized translations $T_{D}$, we extract defect-specific conformal data $(\Delta_{\alpha},s_{\alpha})_{D}$ through coarse-graining (TNR) and spectral analysis; a local unitary moves the defect enabling $T_{D}$ to commute with $M_{D}$ and yield spins. We demonstrate the method for both defects, obtaining precise scaling dimensions and spins deep into the defect towers, and we extend the approach to a generic family of defects $D_{\gamma}$, illustrating how coarse-graining improves finite-size effects. The work also connects to MERA constructions and discusses extensions to other models and alternative conformal-data extraction routes, underscoring tensor networks as a powerful tool for defect CFT data. </p>

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 $s_α$.

Figures (30)

• Figure 1: (a) Partition function $Z$ on a square lattice made of $n \times m$ sites with periodic boundary conditions in both directions (a torus), and the corresponding transfer matrix $M$, such that $Z = \mathop{\mathrm{Tr}}\nolimits \left( M^{m} \right)$. (b) Partition function $Z_D$ on the same torus, where the defect $D$ implements some form of boundary conditions, and the generalized transfer matrix $M_D$, such that $Z_D = \mathop{\mathrm{Tr}}\nolimits \left( (Z_D)^m \right)$.
• Figure 2: The scaling dimensions (vertical axis) and conformal spins (horizontal axis) of the first scaling operators of the Ising CFT obtained from exact diagonalization of a transfer matrix of $n=18$ sites. The scaling operators are divided by their parity, i.e. their eigenvalue under the $\mathbb{Z}_2$ symmetry operator $\Sigma^x$ that commutes with the transfer matrix. The crosses mark the numerical values that can be compared with the circles that are centered at the exact values. Several concentric circles denote the degeneracy $N_\alpha$ of that $(\Delta_\alpha, s_\alpha)$ pair. The primary fields identity$\mathbb{I}$, spin$\sigma$ and energy density$\epsilon$ appear at the basis of their three conformal towers.
• Figure 3: In this figure and throughout the paper we use the usual graphical tensor network language where tensors are represented by various shapes and their indices by legs coming out from them. A leg connecting two tensors is summed over. (a) The partition function $Z$ of a classical two dimensional lattice model on a torus as a tensor network, first using the Boltzmann weights $B_{ij} = e^{- \beta E_{ij}}$ and then in terms of the tensor $A_{ijkl} = B_{ij} B_{jk} B_{kl} B_{li}$. We call the network on the right $Z_{n,m}(A)$. $\delta_{ijk}$ and $\delta_{ijkl}$ are three- and four-way Kronecker deltas that fix all their indices to have the same value. (b) The transfer matrix $M$ as a tensor network. (c) The one-site translation operator $T$. (d) The translation operator composed with the transfer matrix.
• Figure 4: The invariance of tensor $A$ under the symmetry transformation $V$.
• Figure 5: Repeating a coarse-graining produces a series of tensors $A^{(s)}$ and corresponding networks that all contract to approximately the same value. We think of each $A^{(s)}$ as representing a local patch of the system at a different length scale. With a $2 \times 2 \mapsto 1$ coarse-graining like the one we consider, a network $Z_{2^k, 2^k}(A)$ can be coarse-grained to a single tensor in $k$ steps.