Lattice Realizations of Topological Defects in the critical (1+1)-d Three-State Potts Model

TL;DR

This work develops explicit lattice realizations for topological and duality defects in the (1+1)d critical three-state Potts model, via both spin-chain and D4 RSOS formulations. It systematically constructs lattice line operators and defect Hamiltonians for the primitive lines I, η, C, N, and W, analyzes their spectra and fusion in crossed and direct channels, and validates these against CFT expectations. Entanglement entropy, via symmetric blocks and folding, yields defect g-functions that match the Affleck-Ludwig predictions for the topological lines, confirming continuum limits. The results open pathways for lattice-based defect fusion, KW duality, and RSOS generalizations to broader RCFTs. Overall, the paper provides a robust framework marrying integrability, TL algebra, and DMRG/ED techniques to probe topological defects in non-diagonal RCFTs.

Abstract

Topological/perfectly-transmissive defects play a fundamental role in the analysis of the symmetries of two dimensional conformal field theories (CFTs). In the present work, spin chain regularizations for these defects are proposed and analyzed in the case of the three-state Potts CFT. In particular, lattice versions for all the primitive defects are presented, with the remaining defects obtained from the fusion of the primitive ones. The defects are obtained by introducing modified interactions around two given sites of an otherwise homogeneous spin chain with periodic boundary condition. The various primitive defects are topological on the lattice except for one, which is topological only in the scaling limit. The lattice models are analyzed using a combination of exact diagonalization and density matrix renormalization group techniques. Low-lying energy spectra for different defect Hamiltonians as well as entanglement entropy of blocks located symmetrically around the defects are computed. The latter provides a convenient way to compute the $g$-function which characterizes various defects. Finally, the eigenvalues of the line operators in the "crossed channel'' and fusion of different defect lines are also analyzed. The results are all in agreement with expectations from conformal field theory.

Figures (21)

• Figure 1: Above: A graphical illustration of the three-state Potts model, where each blue dot represents a spin site. Nearest-neighbor interactions are represented by horizontal links connecting the spins, while vertical links represent the transverse field. Below: Local modifications involved in the $\eta$, $C$ and $N$ defect Hamiltonians. The $\eta$ and $C$ defects effectively introduce a twist by their appropriate symmetry actions. The N defect Hamiltonian involves removing the transverse field at one site of the defect link, while modifying the nearest-neighbor interaction to mimic the coupling between the original spin and the dual spin.
• Figure 2: The $\mathbf{D}_4$ dynkin diagram and its adjacency matrix.
• Figure 3: The $Y$ operator maps between ${\mathcal{H}}_{\text{even}}$ and ${\mathcal{H}}_{\text{odd}}$.
• Figure 4: Local weights associated with a face with the spectral parameter set to $\text{i}\infty$.
• Figure 5: Expectation values of the $\widehat{W}$ lattice operator for several sets of states. The results for the two groups should converge to the golden ratio $x$ (resp. $- x^{-1}$) as $L\to\infty$, see Table \ref{['table:PottsTDLaction']}.