Continuum Description of the Edge States in a (×Z₃)³-Symmetry-Protected Topological Model

Abstract

We investigate the continuum limit of the edge-state Hamiltonian of a two-dimensional symmetry-protected topological (SPT) model with (×Z₃)³ = Z³ × Z³ × Z³ symmetry. Starting from the lattice Hamiltonian, we first reformulate the model in terms of S₃ generators and subsequently introduce a constrained Schwinger-like fermionic representation. Within this formulation, the relevant operators are expressed as fermionic bilinears, while non-local Jordan–Wigner string operators do not appear explicitly. At the critical point, λ = λ̄, we show that the continuum Hamiltonian supports gapless excitations with a linear dispersion relation, consistent with a conformal description of the edge theory. Using a mean-field treatment of the interaction terms, we further obtain a preliminary estimate of the finite-size correction to the ground-state energy. The corresponding effective central charge, c = √3 ≈ 1.73, is remarkably close to the numerical value previously obtained for the (×Z₃)³ SPT edge model and to the value predicted by the SU(3)₂/SU(2)₂ coset conformal field theory. Although this derivation remains heuristic, our results suggest that the constrained fermionic formulation provides a natural framework for describing the low-energy continuum limit of (Z₃)³ SPT edge states and may serve as a useful starting point for a more systematic derivation of the underlying conformal field theory.