Topological edge states in two-dimensional ${\mathbb{Z}}_4$ Potts paramagnet protected by the ${\mathbb{Z}}_4^{\times 3}$ symmetry

Abstract

We construct a two-dimensional bosonic symmetry-protected topological (SPT) paramagnet protected by an on-site $G={\mathbb{Z}}_4^{\times 3}$ symmetry, starting from a three-component ${\mathbb{Z}}_4$ Potts paramagnet on a triangular lattice. Within the group-cohomology framework, $H^3(G,U(1))\cong {\mathbb{Z}}_4^{\times 7}$, we focus on a "colorless" cocycle representative obtained by antisymmetrizing the basic ${\mathbb{Z}}_4$ three-cocycle, and generate the corresponding SPT Hamiltonian via a cocycle-induced nonlocal unitary transformation followed by symmetry averaging. For open geometry, we derive the boundary theory explicitly: one color sector decouples, while the nontrivial edge reduces to an interacting ${\mathbb{Z}}_4$ chain with next-to-nearest-neighbor constraints that admits a compact dressed-Potts form. Using DMRG we show that the boundary model is gapless, with the lowest gap scaling as $1/L$ and an entanglement-entropy scaling consistent with a conformal field theory of central charge $c=2.191(4)\simeq 11/5$. The rational value $c=11/5$ matches the coset $SU(3{)}_3/SU(2{)}_3$, making it a candidate for the continuum description of the ${\mathbb{Z}}_4^{\times 3}$ edge; we outline spectral and symmetry-resolved diagnostics needed to test this identification at the level of conformal towers beyond the central charge.