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

TL;DR

This paper constructs a two-dimensional bosonic SPT phase protected by a $ ext{Z}_4^{ imes 3}$ symmetry, analyzes its boundary theory, and identifies a gapless edge with a conformal field theory of central charge approximately 11/5.

Contribution

It introduces a novel $ ext{Z}_4^{ imes 3}$ SPT model using group cohomology and explicitly derives its boundary theory, revealing a gapless edge described by a specific conformal field theory.

Findings

Boundary theory is a gapless $ ext{Z}_4$ chain with next-nearest-neighbor interactions.

Entanglement entropy scaling suggests a conformal field theory with $c \, \approx \, 11/5$.

Edge theory matches the coset $SU(3)_3/SU(2)_3$ CFT candidate.

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.