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

Hrant Topchyan, Tigran Hakobyan, Mkhitar Mirumyan, Tigran A. Sedrakyan, Ara Sedrakyan

TL;DR

The work addresses the realization and boundary characterization of two-dimensional bosonic SPT phases protected by $G=\mathbb{Z}_4^{\times3}$, using a decorated $ ext{Z}_4$ Potts paramagnet on a triangular lattice. It constructs a nontrivial SPT via a cocycle-induced nonlocal unitary and $G$-averaging, deriving a boundary theory that reduces to a constrained $\mathbb{Z}_4$ chain. Numerical analysis with DMRG confirms a gapless edge whose entanglement scaling yields a central charge $c=2.191(4) \approx 11/5$, compatible with the coset CFT $SU(3)_3/SU(2)_3$, and identifies a small boundary scaling dimension $h_b^{(1)}\approx 0.04$, motivating further symmetry-resolved towers and periodic-boundary checks. The results provide a concrete lattice realization of a 2D bosonic SPT with a nontrivial edge anomaly and establish a concrete target for continuum edge theories and conformal-tower diagnostics.

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.

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

TL;DR

The work addresses the realization and boundary characterization of two-dimensional bosonic SPT phases protected by , using a decorated Potts paramagnet on a triangular lattice. It constructs a nontrivial SPT via a cocycle-induced nonlocal unitary and -averaging, deriving a boundary theory that reduces to a constrained chain. Numerical analysis with DMRG confirms a gapless edge whose entanglement scaling yields a central charge , compatible with the coset CFT , and identifies a small boundary scaling dimension , motivating further symmetry-resolved towers and periodic-boundary checks. The results provide a concrete lattice realization of a 2D bosonic SPT with a nontrivial edge anomaly and establish a concrete target for continuum edge theories and conformal-tower diagnostics.

Abstract

We construct a two-dimensional bosonic symmetry-protected topological (SPT) paramagnet protected by an on-site symmetry, starting from a three-component Potts paramagnet on a triangular lattice. Within the group-cohomology framework, , we focus on a "colorless" cocycle representative obtained by antisymmetrizing the basic 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 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 and an entanglement-entropy scaling consistent with a conformal field theory of central charge . The rational value matches the coset , making it a candidate for the continuum description of the edge; we outline spectral and symmetry-resolved diagnostics needed to test this identification at the level of conformal towers beyond the central charge.

Paper Structure

This paper contains 7 sections, 26 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The $\mathbb{Z}_4^{\times 3}$ SPT model
  3. General structure of the SPT phase space
  4. Construction of nontrivial SPT phases
  5. The boundary model
  6. Numerical analysis of the boundary model
  7. Conclusion

Figures (4)

  • Figure 1: The initial $\mathbb{Z}_4$ triangular lattice with colored (red, green blue) nodes and yellow links and the colorless composite $\mathbb{Z}_4^{\times3}$ triangular lattice with gray-shaded triangles as the nodes and the black double line links.
  • Figure 2: The schematic depiction of interactions. The gray-shaded blobs are the lattice edge points, with their internal components of colors $A$ (red), $B$ (green) and $C$ (blue). The interacting components are connected by brown or purple lines.
  • Figure 3: The entanglement entropy on reparameterized subsystem size ratio $l/L$ for various system sizes $L$. Central charge $c=2.191 \pm 0.004$ can be determined from the slope.
  • Figure 4: The ground state energy $E_0$ and the first excited state energy $E_1$ on system size $L$, after subtracting the linear term in $L$. The numerical results (dots) and the corresponding fitting curves. $h^{(\mathrm{b})}_1=(0.019 \pm 0.009) c$.