Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian

Abstract

We systematically study gapless edge modes corresponding to ${\mathbb{Z}}_3$ symmetry-protected topological (SPT) phases of two-dimensional three-state Potts paramagnets on a triangular lattice. First, we derive microscopic lattice models for the gapless edge and, using the density-matrix renormalization group (DMRG) approach, investigate the finite-size scaling of the low-lying excitation spectrum and the entanglement entropy. Based on the obtained results, we identify the universality class of the critical edge, namely the corresponding conformal field theory and the central charge. Finally, we discuss the inherent symmetries of the edge models and the emergent winding number symmetry. As a result, one-dimensional chains with this symmetry form a model that supports gapless excitations due to its tricritical symmetry. Numerically, we show that low-energy states in the continuous limit of the edge model can be described by conformal field theory (CFT) with central charge $c=1$, given by the coset $S{U}_k(3)/S{U}_k(2)$ CFT at level k=1.

21 pages, 9 figures

References in corpus (22)

• Topological Order and Conformal Quantum Critical Points
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge
• Fractionalization, topological order, and quasiparticle statistics
• Long-range entanglement from measuring symmetry-protected topological phases
• Composite fermion state of spin-orbit coupled bosons
• Spontaneous formation of a non-uniform chiral spin liquid in moat-band lattices
• Z_3 symmetry-protected topological phases in the SU(3) AKLT model
• Excitonic topological order in imbalanced electron-hole bilayers
• Symmetry-protected sign problem and magic in quantum phases of matter
• Dimensional Reduction and Topological Invariants of Symmetry-Protected Topological Phases
• Categories of quantum liquids I
• Decorated Defect Construction of Gapless-SPT States
• Anomalies in bosonic SPT edge theories: connection to F-symbols and a method for calculation
• Topological spin ordering via Chern-Simons superconductivity
• Theory of competing Chern-Simons orders and emergent phase transitions
• Relative Anomaly in (1+1)d Rational Conformal Field Theory
• Universal finite-size amplitude and anomalous entanglement entropy of $z=2$ quantum Lifshitz criticalities in topological chains
• Chiral spin liquid state of strongly interacting bosons with a moat dispersion: a Monte Carlo simulation
• Universal finite size scaling around tricriticality between topologically ordered, SPT, and trivial phases
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• SPT extension of $Z_2$ quantum Ising model's ferromagnetic phase

Cited by in corpus (1)

• Parafermionic representation of Potts-based cluster chain