Symmetry fractionalization in the topological phase of the spin-$\frac{1}{2}$ $J_1$-$J_2$ triangular Heisenberg model

TL;DR

This study uses advanced numerical methods to analyze the topological and symmetry properties of the spin-liquid phase in a frustrated quantum magnet model, revealing fractionalization and coexistence of different orders.

Contribution

It provides the first detailed characterization of symmetry fractionalization and topological order in the $J_1$-$J_2$ triangular Heisenberg model's spin-liquid phase.

Findings

Identifies four distinct $Z_2$ topologically ordered ground states.

Detects fractionalization of time-reversal and dihedral symmetries.

Suggests gapless edge excitations may develop in large-width limits.

Abstract

Using density-matrix renormalization-group calculations for infinite cylinders, we elucidate the properties of the spin-liquid phase of the spin-$\frac{1}{2}$ $J_1$-$J_2$ Heisenberg model on the triangular lattice. We find four distinct ground-states characteristic of a non-chiral, $Z_2$ topologically ordered state with vison and spinon excitations. We shed light on the interplay of topological ordering and global symmetries in the model by detecting fractionalization of time-reversal and space-group dihedral symmetries in the anyonic sectors, which leads to coexistence of symmetry protected and intrinsic topological order. The anyonic sectors, and information on the particle statistics, can be characterized by degeneracy patterns and symmetries of the entanglement spectrum. We demonstrate the ground-states on finite-width cylinders are short-range correlated and gapped; however some features in the entanglement spectrum suggest that the system develops gapless spinon-like edge excitations in the large-width limit.