Gapless chirality liquid with symmetry-protected edge spins

TL;DR

This paper demonstrates a gapless symmetry-protected topological phase in a spin-1/2 tetrahedral Heisenberg chain, featuring coexistence of critical chirality and protected edge states, characterized by specific discrete symmetries.

Contribution

It introduces a novel gSPT phase with coexisting Tomonaga-Luttinger liquid and edge states, analyzed through numerical and theoretical methods, highlighting symmetry protection mechanisms.

Findings

Coexistence of critical entanglement entropy and degenerate entanglement spectrum

Chirality forms Tomonaga-Luttinger liquid, spins form Haldane state

gSPT phase protected by multiple discrete symmetries

Abstract

We report that a spin-1/2 tetrahedral Heisenberg chain realizes a gapless symmetry-protected topological (gSPT) phase characterized by the coexistence of the Tomonaga-Luttinger-liquid criticality due to chirality degrees of freedom and the symmetry-protected edge state due to spin degrees of freedom. This gSPT phase has an interesting feature that no symmetry forbids the trivial spin gap opening but a discrete symmetry, $\mathbb Z_3\times\mathbb Z_2^T$, forbids the unique gapped ground state. In the first part of the paper, we numerically show the coexistence of a critical entanglement entropy and a nontrivially degenerate entanglement spectrum based on the density-matrix renormalization group (DMRG) method.Next, we clarify that chirality degrees of freedom form the Tomonaga-Luttinger liquid while spin degrees of freedom form the spin-1 Haldane state based on a degenerate perturbation theory. Last but not least, we discuss the Lieb-Schultz-Mattis-type ingappability in the gSPT phase, using a local $\mathbb{Z}_3$ rotation. We can thus characterize our gSPT phase as a symmetry-protected critical phase protected by the $\mathbb{Z}_3$ on-site symmetry, the $\mathbb{Z}_2^T$ time-reversal symmetry, the lattice translation symmetry, and the U(1) spin-rotation symmetry.