Topological Phases in Half-Integer Higher Spin $J_1$-$J_2$ Heisenberg Chains

TL;DR

This paper investigates topological phases in half-integer spin $J_1$-$J_2$ Heisenberg chains, mapping phase diagrams and identifying topological regions that shrink with increasing spin, converging at a critical point.

Contribution

It provides a comprehensive analysis of topological phases in high-spin chains using DMRG, extending the understanding of the Majumder-Ghosh state beyond $S=1/2$.

Findings

Topological phases are identified with alternating valence bonds.

Topological regions decrease inversely with spin $S$ and converge at $rac{J_2}{J_1}=rac{1}{4}$.

Possible presence of the Majumder-Ghosh state in high-spin systems.

Abstract

We study the ground state properties of antiferromagnetic $J_1$-$J_2$ chains with half-integer spins ranging from $S=\frac{3}{2}$ to $S=\frac{11}{2}$ using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of $\frac{J_2}{J_1}$ containing topological phases with alternating $\frac{2S-1}{2}$ and $\frac{2S+1}{2}$ valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those high-$S$ systems in thermodynamic limit (finite size scaling). We find that these topological regions narrow down inversely with $S$ and converge to a single point at $\frac{J_2}{J_1}=\frac{1}{4}$ in the classical limit -- a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for $S=\frac{1}{2}$, and speculate its possible presence as a ground state in half-integer high spin systems over a substantial range of $\frac{J_2}{J_1}$ values.