# Dimerization and effective decoupling in two spin-1 generalizations of the spin-1/2 Majumdar-Ghosh chain

## TL;DR

This paper uses DMRG to explore two spin-1 generalizations of the Majumdar-Ghosh chain, revealing distinct phase behaviors including dimerization and NNN-Haldane phases, with phase transitions characterized by Ising and Kosterlitz-Thouless types.

## Contribution

It provides a systematic DMRG analysis of the spin-1 $J_1-J_2$ and $J_1-J_3$ chains, uncovering their phase diagrams and transitions, highlighting differences from the spin-1/2 case.

## Key findings

- The $J_1-J_2$ chain exhibits effective decoupling into NNN Haldane chains.
- The $J_1-J_3$ chain shows both dimerized and NNN-Haldane phases.
- Transitions between phases are consistent with Ising and Kosterlitz-Thouless types.

## Abstract

We perform a systematic DMRG investigation of the two natural spin-1 generalizations of the spin-1/2 Majumdar-Ghosh chain, the spin-1 $J_1-J_2$ Heisenberg chain, where $J_2$ is a next-nearest neighbor Heisenberg coupling, and the spin-1 $J_1-J_3$ model, where $J_3$ refers to a three-site interaction defined by $J_3\left[({\bf S}_{i-1}\cdot {\bf S}_i)({\bf S}_i\cdot {\bf S}_{i+1})+\mathrm {H.c.}\right]$. Although both models are rigorously equivalent to the Majumdar-Ghosh chain for spin-1/2, their physics appears to be quite different for spin 1. Indeed, when all couplings are antiferromagnetic, the spin-1 $J_1-J_2$ model undergoes an effective decoupling into two next-nearest neighbour (NNN) Haldane chains upon increasing $J_2$, while the $J_1-J_3$ chain undergoes a spontaneous dimerization similar to the spin-1/2 Majumdar-Ghosh chain upon increasing $J_3$. By extending the phase diagram to all signs of the couplings, we show that both the dimerized and the NNN-Haldane phase are actually present in the $J_1-J_3$ model, the former one adjacent to the Haldane one, the latter one to the ferromagneric one, with an Ising transition between them. By contrast, the $J_1-J_2$ chain only has a NNN-Haldane phase between the Haldane phase and the ferromagnetic phase for positive $J_2$. In both cases, our DMRG data are consistent with a continuous Kosterlitz-Thouless transition between the NNN-Haldane and the ferromagnetic phases.