The Heart of Entanglement: Chiral, Nematic, and Incommensurate Phases in the Kitaev-Gamma Ladder in a Field

TL;DR

This study maps the complex phase diagram of the Kitaev-Gamma ladder under a magnetic field, revealing fifteen phases including chiral, nematic, and incommensurate states, and offers insights into potential spin liquids in 2D systems.

Contribution

First comprehensive phase diagram of the Kitaev-Gamma ladder in a magnetic field, identifying novel entangled and chiral phases, and linking ladder results to 2D honeycomb lattice behavior.

Findings

Identified fifteen distinct phases in the Kitaev-Gamma ladder.

Discovered a highly entangled 'heart of entanglement' phase with staggered chirality.

Compared ladder phases with 2D honeycomb cluster to infer possible spin liquids.

Abstract

The bond-dependent Kitaev model on the honeycomb lattice with anyonic excitations has recently attracted considerable attention. However, in solid state materials other spin interactions are present, and among such additional interactions, the off-diagonal symmetric Gamma interaction, another type of bond-dependent term, has been particularly challenging to fully understand. A minimal Kitaev-Gamma (KG) model has been investigated by various numerical techniques under a magnetic field, but definite conclusions about field-induced spin liquids remain elusive and one reason may lie in the limited sizes of the two-dimensional geometry it is possible to access numerically. We therefore focus on the KG model defined on a two-leg ladder which is much more amenable to a complete study, and determine the entire phase diagram in the presence of a magnetic field along [111]-direction. Due to the competition between the interactions and the field, an extremely rich phase diagram emerges with fifteen distinct phases. Focusing near the antiferromagnetic Kitaev region, we identify nine different phases solely within this region: several incommensurate magnetically ordered phases, spin nematic, and two chiral phases with enhanced entanglement. Of particular interest is a highly entangled phase with staggered chirality with zero-net flux occurring at intermediate field, which along with its companion phases outline a heart-shaped region of high entanglement, the heart of entanglement. We compare our results for the ladder with a C3 symmetric cluster of the two-dimensional honeycomb lattice, and offer insights into possible spin liquids in the two-dimensional limit.