# One Proximate Kitaev Spin Liquid in the $K$-$J$-$\Gamma$ Model on the Honeycomb Lattice

## TL;DR

This paper explores the phase diagram of the $K$-$J$-$\Gamma$ model on the honeycomb lattice, identifying a unique proximate Kitaev spin liquid with distinct gapless and gapped topological phases under magnetic fields.

## Contribution

It discovers a single proximate Kitaev spin liquid phase with specific properties and characterizes its behavior under magnetic fields, expanding understanding of Kitaev materials.

## Key findings

- Identified a unique proximate Kitaev spin liquid with 14 Majorana cones.
- Discovered two distinct gapped chiral spin-liquid phases with different Chern numbers.
- Characterized the thermal Hall conductance differences between the phases.

## Abstract

In addition to the Kitaev ($K$) interaction, candidate Kitaev materials also possess Heisenberg ($J$) and off-diagonal symmetric ($\Gamma$) couplings. We investigate the quantum ($S = 1/2$) $K$-$J$-$\Gamma$ model on the honeycomb lattice by a variational Monte Carlo (VMC) method. In addition to the "generic" Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL) phase, while the rest of the phase diagram contains different magnetically ordered states. The PKSL is a gapless Z$_2$ state with 14 Majorana cones, which in contrast to the KSL has a gapless spin response. In a magnetic field applied normal to the honeycomb plane, it realizes two of Kitaev's gapped chiral spin-liquid phases, of which one is non-Abelian with Chern number $\nu = 5$ and the other is Abelian with $\nu = 4$. These two phases could be distinguished by their thermal Hall conductance.