Revealing the phase diagram of Kitaev materials by machine learning: Cooperation and competition between spin liquids

Ke Liu,Jonas Greitemann,Lode Pollet,Nihal Rao,Nicolas Sadoune

doi:10.1103/physrevresearch.3.023016

Abstract

Kitaev materials are promising materials for hosting quantum spin liquids and investigating the interplay of topological and symmetry-breaking phases. We use an unsupervised and interpretable machine-learning method, the tensorial-kernel support vector machine, to study the honeycomb Kitaev-$\Gamma$ model in a magnetic field. Our machine learns the global classical phase diagram and the associated analytical order parameters, including several distinct spin liquids, two exotic $S_3$ magnets, and two modulated $S_3 \times Z_3$ magnets. We find that the extension of Kitaev spin liquids and a field-induced suppression of magnetic order already occur in the large-$S$ limit, implying that critical parts of the physics of Kitaev materials can be understood at the classical level. Moreover, the two $S_3 \times Z_3$ orders are induced by competition between Kitaev and $\Gamma$ spin liquids and feature a different type of spin-lattice entangled modulation, which requires a matrix description instead of scalar phase factors. Our work provides a direct instance of a machine detecting new phases and paves the way towards the development of automated tools to explore unsolved problems in many-body physics.

Highlights

Kitaev materials have attracted immense attention in the search for quantum Kitaev spin liquids (KSLs) [1]
We find that the extension of Kitaev spin liquids and a field-induced suppression of magnetic order already occur in the large-S limit, implying that critical parts of the physics of Kitaev materials can be understood at the classical level
Machine-learning techniques are emerging as promising tools in various disciplines of physics [73]

Summary

INTRODUCTION

Kitaev materials have attracted immense attention in the search for quantum Kitaev spin liquids (KSLs) [1]. Successful applications include representing quantum wave functions [75], learning order parameters [76,77], classifying phases [78,79], designing algorithms [80,81], analyzing experiments [82,83], and optimizing material searches [84] Most of these advances are focused on algorithmic developments and resolving known problems. We employ our recently developed tensorialkernel support vector machine (TK-SVM) [85,86,87] to learn the global phase diagram of the honeycomb Kitaev- model under a [111] field, which remains unsettled even in the (semi)classical large-S case.

MODEL AND METHOD

GLOBAL VIEW OF THE PHASE DIAGRAM

EMERGENT LOCAL CONSTRAINTS

Siγ2 Sγj 1

COOPERATIVE AND COMPETITIVE CONSTRAINT-INDUCED ORDERING

CONCLUSIONS