Learning multiple order parameters with interpretable machines

TL;DR

This paper advances machine learning methods to identify and interpret multiple coexisting phases and their order parameters in complex many-body systems, focusing on multipolar orders and phase diagram exploration.

Contribution

It introduces a multiclass SVM approach with kernel methods for detecting multiple phases and interpretable order parameters, especially for multipolar orders.

Findings

Support vector machines effectively classify multiple phases.

Kernel methods reveal hidden spin and orbital orders.

Bias parameter aids in diagnosing phase transitions.

Abstract

Machine-learning techniques are evolving into a subsidiary tool for studying phase transitions in many-body systems. However, most studies are tied to situations involving only one phase transition and one order parameter. Systems that accommodate multiple phases of coexisting and competing orders, which are common in condensed matter physics, remain largely unexplored from a machine-learning perspective. In this paper, we investigate multiclassification of phases using support vector machines (SVMs) and apply a recently introduced kernel method for detecting hidden spin and orbital orders to learn multiple phases and their analytical order parameters. Our focus is on multipolar orders and their tensorial order parameters whose identification is difficult with traditional methods. The importance of interpretability is emphasized for physical applications of multiclassification. Furthermore, we discuss an intrinsic parameter of SVM, the bias, which allows for a special interpretation in the classification of phases, and its utility in diagnosing the existence of phase transitions. We show that it can be exploited as an efficient way to explore the topology of unknown phase diagrams where the supervision is entirely delegated to the machine.