Machine learning approach to study quantum phases of a frustrated one dimensional spin-1/2 system

TL;DR

This paper employs an unsupervised machine learning technique, PCA, on spin configurations of a frustrated quantum spin chain to accurately identify phase transitions and degeneracies in the ground and excited states.

Contribution

It introduces a novel application of PCA to analyze quantum phases in a frustrated spin-1/2 chain, revealing phase transitions and degeneracies from principal components of spin configurations.

Findings

PCA captures phase transitions via the principal component of the covariance matrix.

The method detects the GSL-dimer phase transition at J2/J1 ≈ 0.241.

Distinct phase bands are visible in the scatter plot of principal components.

Abstract

Frustration driven quantum fluctuation leads to many exotic phases in the ground state and study of these quantum phase transitions is one of the most challenging areas of research in condensed matter physics. Here, a frustrated Heisenberg $J_1-J_2$ model of spin-1/2 chain with nearest exchange interaction $J_1$ and next nearest exchange interaction $J_2$ is studied using the principal component analysis (PCA) which is an unsupervised machine learning technique. In this method most probable spin configurations (MPSC) of ground-state (GS) and first excited state (FES) for different $J_2/J_1$ are used as the input in PCA to construct the co-variance matrix. The `quantified principal component' of the largest eigenvalue of co-variance matrix $p_1(J_2/J_1)$ is calculated and it is shown that the nature and variation of $p_1(J_2/J_1)$ can accurately predict the phase transitions and degeneracies in the GS. The $p_1(J_2/J_1)$ calculated from the MPSC of GS can only exhibit the signature of degeneracies in the GS, whereas, $p_1(J_2/J_1)$ calculated from MPSC of FES captures the gapless spin liquid (GSL)-dimer phase transition as well as all the degeneracies of the model system. We show that jump in $p_1(J_2/J_1)$ of FES at $J_2/J_1 \approx 0.241$, indicates the GSL-dimer phase transition, whereas its kinks give the signature of the GS degeneracies. The scatter plot of first two principal components of FES shows distinct band formation for different phases.