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

Frustration-driven quantum fluctuation leads to many exotic phases in the ground state (GS) and the study of these quantum phase transitions is one of the most challenging areas of research in condensed matter physics. We study a frustrated HeisenbergJ1-J2model of spin-1/2 chain with nearest exchange interactionJ₁and next nearest exchange interactionJ₂using the principal component analysis (PCA) which is an unsupervised machine learning technique. In this method most probable spin configurations (MPSCs) of GS and first excited state (FES) for differentJ2/J1are used as the input in PCA to construct the covariance matrix. The 'quantified principal component'p1(J2/J1)of the largest eigenvalue of the covariance matrix is calculated and it is shown that the nature and variation ofp1(J2/J1)can accurately predict the phase transitions and degeneracies in the GS. Thep1(J2/J1)calculated from the MPSC of GS only exhibits the signature of degeneracies in the GS, whereas,p1(J2/J1)calculated from the 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 the jump inp1(J2/J1)of FES atJ2/J1≈0.241, indicates the GSL-dimer phase transition, whereas its kinks give the signature of the GS degeneracies. The scatter plot of the first two principal components of FES shows distinct band formation for different phases. The MPSCs are obtained by using an iterative variational method (IVM) which gives the approximate eigenvalues and eigenvectors.