Abstract 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 Heisenberg J 1 − J 2 model of spin-1/2 chain with nearest exchange interaction J 1 and next nearest exchange interaction J 2 using 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 different J 2 / J 1 are used as the input in PCA to construct the covariance matrix. The ‘quantified principal component’ p 1 ( J 2 / J 1 ) of the largest eigenvalue of the covariance matrix 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 only exhibits the signature of degeneracies in the GS, whereas, p 1 ( J 2 / J 1 ) 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 in p 1 ( J 2 / J 1 ) of FES at J 2 / J 1 ≈ 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.
Abstract 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 Heisenberg J 1 − J 2 model of spin-1/2 chain with nearest exchange interaction J 1 and next nearest exchange interaction J 2 using 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 different J 2 / J 1 are used as the input in PCA to construct the covariance matrix. The ‘quantified principal component’ p 1 ( J 2 / J 1 ) of the largest eigenvalue of the covariance matrix 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 only exhibits the signature of degeneracies in the GS, whereas, p 1 ( J 2 / J 1 ) 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 in p 1 ( J 2 / J 1 ) of FES at J 2 / J 1 ≈ 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.
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 J1−J2 model of spin-1/2 chain with nearest exchange interaction J1 and next nearest exchange interaction J2 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 J2/J1 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 p1(J2/J1) is calculated and it is shown that the nature and variation of p1(J2/J1) can accurately predict the phase transitions and degeneracies in the GS. The p1(J2/J1) calculated from the MPSC of GS can only exhibit the signature of degeneracies in the GS, whereas, p1(J2/J1) 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 p1(J2/J1) of FES at J2/J1≈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.
We apply unsupervised machine learning techniques, mainly principal component analysis (PCA), to compare and contrast the phase behavior and phase transitions in several classical spin models-the square- and triangular-lattice Ising models, the Blume-Capel model, a highly degenerate biquadratic-exchange spin-1 Ising (BSI) model, and the two-dimensional XY model-and we examine critically what machine learning is teaching us. We find that quantified principal components from PCA not only allow the exploration of different phases and symmetry-breaking, but they can distinguish phase-transition types and locate critical points. We show that the corresponding weight vectors have a clear physical interpretation, which is particularly interesting in the frustrated models such as the triangular antiferromagnet, where they can point to incipient orders. Unlike the other well-studied models, the properties of the BSI model are less well known. Using both PCA and conventional Monte Carlo analysis, we demonstrate that the BSI model shows an absence of phase transition and macroscopic ground-state degeneracy. The failure to capture the "charge" correlations (vorticity) in the BSI model (XY model) from raw spin configurations points to some of the limitations of PCA. Finally, we employ a nonlinear unsupervised machine learning procedure, the "autoencoder method," and we demonstrate that it too can be trained to capture phase transitions and critical points.
In this work, we apply a principal component analysis (PCA) method with a kernel trick to study the classification of phases and phase transitions in classical XY models of frustrated lattices. Compared to our previous work with the linear PCA method, the kernel PCA can capture nonlinear functions. In this case, the Z 2 chiral order of the classical spins in these lattices is indeed a nonlinear function of the input spin configurations. In addition to the principal component revealed by the linear PCA, the kernel PCA can find two more principal components using the data generated by Monte Carlo simulation for various temperatures as the input. One of them is related to the strength of the U (1) order parameter, and the other directly manifests the chiral order parameter that characterizes the Z 2 symmetry breaking. For a temperature-resolved study, the temperature dependence of the principal eigenvalue associated with the Z 2 symmetry breaking clearly shows second-order phase transition behavior.
We assess the quality of a set of machine-learning (ML) techniques for the prediction of the phase diagrams of two frustrated 2D Ising models with competing interactions, that feature phase transitions characterized by critical manifolds rather than a single critical point. Using raw Monte Carlo spin configurations generated for random system parameters, we apply principal-component analysis (PCA) and auto-encoders to achieve dimensionality reduction, followed by clustering using the DBSCAN method and a support-vector machine classifier. The resulting estimates for the critical manifolds of both systems are in excellent agreement with available exact results, with the auto-encoders leading to quantitatively superior results. Notably, both the structure of the optimized auto-encoder as well its corresponding dimensionality-reduced and clustered space correlate with fundamental physical characteristics of the systems under consideration, which is useful when \emph{a priori} theoretical insight is unavailable.
We investigate phase transitions in the two-dimensional Ising model with competing nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions, where the NNN coupling generates a degenerate superantiferromagnetic (SAF) ground state, by combining machine-learning methods with Monte Carlo simulations and finite-size scaling (FSS). Going beyond earlier studies, we show that unsupervised learning can recover order-parameter and Binder cumulant analogs that obey FSS. Principal component analysis (PCA) reveals the two-component structure of the SAF order parameter, and k-means clustering is used to visualize qualitative differences between ordered and disordered configurations in the principal-component space. A fully connected neural network (FCNN), trained without prior knowledge of the order parameter or the critical temperature, learns the strength of SAF stripe ordering and uses it as the internal feature for phase classification. FSS of thermodynamic observables, the PCA-derived quantity, and the FCNN output yields consistent estimates of the critical temperature and correlation-length exponent. These results demonstrate that machine learning can decode ordering patterns, extract meaningful order-parameter analogs, and reproduce critical behavior in frustrated Ising systems.
This work aims at determining whether artificial intelligence can recognize a phase transition without prior human knowledge. If this were successful, it could be applied to, for instance, analyzing data from the quantum simulation of unsolved physical models. Toward this goal, we first need to apply the machine learning algorithm to well-understood models and see whether the outputs are consistent with our prior knowledge, which serves as the benchmark for this approach. In this work, we feed the computer data generated by the classical Monte Carlo simulation for the XY model in frustrated triangular and union jack lattices, which has two order parameters and exhibits two phase transitions. We show that the outputs of the principal component analysis agree very well with our understanding of different orders in different phases, and the temperature dependences of the major components detect the nature and the locations of the phase transitions. Our work offers promise for using machine learning techniques to study sophisticated statistical models, and our results can be further improved by using principal component analysis with kernel tricks and the neural network method.