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.
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.
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.
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.
This work aims at the goal whether the artificial intelligence can recognize phase transition without the prior human knowledge. If this becomes successful, it can be applied to, for instance, analyze data from 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 of this approach. In this work, we feed the compute with 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 principle 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 principle component analysis with kernel tricks and the neural network method.
The Berezinskii-Kosterlitz-Thouless (BKT) transition in magnetic systems is an intriguing phenomenon, and estimating the BKT transition temperature is a long-standing problem. In this work, we explore anisotropic classical Heisenberg XY and XXZ models with ferromagnetic exchange on a square lattice and antiferromagnetic exchange on a triangular lattice using an unsupervised machine learning approach called principal component analysis (PCA). The earlier PCA studies of the BKT transition temperature (TBKT) using the vorticities as input fail to give any conclusive results, whereas, in this work, we show that the proper analysis of the first principal component-temperature curve can estimateTBKTwhich is consistent with the existing literature. This analysis works well for the anisotropic classical Heisenberg with a ferromagnetic exchange on a square lattice and for frustrated antiferromagnetic exchange on a triangular lattice. The classical anisotropic Heisenberg antiferromagnetic model on the triangular lattice has two close transitions: theTBKTand Ising-like phase transition for chirality atTc, and it is difficult to separate these transition points. It is also noted that using the PCA method and manipulation of their first principal component not only makes the separation of transition points possible but also determines transition temperature.
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 various unsupervised machine learning methods for phase classification to investigate the finite-temperature phase diagram of the spinless Falicov-Kimball model in two dimensions. Using only particle occupation snapshots from Monte Carlo simulations as input, each technique, including a straightforward classification based on principal component analysis (PCA), successfully identifies the phase boundary between ordered and disordered phases, independent of the type of phase transition. Remarkably, these techniques also distinguish between the weakly localized and Anderson-localized regimes within the disordered phase, accurately identifying their crossover, which is a challenging task for standard methods. Among the machine learning approaches used, PCA based analysis outperforms more complex methods, such as neural network predictors and autoencoders. These results underscore the effectiveness of simple unsupervised techniques in examining phase transitions and electron localization in complex correlated systems.