We employ the ``learning by confusion'' technique, an unsupervised machine learning approach for detecting phase transitions, to analyze quantum Monte Carlo simulations of the two-dimensional Holstein model---a fundamental model for electron-phonon interactions on a lattice. Utilizing a convolutional neural network, we conduct a series of binary classification tasks to identify Holstein critical points based on the neural network's learning accuracy. We further evaluate the effectiveness of various training datasets, including snapshots of phonon fields and other measurements resolved in imaginary time, for predicting distinct phase transitions and crossovers. Our results culminate in the construction of the finite-temperature phase diagram of the Holstein model.
We employ the ``learning by confusion'' technique, an unsupervised machine learning approach for detecting phase transitions, to analyze quantum Monte Carlo simulations of the two-dimensional Holstein model---a fundamental model for electron-phonon interactions on a lattice. Utilizing a convolutional neural network, we conduct a series of binary classification tasks to identify Holstein critical points based on the neural network's learning accuracy. We further evaluate the effectiveness of various training datasets, including snapshots of phonon fields and other measurements resolved in imaginary time, for predicting distinct phase transitions and crossovers. Our results culminate in the construction of the finite-temperature phase diagram of the Holstein model.
The application of state-of-the-art machine learning techniques to statistical physic problems has seen a surge of interest for their ability to discriminate phases of matter by extracting essential features in the many-body wavefunction or the ensemble of correlators sampled in Monte Carlo simulations. Here we introduce a gener- alization of supervised machine learning approaches that allows to accurately map out phase diagrams of inter- acting many-body systems without any prior knowledge, e.g. of their general topology or the number of distinct phases. To substantiate the versatility of this approach, which combines convolutional neural networks with quantum Monte Carlo sampling, we map out the phase diagrams of interacting boson and fermion models both at zero and finite temperatures and show that first-order, second-order, and Kosterlitz-Thouless phase transitions can all be identified. We explicitly demonstrate that our approach is capable of identifying the phase transition to non-trivial many-body phases such as superfluids or topologically ordered phases without supervision.
The transfer learning of a neural network is one of its most outstanding aspects, and has given supervised learning with neural networks a prominent place in data science. Here we explore this feature in the context of strongly interacting many-body systems. By means of case studies, we show the potential of this deep learning technique to detect phases and their transitions in frustrated spin systems, using dense and convolutional neural networks. In addition, we explore a recently-introduced technique, which is at the middle point of supervised and unsupervised learning. It consists in evaluating the performance of a neural network which has been deliberately 'confused' during its training. To properly demonstrate the capability of the 'confusion' and transfer learning techniques, we apply them to a paradigmatic model of frustrated magnetism in two dimensions, to determine its phase diagram and compare it with high-performance Monte Carlo simulations.
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.
To evaluate the effectiveness of machine learning in systems with competing interactions, we developed a self-learning quantum Monte Carlo (SLQMC) method to simulate the phase transition in the classical Holstein-spin-fermion model. In SLQMC, machine learning techniques are employed to approximate the free energy, thereby bypassing the need for exact diagonalization and significantly reducing computational cost. We assess the performance of SLQMC using both linear regression and neural network models. Our results show that both models are capable of capturing the phase transition from the antiferromagnetic state to the charge-density-wave state. However, the sampling efficiency decreases near the AFM-CDW phase transition, which is attributed to the increased mean-squared-error of the machine learning model. Additionally, the sampling efficiency decreases with increasing lattice size. This suppression is due to the increased root-mean-squared-error as the machine learning model is applied to a large lattice and the finite-size effect, wherein the energy gap between the ground state and low-energy excited states decreases as the lattice grows. Our findings highlight the necessity of highly accurate machine learning models to simulate theoretical models with complex, competing microscopic interactions on a large lattice.
The use of Neural Networks in quantum many-body theory has undergone a formidable rise in recent years. Among the many possible applications, their pattern recognition power can be utilized when dealing with the study of equilibrium phase diagrams. Learning by Confusion has emerged as an interesting and unbiased scheme within this context. This technique involves systematically reassigning labels to the data in various ways, followed by training and testing the Neural Network. While random labeling results in low accuracy, the method reveals a peak in accuracy when the data are correctly and meaningfully partitioned, even if the correct labeling is initially unknown. Here, we propose a generalization of this confusion scheme for systems with more than two phases, for which it was originally proposed. Our construction relies on the use of a slightly different Neural Network: from a binary classifier, we move to a ternary one, which is more suitable to detect systems exhibiting three phases. After introducing this construction, we test it on free and interacting Kitaev chains and on the one-dimensional Extended Hubbard model, consistently achieving results that are compatible with previous works. Our work opens the way to wider use of Learning by Confusion, demonstrating once more the usefulness of Machine Learning to address quantum many-body problems.
To evaluate the effectiveness of machine learning in systems with competing interactions, we developed a self-learning quantum Monte Carlo (SLQMC) method to simulate the phase transition in the classical Holstein spin-fermion model. In SLQMC, machine-learning techniques are employed to approximate the free energy, thereby bypassing the need for exact diagonalization and significantly reducing computational cost. We assess the performance of SLQMC using both linear regression and neural network models. Our results show that both models are capable of capturing the phase transition from the antiferromagnetic (AFM) state to the charge-density-wave (CDW) state. However, the sampling efficiency decreases near the AFM-CDW phase transition, which is attributed to the increased mean-squared error of the machine-learning model. Additionally, the sampling efficiency decreases with increasing lattice size. This suppression is due to the increased root-mean-squared error as the machine-learning model is applied to a large lattice and the finite-size effect, wherein the energy gap between the ground state and low-energy excited states decreases as the lattice grows. Our findings highlight the necessity of highly accurate machine-learning models to simulate theoretical models with complex, competing microscopic interactions on a large lattice.