We investigate theoretically the phase transition in a three-dimensional cubic Ising model utilizing state-of-the-art machine learning algorithms. Supervised machine learning models show high accuracies (99%) in phase classification and very small relative errors ($<{10}^{\ensuremath{-}4}$) of the energies in different spin configurations. Unsupervised machine learning models are introduced to study the spin configuration reconstructions and reductions, and the phases of reconstructed spin configurations can be accurately classified by a linear logistic algorithm. Based on the comparison between various machine learning models, we develop a few-shot strategy to predict phase transitions in larger lattices from a trained sample in smaller lattices. The few-shot machine learning strategy for a three-dimensional (3D) Ising model enables us to study the 3D Ising model efficiently and provides an integrated and highly accurate approach to other spin models.
The power of machine learning algorithms to automatically classify different phases of matter and detect quantum phase transitions without necessity to characterize phases by various quantities like local order parameters or topological invariants as in conventional approaches defined machine learning phases of matter as a new research frontier and basic research tool in condensed matter and statistical physics. We study quantum criticality in the spin-1/2 square-lattice J1-J2 model with additional plaquette structure by combination of reinforcement and supervised machine learning techniques. In our calculations the ground-state spin-spin correlation matrices for several system sizes are first found by restricted Boltzmann machine based variational Monte Carlo method, equivalent to reinforcement learning, and then used as a training data for convolutional neural network based supervised machine learning algorithm for phases classification. The model exhibits a quantum phase transition from paramagnetic plaquette resonating valence bond state to an antiferromagnetic state and has been a topic of great interest because of its close connection to cuprate superconductors and possibility of realization in cold atoms experiments. We consider both frustrated and unfrustrated regimes and compare our results with the results obtained previously with other methods. We find that our results are in good agreement with the results obtained by coupled cluster and real space renormalization group methods for both frustrated and unfrustrated regimes. The quantum Monte Carlo and finite-size scaling result for the unfrustrated case however slightly differs from our result for the critical value of the inter-plaquette coupling strength, although the results are still in reasonable agreement.
We study the ground-state properties of a family of frustrated spin-1/2 Heisenberg models on two- and three-dimensional decorated lattices composed of connected star-shaped units. Each star is built from edge-sharing triangles with an antiferromagnetic interaction on the shared side and ferromagnetic interactions on the others. At a critical coupling ratio, the ideal star model - defined by equal ferromagnetic interactions - exhibits a macroscopically degenerate ground state, which we map onto a site percolation problem on the Lieb lattice. This mapping enables the calculation of exponential ground-state degeneracy and the corresponding residual entropy for square, triangular, honeycomb, and cubic lattices. Remarkably, the residual entropy remains high for all studied lattices, exceeding 60\% of the maximal value ln(2). Despite a gapless quadratic one-magnon spectrum, the low-temperature thermodynamics is governed by exponentially numerous gapped excitations. For a distorted-star variant of the model, the ground-state manifold is equivalent to that of decoupled ferromagnetic clusters, leading to exponential degeneracy with a lower, yet still substantial, residual entropy. At low temperature the system mimics a paramagnetic crystal of non-interacting spins with high spin value (
s=4 for a square lattice). The obtained results establish a structural design principle for engineering quantum magnets with a high ground-state degeneracy, suggesting promising candidates for enhanced magnetocaloric cooling and quantum thermal machines.