Studies On Antiferromagnetic Heisenberg Models By Machine Learning Algorithms

Machine learning algorithms are algorithms to let computers automatically learn rules through analyzing existing data and use the learned rules to make judgments and decisions.Machine learning algorithms include unsupervised learning algorithms and supervised learning algorithms.For example,the principal component analysis algorithm is a simple unsupervised learning algorithm,and the neural network is a widely used supervised learning algorithm.In recent years,machine learning algorithms have been introduced into physics research.In studies of quantum correlated systems,machine learning algorithms are used to characterize many-body wave functions;accelerate existing algorithms and develop new algorithms to solve quantum correlated problems;identify different phases and phase transitions of quantum correlated systems.In this thesis,we introduce machine learning algorithms to study antiferromagnetic Heisenberg models: we use the principal component analysis algorithm to visualize the ground state wave functions of Heisenberg chains,and hence to improve and guide the application of the eigenvector continuation algorithm in solving ground states of Heisenberg chains;we also use a neural network to identify the different phases of a Heisenberg model one a triangular lattice under different external magnetic fields and determine the critical magnetic fields for the phase transitions.We firstly use the principal component analysis algorithm to determine the low-dimensional subspaces of the ground-state many-body wave function vector trajectories of the parameter-dependent antiferromagnetic Heisenberg chains containing 4spins and 6 spins,respectively,and visually plot the trajectories in the subspaces;Furthermore,through determining the dimensions of the ground-state vector trajectory spaces of the Heisenberg chains containing 8,10,12,14 spins,we conclude a relationship between the number of spins in a Heisenberg chain and the number of basis needed to apply eigenvector continuation to calculate its ground-state.Finally,we prove that the relationship may be used to guide eigenvector continuation calculation of ground states of Heisenberg chains with more spins through a sample calculation of the ground-state energies of a Heisenberg chain containing 16 spins.We also use a convolutional neural network to study the phase transitions of an antiferromagnetic Heisenberg model on a triangular lattice under an external magnetic field.At a finite temperature,the model has multiple(more than two)phases under different external magnetic fields.Existing machine learning algorithms have been relatively mature in studying on distinguish two phases,such as neural networks used to distinguish the low-temperature phase and high-temperature phase of an Ising model.By training a convolutional neural network,we distinguish the multiple phases of the Heisenberg model under different external magnetic fields and successfully determine the critical magnetic fields for the phase transitions.