Machine Learning Based Approaches for Classical and Quantum Critical Phenomena

Abstract

We employ machine learning algorithms to investigate critical phenomena of classical and quantum many-body systems. First, we introduce the supervised learning schemes for the classical Ising model and extract critical exponents through finite-size scaling of neural network outputs. To explain the emergence of universal scaling behavior in neural network outputs, we devise the exactly solvable models via scrutinizing the inner structure of a neural network. We show that learnable parameters exhibit a power-law scaling with the Ising scaling dimension, which elucidates that a neural network is only capable of exactly discriminating between phases of systems in the same universality class.

Second, we introduce the reinforcement learning schemes to pave a way for solving quantum many-body problems in the context of quantum criticality. We present neural-network quantum states as an ansatz of variational Monte Carlo (VMC) and enumerate methods for exploiting a graphic processing unit (GPU) to accelerate computation speed. For the quantum Ising chain, we show that multilayer perceptron ansatz is capable to provide sufficient accuracy to extract the critical point and critical exponents. We apply VMC with restricted Boltzmann machine ansatz (RBM) to the long-range antiferromagnetic quantum Ising chain (LRAI) with power-law decaying interactions $r^{-\alpha}$ ($\alpha>0$) and present comprehensive studies for the region $\alpha<2$ in which the universality class remains controversial.