Learning the Renyi entropy of multiple disjoint intervals in transverse-field quantum Ising models with restricted Boltzmann machine

TL;DR

This paper compares improved swapping operations and neural network-based machine learning methods to compute Renyi entropy in the transverse-field quantum Ising model, demonstrating high accuracy and applicability for multiple disjoint intervals.

Contribution

It introduces a novel approach combining improved swapping operations with neural networks to accurately compute Renyi entropy in quantum spin chains.

Findings

Results from both methods agree within errors.

Renyi entropy increases with magnetic field until critical point.

Entropy decreases beyond the critical point in the paramagnetic phase.

Abstract

Renyi entropy with multiple disjoint intervals are computed from the improved swapping operations by two methods: one is from the direct diagonalization of the Hamiltonian and the other one is from the state-of-the-art machine learning method with neural networks. We use the paradigmatic transverse-field Ising model in one-dimension to demonstrate the strategy of the improved swapping operation. In particular, we study the second Renyi entropy with two, three and four disjoint intervals. We find that the results from the above two methods match each other very well within errors, which indicates that the machine learning method is applicable for calculating the Renyi entropy with multiple disjoint intervals. Moreover, as the magnetic field increases, the Renyi entropy grows as well until the system arrives at the critical point of the phase transition. However, as the magnetic field exceeds the critical value, the Renyi entropy will decrease since the system enters the paramagnetic phase. Overall, these results match the theoretical predictions very well and demonstrate the high accuracy of the machine learning methods with neural networks.