Tensor network method for solving the Ising model with a magnetic field

TL;DR

This paper applies tensor network methods to analyze the 2D Ising model with a magnetic field, accurately identifying critical lines and calculating physical quantities like magnetization and entanglement entropy.

Contribution

It introduces a tensor network approach with gauge fixing to study the Ising model under magnetic fields, providing detailed numerical results and critical line determination.

Findings

Tensor network method effectively computes the partition function.

Entanglement entropy accurately locates the critical line.

Distinct symmetry properties for ferromagnetic and antiferromagnetic cases.

Abstract

We study the two-dimensional square lattice Ising ferromagnet and antiferromagnet with a magnetic field by using tensor network method. Focusing on the role of guage fixing, we present the partition function in terms of a tensor network. The tensor has a different symmetry property for ferromagnets and antiferromagnets. The tensor network of the partition function is interpreted as a multiple product of the one-dimensional quantum Hamiltonian. We perform infinite density matrix renormalization group to contract the two-dimensional tensor network. We present the numerical result of magnetization and entanglement entropy for the Ising ferromagnet and antiferromagnet side by side. In order to determine the critical line in the parameter space of temperature and magnetic field, we use the half-chain entanglement entropy of the one-dimensional quantum state. The entanglement entropy precisely indicates the critical line forming the parabolic shape for the antiferromagnetic case, but shows the critical point for the ferromagnetic case.