Entanglement entropy as a marker of phase transition in the Ising model

The uniform two-dimensional tensor network of the partition function is interpreted as a multiple product of the one-dimensional quantum Hamiltonian, which is the one-dimensional transfer operator. We perform infinite density matrix renormalization group with matrix product operator and matrix product state in the actual contraction of the two-dimensional tensor network. In the process of contraction, we focus on the half-chain entanglement entropy of the one-dimensional quantum state linked to the one-dimensional transfer operator. We show that the entanglement entropy correctly exhibits a singularity at the critical temperature in the Ising model. We note that when we try to determine the critical temperature, the entanglement entropy can do more precisely than the magnetization order parameter. The difference between our numerical result and the analytical value of the critical temperature is given by 0.00003. For complex models inaccessible to analytical techniques, our method using the entanglement entropy is suitable as long as the partition function is written as a tensor network.