Finite-temperature fidelity and von Neumann entropy in the honeycomb spin lattice with quantum Ising interaction

The finite temperature phase diagram is obtained for an infinite honeycomb lattice with spin-$1/2$ Ising interaction $J$ by using thermal-state fidelity and von Neumann entropy based on the infinite projected entangled pair state algorithm with ancillas. % The tensor network representation of the fidelity, which is defined as an overlap measurement between two thermal states, is presented for thermal states on the honeycomb lattice. % We show that the fidelity per lattice site and the von Neumann entropy can capture the phase transition temperatures for applied magnetic field, consistent with the transition temperatures obtained via the transverse magnetizations, which indicates that a continuous phase transition occurs in the system. In the temperature-magnetic field plane, the phase boundary is found to have the functional form $(k_BT_c)^2 + h_c^2/2 = a J^2$ with a single numerical fitting coefficient $a = 2.298$, where $T_c$ and $h_c$ are the critical temperature and field with the Boltzmann constant $k_B$. For the quantum state at zero temperature, this phase boundary function gives the critical field estimate $h_c = \sqrt{2a} J \simeq 2.1438 J$, consistent with the known value $h_c = 2.13250(4)\, J$ calculated from a Cluster Monte Carlo approach. The critical temperature in the absence of magnetic field is estimated as $k_BT_c = \sqrt{a}J \simeq 1.5159\, J$, consistent with the exact result $k_BT_c = 1.51865...\, J$.