Finite-temperature crossover in resistively shunted Josephson junctions via crossing approximations

We study the finite temperature behavior of the minimal model of the resistively shunted Josephson junction (RSJJ) using the crossing expansions, particularly the one-crossing approximation (OCA). For a simple benchmark model of the RSJJ, the OCA shows a good agreement with the (numerically) exact method up to deep strong coupling regime. The spin-spin correlation function obtained by the OCA reveals the finite-temperature crossover between the superconductor and the insulator for finite system-bath coupling and local anharmonicity. And the observed insulating regime clearly expands in the parameter space as the temperature is raised. The estimated crossover boundary is qualitatively consistent with the previously reported ground-state phase diagram (K. Masuki et. al, Phys. Rev. Lett. 129, 087001 (2022)), but the quantitative values are sizably different. We discuss the possible origin of the deviations.