Construction of a well-defined free energy from the harmonic approximation for Josephson junctions

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Abstract
The trial free energy of the harmonic approximation, which has been widely used to study Josephson junctions, diverges to negative infinity in the normal state for any finite temperature because a nonperiodic, harmonic Hamiltonian is used to describe a periodic system. In this paper, I employ a periodic, scalloped potential to construct a well-defined free energy, which is then minimized to obtain self-consistent solutions that are close to the harmonic solutions when phase fluctuations are small.