Pokrovsky--Talapov and Berezinskii--Kosterlitz--Thouless Phase Transitions in Bilayer Superconducting Films under an In-Plane Magnetic Field

We study finite-temperature phase transitions in a Josephson-coupled bilayer superconducting film with compact layer phases under an in-plane magnetic field. At zero temperature, where thermally excited layer vortices are absent, the relative-phase sector undergoes a Pokrovsky–Talapov (PT) commensurate–incommensurate (C–IC) transition from a commensurate Fulde–Ferrell (C/FF) state to an incommensurate Bloch superconducting (IC/Bloch SC) state. At finite temperature, compactness separates two distinct defect mechanisms. The C–IC boundary remains a PT soliton-entry line: interlayer Josephson vortex–antivortex-pair solitons enter with the square-root onset ${\rho }_{\mathrm{s}\mathrm{o}\mathrm{l}}\propto [k_0-k_0^c(T){]}^{1\mathrm{/}2}$. Thermal melting is instead Berezinskii–Kosterlitz–Thouless (BKT)-like, with correlation exponent $\eta =1\mathrm{/}4$ at the boundary, but the active vortex channel changes across the phase diagram. Josephson locking suppresses elementary layer vortices in the C/FF state and selects a same-vorticity layer-pair BKT channel, whereas elementary layer vortices control melting of the IC/Bloch SC state.