A tractable framework for phase transitions in phase-fluctuating disordered 2D superconductors: applications to bilayer MoS$_2$ and disordered InO$_x$ thin films

Abstract

Starting from the purely microscopic model, we go beyond conventional mean-field theory and develop a self-consistent microscopic thermodynamic framework for disordered 2D superconductors. It incorporates the fermionic Bogoliubov quasiparticles, bosonic Nambu-Goldstone (NG) quantum and thermal phase fluctuations in the presence of long-range Coulomb interactions, and topological Berezinskii-Kosterlitz-Thouless (BKT) vortex-antivortex fluctuations on an equal footing, to self-consistently treat the superconducting gap and superfluid density. This unified phase-fluctuating description naturally recovers the previously known limiting results: the superconducting gap in the 2D limit can remain robust against long-wavelength NG phase fluctuations at $T=0^+$ due to Coulomb-induced regularization, while the gradual proliferation of BKT fluctuations as the system approaches criticality drives a separation between the global superconducting transition temperature $T_c$ and the gap-closing temperature $T^*$. In contrast to mean-field theory, which predicts 2D superconductivity to be independent of carrier density and non-magnetic disorder (Anderson theorem), the incorporation of phase fluctuations generates a density- and disorder-dependent zero-point gap $Δ(0)$ and consequently $T_c$ and $T^*$. Remarkably, applications to bilayer MoS$_2$ [Nat. Nanotechnol. 14, 1123 (2019)] and disordered InO$_x$ thin films [Nat. Phys. 21, 104 (2025)] quantitatively reproduce key experimental observations in excellent agreement. The framework offers a useful theoretical tool for understanding phase-fluctuation-dominated superconductivity.