First-Order Transitions in Weak Ising Spin-Orbit-Coupled Superconductors

Ising spin-orbit coupling (ISOC) can strongly protect superconductivity against exchange-field-induced depairing, typically leading to critical fields far exceeding the Pauli limit and continuous (second-order) phase transitions. Here, using a free-energy approach, we demonstrate that first-order transitions can emerge in superconductors with weak ISOC under large exchange fields. In this regime, conventional theoretical approaches based on the gap equation fail to determine the thermodynamic critical field and instead yield only the supercooling field. Moreover, we identify two pronounced in-gap coherence peaks in the quasiparticle spectra, which represent the weak-ISOC manifestation of the previously reported mirage-gap states. Our results establish the importance of free-energy analysis in describing the first-order phase transitions in Ising superconductors and reveal distinct spectroscopic signatures of the weak-ISOC regime.