Order of the disorder-driven superconducting transition

Determine whether the ordered-state transition between monopole and conventional pairing is first order or contains an intermediate homogeneous \(s+m\) or time-reversal-breaking \(s+im\) coexistence phase by deriving the point-group-resolved, disorder-dressed two-component Ginzburg–Landau functional.

Background

The calculated crossing is a crossing of linearized transition temperatures and is not necessarily the phase boundary between the fully developed ordered states. The continuum quartic estimate disfavors simple homogeneous coexistence, but it omits discrete-crystal phase-locking terms and does not include disorder-renormalized quartic coefficients.

Consequently, the actual nonlinear phase diagram remains unresolved. Establishing the transition order and determining whether an intermediate coexistence regime appears requires a complete Ginzburg–Landau treatment incorporating the crystalline point group and disorder-dressed quartic vertices.

References

Whether the physical boundary is first order, or whether an s+m coexistence or an s+im time-reversal-breaking window intervenes, requires the two-component Ginzburg--Landau functional with the disorder-dressed quartic vertices and the actual crystalline point group, which we do not compute. A continuum (C_\infty) estimate, given in App.~\ref{app:gl}, yields a biquadratic ratio R_{\rm clean}=2{|f_m|2}/!\sqrt{|f_m|4}=1.83,\,1.63,\,1.48 for J=1,2,3, all exceeding unity, so that simple homogeneous coexistence is disfavored within that estimate. It does not determine the transition order: the continuum expansion drops the phase-locking term \propto e{2iJ\phi} restored by a discrete C_n whenever n\mid2J, and disorder renormalizes all quartic coefficients. We therefore leave the order undetermined and make no claims of hysteresis, latent heat, or rare-region rounding.

— Disorder-tuned crossing of monopole and conventional pairing instabilities in multi-Weyl semimetals  (2608.24587 - Muñoz et al., 25 Aug 2026) in Section 6, “Nature of the transition: crossing of pairing instabilities” (Sec. \ref{sec:order}); see also Appendix, “Continuum Ginzburg–Landau estimate” (App. \ref{app:gl})