Ginzburg-Landau theory for unconventional surface superconductivity in PtBi$_2$

TL;DR

This work tackles the puzzle of unconventional surface superconductivity in the trigonal compound $PtBi_2$ by constructing a symmetry-based Ginzburg–Landau theory for three order-parameter components transforming as the $A_1$, $A_2$, and $E$ irreps of $C_{3v}$. It derives all symmetry-allowed terms up to fourth order, including gradient, magnetic-field, and Lifshitz-invariant couplings, and analyzes how uniform in-plane and out-of-plane fields modify the nodal structure for the $A_1$ and $A_2$ pairings. A key result is the distinct field-induced behavior: $A_2$ has symmetry-imposed nodes that can be lifted by certain field orientations, while $A_1$ may have accidental nodes that can be removed by a field; both cases can host helical states under in-plane fields. The framework provides clear experimental signatures to differentiate $A_1$ vs $A_2$ surface pairing (e.g., magneto-ARPES and STS gap evolution) and guides future studies on vortices and bulk–surface interplay in PtBi$_2$.

Abstract

Recent experimental evidence suggests the presence of an unconventional, nodal surface-su\-per\-con\-duc\-ting state in trigonal PtBi\textsubscript{2}. We construct a Ginzburg--Landau theory for the three superconducting order parameters, which correspond to the three irreducible representations of the point group $C_{3v}$. The irreducible representations $A_1$ and $A_2$ are the most likely. We develop a systematic method to determine the symmetry-allowed terms and apply it to derive all terms up to fourth order in the three order parameters. The Ginzburg--Landau functional also includes coupling to the magnetic field. The functional is employed to determine the effect of an applied uniform magnetic field on the nodal structure for $A_1$ and $A_2$ pairing. The results facilitate clear-cut experimental differentiation between these symmetries. We also predict field-induced helical superconductivity.

Figures (1)

• Figure 1: Polar plots of the lowest-order basis functions showing the symmetries of the superconducting pairing amplitudes of (a) $A_1$ symmetry ($\Delta(\phi) \propto 1$), (b) $A_2$ symmetry ($\sin6\phi$), (c) ${}^1\!E$ symmetry ($\sin2\phi$), and (d) ${}^2\!E$ symmetry ($\cos2\phi$). Red and blue color refer to positive and negative sign, respectively. The angular ranges spanned by the Fermi arcs are sketched as gray sectors, which have been exaggerated for clarity.