Revisiting vestigial order in nematic superconductors: Gauge-field mechanisms and model constraints

An electronic nematic order that originates from superconducting fluctuation but persists above the superconducting transition temperature is often referred to as a vestigial nematic phase. Such a vestigial order belongs to the broader class of composite orders discussed in earlier literature, characterized by ordering in gauge-invariant combinations of superconducting order parameters while the individual superconducting order parameters remain disordered. These states include metallic superfluids, paired phases, and composite (charge-4e) superconductors. Whether and under what conditions such a vestigial phase can emerge in realistic models of nematic superconductors remains an open question. Recent analytical work [P. T. How and S. K. Yip, Phys. Rev. B 107, 104514 (2023)] concluded that vestigial nematic phases--and related mechanisms--do not appear in the widely studied models proposed for, e.g., Bi$_2$Se$_3$-based candidates. To shed light on this question, we perform large-scale Monte Carlo simulations of a three-dimensional Ginzburg-Landau model of a nematic superconductor. Consistent with the findings of How and Yip, our numerical results confirm that commonly considered models do not exhibit vestigial nematic phases or nematic-fluctuation-induced charge-4e superconductivity. Extending the analysis to include coupling to a gauge field, we show that vestigial nematic order can, under restrictive conditions, be stabilized through an alternative mechanism: intercomponent coupling mediated by the gauge field or the effects of strong correlations.