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

Abstract

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.

Figures (10)

• Figure 1: Nematic potential $V_{nem}(\phi_{12}, \gamma)$ in Eq.\ref{['potential_nem']}, plotted around the unitary sphere defined by the azimuthal angle, $\phi_{12}$, and the polar angle, $\gamma$. Upper panel: For $K<0$ and $\lambda=0$, the system has an accidental degeneracy along $\gamma$. Lower panel: The degeneracy is lifted by higher-order potential terms. For $\lambda>0$ the potential define three equivalent minima.
• Figure 2: Monte Carlo numerical results obtained for the case $K=-1$, $\lambda=1$ in the extreme type-II limit, i.e. $e=0$. The three panels show (a) the specific heat, $C_v$, (b) the nematic Binder cumulant, $U_{nem}$ and (c) the helicity modulus sum, $\Upsilon_+$, multiplied by the linear system size $L$ as a function of the inverse critical temperature $\beta=1/T$ and for different values of $L$. Finite-size scaling analysis reveals the presence of a single phase transition: the specific heat exhibits a single peak, and the crossing points of $U_{\rm nem}$ and $L \Upsilon_+$ converge in the thermodynamic limit, indicating $\beta_c^{Z_3} = \beta_c^{U(1)}$. More details on the finite-size scaling analysis and the determination of $\beta_c$ can be found in Appendix A. In all panels, the dashed vertical black line indicates the extrapolated inverse critical temperature $\beta_c$.
• Figure 3: Phase diagram as a function of the nematic coupling $\lambda$ at fixed value of $K=-1$, and in the zero gauge-field limit. The phase diagram reveals that the model at $e=0$ does not show a vestigial phase. For all the values of $\lambda$ investigated the two critical temperatures coincide. In the phase diagram, where not visible, the error bars of the data points are smaller than the marker size. For details on the extraction of critical points, see the Supplemental Material.
• Figure 4: Monte Carlo numerical results obtained for the case $K=-1$, $\lambda=1$ and $e=3.8$. The three panels show (a) the specific heat, $C_v$, (b) the nematic Binder cumulant, $U_{nem}$ and (c) the dual stiffness, $\rho_s$, multiplied by the linear system size $L$, as a function of the inverse critical temperature $\beta=1/T$ and for different values of $L$. These numerical results demonstrate the existence of two distinct phase transitions, with $\beta_c^{Z_3} < \beta_c^{U(1)}$, as determined from the finite-size scaling analysis of $U_{\rm nem}$ and $L\rho_s$, respectively. This separation signals the emergence of an intermediate nematic nonsuperconducting phase in the regime $\beta_c^{Z_3}<\beta<\beta_c^{U(1)}$. Consistently, the specific heat exhibits two anomalies associated with the $Z_3$ and $U(1)$ critical points. While the $Z_3$ transition produces a sharp peak, the $U(1)$ anomaly is only weakly visible at these system sizes due to finite-size effects; it becomes slightly more pronounced for larger values of $e$, where the two transitions are more clearly separated (see Fig. \ref{['fig:e_5']} in Appendix A). The dashed vertical red and blue lines mark the critical temperatures obtained from the crossing points of $U_{\rm nem}$ and $L\rho_s$, respectively. Further details on the finite-size scaling analysis can be found in Appendix A.
• Figure 5: Phase diagram as a function of the electric charge $e$ for the free energy with parameters $\lambda=|K|=1$. Where not visible, the error bars of the data points are smaller than the marker size. This phase diagram reveals that a composite nematic phase can be resolved at large enough values of the electric charge $e>3.5$.