Charge-$4e$ Superconductivity from Nematic Superconductors in Two and Three Dimensions

Charge-$4e$ superconductivity as a novel phase of matter remains elusive so far. Here, we show that charge-4e phase can arise as a vestigial order above the nematic superconducting transition temperature in time-reversal-invariant nematic superconductors. On the one hand, the nontrivial topological defect—nematic vortex—is energetically favored over the superconducting phase vortex when the nematic stiffness is less than the superfluid stiffness; consequently the charge-$4e$ phase emerges by proliferation of nematic vortices upon increasing temperatures. On the other hand, the Ginzburg-Landau theory of the nematic superconductors has two distinct decoupling channels to either charge-$4e$ orders or nematic orders; by analyzing the competition between the effective mass of the charge-$4e$ order and the cubic potential of the nematic order, we find a sizable regime where the charge-$4e$ order is favored. These two analyses consistently show that nematic superconductors can provide a promising route to realize charge-$4e$ phases, which may apply to candidate nematic superconductors such as ${\mathrm{PbTaSe}}_2$ and twisted bilayer graphene.

Figure 1

Schematic phase diagram of the defect theory. $\kappa$, $\rho$, and $T$ denote the nematic stiffness, the superfluid stiffness and temperature, respectively. The solid lines refer to phase boundaries. The threefold anisotropy is relevant at the transition point from the charge-$4e$ superconductor to the nematic superconductor.

Figure 2

The Feynman diagrams that contribute to the effective mass of the charge-$4e$ order and the nematic order, respectively. The difference of (a) and (b) arises from the different vertices denoted by $\mathit{\rho }$ and $\mathit{\tau }$.

Figure 3

The mean-field phase diagram of the charge-$4e$ order (a) and the nematic order (b). $|{\mathrm{\Delta }}_{4e}|$ and $Q=\sqrt{Q_x^2+Q_y^2}$ are the amplitude of the charge-$4e$ and the nematic order, respectively. $\delta r=r_0-r_0^{*}\propto T-T^{*}$, where $r^{*}$ ($T^{*}$) denotes the transition point (transition temperature) without vestigial orders. The parameters are $d_{\parallel }=1$, $d^{\prime }=0.5$, $d_z=0.1$, $\overline{d}=0.1$, $v=1$, $u=4$.