Local condensation of charge-$4e$ superconductivity at a nematic domain wall

In the fluctuation regime that precedes the onset of pairing in multicomponent superconductors, such as nematic and chiral superconductors, the normal state is generally unstable toward the formation of charge-$4e$ order—an exotic quantum state in which electrons form coherent quartets rather than Cooper pairs. However, charge-$4e$ order is often suppressed by other competing composite orders, such as nematics. Importantly, the formation of nematic domains is unavoidable due to the long-range strains generated, leading to one-dimensional regions where the competing nematic order is suppressed. Here, we employ a real-space variational approach to demonstrate that, in such nematic domain walls, charge-$4e$ order is locally condensed via a vestigial-order mechanism. We explore the experimental manifestations of this effect and discuss materials in which it can be potentially observed.

Figure 1

(a) Real-space sketch of our main result, the emergence of charge-$4e$ order at a nematic domain wall. (b) Phase diagram of the vestigial orders supported by the nematic superconducting ground state of a tetragonal system. Here, $u, v,$ and $w$ are the Landau parameters of Eq. (9). The different shaded regions indicate which vestigial channel is attractive: none (green), nematic (blue), and leading nematic with subleading charge-$4e$ (red).

Figure 2

Variational nematic order parameter (${\mathrm{\Phi }}^{B_{1g}}$), inverse superconducting susceptibility ${\chi }^{-1}$ (14), and variational charge-$4e$ order parameter (${\varphi }^{A_{1g}}$) obtained from the numerical minimization of the variational free energy (12) [or Eq. (A9)] across two nematic domains (all in units of the gradient-term stiffness $t_0$). Each panel corresponds to a different temperature, parametrized by $r_0\propto T-T_0$. The parameters used here are $w=-0.5u$ and $v=0.1u$ [red cross in Fig. 1]. Here and in Fig. 3, we set $t_0/\sqrt{u{T}_0/L}=20/7$.

Figure 3

Same variational parameters as in Fig. 2, but evaluated for the parameters $w=-0.05u$ and $v=2.1u$ [purple cross in Fig. 1], corresponding to a subleading attractive charge-$4e$ channel ($u_{A_{1g}}<0$). Local condensation of ${\varphi }^{A_{1g}}$ is observed at the nematic domain wall.