Many-electron characterizations of higher-charge superconductors

Abstract

The theoretical understanding of conventional superconductivity as the phonon-assisted formation and condensation of two-electron Cooper pairs is a significant triumph in condensed matter physics. Here, we propose many-electron characterizations of higher-charge superconductivity with Cooper pairs consisting of more than two electrons, by implementing translation symmetrization on parent pair-density-wave-ordered states. In particular, we demonstrate many-electron constructions with vanishing charge-2e sectors, but with sharp signatures in charge-4e or charge-6e expectation values instead. Such characterizations are consistent with previous phenomenology of vestigial order and Ginzburg-Landau theory, yet, instead of point-group-symmetry presumptions, we show that momentum conservation is both vital and sufficient. Our study thus offers a novel, general, and microscopic route to understand and characterize higher-charge superconductivity, including nontrivial experimental signatures such as fractional magnetic flux and period in interferometry, as well as localized Cooper pairs at lattice topological defects.

Figures (5)

• Figure 1: (a) Illustration of the model in Eq. \ref{['eq:Hpdw']} on a two-dimensional lattice (blue dots). In contrast to the parent state with a stripe PDW (left), the many-electron state following translation symmetrization exhibits vanishing 2e Cooper-pair density $\Delta$. (b) A schematic diagram illustration of the Fermi sea and the $\mathbf{\pm Q}$ momentum transfers induced due to the PDW. The hot spots (purple crosses) are located at the intersections of the original and shifted Fermi surfaces. Correspondingly, the momentum-space electron density $\langle c^\dagger_{\mathbf{k},s} c_{\mathbf{k},s}\rangle$ of the model in Eq. \ref{['eq:Hpdw']} clearly reveals its Fermi sea and the hot spots. Here, $V=0.1$, $Q=1.76$, and $\mu=-1.0$.
• Figure 2: (a) Heat map of the four-point correlation functions $|\langle c^\dagger_{\mathbf{k}} c^\dagger_{-\mathbf{k}+\mathbf{Q}} c^\dagger_{\mathbf{k^\prime}} c^\dagger_{-\mathbf{k^\prime}-\mathbf{Q}} \rangle|$ of the charge-$4e$ superconducting state $P_T|\Psi\rangle$ show a sharp peak at $(k_x, k'_x)=(Q/2, -Q/2)$, which corresponds to the hot spots on the Fermi surface in Fig. \ref{['Fig:model']}(b); we have fixed $k_y=k^\prime_y=1.71$ for simpler display. (b) While the charge-2e Cooper-pair density $\Delta(\mathbf{k})=\braket{c^\dagger_{\mathbf{k}} c^\dagger_{-\mathbf{k}+\mathbf{Q}}}$ flourishes in the parent PDW state $|\Psi\rangle$, (c) it is completely suppressed in the higher-charge superconducting state $P_T|\Psi\rangle$.
• Figure 3: The correlations $|\langle c^\dagger_{\mathbf{k}} c^\dagger_{-\mathbf{k}+\mathbf{Q}} c^\dagger_{\mathbf{k^\prime}} c^\dagger_{-\mathbf{k^\prime}-\mathbf{Q}} \rangle|$ of the $P_T|\Psi \rangle$ states with (a) asymmetric pairing amplitude $V_\mathbf{Q}\ne V_{\mathbf{-Q}}$, (b) asymmetric Fermi sea with non-zero total momentum $\mathbf{K}$, and (c) nonreciprocal PDW momentum $\mathbf{Q'}\ne-\mathbf{Q}$, suggests that the charge-$4e$ superconductivity may exists as long as momentum conserves despite broken inversion symmetry. (d) The electron density $\langle c^\dagger_{\mathbf{k},s} c_{\mathbf{k},s}\rangle$ of a charge-$6e$ superconductor displays hot spots on its Fermi surface. The corresponding $P_T|\Psi\rangle$ exhibits vanishing charge-2e and (e) charge-$4e$ expectation values, yet (f) a sharp signal in charge-$6e$ correlations (we have fixed $k_y=k^\prime_y=k^{\prime\prime}_y=2.0$ ). (g) The charge-$4e$ order parameter $V_{4e}$ scales proportionally to both $V_\mathbf{Q}$ and $V_{\mathbf{-Q}}$; (h) similarly, the charge-$6e$ order parameter scales tri-linearly to each of the three parent PDWs' amplitudes. Here, we set $L=1000$.
• Figure 4: (a) An illustration of our interferometer setup $P_T|\Psi(\phi)\rangle$: translation-symmetrizing the parent PDW model in Eq. \ref{['eq:Hpdw']} with a PDW in the $x$-direction and a magnetic flux $\phi$ in the $y$-direction. (b) Unlike the parent PDW system $|\Psi(\phi)\rangle$ with a $\pi$ periodicity in $\phi$ corresponding to conventional charge-2e Cooper pairs, the susceptibility $\mathcal{L(\phi)}$ in Eq. \ref{['eq:susceptibility']} of $P_T|\Psi(\phi)\rangle$ exhibits an unmistakable charge-$4e$ signature: $\pi/2$ period with respect to the threaded magnetic flux $\phi$. Here, our system size is $L=1000$.
• Figure 5: (a) Charge-2e Cooper pairs re-emerge locally at topological defects of the underlying lattice when the Burgers vector $\mathbf{B}$ aligns with the parent PDW wave vector $\pm \mathbf{Q}$, yet (b) no such modes are visible when $\mathbf{B}$ and $\mathbf{Q}$ are vertical. The upper panels show schematic illustrations of the lattices with a pair of topological defects and the PDWs before translation symmetrization $P_T$. The dashed lines illustrate the local hopping connectivity, which becomes 1-to-2 on the edge-dislocation sites. The lower panels are the charge-2e expectation values $\langle c^\dagger_{\mathbf{r},\uparrow} c^\dagger_{\mathbf{r},\downarrow} \rangle$ for such $P_T|\Psi \rangle$ with various distances $l$ between the two topological defects. Here, we have $L=50$.