Finite-momentum pairing and superlattice superconductivity in valley-imbalanced rhombohedral graphene

Abstract

Inspired by the recent experimental discovery of superconductivity emerging from a time-reversal symmetry-breaking normal state in tetralayer rhombohedral graphene, we here investigate superconducting instabilities in this system. We classify the possible pairing instabilities, including states with commensurate and incommensurate center of mass momenta. As rotational symmetry is broken in the latter type of pairing states, their momentum-space structure is most naturally characterized by a `valley-independent Chern number', measuring the relative chirality between the normal and superconducting state. We further demonstrate that superconductivity can condense at multiple incommensurate momenta simultaneously, leading to the spontaneous formation of a translational-symmetry-breaking superlattice superconductor. Studying multiple different pairing mechanisms and varying the degree of spin and valley polarization in the normal state, we compare the energetics of these superconductors. Our results demonstrate that valley-imbalanced rhombohedral tetralayer graphene can give rise to rich superconducting phenomenologies.

Figures (9)

• Figure 1: Band structure (a) for different $\mu_v$ in $+$ valley (solid lines) and $-$ valley (dashed lines). In b), we illustrate the energetically favorable value of $\boldsymbol{q}$ for a valley-polarized Fermi surface, leading to the maximum approximate overlap of the Fermi surfaces of $\xi_{\boldsymbol{k}+\frac{\boldsymbol{q}}{2},\eta,\sigma}$ and $\xi_{-\boldsymbol{k}+\frac{\boldsymbol{q}}{2},\eta,\sigma}$. (c) Eigenvalue of the leading instability $\lambda(\boldsymbol{q})$ normalized by its maximum $\lambda_0$ for ferromagnetic fluctuations as a function of center-of-mass momentum $\boldsymbol{q}$ for a fully valley-polarized and spin-degenerate normal state with global maxima marked by $\times$. In d), we plot the value of the Cooper pair center of mass momentum $\boldsymbol{q}$ as a function of the amount of valley polarization $\mu_v$ for $\mu_s=0$. The dashed black line denotes where the normal state becomes fully valley polarized.
• Figure 2: We show the highest eigenvalue solutions of the linearized gap equation for spin ferromagnetic fluctuations (a-c) and phonons (d-f) for varying degree of valley polarization $\mu_v$, with $\mu_s=0$. For each solution, we show a single, nonzero magnitude component of $\Delta_{\boldsymbol{k}}^{\boldsymbol{q}}$ in the projected space of the active band, where the intensity and color indicate the the magnitude and phase of $\Delta_{\boldsymbol{k}}^{\boldsymbol{q}}$ respectively. We also indicate $\overline{C}$ and the state's spin structure. We take a displacement field $D=110$ meV. For a discussion of how the gauge of the Bloch wavefunctions is fixed, see App. \ref{['App:LGE']}.
• Figure 3: In a), we show the phase (color) and magnitude (intensity) in real space of a 3-$\boldsymbol{q}$ state order parameter $\Delta_{\boldsymbol{r},\boldsymbol{q}}$. In b) we plot $v/E_F$, relevant to the Landau free energy in Eq. \ref{['eq: Landau th']}, as a function of center of mass momentum $\boldsymbol{q}$ and $w$. We also compare the minimum excitation energy for the single-$\boldsymbol{q}$ (c) and three-$\boldsymbol{q}$ (d) for an intravalley spin-singlet gap function of the form $\Delta_{\boldsymbol{k}}=|\Delta|$.
• Figure 4: We plot the leading solution $\Delta_{\boldsymbol{k}}^{\boldsymbol{q}}$ for ferromagnetic fluctuations $(\rho^i=(s_x,s_y,s_z)\eta_0)$ as a function of varying amounts of spin and valley polarization in the normal state of superconductivity. We also plot the normal state Fermi surfaces for the $+$ (solid lines) and $-$ (dashed lines) valleys and $\uparrow$ (black) and $\downarrow$ (red) Fermi surfaces. We also use "spin singlet" to refer to anti-symmetric pairing between spins and "spin triplet" to refer a symmetric pairing state with respect to spin when $\mu_s\neq 0$ and SU(2)$_s$ spin symmetry is broken. We solve the linearied gap equation with displacement field $D=110$ meV. We observe a tendency towards a state with $\overline{C}=+1$ relative to the normal state with increasing valley polarization.
• Figure 5: We plot the leading solution $\Delta_{\boldsymbol{k}}^{\boldsymbol{q}}$ for ferromagnetic fluctuations $(\rho^i=(s_x,s_y,s_z)\eta_0)$ as a function of varying amounts of spin and valley polarization in the normal state of superconductivity, for a normal state at the same doping of $n_e=0.6\times10^{12}$ cm$^-2$ but now at a higher displacement field $D=140$ meV. We plot the normal state Fermi surfaces for the $+$ (solid lines) and $-$ (dashed lines) valleys and $\uparrow$ (black) and $\downarrow$ (red) Fermi surfaces.