Family of multilayer graphene superconductors with tunable chirality: Momentum-space vortices nucleated by a ring of Berry curvature

TL;DR

This work addresses how Berry curvature and Bloch-wave topology shape superconductivity in spin- and valley-polarized rhombohedral multilayer graphene by introducing a minimal two-band model with a Berry ring of fire (BRF) and a tunable attractive interaction. Using a BCS framework and projections to the conduction band, the authors show that local attraction yields chiral $N$-fold winding for odd $N$, while extended interactions unlock a family of chiral superconductors for both odd and even $N$, with momentum-space vortices nucleated on the BRF. A large-$\alpha$ (extended-range) limit reveals a momentum-space flux quantization condition in which the Berry curvature acts like a magnetic field and the quantum metric modulates the effective attraction, leading to a lattice of BRF vortices and tunable winding. They further show density- and displacement-field-driven transitions between chiral phases and discuss experimental probes (edge modes, thermal conductance, and bulk superfluid stiffness) to detect these topological superconducting states, highlighting the BRF as a controllable platform for tunable chirality and potential Majorana physics.

Abstract

Recent experiments in rhombohedrally-stacked multilayer graphene heterostructures have reported signatures of chiral superconductivity, emerging from a spin and valley-polarized normal state with broken time-reversal symmetry and an associated anomalous Hall effect. These findings bring into focus the role of the electronic Bloch wavefunction and the quantum geometric tensor in determining the superconducting pairing channel. In this work, we examine superconducting instabilities of a model of $N$-layer rhombohedral graphene that possesses an enhanced Berry curvature distribution on an extended ring in momentum space $-$ that we dub the 'Berry ring of fire' $-$ in the presence of an isotropic attractive interaction with a parametrically controlled spatial range. We determine that local interactions favor a $N$-fold winding in the order parameter phase for odd-$N$ layered systems, with even-$N$ layers requiring a spatially extended attraction range to achieve pairing. For generic interaction lengths, we discover a family of chiral superconductors and, remarkably, momentum-space vortices nucleated on the Berry ring of fire. The existence of these vortices can be traced to a momentum-space flux quantization condition involving the Berry curvature, with the phase winding dictated by a combination of the Berry flux and a 'statistical flux' to enforce Fermi-Dirac statistics. Such an order parameter structure allows for the possibility of in-situ tuning between various chiral superconducting phases through changes in the electron density or the displacement field. We discuss ways in which these predictions can be experimentally tested and potentially exploited in future devices.

Figures (12)

• Figure 1: Nucleation of momentum-space vortices on the Berry ring of fire (BRF). To illustrate the central idea of this work, we depict the magnitude of the superconducting order parameter $\Delta_k$ being suppressed at four points on the BRF (yellow disk), with a superconducting phase $\Phi_k$ winding by $\pm2\pi$ around these points, indicated by the color gradient. The vortex at the origin (black dot) is the remnant from a momentum-isotropic bare attraction potential and reflects the odd spatial parity Cooper pair wave function, $p_x+ip_y$ in this case, required for pairs composed of same-spin electrons.
• Figure 2: Continuum conduction band dispersion for spin-valley ($\uparrow K$) polarized electrons in rhombohedral stacking (see right inset) of $N=4,5$ (blue and orange) layers of graphene. Fermi level (dashed line) is set such that a density of $10^{12}$ cm$^{-2}$ is filled about charge neutrality. Displacement energy of $|D| = 2.25$eV for illustration. Inset (Left): Berry curvature distribution in momentum space for $N=5$. The ring deforms into a $C_3$-symmetric shape with more realistic microscopic modelling dong2024stability.
• Figure 3: Superconducting order parameter magnitude (top) and phase (bottom) for [left to right] $N=3,5,7$ layers for on-site local attraction ($\alpha = 0$), and $U = 10^{-6} a$ eV/m.
• Figure 4: Superconducting order parameter solutions magnitude (top) and phase (bottom) for odd-$N$ layers ($N=5$) for non-local attraction ($\alpha = 20.4 a^{2}$). The black dashed circle indicates where Berry curvature is peaked. $U = 10^{-4} a$ eV/m.
• Figure 5: Evolution of magnitude (top) and phase (bottom) of $N$-times phase winding order parameter for $N=5$ layers as a function of attraction length $\alpha$. The black dashed circle indicates where the Berry curvature is peaked. $U = 10^{-4} a$ eV/m. Figures \ref{['fig_evolution_alpha_N_5_Nm2_winding']} and \ref{['fig_evolution_alpha_N_5_single_winding']} in Appendix \ref{['app_r5g_evolution_alpha']} depict the evolution of the $N-2$ and single-winding solutions under increasing $\alpha$.