Band Renormalization, Quarter Metals, and Chiral Superconductivity in Rhombohedral Tetralayer Graphene

TL;DR

The paper tackles the origin of superconductivity in rhombohedral tetralayer graphene arising from a spin-valley-polarized quarter-metal. It develops a screened Hartree-Fock (sHF) framework that combines gate screening and intrinsic internal screening to renormalize the band structure, followed by a Kohn-Luttinger–type analysis to obtain superconductivity from repulsive interactions using the sHF band states. The results predict a parent quarter-metal state, a finite-momentum $\,p$-wave superconducting order that breaks time-reversal symmetry, and critical temperatures consistent with experiments, highlighting the crucial role of band renormalization and trigonal warping in stabilizing exotic superconductivity in graphene multilayers. This approach provides a robust, physically motivated route to understanding electron-mediated pairing in 2D materials and informs future explorations of chiral superconductivity in multilayer graphene systems.

Abstract

Recently, exotic superconductivity emerging from a spin-and-valley-polarized metallic phase has been discovered in rhombohedral tetralayer graphene. To explain this observation, we study the role of electron-electron interactions in driving flavor symmetry breaking, using the Hartree-Fock (HF) approximation, and in stabilizing superconductivity mediated by repulsive interactions. Though mean-field HF correctly predicts the isospin flavors and reproduces the experimental phase diagram, it overestimates the band renormalization near the Fermi energy and suppresses superconducting instabilities. To address this, we introduce a physically motivated scheme that includes internal screening in the HF calculation. Using this formalism, we find superconductivity arising from the spin-valley polarized phase for a range of electric fields and electron dopings. Our findings reproduce the experimental observations and reveal a p-wave, finite-momentum, time-reversal-symmetry-broken superconducting state, encouraging further investigation into exotic phases in graphene multilayers.

Figures (14)

• Figure 1: Screened HF diagram. Diagrammatic representation of the screened Hartree-Fock procedure used for the calculations of superconductivity. Double (single) arrowed lines correspond to interacting (bare) Green functions, wavy lines correspond to interactions mediated by the bare Coulomb potential and the bubbles correspond to electron-hole pairs.
• Figure 2: Hartree-Fock phase diagram. (a) Self-consistently calculated HF phase diagram, showing M$_{1/4}$, M$_{1/2}$ and M$_1$ phases as a function of the external displacement field ($\Delta_{\mathrm{D}}$) and electronic density ($n_e$). The purple region at low densities and high displacement fields highlights schematically where the nem-M$_{1/4}$ is expected to emerge. In (c), we show the calculated phase boundary between M$_{1/4}$ and nem-M$_{1/4}$ in the region indicated by a dashed rectangle in (a). (b) Representative Fermi surfaces at the same point in parameter space indicated by the yellow and blue stars in (c) solutions with and without $\mathcal{C}_3$ symmetry breaking.
• Figure 3: Superconductivity phase diagram. Superconducting critical temperature, $T_c$, as a function of displacement and electron filling in the M$_{1/4}$ phase. The colored regions indicate different Fermi surface shapes. Black dashed lines mark Lifshitz transitions, between different shapes of the fermi surfaces (see Sec. I of SI). A representative Fermi surface is displayed within each region. The yellow star indicates the parameters used for the calculation of the superconducting order parameter shown in Fig. \ref{['fig:Figura3']}(a-b).
• Figure 4: Superconducting order parameter. Order parameter of the leading eigenvalue in the screened Hartree-Fock superconductivity calculations. (a) Normalized magnitude of the order parameters and (b) its corresponding phase. The Fermi surface is indicated by solid white and black lines in each panel, respectively. Calculations were made at values $\Delta_{\mathrm{D}}=137.5$ meV and $\text{n}_e=6.01\times10^{11}\text{cm}^{-2}$ with $T_c~=~80$ mK, marked in Fig. \ref{['fig:Figura2']} as a yellow star.
• Figure S1: Lattice structure of ABCA tetralayer graphene with a representation of the hopping parameters between carbon atoms. The displacement field, $\Delta_{\mathrm{D}}$, acting in each layer is also indicated. $\delta_d$ is the on-site energy at the corresponding site.