Pair-density-wave in quarter-metals from a repulsive fermionic interaction in graphene heterostructures: A renormalization group study

Abstract

Electronic bands in chirally stacked $n$ layer carbon-based honeycomb heterostructures, encompassing rhombohedral ($n \geq 3$), Bernal bilayer ($n=2$), and monolayer ($n=1$) graphene, possess four-fold valley and spin degeneracy. Such systems with $n \geq 2$, when subject to external perpendicular electric displacement fields, feature a fully degenerate metal at high doping, a spin non-degenerate but valley degenerate half-metal at moderate doping, and a non-degenerate quarter-metal at low doping. Due to the fully polarized nature of the quasiparticles in the quarter-metal, realized around one particular valley otherwise chosen spontaneously, it can sustain a single local superconducting ground state, representing a pair-density-wave that is chiral and odd parity in nature. From a leading order renormalization group analysis, here we show that repulsive density-density interaction among such polarized fermionic excitations can foster the pair-density-wave phase at low temperatures. Possible connections with experimentally observed superconducting states in the close vicinity of the quarter-metal in some members of such graphene heterostructures family are discussed.

Figures (4)

• Figure 1: Feynman diagrams for (a) the bare four-fermion interaction $\left(\psi^{\dagger} \tau_0 \psi\right)^2$ and (b) the fermionic self energy correction, whose contribution yielding a renormalization of the chemical potential $\mu$ is, however, ignored within the framework of a leading-order renormalization group (RG) analysis. (c)-(f) Leading order corrections to the four-fermion interaction stem from the Feynman diagrams. Solid (dashed) lines represent fermions (interaction mediating bosons). The red (black) solid lines represent fast (slow) modes with momentum $\Lambda e^{-\ell} < |\boldsymbol{k}| < \Lambda$ ($|{\boldsymbol{k}}| < \Lambda e^{-\ell}$), where $\Lambda$ is the ultraviolet momentum cut-off and $\ell$ is the logarithm of the RG scale.
• Figure 2: Feynman diagrams representing (a) the bare vertex for the source term $\psi^{\dagger} M \psi$ (excitonic) or $\psi^{\dagger} M \psi^\star$ (pairing), where $M$ is a Hermitian matrix, and (b) and (c) its leading order corrections. Rest of the details are the same as in Fig. \ref{['fig:Feynman_Interaction']}.
• Figure 3: Cuts of the global phase diagram for the quarter-metal in chirally-stacked $n$ layer graphene with $n \geq 2$ in the presence of a repulsive density-density interaction ($g_{_0}$) when the normal state features annular Fermi rings at finite chemical potential ($\mu=2 \times 10^{-n}$) for (a) $n=2$, (b) $n=3$, (c) $n=4$, (d) $n=5$, and (e) $n=6$ with $\alpha_0= 10^{1-n}$ and $\alpha_3^n=10^{2n-2}$ (see text for notations). The vertical axis corresponds to temperature ($t$). Here all the quantities are dimensionless (see text for their definition). The region to the right (left) of the colored lines represents a pair-density-wave (a disordered state or a quarter-metal) for different flavor number $N$ that also marks the transition temperature ($t_c$) for the paired state.
• Figure 4: Same as Fig. \ref{['fig:PhaseDiag_Annular']} but in the presence of a simply connected Fermi ring in the normal state. Here we show results for (a) $n=1$ (monolayer graphene) for which we set $\alpha_0= 0.03$ and $\mu=0.08$, and (b)-(f) $n=2$, $n=3$, $n=4$, $n=5$, and $n=6$, respectively, by setting $\alpha_0= 10^{1-n}$, $\alpha_3^n=10^{2n-2}$, and $\mu=2 \times 10^{1-n}$. Rest of the details are the same as in Fig. \ref{['fig:PhaseDiag_Annular']}.