Variational Monte Carlo Optimization of Topological Chiral Superconductors

Abstract

We perform the variational Monte Carlo calculation for recently proposed chiral superconducting states driven by strong Coulomb interactions. We compare the resulting energetics of these electronic phases for the electron dispersion relation $E_k = c_2 k^2+c_4 k^4$. Motivated by the recent discovery of chiral superconductivity in rhombohedral graphene systems, we apply our analysis to relevant parameter regimes. We demonstrate that topological chiral superconducting phases (including a spin-unpolarized state) can be energetically favored over the spin-valley polarized Fermi liquid above the density of Wigner crystal phase. Our results show that the preference for chiral superconductivity is strongest when $c_2$ lies between zero and a negative value, corresponding to a system on the verge of forming a hole pocket around $k=0$. This finding suggests that superconductivity can arise from pure repulsive Coulomb interactions in systems with an almost flat band bottom, without relying on the pairing instability of a Fermi surface. This mechanism opens a new pathway to superconductivity beyond the conventional BCS mechanism.

Figures (8)

• Figure 1: The phase diagram for electron dispersion $E_k = c_2 k^2 + c_4 k^4$ at various $c_2$ and electron density $n_e$. The horizontal axis is $n_e \in [0,10^{12} cm^{-2}]$. The vertical axis is $-c_2$ (which is tuned experimentally by the displacement field). Green is the $K_{2a}$-state, blue is the $K_{2b}$-state, red is the Pfaffian state, and black is the quarter Fermi liquid ( i.e. the spin-valley polarized Fermi liquid). The horizontal line is the $c_2=0$ line. The slanted line is the Fermi-surface transition line (see Fig. \ref{['FStrans']}). The data above the Fermi-surface transition line should be ignored. Here we choose some possible parameters for four-layer graphene HJ240815233: (left) $\epsilon = 10$, $c_4= 549$ m$eV$ n$m^4$, and $c_2 \in [69,-69]$ m$eV$ n$m^2$ ; (middle) $\epsilon = 5$, $c_4= 549$ m$eV$ n$m^4$, and $c_2 \in [69,-69]$ m$eV$ n$m^2$ ; (right) $\epsilon = 5$, $c_4= 366$ m$eV$ n$m^4$, and $c_2 \in [46,-46]$ m$eV$ n$m^2$. (In four-layer samples, Wigner crystal appears for $n_2 \lesssim 0.2\times 10^{12} cm^{-2}$ when $c_2=0$.)
• Figure 2: Electron dispersion $E_k = c_2 k^2 + c_4 k^4$ has a Fermi surface transition when $c_2$ satisfies $c_2 k_F^2 + c_4 k_F^4 = 0$ and the Fermi momentum $k_F$ is determined by the electron density $n_e$. At the Fermi surface transition, a hole pocket is generated at $k=0$ as we make $c_2$ more negative.
• Figure 3: Comparison of two ground state energies (per electron), $E_\text{QFL}$ and $E_\text{QFL0}$, for QFL. $E_\text{QFL}$ is from our numerical calculation and $E_\text{QFL0}$ is from Ref. TC8905. $E_{K_{2a}}$ is the ground state energy (per electron) of $K_{2a}$ state and $E_{Pf}$ is the same for the Pfaffian state. We used the four-layer graphene parameters $c_2 = 70$ m$eV$ n$m^2$ and $\epsilon=5$.
• Figure 4: Ground state energy per electron for $K_{2a}$, $K_{2b}$, and Pfaffian superconductors, minus that of QFL, along the Fermi-surface transition line (the slanted line in Fig. \ref{['fig:phase']}).
• Figure 5: The phase diagram for electron dispersion $E_k = c_2 k^2 + c_4 k^4$ along the Fermi surface transition line ( i.e. for when $c_2 = - 4\pi c_4 n_e$). The horizontal axis is $n_e \in [0,10^{12} cm^{-2}]$. The vertical axis is magnetic field $B$ in teslas.