Chiral skyrmionic superconductivity from doping a Chern Ferromagnet

Abstract

We show that chiral superconductivity can be stabilized by hole doping a Chern ferromagnet. Performing exact diagonalization and density-matrix-renormalization-group calculations on the repulsive Kane-Mele-Hubbard model at hole doping relative to filling $ν=1$ electron per unit cell, we find that a Cooper pair formed by a magnon (spin-flip excitation) bound to two holes is stabilized at sufficiently strong interactions and sufficiently large Ising spin-orbit coupling (SOC). This Cooper pair exhibits both finite spin chirality -- signaling a noncoplanar skyrmionic spin texture -- and chiral $f$-wave symmetry. The pairing and spin chirality are set by the Chern number/polarization of the parent Chern ferromagnet. We further find that interactions between skyrmion Cooper pairs evolve from repulsive to attractive as the Ising SOC increases, revealing an intermediate-SOC region where chiral superconductivity can emerge from the condensation of hole-skyrmion Cooper pairs. Our findings provide a novel microscopic mechanism for chiral superconductivity and may be relevant for the recent observation of superconductivity in the MoTe$_2$ moiré superlattice.

Figures (12)

• Figure 1: Summary of main results. (a) Illustration of the 2h1s skyrmion-bipolaron ground-state. Filled (empty) bands are sketched using full (dashed) lines along a path in the Brillouin zone containing the $\Gamma$ and $\textbf{M}$ points. The picture illustrated here with absolute maxima/minima at the $\textbf{M}$ and $\Gamma$ points is valid for $0.2\lesssim |\lambda| \lesssim 0.575$. Empty (filled) circles represent doped holes (electrons). $\Delta({\bf k})$ corresponds to the pair wave function and the plots illustrate its phase winding for a closed loop around ${\bf k} = 0$. The sign of both the spin and pairing chiralities is set by the Chern number/polarization of the parent Chern ferromagnet. (b) Sketch of the phase diagram as a function of the Ising SOC $\lambda$, for a fixed interaction strength.
• Figure 2: Exact diagonalization results for $\lambda=0.225$ on a 12-unit-cell lattice with $N_1=4$. (a) Spin-flip gap with respect to the fully polarized state as function of interaction strength $U$, for different doping levels indicated in the figure. (b) Ground-state energy vs. $p_z=2 S_z$ for different $U$ and different doping levels.
• Figure 3: DMRG results for $\lambda=0.225$ and $U=100$ in the 2h1s and 4h2s sectors. Top row: charge density $n_i$. Middle: magnetization $m_z$. Bottom: spin chirality $\chi_{ijl}$.
• Figure 4: Energy dispersion and pairing symmetry of the 2h1s Cooper pair. (a) Dispersion for a 16-unit-cell lattice and $N_1=4$. (b) Overlap of the full two-band ground-state and the three one-band projected lowest-energy states as a function of $U$. (c) $\Delta(\bf{k})$ for the $\left| \Uparrow\right\rangle$ Chern ferromagnet parent state, for a 36-unit-cell lattice. The size of the points are proportional to $|\Delta(\bf{k})|$, while the phase $\theta(\bf{k})$ is given by the color map. (d) $\Delta(\bf{k})$ for the $\left| \Downarrow\right\rangle$ Chern ferromagnet parent state.
• Figure S5: Momentum meshes used for the ED calculations.