Superconductivity Proximate to Non-Abelian Fractional Spin Hall Insulator in Twisted Bilayer MoTe$_2$

Abstract

Twisted bilayer MoTe$_2$ near two-degree twists has emerged as a platform for exotic correlated topological phases, including ferromagnetism and a non-Abelian fractional spin Hall insulator. Here we reveal the unexpected emergence of an intervalley superconducting phase that intervenes between these two states in the half-filled second moiré bands. Using a continuum model and exact diagonalization, we identify superconductivity through multiple signatures: negative binding energy, a dominant pair-density eigenvalue, finite superfluid stiffness, and pairing symmetry consistent with a time-reversal-symmetric nodal extended $s$-wave state. Remarkably, our numerical calculation suggests a continuous transition between superconductivity and the non-Abelian fractional spin Hall insulator, in which topology and symmetry evolve simultaneously, supported by an effective field-theory description. Notably, our field-theoretic analysis indicates that superconductivity is driven by the condensation of charge-$e/2$ self-bosonic non-Abelian anyons, thereby providing a concrete realization of anyon superconductivity. Complementarily, when approached from the normal metallic side, superconductivity instead emerges from a Kohn-Luttinger instability enabled by the non-uniform quantum geometry of the flat moiré bands. Our results establish higher moiré bands as fertile ground for intertwined superconductivity and topological order, and point to experimentally accessible routes for realizing superconductivity in twisted bilayer MoTe$_2$.

Figures (4)

• Figure 1: Schematic phase diagram and evolution of many-body spectrum. (a) Schematic phase diagram as a function of $d$ in Eq. \ref{['eq:interaction']}, with $a_M = 1$. As $d$ increases from $0$ to $+\infty$, the system evolves through a ferromagnetic Chern insulator, a superconductor, and finally a fractional spin Hall insulator. (b) The lowest energy of each magnetization subspace for $N=14$. (c,d) Evolution of the many-body spectrum within the paramagnetic sector $M_z =0$ for $N=14$ and 12, respectively. The evolution of the many-body spectrum suggests there are two transition points, a first-order transition $d_{c,1}$ which changes the magnetization, and another continuous transition at $d_{c,2}$ which changes the nature of the ground-state manifold and excitations.
• Figure 2: Many-body spectrum and spectral flow for $N=14$. (a) Many-body spectrum at $d/a_M=0.80$ (blue circle and gray x markers). (b) Spectral flow under time-reversal symmetric flux insertion at $d/a_M=0.80$. We find a $4\pi$-periodic spectral flow which mixes the four lowest states (blue circles), consistent with the fractional spin Hall insulator. (c) Many-body spectrum at $d/a_M=0.28$ (red square, magenta, blue, cyan, and gray x markers). The red square (ground-state) and two blue x markers are the wavefunctions with greatest similarity to three of the four ground-state wavefunctions of the fractional spin Hall insulator. The two cyan x markers both have large contributions from the last fractional spin Hall insulator ground-state wavefunction. The first excited state is the magenta x marked state, which is not adiabatically connected to the fractional spin Hall insulator ground-state wavefunctions. (d) Under time-reversal symmetric flux insertion of (c), the unique ground-state at $d/a_M=0.28$ remains isolated from the excited states. The spectral flow is $2\pi$-periodic.
• Figure 3: Signatures of superconductivity. (a) Intervalley entanglement entropy $S_0^{(v)}$, (b) Binding energy $E_b^{(+)}$, (c) Spectrum of the normalized pair density matrix $\rho_{\mathbf{k}'\mathbf{k}}/N$, and (d) Superfluid stiffness $D_S$ of the ground-state over $d$.
• Figure 4: Pairing symmetry and Kohn-Luttinger instability. (a,b) Cooper-pair order parameter for $N=12$ and $16$, respectively. The size of the circle is proportional to the absolute value of the order parameter at the corresponding momenta, and red and cyan colors indicate opposite signs. (c) Random-phase-approximation-screened intervalley interaction. (d) Corresponding leading pairing instability. All calculations are done for $d/a_M=0.28$.