Topological superconductivity with emergent vortex lattice in twisted semiconductors

TL;DR

The paper investigates how emergent, spatially modulated magnetic fields in twisted transition-metal dichalcogenide bilayers enable a superconducting vortex lattice that coexists with fractional Chern insulator physics. Through a band-projected, numerically exact treatment, it identifies a chiral $f$-wave superconducting state with double vortices per moiré unit cell, and shows an adiabatic path to a weak-coupling BdG description. It reports a half-integer Chern number $C=- frac{1}{2}$ for the superconducting phase, arising from the interplay of band topology and BdG structure, and demonstrates phase transitions and coexistence with FCIs as the moiré geometry is tuned. The work provides a unified mechanism linking fractional quantum anomalous Hall states and topological superconductivity in twisted TMDs, with explicit predictions for Majorana edge modes and real-space vortex patterns observable in experiments.

Abstract

The coexistence of superconductivity and fractional quantum anomalous Hall (FQAH) effect has recently been observed in twisted MoTe$_2$ and theoretically demonstrated in a model of repulsively interacting electrons under an emergent magnetic field arising from the layer pseudospin texture in moiré superlattice. Here, we show that this superconducting state is a chiral $f$-wave superconductor hosting an array of $double$ vortices, which are induced by the emergent magnetic field with $h/e$ flux quanta per moiré unit cell. This superconducting vortex lattice state is topological and features Chern number $-1/2$, giving rise to a half-integer thermal Hall conductance. Our theory provides a common mechanism and unified understanding of FQAH and topological superconductivity, with a rich phase diagram controlled by the spatial modulation of the emergent magnetic field.

Figures (14)

• Figure 1: The twisted heterostructure (a) hosts a skyrmion lattice resulting in an inhomogeneous magnetic field (b) with two superconducting flux quanta per unit cell. For weak inhomogeneity of the emergent magnetic field (top image in panel (c)), anomalous Hall metal and superconducting correlations develop for fillings $2/3<\nu<1$. Increasing the inhomogeneity drives a first-order transition from the fractional Chern insulator to a superconducting phase $\nu=2/3$, allowing superconductivity to extend to lower fillings.
• Figure 2: 2RDM spectrum at $2/3$ and condensate wavefunction for $\mathcal{K}=0.8$: Panel (a) show the spectrum of $\rho^{(2)}$ comparing system sizes $N_s=21,$ 27. Panel (b) shows $\chi_{0({\boldsymbol{k}},-{\boldsymbol{k}})}$ for $N_s=27$ where the size of the dots quantify the absolute value of $\chi_{0({\boldsymbol{k}},-{\boldsymbol{k}})}$ while the color its phase.
• Figure 3: Inhomogeneous magnetic field and the real-space superconducting order: Panel (a) shows the magnetic field $B({\boldsymbol{r}})$ in unit of $\hbar/(e\ell^2_B)$. Panel (b) shows the absolute value of the pair wavefunction, with the tone (from dark to bright) indicating its magnitude, while panel (c) displays the complex phase at short distances within the square region delineated by the dashed white line in panel (b). $\Psi_{\rm pair}$ is shown as a function of the relative coordinate $\Delta {\boldsymbol{r}}={\boldsymbol{r}}-{\boldsymbol{r}}'$ and ${\boldsymbol{r}}'=-a\hat{x}/\sqrt{3}$. Panel (d) shows the order parameter as a function of the center-of-mass coordinate, where the tone (from black to light) indicates its magnitude and the hue corresponds to the complex phase. Parameters: $\mathcal{K}=0.8$, $N_s=27$, and $\nu=2/3$. White lines in panels (a) and (d) mark the unit cell.
• Figure 4: Panels (a) and (b) show the evolution of the many-body energy spectrum as a function of the coupling $g$ for the $f\!-\!if$ ($m=-$) and $f\!+\!if$ ($m=+$) attraction, respectively. Panels (c) and (d) display the single-particle occupation numbers for the unperturbed ground state and for $g/V_1 = 10$ and $f\!-\!if$, respectively. Calculations are performed for $N_s=27$, $N=16$ and $\mathcal{K}=0.8$.
• Figure 5: Chern number and BdG bands. Panel (a): Chern number as a function of the chemical potential (lower axis) and filling factor (upper axis). Panel (b): Representative low-energy BdG band structures of the chiral topological superconductor (SC) with $C=-1/2$$(\nu=0.66)$ and of the Chern insulator (CI) with $C=+1$$(\nu=1.0)$. $W$ is the bandwidth of the lowest Chern band.