Repulsive-Interaction-Driven Topological Superconductivity in a Landau Level Coupled to an $s$-Wave Superconductor

Abstract

A two-dimensional topologically nontrivial state of noninteracting electrons, such as the surface state of a three-dimensional topological insulator, is predicted to realize a topological superconductor when proximity-coupled to an ordinary $s$-wave superconductor. In contrast, noninteracting electrons partially occupying a Landau level, with Rashba spin-orbit coupling that lifts the spin degeneracy, fail to develop topological superconductivity under similar proximity coupling in the presence of the conventional Abrikosov vortex lattice. We demonstrate through exact diagonalization that, at half-filled Landau level, introducing a repulsive interaction between electrons induces topological superconductivity for a range of parameters. This appears rather surprising because a repulsive interaction is expected to inhibit, not promote, pairing, but suggests an appealing principle for realizing topological superconductivity: proximity-coupling a composite Fermi liquid to an ordinary $s$-wave superconductor.

Figures (9)

• Figure 1: Schematic illustration of the hybrid system considered here. An interacting electron system that forms a composite Fermi liquid (bottom layer) is proximity-coupled to the type-II $s$-wave superconductor with an Abrikosov vortex lattice (top layer). The cuboid outlines the magnetic unit cell (MUC) containing two vortices. The system shown corresponds to the largest size used in our exact diagonalization, consisting of $n_x\times n_y=4\times8$ MUCs with total magnetic flux $N_\phi=32$.
• Figure 2: (a) Low-energy spectrum $E_n-E_1$ at $\nu=1/2$ as a function of the pairing strength $\Delta_0$, where $E_n$ is the $n$th lowest energy. Energies are measured in units of the interaction strength $V_C$. Colors indicate the fermion parity $P$. The system size is $N_\phi=n_x\times n_y=4\times4$. A transition occurs near $\Delta_0/V_C\approx0.02$. (b) Spectrum at $\Delta_0/V_C=0.1$, plotted versus total momentum index $j_x+n_xj_y$, where $j_\alpha=0,1,\ldots,n_\alpha-1$. The ground state consists of two states at ${\bm{K}}=(\pi/2,\pi/2)$ and two at ${\bm{K}}=(\pi/2,3\pi/2)$. (c) Same as (b), but at $\nu\approx0.485$. The inset shows a lifting of the twofold degeneracy within each momentum sector.
• Figure 3: (a) Modulus of $\Delta_{\bm{k}}$. The circles indicate nodes. (b) Finite-size scaling of the energy gaps for the fourfold degenerate RID-TSC state at $\Delta_0/V_C=0.1$ and for the two-fold degenerate CFL at $\Delta_0/V_C=0$. (c) Spatial profile of $|\langle c^\dagger_\uparrow({\bm{r}})c^\dagger_\downarrow({\bm{r}})\rangle|$ at $\Delta_0/V_C=0.1$ and $n_x\times n_y=4\times8$. $L_{x(y)}$ denotes the system length. (d,e) Finite size scaling of (d) $\max_{{\bm{r}}}|\langle c^\dagger_\uparrow({\bm{r}})c^\dagger_\downarrow({\bm{r}})\rangle|$ and (e) the critical pairing strength $\Delta_c/V_C$, defined as the $\Delta_0/V_C$ where the phase transition to the RID-TSC phase occurs.
• Figure 4: (a) Filling factor $\nu$ and (b)(c) energy differences $(E_n-E_1)/\Delta_0$ for $n=2$ and $n=5$ respectively, as functions of the interaction strength $V_C/\Delta_0$ and $(\mu-\mu_{1/2})/\Delta_0$, where $\mu_{1/2}$ is the chemical potential yielding $\nu=1/2$. Stars mark the parameters corresponding to Figs. \ref{['fig:CFStoSC']}(b) and \ref{['fig:CFStoSC']}(c). The gapped regions in (b) corresponds to vacuum and $\nu=1$ IQH phases, respectively. The RID-TSC phase in (c) emerges from the critical point $V_C/\Delta_0=(\mu-\mu_{1/2})/\Delta_0=0$ and expands as $V_C/\Delta_0$ increases. The system size is $n_x\times n_y=4\times4$. The composite Fermi liquid appears at $V_C/\Delta_c\approx50$, beyond the plotted range.
• Figure 5: Modulus of the pair amplitude $F_{\sigma_1\sigma_2}({\bm{R}},\bar{{\bm{r}}})$. Panels show (a) $s_z=1$, (b) $s_z=-1$, (c)$s_z=0$ for $p$-wave and (d)$s_z=0$ for $s$-wave pairing, plotted as a function of the relative coordinate $\bar{{\bm{r}}}$. The center-of-mass coordinate is set as (a)(b)${\bm{R}}={\bm{0}}$ and (c)(d) ${\bm{R}}={\bm{a}}\equiv l_B(1,1)$. The insets show the phase near $\bar{{\bm{r}}}=0$, with the winding number of (a) $1$, (b) $-1$, (c) $1$, and (d) $0$. Parameters are set as $\Delta_0/V_C=0.1$ and $n_x\times n_y=4\times8$.