Fractional Topological Insulators of Cooper Pairs Induced by the Proximity Effect

Certain insulating materials with strong spin-orbit interaction can conduct currents along their edges or surfaces owing to the nontrivial topological properties of their electronic band structure. This phenomenon is somewhat similar to the integer quantum Hall effect of electrons in strong magnetic fields. Topological insulators analogous to the fractional quantum Hall effect are also possible, but have not yet been observed in any material. Here we show that a quantum well made from a topological band insulator such as ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ or ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, placed in contact with a superconductor, can be used to realize a two-dimensional topological state with macroscopic many-body quantum entanglement whose excitations carry fractional amounts of an electron’s charge and spin. This fractional topological insulator is a “pseudogap” state of induced spinful $p$-wave Cooper pairs, a new strongly correlated quantum phase with possible applications to spintronic devices and quantum computing.

Figure 1

The heterostructure device that can host fractional TR-invariant quantum states. A topological insulator (TI) quantum well is sandwiched between a conventional superconductor (SC) and a conventional insulator (I). The gate (G) voltage can be used to control the state of the TI, and the topological properties of the TI can be probed via a Hall-bar setup of leads (L).

Figure 2

(a) The qualitative zero-temperature phase diagram of attractively interacting electrons in a quantum well. Gate voltage $V_{\mathrm{G}}$ controls the electron gap in the quantum well and tunes the quantum critical point (QCP) between a superconductor (SC) and a bosonic Mott insulator (MI) of Cooper pairs. The lowest energy excitations in the MI are charge $2e$ bosons, but they disappear at the crossover (dashed line) to the band insulator (BI) where only gapped fermionic quasiparticles with charge $e$ exist. A spin-orbit coupling whose strength is measured by a “cyclotron” energy ${\omega }_{\Phi }$ (defined in the text) introduces a vortex lattice in the superconducting state of spinful $p$-wave Cooper pairs. Quantum melting of such a vortex lattice gives rise to correlated “vortex liquid” (VL) states, which are the prime candidates for fractional TIs. (b) The energy spectrum $E(k)$ of the Hamiltonian Eq. (1) for $m\sim 2.5\times {10}^{-31}\mathrm{kg}$, $v\sim 4\times {10}^5\mathrm{m}/\mathrm{s}$, and $\Delta \sim 100\mathrm{eV}$ ($k=p/\hslash$). This two-orbital example approximates the ARPES spectrum from Fig. 3(d) of Ref. 33 when $\Delta =0$. (c) The Cooper pairing channels in the TI include intraorbital spin singlets (${\Phi }_{\pm }$), interorbital spin singlets (${\Phi }_0$), and interorbital $p$-wave spin triplets (${\eta }_m$, where $m\in \{0,\pm 1\}$ is the $z$-axis spin projection).

Figure 3

A TR-invariant Abrikosov lattice in the helical triplet condensate, ${\eta }_{+}={\eta }_{-}^{*}$, ${\eta }_0=-{\eta }_0^{*}$. Coinciding vortices in ${\eta }_{+}$ (red circles) and antivortices in ${\eta }_{-}$ (blue circles), comprise an equilibrium state without charge currents and spin texture, which gains energy by its Rashba-coupled spin current loops.