Imprint of topological degeneracy in quasi-one-dimensional fractional quantum Hall states

We consider an annular superconductor-insulator-superconductor Josephson junction, with the insulator being a double layer of electron and holes at Abelian fractional quantum Hall states of identical fillings. When the two superconductors gap out the edge modes, the system has a topological ground-state degeneracy in the thermodynamic limit akin to the fractional quantum Hall degeneracy on a torus. In the quasi-one-dimensional limit, where the width of the insulator becomes small, the ground-state energies are split. We discuss several implications of the topological degeneracy that survive the crossover to the quasi-one-dimensional limit. In particular, the Josephson effect shows a $2\pi d$ periodicity, where $d$ is the ground-state degeneracy in the two-dimensional limit. We find that at special values of the relative phase between the two superconductors there are protected crossing points in which the degeneracy is not completely lifted. These features occur also if the insulator is a time-reversal-invariant fractional topological insulator. We describe the latter using a construction based on coupled wires. Furthermore, when the superconductors are replaced by systems with an appropriate magnetic order that gap the edges via a spin-flipping backscattering, the Josephson effect is replaced by a spin Josephson effect.

Figure 1

(a) The first realization we consider is that of an electron annulus (blue) and a hole annulus (red) under the action of a uniform magnetic field. It is evident that coupling the annuli's edges forms the topology of a torus. The second realization we suggest is that of a fractional topological insulator. (b) Shows a possible model for a fractional topological insulator. We have an array of $N$ wires, with a strong spin-orbit coupling. The spin-orbit coupling is linear with the wire index $n$. The similarity of the resulting spectrum [see Fig. 3] to the one corresponding to the wires' construction of quantum Hall states suggests an equivalence to two quantum Hall annuli subjected to opposite magnetic fields (each annulus corresponds to a specific spin). The use of the wires construction enables us to include interaction effects using a bosonized Tomonaga-Luttinger liquid theory for the description of the wires. (c) The edge modes of the two above models can be gapped out by proximity coupling to superconductors. In the case of a thin (quasi-1D) system, the phase difference between the inner and the outer superconductors leads to a Josephson effect mediated by tunneling across the region of a fractional quantum Hall double layer or a fractional topological insulator. The spectrum as a function of the phase difference $\phi$ is depicted in Fig. 2. The edge modes can also be gapped using proximity to magnets, in which case one can measure the spin Josephson effect.

Figure 2

The spectrum of the three low-energy states as a function of the phase difference $\phi$ between the two superconductors (see text for elaboration). The amplitude of oscillations falls exponentially with the number of wires $N$. For a finite $N$, each eigenstate has a periodicity of $6\pi$. At the special points $\phi =\pi n$ the spectrum remains twofold degenerate. If the system is of finite length $L_x$, the degeneracy at these points is lifted by a term that is exponentially small in $L_x$.

Figure 3

(a) The spectrum of a system consisting of three wires [see Fig. 1] with noninteracting electrons subjected to spin-orbit coupling whose magnitude depends on the wire index according to Eq. (14), when tunneling between the wires is switched off. The spectra in blue, red, and green correspond to wires number 1, 2, and 3. Solid lines correspond to spin down, and dashed lines correspond to spin up. (b) The resulting spectrum when a weak spin-conserving tunneling amplitude is switched on between the wires. The bulk is now gapped, with helical modes localized on the edges.

Figure 4

A diagrammatic representation of the spectrum in the case $\nu =1$. Once we linearize the spectrum around the Fermi points, it becomes convenient to plot only the Fermi momenta as a function of the wire index ($n$). The symbol $\odot$ ($\otimes$) represents a right (left) mover. Blue (red) symbols represent the spin-down (spin-up) component. One can observe that single-electron spin-conserving tunneling operators conserve momentum, and can therefore easily gap out the bulk in this case.

Figure 5

A diagrammatic representation of the fractional case $\nu =\frac{1}{3}$. Now, we find that only multielectron processes can gap out the bulk. The processes we consider are represented by colored arrows. In terms of the composite $\tilde{\psi }$ fields, however, the diagram corresponding to the fractional case is identical to the one corresponding to the integer case $\nu =1$ (Fig. 4). In this case, the complicated multielectron processes are transformed into single-$\tilde{\psi }$ tunneling operators. The transformation from $\psi$ to $\tilde{\psi }$ therefore proves very useful in analyzing the fractional case.

Figure 6

(a) The spectrum of the wires model for an electron-hole double layer at filling $\nu =1$ when all the interwire terms are switched off. The spectra in blue correspond to wires in the electron layer, and the spectra in red correspond to wires in the hole layer. (b) The spectrum when tunneling between wires in the same layer is switched on. A gap is formed in the bulk, and we get achiral edge modes. (c) The spectrum in the fractional case $\nu =\frac{1}{3}$.

Figure 7

The spectrum corresponding to $\nu =\frac{2}{5}$ with ${\lambda }_2/{\lambda }_1=0.2$ as a function of the relative phase difference $\phi$. The periodicity of each eigenstate is $10\pi$. At the points $\phi =\pi n$, we find two crossing points whose splitting falls exponentially with $L_x$.