Creation and manipulation of anyons in a layered superconductor–two-dimensional electron gas system

We describe and analyze in detail our recent theoretical proposal for the realization and manipulation of anyons in a weakly interacting system consisting of a two-dimensional electron gas in the integer quantum Hall regime adjacent to a type-II superconducting film with an artificial array of pinning sites. The anyon is realized in response to a defect in the pinned vortex lattice and carries a charge of $\pm e/2$ and has a statistical angle of $\pi /4$. We establish this result, both analytically and numerically, in three complementary approaches: (i) a continuum model of two-dimensional electrons in the vortex lattice of the superconducting film; (ii) a minimal tight-binding lattice model that captures the essential features of the system; and (iii) an effective theory of the superconducting vortex lattice superposed on the integer quantum Hall state. We propose a method for measuring the fractional charge directly in a bulk transport experiment and an all-electric setup for an “anyon shuttle” implementing the braiding operations. We briefly discuss conditions for fabricating the system in the laboratory and its potential applications in quantum information processing with non-Abelian anyons.

Figure 1

(Color online) The schematic diagram of the proposed device. A 2D electron gas is sandwiched between two type-II superconducting films.

Figure 2

(Color online) (a) Charge densities in units of $e/2\pi {\ell }_B^2$ in a system with $N_e=100$ electrons without (dashed blue line) and with (solid black line) a defect $\left(\eta =-\frac{1}{2}\right)$ of the widened flux tube with $\xi =0.5{\ell }_B$. (b) The difference in density $-\delta {\rho }_{\eta }={\rho }_0-{\rho }_{\eta }$. The inset shows the accumulated integrated charge $\delta Q\left(r\right)=-\int \delta {\rho }_{\eta }d\mathbf{x}$ in units of $e$.

Figure 3

(Color online) Magnetic fields (in units of ${\Phi }_0/\pi {\ell }_B^2$) of the pinned Abrikosov lattice, (a) without and (b) with a vacancy defect. The plots are a partial view of the larger system with 64 pinning sites, ${\lambda }_L=0.2$, and ${\xi }_0=0.01$ in units of vortex separation, $\sqrt{\pi }{\ell }_B$.

Figure 4

(Color online) The electronic densities (in units of $e$) in the Abrikosov lattice [(a) and (c)] without and [(b) and (d)] with defects. Panels (c) and (d) are zooms on (a) and (b) where a vacancy is introduced. The difference between (a) and (b) is shown in (e), and that between (c) and (d) is in (f). The system has 64 pinning sites, ${\lambda }_L=0.2$, and ${\xi }_0=0.01$ in units of vortex separation, $\sqrt{\pi }{\ell }_B$.

Figure 5

(Color online) (a) The Peierls factors $e^{i{\chi }_{ij}}$ in the Landau gauge: (green) dashed links are $-1$, (black) solid links are $+1$, and (blue) diagonal links are $+i$ in the direction of the arrow. (b) The string gauge for a defect (black disk): every link that intersects the string (dashed wiggly line) acquires an extra $-1$.

Figure 6

(Color online) The spectra of the lattice model (a) without a gap, $t^{\prime }=0$, and (b) with a gap $t^{\prime }/t=0.1$. The densities of states in the (c) gapless and (d) gapped systems. In all cases $ϵ=0$ is assumed. Charge densities for (e) a single defect and (f) two defects in a $36\times 36$ system and $t^{\prime }/t=0.3$. (g) The integrated charge in a $40\times 40$ system with open boundary conditions. (h) The energy splitting of zero modes of two defects vs their separation in a $164\times 10$ system with periodic boundary conditions.

Figure 7

(Color online) The temporal string gauge. The vertical direction represents the time and style and color are as in Fig. 5. As the defect is moved from a plaquette to the neighboring one, it threads the shaded temporal plaquettes, thus inducing an electric field. In the next time step the string is extended along the defect’s path, which accounts for the induced electric field. The Peierls factors in the vertical direction remain equal to $+1$.

Figure 8

The Berry phase ${\theta }_B/\pi$ and “even-odd effect.” In (a) and (b) there is a single defect in the system that is taken around a loop with (a) an even or (b) an odd number of sites. In (c) and (d) the system contains two defects, of which one is taken around the other on a loop containing (c) an even or (d) an odd number of sites.

Figure 9

(Color online) Experimental setup for measuring the fractional charge. A supercurrent ${\mathbf{J}}_{\text{SC}}$ is induced in the superconducting film, which causes a vortex flow ${\mathbf{J}}_v$ of unpinned defects (white circles) on the background of pinned vortex lattice (black circles) across the constriction along with a corresponding electric current ${\mathbf{J}}_e$ in the 2DEG. A Hall voltage $V_{2\text{DEG}}$ is generated in the 2DEG and a voltage drop $V_{\text{SC}}$ develops across the superconductor. This measures the fractional charge $\delta Q=\left(V_{2\text{DEG}}/2V_{\text{SC}}\right)e$. The small arrows show the chiral edge current in the 2DEG and the thick (green) lines show the wires that carry the electric current in 2DEG.

Figure 10

(Color online) The anyon shuttle. A superlattice of holes with diameter $D$ and separation $d$, both $⪢{\lambda }_L$, is used as source and sink of anyons. The arrows on the lower left hole shows the chiral edge currents concealed on other holes for clarity. The black squares are the leads on the superconducting film.