• Abstract and Figures
• Public Full-text

Anyons in a weakly interacting system C. Weeks, G. Rosenberg, B. Seradjeh and M. Franz Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z1 (Dated: June 30, 2018) In quantum theory, indistinguishable particles in three-dimensional space behave in only two distinct ways. Upon interchange, their wavefunction maps either to itself if they are bosons, or to minus itself if they are fermions. In two dimensions a more exotic possibility arises: upon iθ exchange of two particles called “anyons” the wave function acquires phase e =6 ±1. Such fractional exchange statistics are normally regarded as the hallmark of strong correlations. Here we describe a theoretical proposal for a system whose excitations are anyons with the exchange phase θ = π/4 and charge −e/2, but, remarkably, can be built by filling a set of single-particle states of essentially noninteracting electrons. The system consists of an artificially structured type-II superconducting film adjacent to a 2D electron gas in the integer quantum Hall regime with unit filling fraction. The proposal rests on the observation that a vacancy in an otherwise periodic vortex lattice in the superconductor creates a bound state in the 2DEG with total charge −e/2. A composite of this fractionally charged hole and the missing flux due to the vacancy behaves as an anyon. The proposed setup allows for manipulation of these anyons and could prove useful in various schemes for fault-tolerant topological quantum computation. I. INTRODUCTION particle states. It has been pointed out very recently [9] that a two-dimensional version of the physics underly- Anyons and fractional charges appear in a variety of ing the above effect could be observable under certain theoretical models involving electron or spin degrees of conditions in graphene. freedom in 2D [1, 2, 3, 4, 5, 6]. In all known cases, in- An appealing feature of weakly interacting systems is teractions between the fundamental degrees of freedom that their properties can be often understood from es- are of paramount importance for the emergence of the sentially exact calculations, which is rarely possible for exotic excitations. For example, it is understood that strongly correlated systems in dimension greater than the Coulomb interaction between electrons is crucial to one. It is thus desirable to study weakly interacting stabilize the incompressible ground state in the frac- systems with exotic excitations if there is a chance that tional quantum Hall (FQH) liquids that support frac- they can be realized in a laboratory. Our proposal for tionalized excitations. One can say that such systems such a system is depicted in Fig. 1(a). It consists of a are strongly correlated in the sense that their many-body type-II superconducting film grown on top of a semicon- wavefunctions cannot be constructed from the single- ductor heterostructure (such as GaAs/AlGaAs) hosting particle states of the constituent fundamental degrees of a 2D electron gas. When a magnetic field is applied per- freedom. pendicular to this planar device, an Abrikosov lattice of The question thus arises whether anyons and frac- vortices forms in the superconductor. This results in a tionalization are inextricably connected with the phe- periodic modulation of the magnetic field in the 2DEG nomenon of strong correlations as defined above. There with one-half of a magnetic flux quantum Φ0 = hc/e per are compelling reasons to believe that this is in fact unit cell. Devices like this have in fact been fabricated not so. In one dimension, Su, Schrieffer and Heeger [7] previously [11, 12, 13] and studied theoretically [14, 15]. showed that a half of an electron can be bound to a do- A key new aspect of our proposal is a periodic array of main wall in trans-polyacetylene modeled by a dimerized pinning sites imprinted on the superconductor. Pinning tight binding chain. Interactions play no role in this con- sites are regions of weakened superconductivity, which struction and the many-body wavefunction indeed can attract vortex cores and can be created in a variety of be built as a Slater determinant of single-particle elec- ways [16]. When the number of vortices in the sample tron states. In this case fractionalization is a many-body equals the number of pinning sites, each pin is occupied effect in that it is a cooperative phenomenon of a large by a vortex and a periodic vortex lattice arises. The number of electrons but it arises from the geometry of Magnetic field strength at which this happens is called arXiv:cond-mat/0703001v1 [cond-mat.mes-hall] 28 Feb 2007 the system rather than strong correlations. the matching field BM . When the field is slightly below In three dimensions, Jackiw and Rebbi [8] argued that (above) BM , vacancies (interstitials) appear in the vortex it is consistent to assign a half-integer fermion number lattice as illustrated in Fig. 1(b). Such defects in the to a monopole in the Yang-Mills gauge field coupled to vortex lattice then lead to a localized deficit or surplus relativistic Dirac fermions. In essence, this implies that a of magnetic flux quantized in the units of Φ0/2 shown in half-fermion is permanently bound to such a monopole. Fig. 1(c). Again, interactions between fermions play no role in this Our principal claim, which we justify below by a gen- construction and the half-fermion emerges from a many- eral argument and a detailed computation, is this: When body wavefunction that is a Slater determinant of single- the lowest Landau level of the 2DEG is filled by spin- 2 (a) B polarized electrons, i.e. at filling fraction ν = 1, a defect in the vortex lattice binds fractional electric charge in the 2DEG with the exact value −e/2 for a vacancy and +e/2 for an interstitial. Such a defect constitutes a literal real- type−II SC ization of Wilczek’s model of an anyon as a bound state 2DEG of magnetic flux and electric charge [17]. The exchange phase of this anyon π/4. (b) The flux deficit/surplus can be, at least in principle, created, detected, and manipulated by the suite of scan- ning SQUID and Hall bar probes that have been devel- vacancy oped over the years to image [18, 19, 20] and manipulate [21, 22] vortices in superconducting films and crystals. The proposed device could thus offer a unique opportu- interstitial nity to manipulate anyons with an unprecedented degree of control and advance the quest for a topologically pro- tected quantum computer. In order to execute a useful quantum computation, non-Abelian anyons are needed [4]. We remark that a the device pictured in Fig. 1 could prove useful for the (c) manipulation of such non-Abelian anyons if the 2DEG 2 5 0 were put into the ν = 2 FQH state, which is be- -2 lieved to be described by the Moore-Read “Pfaffian” state [23, 24, 25]. In that situation, vacancies and intersti- 1.2 tials should bind non-Abelian anyons and the scanning BB SQUID probe could be used to locate these and possibly 1.0 perform the braiding operations required to implement a 0.8 fault tolerant quantum algorithm. -2 0 2 II. ANYONS BOUND TO VACANCIES AND INTERSTITIALS: GENERAL ARGUMENTS (d) 2DEG E So why does a vacancy bind fractional charge? The answer can be given at several levels of sophistication and is essentially implicit in the body of work on the j Φ FQH effect. The simplest way to see that this is true is to recall that electrons in the quantum Hall state at ν = 1 form an incompressible fluid. This means that in the ground state the electron charge density ρ(r) tracks the magnetic field strength B(r) in accordance with the defining relation N ρΦ0 FIG. 1: (a) The schematic of the proposed device. Side view ν = e = , (1) of the superconductor-2DEG heterostructure in an applied NΦ eB magnetic field B. (b) Top view. Open circles represent pin- ning sites in the superconductor while crosses denote vortices. where Ne and NΦ denote the total number of electrons Throughout this paper we consider, for simplicity, the square and flux quanta in the sample, respectively. In an in- vortex lattice. All of our conclusions, however, remain valid compressible state the above relation remains valid, to for any arbitrary periodic Bravais lattice, including the most a good approximation, even for a spatially varying mag- natural triangular case. (c) The magnetic field produced by netic field, as long as the variation is weak. In our device, a periodic square lattice of vortices with one vacancy (lower a slowly varying field occurs in the limit when the mag- left corner) and one interstitial (upper right corner). The field netic penetration depth λ ≫ a, a being the intervortex distribution is calculated from a simple London model [10] with the penetration depth λ = a and the vortex core cutoff distance. In this limit, one obtains ξ = 0.05λ. (d) Thought experiment: creating an interstitial B(r) by adiabatically threading flux through the system. ρ(r) ≃ νe . (2) Φ0 This relation, which becomes exact when integrated over the region containing the vacancy, implies that at ν =1 3 a deficit of one half flux quantum produces a deficit of fusion rules for anyons. The relevant rule [28] states that one half of an electron in the 2DEG. the exchange phase Θ of a particle formed by combining To see more clearly how the charge deficit/surplus n identical anyons with exchange phase θ is Θ = n2θ. comes about we may employ a version of Laughlin’s argu- Consider bringing together two interstitials.

Load more

  7

Copy Link