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(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 scanning 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 protected 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 pinning 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. The re- ment [26] similar to that used by Halperin to deduce the sulting object is (e, Φ0), a bound state of an electron and existence of edge states in integer quantum Hall liquids a full flux quantum. The exchange phase of this object [27]. The idea is to start from a perfect vortex lattice is π +2π =3π, the first term reflecting the intrinsic elec- and imagine creating an interstitial by adiabatically tun- tron exchange phase and the second the Aharonov-Bohm ing the flux of the extra vortex from zero to Φ0/2. We phase. But a 3π phase is equivalent to π, i.e. the bound again consider the limit λ ≫ a. state is a fermion. The above fusion rule with n = 2 thus According to Faraday’s law, ∇× E = −(1/c)(∂B/∂t), determines the exchange phase of an interstitial to be the time dependent magnetic field associated with in- θ = π/4. A similar argument leads to the identical result troducing the extra vortex induces an electric field with for a vacancy. These results will be confirmed below by concentric field lines as shown in Fig. 1(d). In the quan- explicit model calculations. tum Hall liquid such an electric field produces current strictly perpendicular to the field, III. CONTINUUM MODEL j = σxy(E × zˆ), (3) We first consider a very simple continuum model, where σ = νe2/h is the quantized Hall conductance. xy which is exactly soluble and captures the essence of Using the continuity equation for the electric charge, one the effect discussed above. Consider the following 2- can integrate this current to find the rate of change of dimensional electron Hamiltonian, the total charge Q in the region bounded by an arbitrary closed contour C enclosing the flux, 1 e 2 H = p − A , (6) dQ 2me c = − dl n · j = −σxy dl · E, (4) dt IC IC with me the electron mass, p the momentum operator in 1 the x-y plane and A = A0 + δA. Here A0 = 2 B0(r × zˆ) where n is a unit vector normal to the contour. Using represents a uniform field in thez ˆ direction and Stokes theorem, the last integral can then be writen as an integral of ∇× E over an area bounded by C. This ηΦ0 δA = (r × zˆ). (7) we can evaluate with help of Faraday’s law to obtain 2πr2 dQ/dt = (σxy/c)dΦ/dt. It follows that the charge δQ accumulated inside C during the process of adiabatic flux The total field seen by an electron is then insertion is B(r)= ∇× A =zB ˆ 0 +ˆzηΦ0δ(r). δΦ δQ = eν . (5) The δ-function serves as a crude representation of the Φ0 1 flux added by an interstitial (η = 2 ) or removed by a 1 For ν = 1 and δΦ = ±Φ0/2 the accumulated charge is vacancy (η = − 2 ) in the vortex lattice. ±e/2. It is to be noted that this argument depends only It is straightforward to find the single particle eigen- on the fundamental property (3) and the above result (5) states ψkm(r) of H for arbitrary real η by working in the should thus be valid unconditionally as long as the 2DEG polar coordinate basis. They are labeled by the principal remains in the quantum Hall plateau. quantum number k = 0, 1, 2,... and an integer angular Vacancies and interstitials, being bound states of momentum m. The allowed energy levels read charge and flux, (−e/2, −Φ0/2) and (e/2, Φ0/2) respec- 1 tively, acquire nontrivial Aharonov-Bohm phases upon ǫkm = ¯hωc [2k +1+ |m − η|− (m − η)] , (8) adiabatic exchange, and can thus be thought of as 2 anyons. The standard textbook counting procedure [28], where ωc = eB0/mec is the cyclotron frequency. The according to which a charge q taken around a flux Φ ac- spectrum (8) is depicted in Fig. 2(a). For non-integer η, quires a phase 2π(q/e)(Φ/Φ0), would imply a “semionic” in addition to the usual Landau levels, there also exist exchange phase θ = π/2 for both vacancies and intersti- bound states associated with the extra flux. tials, and a mutual exchange phase θM = −π/2. This The eigenstates ψ0m in the lowest Landau level have a conclusion, however, is incorrect for the following reason. simple form The charge ±e/2 is built up from fermionic degrees of 2 −η m −|z| /4 freedom and thus has attached to it half of an electron ψ0m(z)= Am|z| z e , (9) spin, or more precisely, half of the electron statistical phase, which needs to be taken into account when com- where z = (x+iy)/ℓB is the dimensionless complex coor- 2 m−η −1/2 puting the exchange phase of the composite object. The dinate, Am = [ℓB2π2 Γ(1+m−η)] is the normal- simplest way to arrive at the correct result is to consider ization constant and ℓB = ¯hc/eB the magnetic length. p 4

4.0 b) 1.0 c) a) 4 1.0 3.5 0.5 0.5 0.4 3.0 3 0.8 0.3

2.5 0.0 Q(r) 0.6 δ 0.2 (r) km 2.0 2 (r) ε ρ δρ 1.0 0.1 1.5 0.4 0.0 0 5 10 15 20 1.0 1 0.5 r/l 0.2 B 0.5 0.0 0.0 0.0 0 5 10 15 20 -0.5 -0.25 0 0.25 0.5 0 5 r/l 10 15 20 r/l η B B

FIG. 2: (a) The energy levels (in units ofhω ¯ c) of Hamiltonian (6) as a function of fractional flux η. The numerals denote degeneracy of bound state levels represented by dashed lines. Degeneracy of the extended states (solid lines) is proportional to 2 the area of the system. (b) Charge density ρ(r), in units of e/2πℓB , for N = 100 as a function of distance r from the origin. Solid line represents density ρ0(r) for uniform magnetic field, dashed line has a flux tube carrying −Φ0/2 at the origin. (c) The r ′ ′ ′ excess charge density δρ(r)= ρ0(r) − ρ(r). Inset shows the accumulated excess charge δQ(r) = 2π R0 r dr δρ(r ) in units of e.

If we fill the lowest Landau level with electrons, then the positive x axis. The effect of the gauge transformation many-body wavefunction can be constructed as a Slater is to multiply the eigenstates (9) by e−iηϕ which has the −η −η determinant of ψ0m(zi), where zi is a complex coordinate effect of replacing |z| by z . In this “string gauge” the of the i-th electron. The usual simplifications using the entire phase information is encoded in the wavefunction. 1 properties of Vandermonde determinants then give The many-body wavefunction for a vacancy (ν = − 2 ) 2 located at the complex coordinate w can thus be written −η − P |zi| /4 Ψ({zi})= N |zi| (zi − zj)e i . (10) as Yi i

ηΦ0 If we now carry wa along a contour C that encloses wb, δA → δA − ∇ϕ (12) 2π the latter held static, then Eq. (14) gives the Berry phase where z = |z|eiϕ. The resulting vector potential can be Φ 1 γ(C)= −π − . (17) seen to vanish everywhere, except for a cut along the Φ0 2 5

The first term reflects the Aharonov-Bohm phase as be- points, as illustrated in Fig. 3(a). We are interested in fore. Since the encircling operation can be thought of the half-filled case with the lower band filled and the up- as two consecutive exchanges we must interpret the sec- per band empty. This situation is analogous to the ν =1 2 ond term as twice the exchange phase of the vacancy, continuum case with Hall conductance σxy = e /h dis- θ = π/4. cussed previously. A vacancy in the vortex lattice removes flux Φ0/2 from the plaquettes adjacent to the affected site, modifies its IV. LATTICE MODEL on-site energy ǫi and possibly also those of the nearby sites. While the total flux removed is exactly Φ0/2, the Thus far we have ignored the periodic variation of the change of the on-site energies δǫi depends on the details magnetic field induced by the vortex lattice. A question of the underlying microscopic model for the Zeeman in- thus arises whether this could affect the emergence of teraction. We find that, as expected by the general argu- anyons in realistic systems. To see that this is not the ment, the total charge deficit associated with a vacancy case we now consider a model which describes the limit depends critically on the total flux removed but is inde- of strong field variation and also includes the effect of a pendent of δǫi or the details of the flux distribution. Zeeman interaction. We have solved the model specified by Hamiltonian If the effective Zeeman coupling in a 2DEG is suffi- (18) by exact numerical diagonalization for lattice sizes ciently strong, then, in addition to a quasi-periodic vector up to 50 × 50 sites. We have considered various configu- potential, the electrons also feel a periodic scalar poten- rations of δǫi. In all cases we found that the total charge tial. This effect becomes important in diluted magnetic induced by an interstitial/vacancy is ±e/2, to within ma- semiconductors, such as Ga1−xMnxAs or Cd1−xMnxTe, chine accuracy. This result continues to hold even in the where the effective gyro-magnetic ratio can be of the or- presence of randomness in the on-site energies, as long der ∼ 102 − 103. As argued in Ref. [15], in the limit as the randomness is weak compared to the gap. This is λ ≪ a, electrons in a 2DEG become almost localized illustrated in Fig. 3(b-f). near the maxima of the Zeeman field and can be described by a tight binding model with one-half magnetic flux quantum per unit cell. V. SUMMARY AND CONCLUSIONS This motivates our consideration of a tight binding Hamiltonian Our foregoing discussion shows how two simple and H = − (t eiθij c†c + h.c.)+ ǫ c†c , (18) well understood systems, a type-II superconductor and a ij j i i i i 2-dimensional electron gas in the integer quantum Hall Xij Xi regime, can exhibit some rather unusual properties when † brought into close proximity. These include nontrivial ex- where c creates an electron on a site ri of a square lat- i change statistics and charge fractionalization, properties tice, and tij and ǫi are real hopping amplitudes and on- site energies, respectively. The effect of magnetic field is that are normally thought of as hallmarks of strong corre- included by the Peierls phase factors lation physics. In our proposed device these phenomena occur in what must be characterized as a weakly corre- r 2π j lated system: the many-body wavefunctions of electrons θij = A · dl. (19) Φ0 Zr in both subsystems are conventional Slater determinants i of single-particle states. In the following we consider the case of nearest and next- There are two key ingredients that give rise to the nearest neighbor hopping amplitudes t and t′ respec- above unusual properties: (i) precise flux quantization tively, but note the addition of longer range hoppings in a type-II superconductor and, (ii) incompressibility of does not affect our results. the 2DEG at integer filling ν. The device displayed in In the absence of defects we have ǫi = ǫ =const and the Fig. 1 is designed to project a precisely quantized deficit magnetic flux through each plaquette is Φ0/2. The spec- or surplus of magnetic flux onto the 2DEG, which re- trum of the Hamiltonian (18) consists of two branches, sponds by adjusting its many-body wavefunction to pro- reflecting the fact that the unit cell now contains two duce a localized deficit or surplus of electric charge with lattice sites. In the Landau gauge, A = B0yxˆ with precise value e/2. This phenomenon can be regarded as a 2 B0 =Φ0/2a , the allowed energies read consequence of gauge invariance and is therefore robust against any weak but otherwise arbitrary perturbation 2 2 2 2 2 1/2 Ek = ǫ ± 2t cos kx + cos ky +4γ sin kx sin ky , imposed on the system. (20) A question that we must ask is to what extent are the where k is a wavevector drawn from the first magnetic fractional charges associated with the vacancies and in- Brillouin zone, γ = t′/t, and we have set a = 1. For t′ =0 terstitials legitimate quasiparticle excitations of the sys- the spectrum exhibits two Dirac points at (±π/2,π/2). tem. Surely, from the point of view of a 2DEG at integer Inclusion of next-nearest neighbor hopping breaks the filling, such objects are not natural excitations. Indeed, time reversal symmetry and produces a gap at these two in our device they occur in response to the local change 6

FIG. 3: a) Band structure of the tight binding model on a square lattice with one half flux quantum threading each plaquette ′ ′ and t = 0.05t. b,c) Charge density ρ at half filling with random on-site energies (Gaussian distribution with ∆ǫ = 0.05t ) and a full flux quantum modeling the interstitial piercing the central plaquette. d) Charge density ρ0 with the same realization of randomness but without the extra flux. e,f) The charge density δρ = ρ − ρ0 induced by the interstitial. The induced charge δQ integrates to e/2 to within machine accuracy. in the magnetic flux, which is external to the 2DEG. The tum computation [4, 5, 6] involve execution of braiding key point is that fractional charges bound to defects are and fusion operations on anyons. In the present setup excitations of the entire system in the same sense as frac- these could be facilitated by the fact that our anyons tional charges bound to domain walls are excitations of are permanently bound to physical magnetic fluxes and a dimerized polyacetylene chain [7]. well developed techniques exist to detect and manipulate Fractional charges should be relatively easy to detect these. As already remarked above if the 2DEG in our 5 in a transport experiment in the regime where the longi- device were tuned into the ν = 2 Moore-Read Pfaffian tudinal conductivity of the 2DEG is furnished solely by state then a defect in the vortex lattice would bind a the e/2 defects. Fractional statistics, on the other hand, quasiparticle with non-Abelian exchange statistics. Hav- will be more difficult to establish since this requires an ing a “handle” on such a quasiparticle in the form of the interference experiment; in essence we need a defect to attached magnetic flux could prove useful for realizing quantum-delocalize and interfere with itself after encir- the requisite braiding and fusion operations. cling another defect. As noted recently, non-trivial statis- 5 tics might be easier to detect in the ν = 2 non-Abelian Acknowledgments — The authors are indebted to M. case [29, 30]. Berciu and A. Vishwanath for some key suggestions and Aside from demonstrating the possibility of uncon- wish to thank I. Affleck, D. Fisher, D. Haldane, C. Kallin, ventional phenomena in a weakly interacting system, A. Kitaev, K. Shtengel, D. Scalapino and Z. Tesanovic for our proposed device could allow for the manipulation of stimulating discussions and correspondence. The work anyons in the context of quantum information processing. reported here was supported by NSERC, CIAR and the Indeed, some schemes for fault tolerant topological quan- Killam Foundation.

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