Acta Polytechnica Vol. 50 No. 5/2010 Aharonov-Bohm Effect and the Supersymmetry of Identical Anyons V. Jakubsk´y Abstract We briefly review the relation between the Aharonov-Bohm effect and the dynamical realization of anyons. We show how the particular symmetries of the Aharonov-Bohm model give rise to the (nonlinear) supersymmetry of the two-body system of identical anyons. Keywords: nonlinear supersymmetry, Aharonov-Bohm effect, Anyons. Φ (− sin ϕ, cos ϕ,0) , 1 Aharonov-Bohm effect 2πr More than fifty years ago, Aharonov and Bohm ar- where x1 = r cos ϕ, x2 = r sin ϕ, −π<ϕ≤ π, gued in their seminal paper [1] that the fundamen- and Φ is the flux of the singular magnetic field, tal quantity in a description of the quantum system 2 B3 =Φδ (x1,x2). As we will work mostly in po- is the electromagnetic potential and not the elec- lar coordinates, let us present the explicit form of tromagnetic field. They proposed an experiment in the Hamiltonian in this coordinate system which two beams of electrons are guided around a thin solenoid that is shielded completely from the 1 1 1 H = −∂2 − ∂ + (−i∂ + α)2 + α δ(r)σ , (1.3) electrons. Despite the absence of the magnetic field α r r r r2 ϕ r 3 outside the solenoid, the wave functions are affected 1 α = Φ . by the non-vanishing electromagnetic potential and 2π acquire an additional phase-factor which is mani- 2 1 fested in the altered interference of the beams. The Here we used the identity δ (x1,x2)= δ(r)forthe so-called Aharonov-Bohm (AB) effect has been ob- πr two dimensional Dirac delta function2. served experimentally [2] and has found its applica- To specify the system uniquely, we have to deter- tion in numerous fields of physics. In the present mine the domain of the Hamiltonian. We require the article, we will review its relation to anyons, two- operator (1.3) to act on 2π-periodic functions Ψ(r, ϕ), dimensional particles of exotic statistics. We will i.e. Ψ(r, ϕ +2π)=Ψ(r, ϕ). Using the expansion in present the recent results on the nonlocal symmetries partialwaves,wecanwrite of the AB system and their relation to the supersym- metry of two-body anyon models. Ψ(r, ϕ)= eijϕf (r). (1.4) Let us consider a spin−1/2particlewhichismov- j ing in a plane. The plane is punctured perpendicu- j larly in the origin by an infinitely thin solenoid. The The functions fj(r) should be locally square- solenoid is impenetrable for the particle. Hence, the integrable (i.e. fj(r) should be square integrable on origin is effectively removed from the space where the any finite interval). The partial waves fj(r) are reg- particle lives. The Pauli Hamiltonian of the system ular at the origin up to the exception specified by the 1 acquires the following simple form following boundary condition 1 e¯h H P2 − (1 + eiγ )2−αΓ(1 − α)r−1+αe−iϕ = j B3 σ3 , (1.1) ∼ 2m 2mc lim Ψ − − (1.5) j=1,2 r→0+ (1 − eiγ )2 1+αΓ(α)r α e where Pj = −i¯h∂j − Aj, B3 = ∂1A2 − ∂2A1.The where parameter γ can acquire two discrete values 0 c non-vanishing electromagnetic potential in the sym- and π. The boundary condition (1.5) is related to metric gauge reads the self-adjoint extensions of the Hamiltonian. Let us note that the boundary condition (1.5) just fixes Φ − x2 x1 two self-adjoint extensions (one for γ = 0, the second A = 2 2 , 2 2 , 0 = (1.2) H 2π x1 + x2 x1 + x2 one for γ = π) of the formal operator α that are 1We set m =1/2, ¯h = c = −e = 1 from now on. 2 In fact, the Dirac delta term in the Hamiltonian is quite formal. It can be omitted when the domain of Hα is specified correctly. 46 Acta Polytechnica Vol. 50 No. 5/2010 compatible with the existence of N =2supersym- of Q, Q˜ and W on the kets |E,l,s is illustrated in metry, see [3]. To keep our presentation as simple as Fig. 1. possible, we fix from now on γ =0. (1.6) We modify the actual notation to indicate the domain H →H0 of the Hamiltonian, i.e. we will write α α.For readers who are eager for a more extensive analysis of the problem we recommend [3] for reference. H0 The Hamiltonian α commutes with the angular momentum operator J = −i∂ϕ + α and the spin pro- 1 jection s3 = σ3. Hence, one can find the vectors Fig. 1: The action of operator W (thick dotted arrows) on 2 | | |E,l,s such that the states E,2l, 1/2 and E,2l +1, 1/2 as a sequential action of Q˜ (solid arrows) and Q (thindottedarrows). H0 | | Black squares represent the eigenstates |E,l,s with cor- α E,l,s = E E,l,s , J|E,l,s =(l + α)|E,l,s, (1.7) responding values of l and s | | s3 E,l,s = s E,l,s . 2Anyons We define the following two additional integrals of Quantum theory has classified particles into two dis- motion joint families; there are bosons with integer spin and fermions with half-integer spin. The wave functions Q = σ P + σ P = q σ + q−σ− , 1 1 2 2 + + of indistinguishable bosons or fermions reflect the ∓iϕ 1 q± = −ie ∂r ± (−i∂ϕ + α) , (1.8) specific statistical properties of the particles. When r we exchange two bosons, the wave function remains 1 the same. When we exchange two fermions, the cor- σ± = (σ1 ± iσ2) 2 responding wave function changes the sign. The wave functions respect either Bose-Einstein or Fermi-Dirac and statistics in this way. However, when one makes a quantum system be Q˜ = P11 + iRσ3P2, two-dimensional, there emerges an alternative to the RrR = r, (1.9) classification. RϕR = ϕ + π, As predicted by Wilczek [5], there can exist ex- otic particles in two-dimensional space that are called where 1 is a unit matrix and anyons. Anyons interpolate between bosons and fermions in the sense that when we exchange two P + iRP = q Π + q−Π− , (1.10) 1 2 + + of them in the system, the associated wave function 1 Π± = (1 ± R) . acquires a multiplicative phase-factor of unit ampli- 2 tude but distinct from ±1. The prediction of these We can make a qualitative analysis of how these op- particles is physically relevant for various condensed erators act on the wave functions |E,l,s just by ob- matter systems where the dynamics is effectively two- serving their explicit form. For instance, we have dimensional. Wilczek proposed a simple dynamical realization Q|E,l,1/2∼|E,l +1, −1/2, (1.11) of anyons with the use of “composite” particles. Let Q˜|E,2l, s∼|E,2l − 2s, s. us explain the idea on a simple model of two identi- cal particles [6]. Take either two bosons or fermions. Hence, neither Q nor Q˜ commutes with the angu- Then, glue each of the particles together with a mag- lar momentum J and the parity R. However, the netic vortex, i.e. with infinitely thin solenoids of the operator Q˜ preserves spin of the wave functions, i.e. same magnetic flux α. As a result, we get two identi- [Q,˜ s3]=0. cal composite particles. Each particle can “see” just The operators Q and Q˜ are related by nonlocal the potential generated by the solenoid of the other unitary transformation, see [4]. In addition, we can particle. The Hamiltonian corresponding to this two- define body system has the following form W = Q Q˜ = QQ.˜ (1.12) 2 This operator alters both the angular momentum and 2 Hany =2 (pI − aI (r)) . (2.1) the spin of the wave functions. The explicit action I=1 47 Acta Polytechnica Vol. 50 No. 5/2010 where pI = −i∂/∂xI , r = x1 − x2 and the index formalism, the Hamiltonian reads I ∈{1, 2} labels the individual particles. The poten- Hγ=0 0 0 AB AB tial α =diag(Hα,+,Hα,−,Hα,+,Hα,−) , l k k 1 kl r 0 0 a (r)=−a (r)= α (2.2) H ± = H Σ+Π±, (2.5) 1 2 2 r 2 α, α AB H0 encodes the “statistical” interaction of the particles. Hα,± = αΣ−Π± . In this sense, we call aI the statistical potential. Let us make a few comments on the elements of When we write the Hamiltonian in center-of-the-mass AB coordinates, the relative motion of the particles is (2.5). Consider Hα,+ in more detail first. It acts on governed by the effective Hamiltonian the wave functions that are periodic in π. Hence, it can be interpreted as the Hamiltonian of the relative − 2 − 1 1 − 2 motion of two identical anyons based on bosons. Its Hrel = ∂r ∂r + ( i∂ϕ + α) , (2.3) r r2 wave functions are regular at r → 0, which can be where r is the distance between the particles and ϕ interpreted as a consequence of a hard-core interac- measures their relative angle. tion between the anyons. It is worth noting that the AB The Hamiltonian (2.3) manifests the relation be- system represented by Hα,+ coincides with the sys- 0 tween the two-body model of identical anyons and tem represented by Hα,+. Indeed, the Hamiltonians H0 the AB system; formally,itcoincideswith α up to coincide not only formally but in their domains as the irrelevant Dirac delta term.