ap-5-10.dvi Acta Polytechnica Vol. 50 No. 5/2010 Aharonov-Bohm Effect and the Supersymmetry of Identical Anyons V. Jakubský Abstract We briefly review the relation between the Aharonov-Bohm effect and the dynamical realization of anyons. We show how the particular symmetries of theAharonov-Bohmmodel give rise to the (nonlinear) supersymmetry of the two-body system of identical anyons. Keywords: nonlinear supersymmetry, Aharonov-Bohm effect, Anyons. 1 Aharonov-Bohm effect More than fifty years ago, Aharonov and Bohm ar- gued in their seminal paper [1] that the fundamen- tal quantity in a description of the quantum system is the electromagnetic potential and not the elec- tromagnetic field. They proposed an experiment in which two beams of electrons are guided around a thin solenoid that is shielded completely from the electrons. Despite the absence of the magnetic field outside the solenoid, the wave functions are affected by the non-vanishing electromagnetic potential and acquire an additional phase-factor which is mani- fested in the altered interference of the beams. The so-called Aharonov-Bohm (AB) effect has been ob- served experimentally [2] and has found its applica- tion in numerous fields of physics. In the present article, we will review its relation to anyons, two- dimensional particles of exotic statistics. We will present the recent results on the nonlocal symmetries of theAB systemand their relation to the supersym- metry of two-body anyon models. Let us consider a spin−1/2particlewhich ismov- ing in a plane. The plane is punctured perpendicu- larly in the origin by an infinitely thin solenoid. The solenoid is impenetrable for the particle. Hence, the origin is effectively removed fromthe spacewhere the particle lives. The Pauli Hamiltonian of the system acquires the following simple form 1 H = 1 2m ∑ j=1,2 P2j − eh̄ 2mc B3 σ3 , (1.1) where Pj = −ih̄∂j − e c Aj, B3 = ∂1A2 − ∂2A1. The non-vanishing electromagnetic potential in the sym- metric gauge reads �A = Φ 2π ( − x2 x21 + x 2 2 , x1 x21 + x 2 2 ,0 ) = (1.2) Φ 2πr (−sinϕ , cosϕ ,0) , where x1 = rcosϕ, x2 = rsinϕ, −π < ϕ ≤ π, and Φ is the flux of the singular magnetic field, B3 = Φδ 2(x1, x2). As we will work mostly in po- lar coordinates, let us present the explicit form of the Hamiltonian in this coordinate system Hα = −∂2r − 1 r ∂r + 1 r2 (−i∂ϕ + α)2 + α 1 r δ(r)σ3,(1.3) α = 1 2π Φ . Hereweused the identity δ2(x1, x2)= 1 πr δ(r) for the two dimensional Dirac delta function2. To specify the system uniquely, we have to deter- mine the domain of theHamiltonian. We require the operator (1.3) toacton2π-periodic functionsΨ(r, ϕ), i.e. Ψ(r, ϕ +2π) = Ψ(r, ϕ). Using the expansion in partial waves, we can write Ψ(r, ϕ)= ∑ j eijϕfj(r). (1.4) The functions fj(r) should be locally square- integrable (i.e. fj(r) should be square integrable on any finite interval). The partial waves fj(r) are reg- ular at the originup to the exception specifiedby the following boundary condition lim r→0+ Ψ ∼ ( (1+ eiγ)2−αΓ(1 − α)r−1+αe−iϕ (1 − eiγ)2−1+αΓ(α)r−α ) (1.5) where parameter γ can acquire two discrete values 0 and π. The boundary condition (1.5) is related to the self-adjoint extensions of the Hamiltonian. Let us note that the boundary condition (1.5) just fixes two self-adjoint extensions (one for γ =0, the second one for γ = π) of the formal operator Hα that are 1We set m = 1/2, h̄ = c = −e = 1 from now on. 2In fact, theDirac delta term in the Hamiltonian is quite formal. It can be omittedwhen the domain of Hα is specified correctly. 46 Acta Polytechnica Vol. 50 No. 5/2010 compatible with the existence of N = 2 supersym- metry, see [3]. To keep our presentation as simple as possible, we fix from now on γ =0. (1.6) Wemodify theactualnotation to indicate thedomain of the Hamiltonian, i.e. wewill write Hα → H0α. For readers who are eager for a more extensive analysis of the problem we recommend [3] for reference. The Hamiltonian H0α commutes with the angular momentum operator J = −i∂ϕ +α and the spin pro- jection s3 = 1 2 σ3. Hence, one can find the vectors |E, l, s〉 such that H0α|E, l, s〉 = E|E, l, s〉, J |E, l, s〉 = (l + α)|E, l, s〉, (1.7) s3|E, l, s〉 = s|E, l, s〉. We define the following two additional integrals of motion Q = σ1P1 + σ2P2 = q+σ+ + q−σ− , q± = −ie∓iϕ ( ∂r ± 1 r (−i∂ϕ + α) ) , (1.8) σ± = 1 2 (σ1 ± iσ2) and Q̃ = P11+ iRσ3P2, RrR = r, (1.9) RϕR = ϕ + π, where 1 is a unit matrix and P1 + iRP2 = q+Π+ + q−Π− , (1.10) Π± = 1 2 (1 ± R) . We can make a qualitative analysis of how these op- erators act on the wave functions |E, l, s〉 just by ob- serving their explicit form. For instance, we have Q|E, l,1/2〉 ∼ |E, l +1, −1/2〉, (1.11) Q̃|E,2l, s〉 ∼ |E,2l − 2s, s〉. Hence, neither Q nor Q̃ commutes with the angu- lar momentum J and the parity R. However, the operator Q̃ preserves spin of the wave functions, i.e. [Q̃, s3] = 0. The operators Q and Q̃ are related by nonlocal unitary transformation, see [4]. In addition, we can define W = Q Q̃ = Q̃ Q. (1.12) This operatoraltersboth theangularmomentumand the spin of the wave functions. The explicit action of Q, Q̃ and W on the kets |E, l, s〉 is illustrated in Fig. 1. Fig. 1: Theaction of operator W (thickdottedarrows) on the states |E,2l,1/2〉 and |E,2l +1,1/2〉 as a sequential action of Q̃ (solid arrows) and Q (thin dotted arrows). Black squares represent the eigenstates |E, l, s〉 with cor- responding values of l and s 2 Anyons Quantumtheory has classifiedparticles into twodis- joint families; there are bosons with integer spin and fermions with half-integer spin. The wave functions of indistinguishable bosons or fermions reflect the specific statistical properties of the particles. When we exchange two bosons, the wave function remains the same. When we exchange two fermions, the cor- respondingwave function changes the sign. Thewave functions respect eitherBose-EinsteinorFermi-Dirac statistics in this way. However, when one makes a quantum system be two-dimensional, there emerges an alternative to the classification. As predicted by Wilczek [5], there can exist ex- otic particles in two-dimensional space that are called anyons. Anyons interpolate between bosons and fermions in the sense that when we exchange two of them in the system, the associated wave function acquires a multiplicative phase-factor of unit ampli- tude but distinct from ±1. The prediction of these particles is physically relevant for various condensed matter systemswhere thedynamics is effectively two- dimensional. Wilczek proposed a simple dynamical realization of anyons with the use of “composite” particles. Let us explain the idea on a simple model of two identi- cal particles [6]. Take either two bosons or fermions. Then, glue each of the particles togetherwith amag- netic vortex, i.e. with infinitely thin solenoids of the samemagnetic flux α. As a result, we get two identi- cal composite particles. Each particle can “see” just the potential generated by the solenoid of the other particle. TheHamiltonian corresponding to this two- body system has the following form Hany =2 2∑ I=1 (�pI − �aI(�r)) 2 . (2.1) 47 Acta Polytechnica Vol. 50 No. 5/2010 where �pI = −i∂/∂�xI, �r = �x1 − �x2 and the index I ∈ {1,2} labels the individual particles. The poten- tial ak1(�r)= −a k 2(�r)= 1 2 α�kl rl �r 2 (2.2) encodes the “statistical” interaction of the particles. In this sense, we call �aI the statistical potential. Whenwewrite theHamiltonian incenter-of-the-mass coordinates, the relative motion of the particles is governed by the effective Hamiltonian Hrel = −∂2r − 1 r ∂r + 1 r2 (−i∂ϕ + α)2 , (2.3) where r is the distance between the particles and ϕ measures their relative angle. The Hamiltonian (2.3) manifests the relation be- tween the two-body model of identical anyons and the AB system; formally, it coincides with H0α up to the irrelevantDirac delta term. However, its domain of definition is quite different. When anyons (com- posite particles) are composed of bosons, the wave function has to be invariant under the substitution ϕ → ϕ + π that corresponds to the exchange of the particles. When anyons are composed of fermions, the wave function has to change the sign after the substitution. Hence, the wave functions are of two types ψα(r, ϕ)= ∑ l eilϕfα,l(r), (2.4) l ∈ { 2Z for anyons based on bosons, 2Z +1 for anyons based on fermions. We shall explain how the considered model ex- plains anyons as the interpolation between bosons and fermions. We can transform the system by a unitary mapping U = eiϕα and describe alterna- tively the system of two identical anyons by the Hamiltonian H̃rel = U HrelU −1 = −∂2r − 1/r∂r + (−i∂ϕ)2/r2. It coincides with the energy operator of the free motion. The simplicity of the Hamilto- nian is traded for the additional gauge factor that ap- pears in the wave functions, ψ̃α(r, ϕ) = U ψα(r, ϕ) = eiϕα ∑ l eilϕfα,l(r). The wave functions ψ̃α(r, ϕ) ac- quire the phase eiπα after the substitution ϕ → ϕ+π and, hence, interpolate between the values corre- sponding toBose-EinsteinandFermi-Dirac statistics. We are ready to reconsider the AB system and its symmetries in the framework of identi- cal anyons. The Hamiltonian H0α can be rewrit- ten as a direct sum with subsystems of fixed value of spin s3 and parity R. It is convenient to use the notation that reflects the decomposi- tion of the wave functions into these subspaces, Ψ̃= (ΨΣ+Π+,ΨΣ+Π−,ΨΣ− Π+,ΨΣ− Π−) T whe- re Σ± = 1 2 (1 ± σ3) and Π± = 1 2 (1 ± R). In this formalism, the Hamiltonian reads Hγ=0α = diag(H 0 α,+, H 0 α,−, H AB α,+, H AB α,−) , H0α,± = H 0 αΣ+Π±, (2.5) HABα,± = H 0 αΣ−Π± . Let us make a few comments on the elements of (2.5). Consider HABα,+ in more detail first. It acts on the wave functions that are periodic in π. Hence, it can be interpreted as the Hamiltonian of the relative motion of two identical anyons based on bosons. Its wave functions are regular at r → 0, which can be interpreted as a consequence of a hard-core interac- tion between the anyons. It is worth noting that the system represented by HABα,+ coincides with the sys- tem represented by H0α,+. Indeed, the Hamiltonians coincide not only formally but in their domains as well (there are no singularwave functions in their do- mains, see (1.5)). Hence, we canwrite H0α,+ = H AB α,+. The operators HABα,− and H 0 α,− describe the sys- tems of two identical anyons based on fermions. The operator HABα,− prescribes hard-core interaction be- tween anyons. By contrast, the system described by H0α,− allows singular wave functions. It can be un- derstood as a consequence of a nontrivial contact in- teraction between the composite particles. The integrals of motion Q, Q̃ and W shall be rewritten in the 4 × 4-matrix formalism. They read explicitly Q = ⎛ ⎜⎜⎜⎜⎝ 0 0 0 q+ 0 0 q+ 0 0 q− 0 0 q− 0 0 0 ⎞ ⎟⎟⎟⎟⎠ , Q̃ = ⎛ ⎜⎜⎜⎜⎝ 0 q− 0 0 q+ 0 0 0 0 0 0 q+ 0 0 q− 0 ⎞ ⎟⎟⎟⎟⎠ , (2.6) W = ⎛ ⎜⎜⎜⎜⎝ 0 0 q+q− 0 0 0 0 q2+ q−q+ 0 0 0 0 q2− 0 0 ⎞ ⎟⎟⎟⎟⎠ , where q± was defined in (1.8). Substituting (2.5) and (2.6) into the relations [Q, Hγα] = 0, [Q̃, H γ α] = 0 and [W, Hγα] = 0we get the following set of indepen- dent intertwining relations H0+q− = q−H 0 − , q+H 0 + = H 0 −q+ , (2.7) H0+q+ = q+H AB − , q−H 0 + = H AB − q− , (2.8) HAB− q 2 − = q 2 −H 0 − , q 2 +H AB − = H 0 −q 2 − . (2.9) 48 Acta Polytechnica Vol. 50 No. 5/2010 Let us focus on the first set (2.7). They can be rewritten as [ q(1)a ,h (1) ] =0, (2.10) where we used the operators h(1) = ( H0+ 0 0 H0− ) , q(1)1 = ( 0 q− q+ 0 ) , (2.11) q(1)2 = i ( 0 −q− q+ 0 ) . The operators (2.11) close for N = 2 supersym- metry3. Indeed, they satisfy the commutation re- lation{ q(1)a ,q (1) b } =2δa,bh (1) , a, b =1,2 . (2.13) Hence, operatorh(1) canbe understoodas the super- extended Hamiltonian of the two-body anyonic sys- tems. The system represented by H0+ is based on bosons (thewave functions are π-periodic), the other system (represented by H0−) is based on fermions with nontrivial contact interaction. The super- charges q(1)a provide the mapping between these two systems. They exchange the bosons with fermions within the composite particles. Besides, they switch on (off) the nontrivial contact interaction between the anyons. The relations (2.8) can be analyzed in the same vein, giving rise to the N =2 supersymmetric system of the pair of two-body anyonicmodels. For the sake of completeness, we present the corresponding oper- ators and the algebraic relations of the superalgebra h(2) = ( H0+ 0 0 HAB− ) , q(2)1 = ( 0 q+ q− 0 ) , (2.14) q(2)2 = i ( 0 −q+ q− 0 ) , [ q(2)a ,h (2) ] = 0,{ q(2)a ,q (2) b } = 2δa,bh (2) , (2.15) a, b = 1,2 . The only difference appears in the contact interac- tion between the anyons. This time, the hard-core interaction appears in both systems (neither H0+ nor HAB− has singular wave functions in its domain). A qualitatively different situation occurs in the last case (2.9). The intertwining relations define the N =2nonlinear supersymmetry4 representedby the operators h(3) = ( H0− 0 0 HAB− ) , q(3)2 = ( 0 q2+ q2− 0 ) , (2.16) q(3)2 = i ( 0 −q2+ q2− 0 ) . They satisfy the following relations [ q(3)a ,h (3) ] = 0,{ q(3)a ,q (3) b } = 2δab ( h(3) )2 , (2.17) a, b = 1,2 . The supercharges q(3) alter the contact interaction between the anyons (hard-core in HAB− to nontrivial in H0− and vice versa) but do not alter the nature of the composite particles. 3 Comments In this paper, we have utilized the intimate relation between the Aharonov-Bohmmodel and the dynam- ical realization of anyons in order to construct three different N = 2 supersymmetric systems of identi- cal anyons. The origin of the supersymmetry can be attributed to the symmetries Q, Q̃ and W of the 3Let us suppose that we have a quantum mechanical system described by aHamiltonian H. There are N additional observables, represented by the operators Qa, a ∈ {1, . . . , N}. It is said that the system has supersymmetry, as long as operators Qa together with the Hamiltonian satisfy the following algebraic relations {Qa, Qb} ∼ Hδab (2.12) If this is the case, operators Qa are called supercharges. As a direct consequence of (2.12), they satisfy the relations Q2j ∼ H, [Qj, H] = 0. 4The system has nonlinear supersymmetry when the supercharges Qa, a ∈ {1, . . . , N} satisfy the generalized anticommutation relation [7] {Qa, Qb} = δabf(H) where f(H) is a function of the Hamiltonian H. Usually, f(H) is considered to be a higher-order polynomial. 49 Acta Polytechnica Vol. 50 No. 5/2010 spin−1/2 particle in the field of a magnetic vortex. Reduction of these operators into the specific sub- spaces (of the fixed value of spin and parity) gave rise to supersymmetry of the anyon systems. A simi- lar constructionwas recently employed in the case of the reflectionless Poschl-Teller system [8]. Its super- symmetric structure originated from the geometrical symmetries of a higher-dimensional system living in AdS2 space after the reduction to the subspaceswith a fixed angular momentum value. Acknowledgement The author was supported by grant LC06002 of the Ministry of Education, Youth and Sports of the Czech Republic. References [1] Aharonov, Y., Bohm, D.: Significance of electro- magnetic potentials in the quantumtheory,Phys. Rev. 115, 485 (1959). [2] Endo, J., Kawasaki, T., Matsuda, T., Osak- abe, N., Tonomura, A., Yamada, H., Yano, S.: Evidence for Aharonov-Bohm effect with mag- neticfield completely shielded fromelectronwave, Phys. Rev. Lett. 56, 792 (1986); Endo, J., Kawasaki, T., Matsuda, T., Osak- abe, N., Tonomura, A., Yamada, H., Yano, S.: Experimental confirmationofAharonov-Bohmef- fect using a toroidal magnetic field confined by a superconductor, Phys. Rev. A 34, 815 (1986). [3] Correa, F., Falomir, H., Jakubský, V., Plyush- chay, M. S.: Supersymmetries of the spin−1/2 particle in the field of magnetic vortex, and anyons, arXiv:1003.1434 [hep-th]; Correa, F., Falomir, H., Jakubský, V., Plyush- chay,M. S.: Hidden superconformal symmetry of spinless Aharonov-Bohm system, J. Phys. A 43, 075202 (2010). [4] Jakubský, V., Nieto, L. M., Plyushchay, M. S.: The origin of the hidden supersymmetry, arXiv:1004.5489 [hep-th]. [5] Wilczek, F.: Magnetic flux, angular momentum, and statistics, Phys. Rev. Lett. 48, 1144 (1982); Quantum mechanics of fractional spin particles, Phys. Rev. Lett. 49, 957 (1982). [6] Wilczek, F.: Fractional statistics and anyon superconductivity, World Scientific, Singapore (1990); Khare,A.: Fractional statistics andquantumthe- ory, World Scientific, Singapore (1997). [7] Andrianov,A. A., Ioffe, M. V., Spiridonov,V. P.: Higher derivative supersymmetry and theWitten index,Phys. Lett. A 174, 273 (1993), [arXiv:hep- th/9303005]. [8] Correa, F., Jakubský, V., Plyushchay, M. S.: Aharonov-Bohmeffect onAdS2 andnonlinear su- persymmetry of reflectionless Poschl-Teller sys- tem, Annals Phys. 324, 1078 (2009), [arXiv:0809.2854 [hep-th]]. Ing. Vít Jakubský, Ph.D. E-mail: jakubsky@ujf.cas.cz Nuclear Physics Institute of the ASCR, v. v. i. Řež 130, 250 68 Řež, Czech Republic 50