PHYSICAL REVIEW LETTERS 123, 164801 (2019) Relativistic Anyon Beam: Construction and Properties Joydeep Majhi, Subir Ghosh, and Santanu K. Maiti* Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata-700 108, India (Received 13 February 2019; revised manuscript received 9 August 2019; published 17 October 2019) Motivated by recent interest in photon and electron vortex beams, we propose the construction of a relativistic anyon beam. Following Jackiw and Nair [R. Jackiw and V. P. Nair, Phys. Rev. D 43, 1933 (1991).] we derive an explicit form of the relativistic plane wave solution of a single anyon. Subsequently we construct the planar anyon beam by superposing these solutions. Explicit expressions for the conserved anyon current are derived. Finally, we provide expressions for the anyon beam current using the superposed waves and discuss its properties. We also comment on the possibility of laboratory construction of an anyon beam. DOI: 10.1103/PhysRevLett.123.164801 We propose construction of a relativistic beam of anyons beam [1,2]. But, in 2D a monoenergetic wave beam will in a plane. Anyons are planar excitations with arbitrary necessary have dispersion in momentum along propagation spin and statistics. The procedure is similar to three- direction and hence cannot be completely delocalized. The dimensional (3D) optical (spin 1) and electron (spin 1=2) delocalization along propagation direction will be more for vortex beams. a narrower angle of superposition. Vortex beams, of recent interest, carrying intrinsic orbital Apart from earlier interest [6], graphene also involves angular momentum are nondiffracting wave packets in anyons [8]. Non-Abelian anyons [9] are touted as theo- motion. In 3D, vortex beams for photons and electrons were retical building blocks for topological fault-tolerant quan- proposed by Durnin et al. [1] and by Bliokh et al. [2], tum computers [10]. An exact chiral spin liquid with respectively (see [3]). Properties and experimental signatures non-Abelian anyons has been reported [11]. The direct of twisted optical and electron vortex beams were studied observational status of anyons has shown promising devel- respectively in [4] (for spinless electron [2])and[5] (spinor opment [12]. electron [2]). Wave packets were constructed out of free plane The minimal field theoretic and relativistic model of a wave, orbital angular momentum solutions having vorticity. single anyon was constructed by Jackiw and Nair (JN) [13]. In 3D, the nontrivial Lie algebra of rotation generators along JN anyon is the most suitable one that serves our purpose with finite-dimensional representations of angular momen- since, being first order in spacetime derivatives, it closely tum eigenstates restrict the angular momentum to quantize in resembles the Dirac equation used by [2]. A major differ- units of ℏ=2. The absence of such an algebra in two ence is that unitary representation for an arbitrary spin JN dimensions (2D) allows excitations (anyons [6,7])with anyon requires an infinite component wave function [14] arbitrary spin and statistics with infinite-dimensional repre- (to maintain covariance), along with subsidiary constraints sentations. Here, we construct a directed transversely local- that restrict the number of independent components to a ized anyon wave packet or wave beam in 2D. single one [13]. Geometry imposes a qualitative difference between We propose (i) an explicit form of the single anyon monoenergetic wave packet and wave beam both in 3D solution of the JN anyon equation [13] (which is a new and 2D. In 3D the wave packet is described by three result), (ii) an anyon conserved current (again a new result), discrete quantum numbers as it is localized in three (iii) an anyon wave packet in 2D, (iv) an anyon wave packet directions whereas ideally an energy dispersionless wave in the conserved current that constitutes the anyon beam, packet where wave components having identical momen- (iv) anyon charge and current densities for the anyon beam tum along the direction of propagation will be completely numerically since closed analytic expressions could not be delocalized along that direction and will constitute a obtained, (v) a possible laboratory setup for observing anyon beams, and (vi) finally, future prospects. The new results [(i) and (ii)] lead us to our principal outputs [(iii) and (iv) and hopefully (v)]. Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. (i) Jackiw-Nair anyon equation and its solution: We start Further distribution of this work must maintain attribution to with a familiar system, free spin one particle in 2 þ 1 the author(s) and the published article’s title, journal citation, dimensions [13]. The dynamical equation in coordinate and 3 and DOI. Funded by SCOAP . momentum space (i∂a ¼ pa)isgivenby 0031-9007=19=123(16)=164801(6) 164801-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 123, 164801 (2019) abc b a b a ∂aϵ Fc Æ mF ¼ 0; ðpjÞbF þ msF ¼ 0: ð1Þ are in Supplemental Material A [18]. The free anyon solution is Fa a 0 The solution of the three-vector , ¼ , 1, 2 [we use sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Minkowski metric ημν ¼ diagð1; −1; −1Þ] is given by 2m λ Γ 2λ n px ipy n aþ ð þ Þ þ fn ¼ 22 3 2 33 E þ m n!Γð2λÞ E þ m 0 E þ m rffiffiffiffi 1 66 7 px þ ipy 6 77 m Fa p e−ipx; FaðpÞ¼pffiffiffi 44 1 5 þ 4 px 55 : ð2Þ × ð Þ ð7Þ 2 mðE þ mÞ E i py a where F ðpÞ is the same as the spin 1 case defined in (2). This is our primary result that we exploit to construct the Fa The same can also be expressed as a Lorentz boosted wave packet and subsequent anyon beam. form, (ii) Conserved current for a single anyon: Next we 2 pffiffiffi 3 derive the conserved probability current for the anyon 2 00 μ ðs¼1−λÞ s ∂ jμ 0, where j is the probability density. Since a a b c b 1 6 7 ¼ 0 F ðpÞ¼B ðpÞNcF ðpÞ;Nc ¼ pffiffiffi 4 01i 5; b 0 2 the anyon model of [13] is an extension of the spin 1 case 0 −ii we can take a cue from the latter where the conservation ∂μjðs¼1Þ 0 ð3Þ law μ ¼ reads μ ðs¼1Þ 0 0† 0 x† x y† y where the boost is expressed by the generators ja, ∂ jμ ¼ ∂ ½F F þ F F þ F F − ∂x F0†Fx Fx†F0 − ∂y F0†Fy Fy†F0 a ½ þ ½ þ BðpÞ¼eiΩaðpÞj ; ½BðpÞa ¼½B−1ðpÞa b b 0: a a a ¼ ð8Þ a ðp þ η mÞðpb þ ηbmÞ 2p ηb ¼ δb − þ ; ð4Þ mðpη þ mÞ m Considering the Fourier transform of the anyon equation of a motion (5) in position space, a long calculation yields the with ηa ¼ð1; 0; 0Þ. It is straightforward to check that F ð1−λÞ conserved free (single) anyon current jμ , describes a spin one particle (s ¼ 1) of mass m. This construction has been extended in an elegant way to X∞ 0 0† 0 x† x y† y the JN anyon equation [13] to describe an anyon of ∂ ½ðfn fn þ fn fn þ fn fnÞ arbitrary spin s ¼ 1 − λ, whose dynamics in momentum n¼0 space is given by, − i fy†K0 fx − fx†K0 fy ð n nn0 n0 n nn0 n0 Þ ∞ a0 a X P K j f msf 0; D f 0; x 0† x x† 0 y† x x x† x y · ð þ Þana0n0 n0 þ an ¼ ð a Þn ¼ ð5Þ − ∂ f f f f − i f K f − f K f ½ð n n þ n nÞ ð n nn0 n0 n nn0 n0 Þ n¼0 Da ϵa PbKc ( nn0 ¼ bc nn0 ) where the second equation is the X∞ λ 0 − ∂y f0†fy fy†f0 − i fy†Ky fx − fx†Ky fy subsidiary (constraint) relation. For ¼ the anyon ½ð n n þ n nÞ ð n nn0 n0 n nn0 n0 Þ reduces to the spin one model discussed earlier [13]. n¼0 a a The actions of j , K are given by ¼ 0; ð9Þ PKana0n0 ¼ PKnn0 δaa0 ;Pjana0n0 ¼ Pjaa0 δnn0 ; where we have explicitly shown the summation over n, the a a a a ð1−λÞ K λ;nK λ;n0 ; j iϵ : λ 0 jμ nn0 ¼h j j i ð Þa0a00 ¼ a0a00 anyonic index. For ¼ the current reduces to the ð1Þ spin 1 current jμ of (8). A nontrivial check of the con- Generalizing the spin 1 case (3), the free anyon solution sistency of the expressions for the anyon current (9) is to a is formally given by [13] substitute fn from (7) to yield aðÞ a b cðÞ 0 x x fn ðpÞ¼Bn0ðpÞBbðpÞNcf ðpÞ; ð6Þ j ¼ð1 − λÞE=m; j ¼ð1 − λÞp =m; pμ pμ B p Ba p λ jy 1 − λ py=m → jμ 1 − λ s : 10 where nn0 ð Þ, bð Þ are the spin and spin 1 represen- ¼ð Þ ¼ð Þ m ¼ m ð Þ b tations of the boost transformation respectively and Nc is the same numerical matrix as in (3). For computational details see Supplemental Material B [18]. Exploiting coherent state formalism for SUð1; 1Þ [15,16] (iii) Anyon wave packet: Our aim is to construct the along with the SUð1; 1Þ ∼ SOð2; 1Þ connection, we have anyon current, not for a single anyon as done above, but for constructed an explicit form of Bn0ðpÞ [in a matrix repre- an anyon wave packet that can be amenable to experimental sentation of Ka [13] (see, e.g., [17])].