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 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])]. Computational details verification. Let us now construct the anyon wave packet

164801-2 PHYSICAL REVIEW LETTERS 123, 164801 (2019) that we want to move towards, say, þx direction. Since we have superposed plane waves, later figures reveal that the current density has a sharply peaked profile with the y com- ponent of current density having a comparatively reduced value. Note an important difference in geometry between our construction and that of the three-dimensional vortex beam [2]. In the latter case the free monoenergetic plane wave solutions (to be superposed) are distributed over the surface of a right circular cone with identical momentum amplitude in the propagation direction. However, for our anyon wave packet, in a planar geometry the above is not possible. Instead we use the superposition scheme where the azimu- thal angle ϕ of momenta of the plane wave is integrated symmetrically from ϕ0 ¼ −π=2 to ϕ0 ¼þπ=2.InFig.1 we λ have shown profiles for charge density of anyon beam, J0, for λ ¼ 0.6 → s ¼ 1 − λ ¼ 0.4 for ϕ0 ¼π=2. Hence, considering superpositions of anyon plane waves FIG. 1. (a) J0 − θ for different values of ρ. (b) Polar plot for J0 θ ρ J0 with fixed spinpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis ¼ 1 − λ, fixed energy E and fixed kinetic with for fixed where the radial distance is the magnitude of . J0 J0 λ 0 6 energy p ¼ ðpxÞ2 þðpyÞ2, for the special case of inte- (c) Density plot of . (d) 3D plot of . We set ¼ . and the a integration ranges from −π=2 to þπ=2. gration limits ϕ0 ¼π=2, the superposition Fn appears as

2 3 p einαeiα 6 m 7 Z 6   7 π=2 p2 a 6 1 αeiα einα 7 i x α y α i x α−y α F p; x A 6 þ M cos 7 e ð cos þ sin Þ e ð cos sin Þ dα; nð Þ¼ n 6 7ð þ Þ ð11Þ −π=24   5 p2 iα inα i þ M sin αe e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffi 1 λ n where An ¼ 2 ð2m=E þ mÞ ½Γð2λ þ nÞ=n!Γð2λÞ m=Eðp=E þ mÞ , x ≡ px, y ≡ py, and M ¼ mðE þ mÞ. The expression is symmetric separately under x → −x and y → −y. Conserved current for the anyon wave packet: The final analytical task is to substitute the anyon wave packet (11) in the a expression of the anyon current (9). Since the current components are quadratic in the packet wave functions Fn, the final expressions are quite long and involved. We have shown only the expression for probability density J0 and have relegated Jx and Jy to Supplemental Material C [18] together with a few computational steps. The cherished form of anyon beam probability density, in polar coordinates x ¼ ρ cos θ, y ¼ ρ sin θ,is   Z Z 2m 2λ m π=2 π=2 J0ðρ; θÞ¼ dα dβ½e−2iρp sin½θ−ðαþβÞ=2 sin½ðα−βÞ=2 þ e−2iρpfsin½θ−ðα−βÞ=2 sin½ðαþβÞ=2 E þ m E −π=2 −π=2  þ e−2iρpfsin½θþðα−βÞ=2g sin −½ðαþβÞ=2 þ e−2iρpfsin½θþðα−βÞ=2g sin½ð−αþβÞ=2 ½1 − e−iðα−βÞσ2−2λ       p 2 1 2p2 p4 e−iðα−βÞ 1 λ 2 e−iðα−βÞ α − β iλ α − β × 2m þ 4 ð þ Þ þ M þ M2 ½cosð Þþ sinð Þ    λ p2 p4 − 1 − e−iðα−βÞσ2 −2λ−1 2 2 i α − β e−iðα−βÞ ; 2 ½ þ M þ M2 sinð Þ ð12Þ

where σ ¼ðp=E þ mÞ. and 3, for the above values of s, ϕ0, we have plotted the (iv) Visualizing the anyon beam: Unfortunately, closed profiles of Ja, a ≡ x; y respectively, in three ways: a two- form expressions for the anyon beam current components dimensional plot of Ja against the polar angle θ for a few 0 x x J , J , and J are not possible to obtain. For different values of the (planar) radial distance ρ, panel (a) in each 0 choices of λ and ϕ0 the profiles of J are given in group of figures. Another two-dimensional graph of the Supplemental Material D [18]. Subsequently, in Figs. 2 same data as panel (a) with magnitude of Ja against θ for

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nature since the axial symmetry is manifestly broken while constructing a propagating anyon beam. This is also corroborated in the figures that do not have any destructive interference at the origin, a characteristic feature of vortex beams [2]. Hence the anyon beams are characterized by the spin value s of the wave packet, which is the same as that of an individual plane wave single anyon component. An important observation is that in the cases we have considered, J0 is always positive, which has to be the case since it is the probability density. But Jx is also positive throughout, whereas Jy has positive and negative values in equal amount. Furthermore, maximum value of Jy is far lower than each of J0 and Jx. These reflect the nature of our construction of the anyon beam where all the plane waves have positive velocity along the x direction but have pairwise opposite (both þve and −ve) velocities along x FIG. 2. J (λ ¼ 0.6) with integration range π=2, where the y direction. Hence, the anyon beam predominantly (a)–(d) correspond to the similar meaning as described in Fig. 1. moves in the positive x direction with the y component effectively canceled out. the same values of ρ is shown in panel (b). A density plot in (v) Experimental possibilities: Anyons were detected in coordinate plane x-y is given in panel (c). Finally, a three- quantum antidot experiments [19] and in Laughlin quasi- dimensional plot of Ja in coordinate plane x-y is provided particle interferometer [20]. In relation to simulation of T in panel (d). Note that in panel (a), each continuous curve high c superconductivity by charged anyon fluid [21] their Josephson frequency has been observed [22] in 2D elec- represents a fixed polar distance in coordinate plane x-y Ja trons in high magnetic field. with the height being a measure of whereas in panel (b), External electromagnetic field affects the anyon with Ja the radial distance is a measure of the intensity of . Hence charge e and magnetic moment (see, e.g., [2]) the curves that are farther away from the center in panel Z Z (b) represent points that are closer to the coordinate M e dA ϵ rijk = dAj0 plane x-y. ¼ ð ik sÞ s ð13Þ As expected, all the wave packet profiles are symmetric s a about the abscissa since the packets are a superposition of for the single anyon current ja (8) using (7). We replace j plane waves that are symmetrically placed about the x axis. by Ja using the wave packet (11). The nonstationary Contrasting with the three-dimensional wave packets [2] it quantum superposition state (wave packet) can be created is clear that the planar anyon beams do not possess a vortex by exciting matter coherently with an ultrafast laser pulse, which is composed of eigenstates spanned by the frequency bandwidth of the laser [23]. The possibility of fault tolerant quantum computation by (non-Abelian) anyons has generated interest in controlled production of anyonic [12]: collective anyon excitations from electrons in the fractional quantum hall (FQH) systems or from atoms in 1D optical lattices, creation of the FQH effect for photons [using one-dimensional (1D) or 2D cavity array], using a nonlinear resonator lattice subject to dynamic modulation for creating anyons from photons, among others. Topologically ordered many-body states (with quenched kinetic energy) are generated with strongly interacting particles in magnetic field. The system minimizes interac- tion energy forming intricate patterns of long-range entan- glement as observed in the FQH (semiconductor heterojunction, graphene [24,25], and van der Waals bilayers [26]). Formation of Laughlin states in synthetic quantum systems (ultracold atoms [27,28] and photons [29– FIG. 3. Jy (λ ¼ 0.6) with integration range π=2, where 31]) has been developed. Recently in anyon optics, Laughlin (a)–(d) represent a similar meaning as described in Fig. 1. states have been made out of photon pairs (in a synthetic

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