An Introduction to Anyons

An Introduction to Anyons Avinash Deshmukh University of British Columbia In this paper, we introduce anyons, particles that pick up a non-trivial phase under exchange. We start by considering the topological conditions that allow anyons to be created and construct the braid group, which defines anyonic exchange statistics. We proceed to discuss abelian and non- abelian exchange representations and review cyons, fibonacci anyons, and Ising anyons as illustrative examples. Finally, we briefly examine solved condensed matter models in which anyons are produced and review the exchange statistics of these systems. I. INTRODUCTION rigorous version of this argument. It is well known that in three-dimensions, there are two One important consequence of θ being any value is that it is iθ iθ types of particles; bosons are symmetric under exchange and not generally true that e “ e´ . We must then distinguish fermions are anti-symmetric under exchange. However, in between \counterclockwise" exchanges and \clockwise" ex- two-dimensions, it is possible to have particles that are nei- changes, which correspond to the accumulated phases θ and ther anti-symmetric nor symmetric under exchange. These ´θ respectively. particles are known as anyons. It has been shown that the exotic statistics of anyons can be used to encode information for topological quantum computation [1], and it is believed B. The Braid Group that they can be used in the construction of superconductors [2]. Earlier this year, abelian anyons were directly observed Consider a system of n particles that follow θ-statistics in a two-dimensional electronic gas collider [3] and in an elec- 2 with the positions r1; :::; rn P R in the center of mass tronic Fabry-Perot intereferometer [4], showing that these frame, described by the wave function pr1; :::; rnq. Suppose particles actually exist { they are not mere mathematical now that we perform some arbitrary exchange of these oddities. particles such that the particles have the final positions rP p1q; :::; rP pnq, where P is some permutation of n particles. We can perform this exchange through the composition II. ANYON EXCHANGE STATISTICS of exchanges of numerically adjacent particles, each of which may be a clockwise exchange or a counterclockwise ´1 A. Two Particle Exchange exchange. In light of this, we define σi and σi for 1 ¤ i ¤ n ´ 1 to be the counterclockwise and clockwise th th Consider a system of two indistinguishable particles with exchanges respectively of the i and pi ` 1q particle. Notice that a clockwise exchange of two particles undoes hard-core repulsion at the positions r1 and r2 described by the effect of a counterclockwise exchange, so they are indeed the wave function pr1; r2q. We define the exchange of two particles as the arbitrary adiabatic transport of each par- inverses [5]. ticle to the position that was previously occupied by the other particle. Because the particles are indistinguishable, The exchange of two disjoint pairs of particle should not af- the probability density of the system must be unchanged, fect one another, so the order in which those exchanges occur so the wave-function must be unchanged up to accumulat- does not matter. This adds the constraint that σiσj “ σjσi ing some phase eiθ. Suppose we exchange the particles once, for ji ´ jj ¥ 2. We also add the mathematically imposed and then exchange the particles again. Notice that we have Yang Baxter constraint, that σiσi`1σi “ σi`1σiσi`1, which moved the particles in a closed loop about each other back is necessary for a valid commutator structure [6]. to their initial positions. In three dimensions, such an \exchange loop" is topologically equivalent to the trivial loop Using our construction of the σi, we see that exchange (which is no loop at all), so the wave function must be invari- in a system of n-particles with θ-statistics has the same 2iθ ant under this double exchange. This means the e phase structure as the group generated by σ1; ::; σn. This group accumulated by the exchanges must satisfy the constraint: is known as the Braid group Bn. It is not generally true 2 that σi “ 1 for each of the σi, which means that the braid 2iθ pr1; r2q “ e pr1; r2q (1) group Bn is an infinite group. Notice that if we repeat this construction in three dimensions, we obtain the same result 2 so θ “ nπ for some integer n. When n is an even integer, with the additional topological constraint that σi “ 1. the wave function is invariant under exchange, which gives We recognize this as being the symmetric group Sn, which us boson exchange statistics. When n is an odd integer, agrees with our understanding that particle exchange in the wave function is multiplied by a negative sign under three dimensions is equivalent to permuting the particles exchange, which gives us fermi statistics. [6]. A reader interested in a useful visual representation of the braid group can consult the appendix. In two-dimensions, however, not all exchange loops are topologically equivalent to the trivial loop, so double exchanges To understand how the braid statistics impact the quantum do not restore the initial system. Then, we do not have mechanical behavior of an anyonic system, we must deter- any constraint on the phase θ under two exchanges, and θ mine the action of the braid group on the Hilbert space H. could be any value. We say that particles that accumulate This requires us to determine a way to write the elements of a factor of θ obey this exchange rule follow θ-statistics. the braid group as operators on the Hilbert space that leave Bosons obey 0-statistics, fermions obey π-statistics, and we probabilities unchanged, that is we need a unitary represen- refer to particles obeying θ-statistics for θ ‰ 0; π as anyons tation π : Bn ÝÑ UpHq, where UpHq denotes the space of [5]. The interested reader can consult the appendix for a unitary operators acting on the Hilbert space. 2 III. ABELIAN ANYONS B. Physical Realization { Cyons A. The One-Dimensional Representation of the Braid Despite the fact that particles constrained to two dimen- Group sions are able to have anyon statistics, electrons, atoms, and photons constrained to two dimensions retain the Perhaps the simplest non-trivial representation of the braid same fermionic/bosonic statistics that they obey in three group Bn is the one-dimensional representation: dimensions. However, it is possible to construct systems with quasiparticles that obey anyon statistics. iθ πpσkq “ e (2) Consider an infinite and very thin solenoid along the z-axis It is easy to see that this representation is a unitary with magnetic flux Φ along with a spinless particle of charge homomorphism that has all of the properties required of q and mass m orbiting the solenoid at a radius r constrained to some plane perpendicular to the solenoid. The electric the generators. Furthermore, the πpσkq mutually commute, which tells us that the resultant statistics are abelian. field E experienced by the charged particle is: Anyons described by these statistics are called abelion Φ9 anyons. E “ ´ φ^ (8) 2πjrj We can write an arbitrary exchange of particles r ; :::; r p 1 nq ÞÑ Using Coulomb's Law and Newton's Equation of motion, we r ; :::; r as some element b B . Writing this in p P p1q P pnqq P n find that the contribution of the electric field to the angular terms of the generators, we have: momentum of the particle is: n qΦ mk ^ b “ σ (3) JE “ φ (9) k 2πc k“1 ¹ The canonical angular momentum JC “ r ˆ pp ` eAq of for m1; :::; m P . Applying the the representation π, we n Z the particle is some integer multiple of ~. Then, using the find that b operates on the Hilbert space as the following fact that the canonical angular momentum is the sum of the operator: electronic angular momentum JE and the kinetic angular momentum JK “ r ˆ p, we have: πpbq “ eipm1`¨¨¨`mnqθ (4) ^ qΦ ^ JK “ n φ ´ φ (10) The exchanged wave-function is given by applying the oper- ~ 2πc ator πpbq to the wave function, so we have: for n P Z. Now, consider the limit where r Ñ 0 and the ipm1`¨¨¨`mnqθ solenoid become infinitely thin. The resulting quasiparticle prP 1 ; :::; rP n q “ e pr1; :::; r q (5) p q p q n is called a cyon. The spin S is defined to be the kinetic angular momentum at n “ 0 divided by , so the cyon is a Notice that m m is an effective degree of the ~ 1 ` ¨ ¨ ¨ n quasiparticle with mass m, charge q, and fractional spin [7]: exchange; each mk is the number of counter-clockwise exchanges of the k and k ` 1 minus the number of clockwise qΦ exchanges of these particles required to go from the initial S “ (11) 2π~c state to the final state. Consider a system of two cyons at the positions r1 and r2. Next, consider a system of two particles, each of which are In center of mass and relative coordinates R “ pr1 ` r2q{2 composite bound states of n abelian anyons that obey θ- and r “ r1 ´ r2, the wave function describing this system statistics. A counter-clockwise exchange σ of these particles (see the Appendix for a derivation) is: involves n2 counter-clockwise anyon exchanges { each anyon ikR¨R ip`´qΦ{2πqφ in one bound state has a counter-clockwise exchange with ΨpR; rq “ c`;kR e J`pkr rqe (12) `;k each anyon in the other bound state. Then, it follows that: ¸R 2 for the constants c`;kR which are such that the wave-function π σ ein θ (6) th p q “ is normalized, where r “ pr; φq, and where J` is the ` bessel function of the first kind.

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