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Anyons: Field Theory and Applications

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Anyons: Field Theory and Applications August 4, 2010 Anyons: Field Theory and Applications Brian Williams 1 University of Florida Abstract This paper presents the concept of anyons and how they arise in d = 2 + 1 physical theories. Classical properties are briefly discussed, but the main applications are a consequence of the quantum nature of these strange particles. Anyons are particles that occur in planar physics which carry neither integer (bosons) or half integer (fermions) spin. In fact, they can carry any positive real number spin and this paper shows how this manifests. The main physical application addressed is the fractional quantum hall effect (FQHE). Problems with the non-relativisic formulations of anyons is motivation to come up with a relativistic field theoretic way of treating anyons. 1brianwilliams@ufl.edu Advisor: Pierre Ramond, Distinguished Professor, University of Florida. 1 Introduction This paper is intended to convey the peculiarities of physical systems constrained to two space dimensions and explain how these properties come up in physical applications. The motivation for this comes from the applications to the integer quantum hall effect (IQHE) and, more specifically for this paper, the fractional quantum hall effect (FQHE). The latter effect is less understood in terms of solid state techniques, and one may look to the quantum relativist's tool of field theory to understand such an effect. In 2 + 1 dimensions, strange particle states called anyons are present which can, and will, be seen from a straightforward symmetry argument. These particles, unlike bosons or fermions, can have any real value for spin and lead to the idea of generalized statistics in quantum mechanics. Anyons prove to be very useful in describing phenomena like the FQHE. Anyons can be described in a variety of ways. One of the ways is to look at anyons from a model independent framework and consider them as spinning particles with fractional spin (note here, and for the remainder of the paper, we will stick to the awkward convention of fractional depicting the fact that a value may take on any real number). This can be done in a completely classical setting with no reference to statistics. One may also look at anyons from a field theoretic standpoint as quantum excitations of a classically bosonic or fermionic system. We will primarily be focused on the latter description which will restrict our attention to the celebrated Chern-Simons theories in 2+1 dimensions. Widely known for many interesting topological properties, Chern-Simons is important in this setting as it is the covariant generalization of a particle with generalized statistics and can admit anyon-like excitations at the quantum level. We will look at Chern-Simons theory from various perspectives and offer few different methods of quantization, each of which have some advantages and disadvantages. Lastly, we will look into the somewhat ill-understood realm of fractional supersymmetry and how this could possibly be used to construct a relativistic field theory of anyons. No new 2 results are given in this paper, but the methods highlighted above are a relatively new way of solving this problem and will be the topic of future research. 1.1 Conventions We will primarily be working in 2 + 1 dimensions with the choice of metric signature gµν = diag(+1; −1; −1). Repeated indices are always summed over unless otherwise stated. The standard three vector is labeled with a Greek index: xµ = (x0; x1; x2) = (t; x). Latin indices xi go over i = 1; 2 and are the spatial parts of the vector. Shorthand for derivatives @ is @µ = @xµ . 2 Symmetries in the plane and classical anyons Here we analyze the symmetries present in 2 + 1 dimensions. We start with the Poincar´e group and see how its representation leads to the concept of arbitrary spin in the plane. Next, we take a digression to a non-relativstic interpretation of anyons and how it relates to the relativistic notion. This section is devoted to a classical description of anyons which will lead into the quantum concept of anyonic behavior. 2.1 The Poincar´eGroup The set of space-time symmetries of quantities in arbitrary dimension are formally defined to be the set of all real linear transformations µ µ µ 0µ µ ν µ (a ; Λν ): x 7! x = Λν x + a (1) 2 µ ν 0 2 1 2 2 2 which leave s ≡ gµνx x = (x ) − (x ) − (x ) invariant. They actually form a group called the Poincar´egroup. The above transformation law indicates that π is actually ∼ 3 a semidirect product of a translation group N = R and the three-dimensional Lorentz group SO(2; 1). As with any Lie group, we refer to the infinitesimal generators to make 3 the connection to the associated Lie algebra. The Hermitian generators of the Lorentz subgroup are Mµν = i(xµ@ν − xν@µ) + Sµν, and they obey the usual SO(d; 1) Lie algebra, [Mµν;Mρσ] = i(gµσMνρ − gνσMµρ − gµρMνσ + gνρMµσ): (2) As we shall see in a moment, the spin part Sµν is the defining factor for the uniqueness of 2 + 1 dimensional theories. The generators of the translational symmetries are Pµ = −i@µ, which of course commute with each other, yet have non-trivial bracket with the rotation generators. We can summarize the commutation relations by introducing the 0 1 2 (non-Hermitian) generators K = −iM01, K = −iM02, and K = M12. The brackets then read, µ ν µνρ µ ν µνρ [K ;K ] = i Kρ ; [K ;P ] = i Pρ ; [Pµ;Pν] = 0 (3) which summarize the Poincar´ealgebra in 2 + 1 dimensions. As with the 3 + 1 dimensional case, we have two Casimir operators, yet they take a slightly different form. The first is 2 µ P ≡ P Pµ, and the second is the Pauli-Lubanski scalar µνρ W ≡ P · K = PµMνρ: (4) For single particle representations Φ we have the eigenvalue equations P 2Φ = m2Φ and W Φ = −msΦ (5) In contrast to the 3 + 1 case, there is no a priori restriction to the values that the spin s can take on. This conclusion can also be seen from a more formal covering group argument and this is presented in Appendix A[1]. 2.2 The Galilei Group and Anyons We now take a slight detour to demonstrate how the concept of anyons from the clas- sical perspective can lead to unusual implications. The Galilei group G2j1 in three di- mensions is the group of symmetries which simultaneously leaveg ~µν =diag(1; 0; 0) and 4 µν g~ =diag(0; 1; 1) invariant. We have six generators: J, Ki, H, Pi of rotations, boosts, time translations, and spacial translations, respectively. They satisfy the Poisson brackets (only non-vanishing commutators listed), fJ; Kig = ijKj ; fJ; Pig = ijPj ; fKi;Hg = Pi: (6) It is known that G2j1 admits a two-fold central extension via the modifications, fKi;Pjg = mδij and fKi;Kjg = −κij: (7) We have two Casimir invariants, 1 C = mJ + κH − K P and C = mH − P P : (8) 1 ij i j 2 2 i i For the case C1 = C2 = 0 we have the following realization of the algebra, Pi = pi, 1 1 2 Ki = mxi − tpi + mθijpj , J = ijxipj + 2 θpipi, and H = 2m pipi where θ ≡ κ/m . pi With this, we have the usual time evolvementx _ i = fxi;Hg = m , yet coordinates do not commute [2]: fxi; xjg = θij: (9) We can get to this same structure among coordinates by starting with Poincar´esymmetry and by placing no restrictions on the spin. We obtain the result after demanding covariance, pρ fx ; x g = s (10) µ ν µνρ (p2)3=2 2 pipi 1 [3]. We can make the non-relativistic expansion p0 = mc + 2mc + O c2 to get s fx ; x g ≈ ij (11) µ ν m2 in the non-relativistic limit [4]. We conclude there is a relationship between these two situations with the correspondence κ $ s. 3 Quantum Anyons We now look at anyons as particle states that have fractional spin in a statistical sense which is a consequence of their quantum nature. 5 3.1 Fractional spin through quantum solenoids In this example, we show how the spectrum of angular momentum can be shifted from integer or half integer multiples of the fundamental constants [5]. This is not an actual realization of fractional spin, but with some modification we can exhibit such an effect. Consider the planar motion of a charged particle and let there be a solenoid centered at the origin. The Lagrangian is, 1 L = mx_ 2 + e _x · A(x) (12) 2 where we can gauge the potential to be Φ xj Ai(x) = − ij (13) 2π x2 ij (2) [6]. The magnetic field is the scalar quantity given by B = @iAj = Φδ ; therefore, the quantity Φ can just be interpreted as the magnetic flux. The momenta are found to be pi = m · xi + eAi and the Hamiltonian has the form 1 2 1 H = pi − eAi = mx_ 2 (14) 2m 2 which, as expected, is the same as that of a free particle. The presence of the singular solenoid at the origin does not affect the classical dynamics of the particle, but it does give rise to quantum effects. Note that, because the potential outside the solenoid is constant 0 φ we can set it to zero via a gauge transformation Ai = Ai − @iΩ where Ω = 2π Φ and φ is the polar angle. The wave function becomes 0 = e−iΩ . If we demand to satisfy periodic boundary conditions: (φ) = (φ + 2π), then 0 satisfies "twisted" boundary conditions: (φ + 2π) = e−iΩ (φ). The mechanical angular momentum operator is the usual Jm = −i@φ and we see that Φ J 0 = l − 0 (15) m 2π where Jm = l is the usual spectrum.
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