Anyon kuantum mekaniği Chern-Simons modeli

Abstract

Üç boyutlu uzay- zamanda herhangi bir spin değeri alan parçacıklara anyon adı verilir. Genelleştirilmiş spin-istatistik teoreme göre bu parçacıklar sıradan istatistiğe uymazlar. Kesirli istatistiğe uyan bu parçacıklar ne bozonların ne de fermiyonlann özelliklerini tam olarak gösterirler. Anyonları, anyon ayarında ve Chern-Simons ayarında incelemek mümkündür. Bu çalışmada daha çok üzerinde durulan Chern- Simon modelidir. Ve modele uyarak yüzey üzerindeki anyonların manyetik alan yokken, yüzeye dik bir manyetik alan varken nasıl davrandıkları; dalga fonksiyonu, açısal momentum ve Hamiltoniyen ifadeleri N parçacık için elde edilmiştir. Son bölümde kuantum Hail olayında bir kesirli etkinin oluşabildiği durumlar için doldurma faktörünün nasıl hesaplanacağı gösterilmiştir.

In rotation-invariant quantum systems in D space-time dimensions the spin S, of massive particles labels the irreducible unitary representations of the covering group SO (D-l). For D=4 these representations are labelled by integers and half- integers. But in three space-time dimensions the rotation group is SO(2) ~ U(l). Therefore the spin S can be "any" real number. This is the reason why Wilczek called particles with arbitrary real spin in (2+1) dimensions "anyons." In analogy with spin-statistics theorem in four dimensions it is then expected that these particles obey fractional statistics; under interchange of two particles the quantum mechanical wave function can pick up an arbitrary phase. In (2+1) dimensions fictitious gauge potential Ap and a matter current J can be written together with the action S = JL (d3xerkA d Ax + [d\xA J". The first term in this expression is called Chern-Simons action. The zeroth component of the equations of motion gives the constraint aA " aA - -Ş^o. In quantum mechanics J0 becomes a sum of 8 -functions and we can solve this equation for A. The /th particle feels then a vector potential a, = iv, x 5>u--.v.| and the ^/-particle Hamiltonian becomes 1 £_ --4 /A, w = -i E0'2 ' *W,-~, im w This gauge can be eliminated by a gauge transformation at the expense of introducing a multivalued wave function (,- -. V ¥u" = n ' = e^jd2y dnP(x-yW(y)v(y) ? We can deduce the Hamiltonian of the system H = J_ U2x U:(x)Un(x). 2m J Here II» = [dnt-Am{x)}xf{x). We can now explain the quantum mechanics of a system of anyons on a plane in the presence of a constant external magnetic field B orthogonal to it. We choose to describe the anyons as fermions interacting with the Chern-Simons field. By working in the Chern-Simons gauge we can write the Hamiltonian as m ı km jZj 2 D. = 3. + - V - + _ J*' -ij 2A-tf z, 4 ' We- first point out the possible ways of local behaviour for z = zv - > 0. We consider just two anyons and neglect locally the magnetic field. If we assume that the wave function carries an angular momentum n, that is yn(r,§) - e mf\r), we get the eigenvalue equation [-9; - hr + -L(»+-L)] /.(/.) = Emfjjr) r r ' k - Then the two possible ways of behaviour are fn ~ r " T. We will take the wave function behaving as -i"+4l \\f ~ r T. If we consider the problem of just two anyon (İV=2) in a magnetic field, we can split again the center of mass and the relative motion, H = HCM + Hr, where Vll HrM = -A +A + Ico, H = -ia *a + loo + İÜ82(z) A = 3 + _mcaZ a = 3 + + - mcoz 2k z 8 Z = i-Cz, + z2) 7 = 7 - "1 //CM describes the standard Landau levels problem, so we will concentrate on Hr. From the above wave function, we know that we have two types of solutions, type I behaving for small r as r'~m with / > 0; and type II behaving as r'*uk with / > 0. For two types we can obtain the ground state wave functions, the general wave functions and energy eigenvalues. To summarize the essential features of the explanation of the fractional quantum Hall effect let us recall the form of the "free" particle Hamiltonian (we neglect the spin) h0 = 1d+d+J!1 m 2m The covariant derivative is given D = 8 + LeBz, and satisfies [DJD *] = ^eB. The energy levels are given by * m and related eigenf unctions are \jrn = (D *y \|/0, where the ground state is determined through the equation D \|/0 = 0. ¥0 = g(I) exp(-±!l) where g is an arbitrary antiholomorphic function. For fractional fillings of the form / = (2/ + l)"1 with / integer. Laughlin proposed as minimizing solution, according to his "one-component plasma analogy," the state Vlll VJ = n^-^)^1 exp(-_L5:|z/.p) which describes a circular, incompressible droplet of fluid. Laughlin gave also an (approximate) analytic expression for the elementary excitations of \\fJ, localized in z0, which corresponds to piercing the fluid at z0 with an infinitely thin solenoid and passing through it a flux quantum A§ = ÜL adiabatically. Accordingly, approximate quasiholes and quasiparticles are given respectively, by vi = exP(-_LEiz;i2) n^-g n^-^r1 vt = expi.-Lziz^) nd-4> n^-9^1 In the simple case we have a two-component fluid with an electron component and one fractionally charged say quasiparticle component. If the effective filling of the quasiparticle component is the form (2/' + l)"1, for some integer /', the collective ground state of the excitations fluid is analogous to the ground state of the electron fluid and hence particularly stable. In this way the filling factors at which a fractional effect can occur can be expressed by a continued fraction /- 1 2/+1+. a. 2A- a. a, 2/,+_!,-1 j where a. = ±1 according to whether the corresponding component is made out of quasiholes or quasiparticles.