Anyon kuantum mekaniği chern simons modeli
Anton quantun mechanics and chern simons model
- Tez No: 39758
- Danışmanlar: PROF.DR. MAHMUT Ö. HORTAÇSU
- Tez Türü: Yüksek Lisans
- Konular: Fizik ve Fizik Mühendisliği, Physics and Physics Engineering
- Anahtar Kelimeler: Anyonlar, Chern-Simons yöntemi, Kuantum mekaniği, Anions, Chern-Simons method, Quantum mechanics
- Yıl: 1994
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET 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. iv
Özet (Çeviri)
SUMMARY ANYON QUANTUM MECHANICS AND CHERN SIMONS MODEL 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 '</ (z-z.) z.-z. T ¥ where y obeys ordinary statistics and z = x + iy. A gauge-invariant minimal coupling of the Chern-Simons action to a conserved current is most interesting and natural procedure to investigate the effects of such ”anomalous“ kinetic terms for a gauge field in a system of interacting charges. A theory in which a matter field is minimally coupled to a Chern-Simons gauge field corresponds to a low-energy approximation of this more general theory, where the lower derivative Chern-Simons term dominates the higher-derivative Maxwell term. The minimal coupling of a Chern-Simons field to a nonrelativistic matter field \\f {x ), commuting or anticommuting, can be described by the following action: S = - [dh e”vX A dvAx + (dh (/\|/+D+\|/ + J_\|/+(Dt2+D/)\|/) The covariant derivative is defined as usual by D = d - iA, where u goes from zero to two. VIThe constraint on the action determines the space components AN of the gauge potential as a function of the density K> = 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 VllHrM = -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 VlllVJ = 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. IX
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