Statistics of Quasiparticles in Fractional Quantum Hall States

II. FRACTIONAL QUANTUM HALL STATES: A BERRY PHASE APPROACH
In some earlier papers [7, 8] we have analyzed the sequence of quantum Hall states from the viewpoint of chiral anomaly and Berry phase. To this end, we have taken quantum Hall states on the two dimensional surface of a 3D sphere with a magnetic monopole of strength µ at the centre. In this spherical geometry, we can analyze quantum Hall states in terms of spinor wave functions and take advantage of the analysis in terms of chiral anomaly which is associated with the Berry phase. In this geometry the angular momentum relation is given by J = r × p − µˆ r, µ = 0, ± 1/2, ± 1, ± 3/2, ......... Yℓm,µ (1)
arXiv:cond-mat/0402407v1 [cond-mat.mes-hall] 16 Feb 2004
Statistics of Quasiparticles in Fractional Quantum Hall States
B. Basu∗ and P. Bandyopadhyay†
Physics and Applied Mathematics Unit Indian Statistical Institute Kolkata-700108 Abstract We have considered here the statistics parameter for quasiparticles in FQH states from an analysis of these states in the framework of chiral anomaly and Berry phase. It is shown that we have a generalized relation such that the statistical phase of a quasiparticle for FQH states at ν = p/q with p(q ) even or odd, is given by eiπθ with θ = p/q and the charge is −e p . The statistics parameter for q a quasihole is identical with that of a quasiparticle and the charge is of opposite sign.
∗ Electronic † Electronic
address: banasri@isical.ac.in address: pratul@isical.ac.in
2 Here we shall address the problem of statistics in a heuristic manner on the basis of analysis of FQH states in the framework of chiral anomaly and Berry phase. From a calculation of the Berry phase of a quasiparticle (quasihole) encircling another quasiparticle (quasihole) we shall show that the charge and n statistics parameter for quasiparticles at ν = 2mn ±1 with m, n posotive integers are given by - νe and n respectively. For quasiholes the statistics parameter is identical with that of the quasiparticle and 2mn±1 the charge has the opposite sign. In sec. 2 we shall recapitulate certain features of the FQH states analysed from the Berry phase approach. In sec. 3 we shall calculate the statistics parameter for quasiparticles (quasiholes) for FQH states at different filling factors.
PACS numbers: 73.43.-f; 73.43.Cd; 03.65.Vf
I.
INTRODUCTION
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In a seminal paper [1] Arovas, Schrieffer and Wilczek derived the statistics of the quasiholes at ν = 1/m, m odd in a Berry phase calculation. They examined the Berry phase corresponding to one quasihole encircling the origin and interpreted this as the Aharanov-Bohm phase. The charge was found to be e/m. Considering a pair of quasiholes encircling one another they found the statistics parameter to have the value 1/m. The Laughlin quasielectrons were examined along the same line [2] and within certain approximations, the charge and statistics parameter were found to have the values −e/m and 1/m respectively. Kjonsberg and Myrheim [3] have argued that for FQH states the quasiparticles(qp) at ν = 1/m, m odd, do not possess well defined statistics. Their numerical studies show that at ν = 1/3 the charge and statistics parameter for quasiholes(qh) are e/3 and 1/3 respectively. However, for the quasiparticles though a slow convergence towards the expected value of the charge −e/3 is observed with a finite size correction for N electrons, the statistics parameter has no well defined value even for 200 electrons but might possibly converge to 1/3. Johnson and Canright [4] examined the exclusion statistics parameter considering state counting based on numerical simulations for interacting electrons on a sphere. It is found that the one dimensional 1 ). The exclusion statistics parameter for quasiholes is 1/m while for quasielectrons the value is (2 − m exclusion statistics parameter is the same parameter as is obtained from Berry phase analysis with an opposite sign for the quasielectrons since their charge is negative. This implies that the exclusion statistics 1 1 parameter is 1/m for quasiholes and (−2 + m ) for quasielectrons. Though the values 1/m and −2 + m 1 represent the same particle statistics, their states are different, 1/m state singular and −2 + m state nonsingular. In the composite fermion (CF) scenario Kjonsberg and Leinaas [5] calculated the statistics of the unprojected CF quasiparticle at ν = 1/m, the wave function of which is different from Laughlin function, and observed a definite value though the sign is inconsistent with general considerations. Jeon, Graham and Jain [6] have confirmed that the statistics is robust to projection into the lowest Landau level and argued that the sign enigma has its origin in very small perturbations in the trajectory due to the insertion of an additional CF quasiparticle. These authors have numerically studied the statistics of the composite fermion quasiparticles at ν = 1/3 and ν = 2/5 by evaluating the Berry phase for a closed loop encircling 2 another CF quasiparticle and the statistics parameters were found to be − 2 3 and − 5 respectively. The negative sign has been attributed to the fact that there is an inter-CF interaction which is weak and often attractive and involves a significant overlap of CF quasiparticles causing small perturbation in the trajectory.