Anyon Wave Functions Research Articles

Uncertainty relations for the relativistic Jackiw-Nair anyon: A first-principles derivation

In this paper we have explicitly computed the position-position and position-momentum (Heisenberg) uncertainty relations for the model of relativistic particles with arbitrary spin, proposed by Jackiw and Nair (Jackiw R. and Nair V. P. Phys. Rev. D, 43 (1991) 1933) as a model for anyon, in a purely quantum mechanical framework. This supports (via Schwarz inequality) the conjecture that anyons live in a 2-dimensional non-commutative space. We have computed the non-trivial uncertainty relation between anyon coordinates, , using the recently constructed anyon wave function (Majhi J. et al., Phys. Rev. Lett., 123 (2019) 164801), in the framework of Bialynicki-Birula I. and Bialynicka-Birula Z., New J. Phys., 21 (2019) 07306. We also compute the Heisenberg (position-momentum) uncertainty relation for anyons. Lastly we show that the identical formalism when applied to electrons, yield a trivial position uncertainty relation, consistent with their living in a 3-dimensional commutative space.

An anyon wavefunction for the fractional quantum Hall effect

An anyon wavefunction (characterized by the statistical factorn)projected onto the lowest Landau level is derived for the fractional quantum Hall effect states at fillingfactor ν = n/(2pn+1) (p and n are integers). We study the properties of the anyon wavefunction by using detailed MonteCarlo simulations in disc geometry and show that the anyon ground-state energy is a lowerbound to the composite fermion one. Our results suggest that the composite fermionscan be viewed as a combination of anyons and a fluid of charge–neutral dipoles.

On the 3-anyon harmonics

The 3-anyon problem is studied using a set of variables recently proposed in an anyon gauge analysis by Mashkevich, Myrheim, Olaussen, and Rietman (MMOR). Boundary conditions to be satisfied by the wavefunctions in order to render the Hamiltonian self-adjoint are derived, and it is found that the boundary conditions adopted by MMOR are one of the ways to satisfy these general self-adjointness requirements. The possibility of scale-dependent boundary conditions is also investigated, in analogy with the corresponding analyses of the 2-anyon case. The structure of the known solutions of the 3-anyon in the harmonic potential problem is discussed in terms of the MMOR variables. With an explicit analysis of the fermionic-end ground state it is shown that, within a series expansion in a boson gauge framework, the problem of finding anyon wavefunctions can be reduced to a (possibly infinite) set of algebraic equations, whose numerical analysis is proposed as an efficient way to study anyon physics.

Exactly soluble model of multi-species anyons and the braid group on a torus

Exactly soluble models of multi-species anyons are studied on a torus and ground-state wave functions are explicitly obtained. These models describe quasi-excitations of recently discovered fractional quantum Hall states in double-layer electron systems. The representation of the braid group on a torus by the multi-species anyons is examined, and its relationship with ground-state wave functions is clarified. The braid group on a torus requires that the anyon wave function must be multi-component. However, the gauge-covariant wave functions are single-component in the present formalism. We find that zero modes of the Chern-Simons (CS) gauge fields play an essentially important role to link the anyon wave function and the braid group on a torus. Elements of the braid group of noncontractible cycles on a torus induce a “fractional” large gauge transformation of the zero modes, and the canonical commutation relation of the CS zero modes induces the braid-group algebra on a torus. The quantum Hall state in anyon systems is also discussed by comparing the anyon wave functions with the Laughlin and Halperin states.

HIERARCHICAL WAVE FUNCTIONS AND FRACTIONAL STATISTICS IN FRACTIONAL QUANTUM HALL EFFECT ON THE TORUS

One kind of hierarchical wave functions of Fractional Quantum Hall Effect (FQHE) on the torus are constructed. The multi-component nature of anyon wave functions and the degeneracy of FQHE on the torus are very clearly reflected in this kind of wave functions. We also calculate the braid statistics of the quasiparticles in FQHE on the torus and show they fit the picture of anyons interacting with magnetic field on the torus obtained from braid group analysis.

On the origin of multi-component anyon wave functions

In this paper I discuss how the component structure of anyon wave functions arises in theories with non-relativistic matter coupled to a Chern-Simons gauge field on the torus. It is shown that there exists a singular gauge transformation which brings the hamiltonian to a free form. The gauge transformation removes a degree of freedom from the hamiltonian. This degree of freedom generates only a finite-dimensional Hilbert space and is responsible for the component structure of free anyon wave functions. This gives an understanding of the need for multiple component anyon wave functions from the point of view of Chern-Simons theory.

On the electromagnetic interactions of anyons

Using the appropriate representation of the Poincaré group and a definition of minimal coupling, we discuss some aspects of the electromagnetic interactions of charged anyons. In a nonrelativistic expansion, we derive a Schrödinger-type equation for the anyon wave function which includes spin-magnetic field and spin-orbit couplings. In particular, the gyromagnetic ratio for charged anyons is shown to be 2; this last result is essentially a reflection of the fact that the spin is parallel to the momentum in (2 + 1) dimensions.

Analytic wave functions for understanding the spectrum of the three-anyon problem

We show that the low-lying spectrum of three noninteracting anyons in a harmonic potential can be qualitatively and quantitatively understood on the basis of a simple set of analytic wave functions. These wave functions are exact in both the boson and fermion limits and satisfy the anyon wave-function phase condition at all intermediate values of the statistical parameter 0<α<1. The resulting nonlinear spectrum agrees with those of exact numerical calculations to better than a few parts in a thousand

Many anyon wavefunctions in a constant magnetic field

We obtain exact energy eigenstates algebraically for N-anyon system in a constant external magnetic field (and with harmonic interaction). For a fractional statistical parameter α, there exist two classes of states which are obtained by applying regular operations on two different bottom states. It is demonstrated that the operators for excited states are given in terms of one-step energy excitation operators only and their multiplicity table is established.

Many anyon wavefunctions in a constant magnetic field: K. H. Cho and Chaiho Rim. Department of Physics, Chonbug National University, Chonju, 560–756, Korea

Many anyon wavefunctions in a constant magnetic field: K. H. Cho and Chaiho Rim. Department of Physics, Chonbug National University, Chonju, 560–756, Korea

Multi-anyon quantum mechanics and fractional statistics

We study a many-body system consisting of N identical anyons in an external harmonic oscillator potential. We obtain its (partial) analytic solutions by using Jacobi coordinates. Our study sheds considerable light on symmetrization of anyon wave functions, and it also suggests that the anyon wave functions may be generally constructed from the one-body wave functions, by considering the statistical symmetries as is conventional for bosonic and fermionic systems. To see this, we explicitly construct two families of anyon wave functions and energy spectra in terms of those for fermions and bosons.

ON SINGULAR ANYON WAVEFUNCTIONS

We show that quantum mechanics allows anyon wavefunctions that are singular as the particles approach each other. An extra real parameter, which determines the nature of this singular behaviour, is necessary to have a well defined quantum mechanical problem. In realistic models, anyons are expected to have an internal structure, corresponding to a smearing of the statistical flux. We study such models and find that, in generic cases, only regular wavefunctions are allowed.

A large hybrid electro-optic computer for neural network applications

The 3-anyon problem is studied using a set of variables recently proposed in an anyon gauge analysis by Mashkevich, Myrheim, Olaussen, and Rietman (MMOR). Boundary conditions to be satisfied by the wavefunctions in order to render the Hamiltonian self-adjoint are derived, and it is found that the boundary conditions adopted by MMOR are one of the ways to satisfy these general self-adjointness requirements. The possibility of scale-dependent boundary conditions is also investigated, in analogy with the corresponding analyses of the 2-anyon case. The structure of the known solutions of the 3-anyon in the harmonic potential problem is discussed in terms of the MMOR variables. With an explicit analysis of the fermionic-end ground state it is shown that, within a series expansion in a boson gauge framework, the problem of finding anyon wavefunctions can be reduced to a (possibly infinite) set of algebraic equations, whose numerical analysis is proposed as an efficient way to study anyon physics.