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arXiv:quant-ph/9809027 · Journal of Mathematical Physics

Two-particle scattering theory for anyons

C. Korff, G. Lang, R. Schrader

Published Apr 1, 1999• 4 citations
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Abstract

We consider potential scattering theory of a nonrelativistic quantum mechanical 2-particle system in R2 with anyon statistics. Sufficient conditions are given which guarantee the existence of Møller operators and the unitarity of the S-matrix. As examples the rotationally invariant potential well and the δ-function potential are discussed in detail. In case of a general rotationally invariant potential the angular momentum decomposition leads to a theory of Jost functions. The anyon statistics parameter gives rise to an interpolation for angular momenta analogous to the Regge trajectories for complex angular momenta. Levinson’s theorem is adapted to the present context. In particular we find that in case of a zero energy resonance the statistics parameter can be determined from the scattering phase.

DOI
10.1063/1.532837

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Two-Particle Scattering Theory for Anyons

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We consider potential scattering theory of a nonrelativistic quantum mechanical 2-particle system in R"2 with anyon statistics. Sufficient conditions are given which guarantee the existence of Moeller operators and the unitarity of the S-matrix. As examples the rotationally invariant potential well and the #delta#-function potential are discussed in detail. In case of a general rotationally invariant potential the angular momentum decomposition leads to a theory of Jost functions. The anyon statistics parameter gives rise to an interpolation for angular momenta analogous to the Regge trajectories for complex angular momenta. Levinson's theorem is adapted to the present context. In particular we find that in case of a zero energy resonance the statistics parameter can be determined from the scattering phase. (orig.)

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We consider potential scattering theory of a nonrelativistic quantum mechanical 2-particle system in R2 with anyon statistics. Sufficient conditions are given which guarantee the existence of Møller operators and the unitarity of the S-matrix. As examples the rotationally invariant potential well and the δ-function potential are discussed in detail. In case of a general rotationally invariant potential the angular momentum decomposition leads to a theory of Jost functions. The anyon statistics parameter gives rise to an interpolation for angular momenta analogous to the Regge trajectories for complex angular momenta. Levinson’s theorem is adapted to the present context. In particular we find that in case of a zero energy resonance the statistics parameter can be determined from the scattering phase.
We consider potential scattering theory of a nonrelativistic quantum mechanical 2-particle system in R"2 with anyon statistics. Sufficient conditions are given which guarantee the existence of Moeller operators and the unitarity of the S-matrix. As examples the rotationally invariant potential well and the #delta#-function potential are discussed in detail. In case of a general rotationally invariant potential the angular momentum decomposition leads to a theory of Jost functions. The anyon statistics parameter gives rise to an interpolation for angular momenta analogous to the Regge trajectories for complex angular momenta. Levinson's theorem is adapted to the present context. In particular we find that in case of a zero energy resonance the statistics parameter can be determined from the scattering phase. (orig.)
While elementary particles obey either bosonic or fermionic exchange statistics, generalized exchange statistics that interpolate between bosons and fermions -- applicable to quasi-particles -- constitute an intriguing topic, both from the fundamental and practical points of view. This work develops a scattering framework for two identical 1D bosonic anyons and two identical 1D fermionic anyons with zero-range contact interactions. The two-body system with zero-range interactions, both in free space and under external confinement, is used to illustrate the recently proposed bosonic-anyon -- fermionic-anyon mapping~(R. Hidalgo-Sacoto {\em{et al.}}, arXiv:2505.17669), which connects the eigenstates of bosonic anyons to those of fermionic anyons and vice versa. Performing explicit calculations for two-particle systems, the momentum distributions and the off-diagonal correlations of the single-particle density matrix for bosonic anyons and fermionic anyons are confirmed to be distinct. We also confirm the previously derived asymptotic coefficients of the momentum distribution tail at orders k−2k^{-2}k−2 and k−3k^{-3}k−3 for two harmonically confined anyons. Non-universal contributions at order k−4k^{-4}k−4 are discussed.
While elementary particles obey either bosonic or fermionic exchange statistics, generalized exchange statistics that interpolate between bosons and fermions---applicable to quasiparticles---constitute an intriguing topic from both the fundamental and practical points of view. This work develops a scattering framework for two identical one-dimensional (1D) bosonic anyons and two identical 1D fermionic anyons with zero-range contact interactions. The two-body system with zero-range interactions, in both free space and under external confinement, is used to illustrate the recently proposed bosonic-anyon---fermionic-anyon mapping (R. Hidalgo-Sacoto et al., Phys. Rev. A 112, 063309 (2025)), which connects the eigenstates of bosonic anyons to those of fermionic anyons and vice versa. Performing explicit calculations for two-particle systems, the momentum distributions and the off-diagonal correlations of the single-particle density matrix for bosonic anyons and fermionic anyons are confirmed to be distinct. We also confirm the previously derived asymptotic coefficients of the momentum distribution tail at orders ${k}^{\ensuremath{-}2}$ and ${k}^{\ensuremath{-}3}$ for two harmonically confined anyons. Nonuniversal contributions at order ${k}^{\ensuremath{-}4}$ are discussed.
The connection between an observable invariance property of the scattering matrix and the interaction, VVV, generating the scattering, is studied for a particlar case: rotational invariance in nonrelativistic "potential" scattering. It is shown that a determination that the scattering cross section depends only on the angle between the incident and outgoing beams by no means implies that VVV is invariant under rotation. This is true even if coherent incident beams are used to probe the target. Some aspects of these results and their relevance for the construction of physical theories are briefly discussed.
The quantum-mechanical description of assemblies of particles whose motion is confined to two (or one) spatial dimensions offers many possibilities that are distinct from bosons and fermions. We call such particles anyons. The simplest anyons are parameterized by an angular phase parameter θ. θ = 0, π correspond to bosons and fermions, respectively; at intermediate values, we say that we have fractional statistics. In two dimensions, θ describes the phase acquired by the wave function as two anyons wind around one another counterclockwise. It generates a shift in the allowed values for the relative angular momentum. Composites of localized electric charge and magnetic flux associated with an abelian U(1) gauge group realize this behavior. More complex charge-flux constructions can involve nonabelian and product groups acting on a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual statistics. Interchanges of nonabelian anyons implement unitary transformations of the wave function within an emergent space of internal states. Anyons of all kinds are described by quantum field theories that include Chern–Simons terms. The crossings of one-dimensional anyons on a ring are unidirectional, such that a fractional phase θ acquired upon interchange gives rise to fractional shifts in the relative momenta between the anyons. The quasiparticle excitations of fractional quantum Hall states have long been predicted to include anyons. Recently, the anyon behavior predicted for quasiparticles in the ν = 1/3 fractional quantum Hall state has been observed in both scattering and interferometric experiments. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.
A self-contained discussion of nonrelativistic quantum mechanical potential scattering in two dimensions is presented. The discussion includes, among other topics, partial wave decomposition in coordinate and momentum space, Lippmann–Schwinger integral equations of scattering for the scattering wavefunction and the transition operator, optical theorem, and the unitarity relation for the transition operator. The present definition of the scattering amplitude in terms of the asymptotic wavefunction differs from the usual definition. The present definition has certain advantages, for example, in writing the optical theorem and in studying the analytical properties of the scattering amplitude.