Topological two-dimensional quantum mechanics

Abstract

We define a Chern-Simons Lagrangian for a system of planar particles topologically interacting at a distance. The anyon model appears as a particular case where all the particles are identical. We propose exact N-body eigenstates, set up a perturbative algorithm, discuss the case where some particles are fixed on a lattice, and also consider curved manifolds.

Quantum impurity model for anyons

“…In general, one can derive the statistics gauge field within Chern-Simons theory [49][50][51][52]. The action of a system of non-relativistic charged particles with mass M coupled to the Abelian Chern-Simons gauge field A µ is given by…”

“…This problem can be overcome with the similarity transfor-mationĤ 2-anyon = r −αĤ 2-anyon r α , which is a self-adjoint Hamiltonian in the L 2 space weighted by the measure r 2α+1 dr. In the transformed Hamiltonian the divergent term vanishes [52]. In fact, this transformation corresponds to a 'real gauge transformation' leading to an imaginary vector potential.…”

A Quantum Impurity Model for Anyons

“…In general, one can derive the statistics gauge field within Chern-Simons theory [49][50][51][52]. The action of a system of non-relativistic charged particles with mass M coupled to the Abelian Chern-Simons gauge field A µ is given by…”

“…This problem can be overcome with the similarity transformation Ĥ2-anyon = r −α Ĥ2-anyon r α , which is a self-adjoint Hamiltonian in the L 2 space weighted by the measure r 2α+1 dr. In the transformed Hamiltonian the divergent term vanishes [52]. In fact, this transformation corresponds to a 'real gauge transformation' leading to an imaginary vector potential.…”

Anyonic quasiparticles of hardcore anyons

“…The point flux tubes are included in the "statistical" part of the magnetic vector potential in the many-body Hamiltonian with the help of a singular gauge transformation. One can obtain the exact ground-state basis for such a continuum many-anyon Hamiltonian, which is entirely confined to the lowest Landau level [16][17][18]. A recent work has introduced variational ansätze for the ground state of the above Hamiltonian [19,20].…”