Molecular Impurities as a Realization of Anyons on the Two-Sphere

Emergence of Anyons on the Two-Sphere in Molecular Impurities

“…In reference [35], approximately constant α (depending on Ω) is indeed realized for a suitable, and experimentally feasible, choice of ω and c l . Particularly, ω is chosen at the roton minimum of the dispersion relation of superfluid helium, which allows us to achieve a gapped dispersion, and the coupling c l is described by the model interaction used in order to describe angulon instabilities and oscillations observed in the experiment.…”

“…Lastly, we stress that the Born-Oppenheimer approximation itself and the emergence of the exact anyonic spectrum, i.e., the spectrum of the Hamiltonian (1), was justified both analytically and numerically for simpler but highly representative models in [25,35].…”

“…A sufficient condition for Φ(q 1 , q 2 )|b † • Λ e • b|Φ(q 1 , q 2 ) being continuous would be the following growth condition on the coefficients: |c l | ≤ C l and ω l ± Ωl ≥ l 1+ with > 0. While the former condition can be fulfilled with the model parameter describing the molecule-helium interaction, see reference [35], the latter condition can be satisfied by considering a strong magnetic field. Then, Ã −→ q 1 ,q 2 →S 0 vanishes for q 1 , q 2 close to the south pole, in contrast to the singular gauge field ∇ j φ which has a pole at S.…”

“…However, since the statistical behaviour of anyons depends on the topology, and even more importantly on the geometry and symmetry, of the underlying space, investigations on curved spaces can demonstrate novel properties of quantum statistics [2,[26][27][28][29][30][31][32] (see also graph geometries [33,34]). Indeed it has been recently demonstrated that the emerging fractional statistics for particles restricted to move on the sphere, instead of on the plane, arises naturally in the context of quantum impurity problems, particularly, in the context of molecular impurities [35]. There, it has been shown that the emerging statistical interaction manifests itself in the alignment of molecules, which could also be of use as a powerful technique to measure the statistics parameter.…”

“…Time-reversal symmetry is broken by using an additional external magnetic field and applying rotation, while the Born-Oppenheimer approximation is satisfied by considering heavy molecules. We further discuss and supply some technical details of the argument that had been left out in [35]. Note that the phenomenon of quantum statistics transmutation typically involves emergent scalar interaction potentials and non-statistical gauge fields as well, and it is necessary to have sufficient control of these effects in order to provide robust signatures of anyons.…”