Braid group representations and cold Fermi gases in the fast pairing regime

TL;DR

This paper explores using cold Fermi gases and braid group representations to develop topologically protected quantum information processing methods, connecting hyperelliptic functions, Hecke operators, and quantum Hamiltonian limits.

Contribution

It introduces a novel approach employing hyperelliptic functions and braid group representations in cold Fermi gases for quantum information processing, linking algebraic structures with physical systems.

Findings

Homological structure of complex curves relates to braid group representations.

Hecke operators induce detectable singularities in semiclassical oscillations.

The Richardson-Gaudin Hamiltonian limit connects to level-k ff(sl(2)) with fixed q.

Abstract

It is widely recognized that the main difficulty in designing devices which could process information using quantum states is due to the decoherence of local excitations about a ground state. A solution to this problem was suggested in \cite{Kitaev}, relying on (non-local) topological excitations, structurally protected against local noise. However, a practical implementation of this proposal using special Landau levels in fractional quantum Hall effect systems (FQHE) \cite{QHE} has proven elusive, while accessible FQHE states are theoretically not optimal because their representations in the Hilbert space of states are not dense. We propose using a different physical system (cold Fermi atoms), whose semiclassical dynamics is described by a hyperelliptic function in the Sklyanin formalism. The homological structure of the complex curve corresponds to representations of the braid group, with the action of Hecke operators leading to singularities detectable in the semiclassical oscillations. We argue that, for a fixed genus of the hyperelliptic curve, the Richardson-Gaudin pairing Hamiltonian problem is the singular limit $m \to \infty$ of level-$k$ $\widehat{sl}(2)$, with $k+2 = \frac{4}{8m+1}\to 0$, so that the level $k$ is admissible in the sense of Kac and Kazhdan \cite{KK}, but the corresponding Hecke algebra is a $q-$deformation of the symmetric group with fixed $q = e^{i\pi/4}$, as $m \to \infty$.