Application of Braid Groups in 2D Hall System Physics

Bilayer Graphene as the Material for Study of the Unconventional Fractional Quantum Hall Effect

“…The strong Coulomb repulsion of electrons determines the steady uniform distribution of 2D electrons-in the form of triangle Wigner crystal lattice as the classical lowest energy state. Such a distribution of electrons rigidly fixed (in classical sense) by interaction is the starting point for quantization in terms of Feynman path integral including summation on braid group elements assigned with appropriate unitary weight supplied by 1DUR of the braid group [5]. The latter is necessary in the case of multiparticle system as to any path in the multiparticle configuration space an arbitrary closed loop from the braid group can be attached and the resulting paths with such loops fall into geometrically disjoint sectors (topologically non-equivalent) of the path space.…”

“…The loop size may be accommodated to xth-order next-nearest neighbors independently. In particular, it results in the additional hierarchy in all subbands of the first LL, ν ¼ 2ð3; 4; 5Þþ 5 . This opportunity for FQHE pretty well agrees with the recent observations of FQHE in three first subbands of the n = 1 LL in monolayer graphene at ultra-low temperatures [16].…”

“…In order to decipher this enigmatic character of CFs and the topological correlations in FQHE, the algebraic-topology braid group-based approach has been developed and applied to FQHE [5]. Within this approach, the trajectories of interchanging positions of particles on the 2D plane in the system of N charged particles upon strong perpendicular magnetic field are considered including quantum indistinguishability of identical particles.…”

“…1DUR = 1 defines bosons and 1DUR = -1 defines fermions in 3D multiparticle systems. For 2D multiparticle systems, the braid group is different than the permutation group [5][6][7] and 1DURs are different σ j ! e iα ; α ∈ ðÀπ;π and define 2D anyons (including 2D fermions for α = π and bosons for α = 0).…”

“…In the braid group are, however, braids σ q j , which for q odd integer similarly as σ j define exchanges of jth and (j+1)th particles. In distinction to σ j , the braids σ q j realize exchange with additional qÀ1 2 loops [5]. This additional loops "take away" qÀ1 2 flux quanta.…”