Edges of Fractional Quantum Hall Phases in a Cylindrical Geometry
Paul Soulé, LPTMS
Fractional Quantum Hall (FQH) phases are exotic incompressible fluids which support gapless chiral edge excitations. I will present a microscopic study of those edges states in a cylindrical geometry where quasiparticles are able to tunnel between edges.
We first study the principal FQH phase at the filling fraction 1/3 whose ground state is well described by the Laughlin wave function. For an energy scale lower than the bulk gap, the effective theory is given by a very special one dimensional electron fluid localized at the edge: a chiral Luttinger liquid. Using numerical exact diagonalizations, we study the spectrum of edge modes formed by the two counter-propagating edges on each side of the cylinder. We show that the two edges combine to form a non-chiral finite-size Luttinger liquid, where the current term reflects the transfer of quasiparticles between edges. Then, we estimate numerically the Luttinger parameter for a small number of particles and find it coherent with the one predicted by X. G. Wen theory.
We then analyse edge modes of the FQH phase at filling fraction 5/2. From a Conformal Field Theory (CFT) based construction, Moore and Read (Nucl. Phys. B, 1991) proposed that this phase is well described by a P-wave paired state of composite fermions. A striking property of this state is that emergent excitations braid with non-abelian statistics. When localized along the edge, those excitations are described through a chiral boson and a Majorana fermion. In the cylinder geometry, we show that the spectrum of edge excitations is composed of all conformal towers of the IsingXU(1) model. Interestingly, the non-abelian tower is naturally observed as opposed to the usual disk geometry. In addition, with a Monte Carlo method, we estimate the various scaling dimensions for large systems (about 50 electrons), and find them consistent with the CFT predictions.