Many-body study of a quantum point contact in the fractional quantum Hall regime atν=5/2

Abstract

We study a quantum point contact in the fractional quantum Hall regime at Landau level filling factors ν = 1/3 and ν = 5/2. By using exact diagonalizations in the cylinder geometry we identify the edge modes in the presence of a parabolic confining potential. By changing the sign of the potential we can access both the tunneling through the bulk of the fluid and the tunneling between spatially separated droplets. This geometry is realized in the quantum point contact geometry for two-dimensional electron gases. In the case of the model Moore-Read Pfaffian state at filling factor ν = 5/2 we identify the conformal towers of many-body eigenstates including the non-Abelian sector. By a Monte-Carlo technique we compute the various scaling exponents that characterize the edge modes. In the case of hard-core interactions whose ground states are exact model wavefunction we find equality of neutral and charged velocities for the Pfaffian state both bosonic and fermionic.

Absence of a gap in the Gaffnian state

“…These towers are analogous to those found in the Pfaffian case [31] in the sense that the left and right moving modes are generated by the same operator. In these formulas n is an integer that varies as we go from one tower to its neighbor.…”

“…The towers are based on all primary operators of the theory and in the case of an odd number of particles they are heterotic, i.e., their left and right movers are created by different operators, at variance with the Laughlin [38] or Moore-Read case [31]. These towers appear however only in the thin-torus limit which has no thermodynamic limit.…”

“…Indeed, we show that there are well-defined edge modes that can be classified according to the underlying CFT. As in the case of the Pfaffian [31] the edge modes form conformal towers of the Virasoro algebra. These special sets of states are generated by primary operators of the CFT.…”

“…If we now add a shallow confining potential along the axis of the cylinder the degeneracy is lifted and it is possible to identify the edge excitations as conformal towers in the low-energy points of the spectrum. This is possible in the Laughlin [38] and Pfaffian [31] cases. This picture is valid as long as there is a clear separation between the set of quasihole-derived states and the bulk states.…”

Bubble phase at ν= 13 for spinless hollow-core interaction

“…where we have defined This formulation of the FQHE has already been studied by exact diagonalization [25][26][27] . The thermodynamic limit is reached for large number of particles but also by sending L to infinity at the same time.…”