Path-dependent conductivity in the regime of floating delocalized states

We present measurements of the longitudinal conductivity minima in the density--magnetic-field plane of a low-density, strongly disordered two-dimensional electron gas in a gated GaAs/Alₓ{Ga}_{1{{-}}x}$As heterostructure. We find the position in the n-B phase diagram of the minimum in ${{{σ}}}ₓₓ$ to be strongly path dependent in the region where the floating of delocalized states is observed. While the position of the minimum in the n sweep is not universal, the position in the B sweep is always found to correspond to an integer value of nh/eB. In an attempt to understand the origin of the floating, we offer a simple model, consistent with experiment, that directly links floating to Landau-level mixing.