Cascade of Quantum Phase Transitions in Tunnel-Coupled Edge States

Abstract

We report on the cascade of quantum phase transitions exhibited by tunnel-coupled edge states across a quantum Hall line junction. We identify a series of quantum critical points between successive strong and weak tunneling regimes in the zero-bias conductance. Scaling analysis shows that the conductance near the critical magnetic fields Bc is a function of a single scaling argument |B − Bc|T −κ , where the exponent κ = 0.42. This puzzling resemblance to a quantum Hall-insulator transition points to importance of interedge correlation between the coupled edge states. PACS numbers: 73.43.Jn, 73.43.Nq Edge states in the quantum Hall effect provide a highly tunable system for the study of quantum transport in one-dimension [1]. Following the prediction of chiral Luttinger liquids in the fractional quantum Hall effect [2,3], extensive effort has been devoted to the study of tunneling between quantum Hall edge states [4,5,6,7,8,9,10,11]. Tunneling of an electron into a Luttinger liquid is strongly suppressed and theories predict a powerlaw tunneling conductance with a universal exponent related to the quantum number of the bulk quantum Hall liquid. Experimental studies of tunneling between edge states across a quantum point contact [8] and tunneling between an edge state and a three-dimensional metal [9,10] have generally tended to support the predicted Luttinger liquid behavior. However, there remain important open questions regarding the experimentally observed exponent and its correlation to the bulk quantum Hall states [11].A different and perhaps more intriguing geometry for the study of edge state tunneling involves a line junction that juxtaposes two parallel, counterpropagating edge modes against each other. Such a junction has been initially envisioned as a Hall bar with a long narrow gate that couples two right and left moving edge channels of fractional quantum Hall liquids [12,13]. In the limit of weak bias, the conductance across the line junction remains quantized as backscattering between the edge states is negligible. For strong bias, interedge backscattering is suppressed and the conductance across the line junction vanishes. In between the two limits, the inter-mode backscattering is mediated by defects in the line junction and a metal-insulator transition is predicted [12,13]. The transition is characterized by a temperature dependent conductivity that vanishes in the insulating phase and diverges in the metallic state in the limit of zero temperature.Confirmation of the predicted metal-insulator transition has remained elusive as lithographic limitations and vertical offset of the gates from the plane of twodimensional electrons complicate the realization of a linejunction. An alternate approach to a line junction involves taking advantage of the inherent atomic precision of molecular beam epitaxy (MBE) and inserting a precisely defined semiconductor barrier in the plane of two-dimensional electron system through the technique of cleaved edge overgrowth [14,15]. Such a junction strongly couples...

Shubnikov–de Haas oscillations, peaks, and different temperature regimes of the diagonal conductivity in the integer quantum Hall conductor

“…This hopping mechanism remains in effect if the Landau orbits lay within the same conducting plane or belong to the different tunnel-coupled 2D conductors. The latter is important in view of recent observation of a typical IQHE behavior in the tunneling conductance of a two coupled Hall bars reported in [7]. The tunneling conductance in this experiment displays the same scaling features as those usually observed in the bulk Hall sample and, therefore, can not be explained by the tunneling between the two counter-propagating edge states.…”

“…It explains why the square-root exponent (corresponding to a 1D system) appears in the VRH conductivity s xx of a 2D system. The tunneling-conductance oscillations in a two coupled Hall bars observed in [7] display a standard IQHE behavior which can not be understood as a tunneling between the two counter-propagating edge states. The puzzle resolves naturally in our model.…”

“…On the other hand, electron tunneling between Landau orbits from different Hall bars and within the same sample equally contribute into the total conductance. In view of that one can anticipate the standard IQHE behavior in the tunneling conductance observed in experiment [7].…”

Tunneling in fractional quantum Hall line junctions

“…Recently, progress in the technique of cleaved-edge overgrowth [2,3] has led to the fabrication of samples in which the tunneling now occurs along a barrier of mesoscopic extent between two spatially separated two-dimensional electron gases. Kang et al [2,4] have performed detailed studies of the conductance of these new structures. Their samples consist of two-dimensional electron gases (2DEGs) separated by an atomically precise barrier of length 100 µm and of width 8.8 nm.…”

AC conductivity of a quantum Hall line junction

“…A LJ is formed by using a gate voltage to create a narrow barrier which divides a fractional QH state such that there are two chiral edges flowing in opposite directions (counter propagating) on the two sides of the barrier [13,14,15,16,17]. For a QH system corresponding to a filling fraction which is the inverse of an odd integer such as 1, 3, 5, · · · , the edge consists of a single mode which can be described by a chiral bosonic theory [18].…”