Weak Field Phase Diagram for an Integer Quantum Hall Liquid

Abstract

We study the localization properties in the transition from a two-dimensional electron gas at zero magnetic field into an integer quantum Hall (QH) liquid. By carrying out a direct calculation of the localization length for a finite size sample using a transfer matrix technique, we systematically investigate the field and disorder dependences of the metal-insulator transition in the weak field QH regime. We obtain a different phase diagram from the one conjectured in previous theoretical studies. In particular, we find that: (1) the extended state energy Ec for each Landau level (LL) is always linear in magnetic field; (2) for a given Landau level and disorder configuration there exists a critical magnetic field Bc below which the extended state disappears; (3) the lower LLs are more robust to the metal-insulator transition with smaller Bc. We attribute the above results to strong LL coupling effect. Experimental implications of our work are discussed.PACS numbers: 71.30.+h, 73.20.Jc, 73.40.Hm It is very important to understand the localization properties in the transition from two-dimensional electron gas (at zero magnetic field) into an integer quantum Hall liquid [1]. According to the scaling theory of localization [2] all electrons in a two-dimensional system are localized in the absence of magnetic field. When the twodimensional electron system is subject to a strong perpendicular magnetic field, the energy spectrum becomes a series of impurity broadened Landau levels. Extended state appears in the center of each Landau band, while states at other energies are localized. This gives rise to the integer quantum Hall effect. The interesting issue is to understand the evolution of the extended states in the weak field regime as the magnetic field goes to zero where all extended states disappear.There could be two scenarios for the fate of the extended states as B → 0. The first one was proposed by Kivelson, Lee, and Zhang [3] in their global phase diagram of the quantum Hall effect. According to this phase diagram, in a strongly disordered quantum Hall system, the extended states stay with the center of the Landau bands at strong magnetic field, but float up in energy at small magnetic field and go to infinity as B → 0. This phase diagram is consistent with the semiclassical argument put forth by Khmelnitskii [4] and Laughlin [5].In this letter, we are proposing an alternative scenario for the behavior of the extended states at weak magnetic field limit. In our picture, each extended state is simply destroyed by strong disorder at a critical magnetic field instead of floating up in energy. By carrying out a direct calculation of the localization length for a finite size sample using a transfer matrix technique, we systematically investigate the field and disorder dependence of the metal-insulator transition in the weak field quantum Hall regime. We find that: (1) the extended state energy E c for each Landau level (LL) is always linear in magnetic field; (2) for a given Landau level and disorder configurat...

Anti-levitation in integer quantum Hall systems

“…Numerical simulations on a lattice are also not conclusive. For whitenoise random on-site energy, levitation is not substantiated [7] while for finite range correlated disorder, weak levitation is predicted [22][23][24]. In this paper we revisit this issue by focusing on the evolution of extended states in the lowest LB (n = 0) using a square lattice geometry and white-noise random on-site energy.…”

“…In order to add more quantitative perception, it is vital to elucidate the behavior of extended states in LBs as the magnetic field gradually decreases to zero, or as the disorder gradually increases. Different answers to this question lead to different global phase diagrams [6][7][8] for the IQHE.…”

“…In Refs. [6][7][8] the scenario of Chern number annihilation as B → 0 has been discussed on a qualitative level. In order to add more quantitative perception, it is vital to elucidate the behavior of extended states in LBs as the magnetic field gradually decreases to zero, or as the disorder gradually increases.…”

Integer quantum Hall effect in double-layer systems

“…(Here ω c = eB/m * c is the cyclotron frequency.) The loss of an observable quantum Hall effect in these systems appears to be associated with a dramatic increase in the localization length in the middle of the Hall plateaus, rather than with the 'floatation' of extended state energies [31][32][33][34][35][36][37] which occurs in more strongly disordered systems. It seems likely that the same dramatic increase in localization lengths on Hall plateaus will occur in doublelayer systems when the Landau levels in the two-layers are strongly mixed.…”

Possible existence of a band of extended states induced by inter-Landau-band mixing in a quantum Hall system

“…The integrally quantized Hall plateaus (IQHP) are observed when the Fermi level lies in localized states, with the value of the Hall conductance, σ xy = ne 2 /h, related to the number of occupied extended states(n). Many previous studies 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23 have been focused on so-called plateau transitions. The issue there is how the Hall conductance jumps from one quantized value to another when the Fermi level crosses an extended state.…”