Localization and Metal-Insulator Transition in Multi-Layer Quantum Hall Structures

We study the phase structure and Hall conductance quantization in weakly coupled multi-layer electron systems in the integer quantum Hall regime. We derive an effective field theory and perform a two-loop renormalization group calculation. It is shown that (i) finite interlayer tunnelings (however small) give rise to successive metallic and insulating phases and metal-insulator transitions in the unitary universality class. (ii) The Hall conductivity is not renormalized in the metallic phases in the 3D regime. (iii) The Hall conductances are quantized in the insulating phases. In the bulk quantum Hall phases, the effective field theory describes the transport on the surface. PACS numbers: 73.50.Jt, 73.40.Hm, 72.15.Rn, 71.30.+h Typeset using REVTEX 1 The quantum Hall effect (QHE) in a two-dimensional electron gas (2DEG) has led to many new physical concepts and principles [1]. The main part of the phenomenology is that in strong magnetic fields, a 2DEG exhibits continuous zero-temperature phase transitions between successive quantum Hall states of vanishing dissipation and quantized Hall resistances. It is interesting to ask what happens to the physics associated with the QHE in dimensions greater than two [2]. Experimentally, two classes of quasi-three-dimensional materials have been found to show integer-quantized Hall plateaus: multi-layer quantum wells or superlattices formed by GaAs/AlGaAs graded heterostructures [3,4] and molecular crystals (TMTSF)2X (X=PF6,ClO4) [5]. In this paper, we concentrate on the former which is a natural generalization of the QHE above two dimensions. Thus, the changes in the phase structure and the properties of the phase transitions in quantum Hall layers coupled by weak interlayer tunnelings are the concerns of the present paper. Moreover, we focus on the integer quantum Hall regime and ignore the effects of electron-electron interactions [6]. We shall follow the approach of Chalker and Dohmen who generalized the ChalkerCoddington network model [7] for the integer QHE (IQHE) in a 2DEG to layers of networks coupled by interlayer tunnelings [8]. The advantage of this approach is that the single layer network model is known to correctly describe the universality class of the 2D integer quantum Hall transitions. Chalker and Dohmen performed numerical transfer matrix calculations and demonstrated the existence of three phases: insulator, metal, and quantized Hall conductor, and extended surface states in the quantized Hall state. In this paper we provide an analytical treatment using the effective field theory representation of the network model [9–11]. We first show that the long wavelength transport properties are governed by a 3D anisotropic unitary nonlinear σ-model (NLσM). The anisotropic couplings are the dissipative conductivities, whereas the Hall conductivity enters as a coupling to the layered sum of the 2D topological term. The renormalization group (RG) is then used to study the crossover between two and three dimensions. In the 3D regime, the RG flow equations for the conductances are calculated to two-loop order to determine the phase structure in the plane spanned by the Fermi energy and the interlayer tunneling. The results show that a 2 finite interlayer coupling (however weak) leads to metallic and insulating phases and metalinsulator transitions in the unitary universality class. Furthermore, the Hall conductivity is found to be unrenormalized by localization effects in the 3D regime. We show that the Hall conductance is quantized in the insulating phases provided that the above results hold to all orders in the RG. Finally, we demonstrate that in the quantum Hall phases, the field theory reduces to the one appropriate for the coupled edge states on the surface. Following Ref. [10], the Hamiltonian for N layered networks in the (x, y)-plane coupled in the z-direction by interlayer tunneling t⊥ is, H0 = ∑