Band of Critical States in Anderson Localization in a Strong Magnetic Field with Random Spin-Orbit Scattering

Abstract

Anderson localization problem for non-interacting two-dimensional electron gas subject to strong magnetic field, disordered potential and spin-orbit coupling is studied numerically on a square lattice. The nature of the corresponding localization-delocalization transition and the properties of the pertinent extended states depend on the nature of the spin-orbit coupling (uniform or fully random). For uniform spin-orbit coupling (such as Rashba coupling), there is a band of extended states in the center of a Landau band as in a "standard" Anderson metal-insulator transition. However, for fully random spin-orbit coupling, the familiar pattern of Landau bands disappears. Instead, there is a central band of critical states with definite fractal structure separated at two critical energies from two side bands of localized states. Moreover, finite size scaling analysis suggests that for this novel transition, on the localized side of a critical energy Ec, the localization length diverges as ξ(E) ∝ exp(α/ |E − Ec|), a behavior which, along with the band of critical states, is reminiscent of a Berezinskii-Kosterlitz-Thouless transition. Traditionally, non-interacting disordered electronic systems subject to disordered potential are classified according to the symmetries of their Hamiltonian with respect to time reversal (TR) and spin rotation (SR) transformations. Considering the Hamiltonian as a random matrix [1-3], its symmetries determine to which random matrix Gaussian ensemble (also referred to as universality class) it belongs, orthogonal (both TR and SR symmetries are satisfied), symplectic (only TR symmetry) or unitary.This classification is intimately related to one of the most fundamental concepts in the physics of disordered electronic systems: the Anderson localization transition (ALT) [4,5] that is a quantum phase transition between localized and extended states in a disordered system. The critical dimension for existence or non-existence of ALT is d = 2. For d < 2 there is no ALT while for d > 2 there is always ALT. Hence, for two-dimensional electron gas (2DEG), ALT (if it exists) is of special interest. The scaling theory of localization [6] (developed before the discovery of the quantum Hall effect) together with calculations based on nonlinear sigma model [7,8], established that for d = 2, ALT does not exist for the orthogonal and unitary classes (zero or finite magnetic field, respectively) and does exist for the symplectic class (finite spin-orbit scattering and zero magnetic field). After the discovery of the integer Quantum Hall effect (IQHE) [9], topology was also recognized as a property determining the pertinent universality class [10]. Mathematically, the role of topology in the IQHE is quantified by the occurrence of a topological term in the action of the corresponding non-linear sigma model [11]. In the presence of the topological term it is established that if SR invariance is respected, the system is in the IQHE universality class characterized by a Hall transition between localized and cr...

Level statistics of extended states in random non-Hermitian Hamiltonians

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Metal to marginal-metal transition in two-dimensional ferromagnetic electron gases

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Spin-mixing-tunneling network model for Anderson transitions in two-dimensional disordered spinful electrons

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