Tails of the density of states in a random magnetic field

Abstract

We study the tails of the density of states of fermions subject to a random magnetic field with non-zero mean with the Optimum Fluctuation Method (OFM). Closer to the centres of the Landau levels, the density of states is found to be Gaussian, whereas the energy dependence is non-analytic near the lower bound of the spectrum.PACS numbers: 71.23. An, 73.43.Cd, 73.43.Nq The problem of a charged quantum particle constrained to move in a two dimensional (2D) static random magnetic field (RMF) has attracted considerable theoretical and experimental interest in the past few years. The model plays an important role within the composite fermion picture of the fractional quantum Hall effect [1]. Furthermore, it is supposed to describe states with spin-charge separation in high-T c superconductors [2]. It is also relevant to the understanding of the properties of a two dimensional electron gas (2DEG) in latticemismatched InAs/InGaAs heterostructures in magnetic fields [3]. In the latter systems the electron gas is nonplanar due to the lattice-mismatched epitaxial growth. When a uniform magnetic field B is applied, the electrons experience an effective inhomogeneous field perpendicular to the non-planar 2DEG [3]. In addition, a static RMF in 2D inversion layers can be experimentally realized in several ways. One possibility is to use a type-II superconductor with a disordered Abrikosov flux lattice in an external magnetic field as the substrate for the 2DEG [4]. Alternatively, a magnetically active substrate such as a demagnetized ferromagnet with randomly oriented magnetic domains may be used [5]. Recently, static RMFs in 2DEGs were created by applying strong magnetic fields parallel to GaAs Hall-bars decorated with randomly patterned magnetic films [6].The most fundamental quantity for understanding the electonic properties of a random system is the density of energy levels. The standard method to estimate the density of states (DOS) is to calculate the imaginary part of the trace of the single-particle Green function by diagrammatic techniques. However, this approach fails in the tails of an energy band where multiple scattering up to infinite order has to be considered in order to take into account correctly the effect of localisation of electrons. Also, numerical approaches are bound to fail in the asymptotic tails since here the eigenstates are determined by rare statistical fluctuations of the randomness. Moreover, in the case of RMF, the perturbative approach is also fundamentally problematic since one has to deal with the non-diagonal part of the Green function, which is not gauge invariant. In addition, the calculation of the Green function is plagued by infrared divergencies [7,8,9,10,11] that are due to the long-range nature of the correlations of the vector potential, even if the spatial correlations in the RMF are short-ranged. It has been suggested that these divergencies are due to the non-gauge-invariance of the Green function and therefore unphysical [9], although, recently, a physical interpretation h...

Signatures of spin in theν=1∕3fractional quantum Hall effect

“…29 Only recently, the tails of the Landau bands of fermions in a RMF with nonvanishing average have been studied. 30 In our case, where ᐉ CF Ӷ , the fermionic states are essentially cyclotron orbits in presence of the local ͑smoothly varying͒ effective magnetic field B * ͑r͒ = B * + ␦B͑r͒, with ␦B͑r͒ the RMF. The related typical energy shift ⌫ RMF is the cyclotron energy បe␦B͑r͒ / m * and the RMF acts pretty much like a scalar potential.…”