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Register a new userEnergy-level statistics and localization of 2d electrons in random magnetic fields
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M. Batsch
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Using the method of energy-level statistics, the localization properties of electrons moving in two dimensions in the presence of a perpendicular random magnetic field and additional random disorder potentials are investigated. For this model, extended states have recently been proposed to exist in the middle of the band. In contrast, from our calculations of the large-$s$ behavior of the nearest neighbor level spacing distribution $P(s)$ and from a finite size scaling analysis we find only localized states in the suggested energy and disorder range.
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The longitudinal resistivity of two dimensional (2D) electrons placed in strong magnetic field is significantly reduced by applied electric field, an effect which is studied in a broad range of magnetic fields and temperatures in GaAs quantum wells with high electron density. The data are found to be in good agreement with theory, considering the strong nonlinearity of the resistivity as result of non-uniform spectral diffusion of the 2D electrons. Inelastic processes limit the diffusion. Comparison with the theory yields the inelastic scattering time of the two dimensional electrons. In the temperature range T=2-10(K) for overlapping Landau levels, the inelastic scattering rate is found to be proportional to T^2, indicating a dominant contribution of the electron-electron scattering to the inelastic relaxation. In a strong magnetic field, the nonlinear resistivity demonstrates scaling behavior, indicating a specific regime of electron heating of well-separated Landau levels. In this regime the inelastic scattering rate is found to be proportional to T^3, suggesting the electron-phonon scattering as the dominant mechanism of the inelastic relaxation.
We study level statistics of a critical random matrix ensemble of a power-law banded complex Hermitean matrices. We compute numerically the level compressibility via the level number variance and compare it with the analytical formula for the exactly solvable model of Moshe, Neuberger and Shapiro.
This paper provides a review of developments in the physics of two-dimensional electron systems in perpendicular magnetic fields.
We develop a comprehensive theory for the effective dynamics of Bloch electrons based on symmetry. We begin with a scheme to systematically derive the irreducible representations (IRs) characterizing the Bloch functions. Starting from a tight-binding (TB) approach, we decompose the TB basis functions into localized symmetry-adapted atomic orbitals and crystal-periodic symmetry-adapted plane waves. Each of these subproblems is independent of the details of a particular crystal structure and it is largely independent of the other subproblem, hence permitting for each subproblem an independent universal solution. Taking monolayer MoS$_2$ and few-layer graphene as examples, we tabulate the symmetrized $p$ and $d$ orbitals as well as the symmetrized plane wave spinors for these systems. The symmetry-adapted basis functions block-diagonalize the TB Hamiltonian such that each block yields eigenstates transforming according to one of the IRs of the group of the wave vector $G_k$. For many crystal structures, it is possible to define multiple distinct coordinate systems such that for wave vectors $k$ at the border of the Brillouin zone the IRs characterizing the Bloch states depend on the coordinate system, i.e., these IRs of $G_k$ are not uniquely determined by the symmetry of a crystal structure. The different coordinate systems are related by a coordinate shift that results in a rearrangement of the IRs of $G_k$ characterizing the Bloch states. We illustrate this rearrangement with three coordinate systems for MoS$_2$ and tri-layer graphene. Using monolayer MoS$_2$ as an example, we combine the symmetry analysis of its bulk Bloch states with the theory of invariants to construct a generic multiband Hamiltonian for electrons near the $K$ point of the Brillouin zone. The Hamiltonian includes the effect of spin-orbit coupling, strain and external electric and magnetic fields.
We numerically study the level statistics of the Gaussian $beta$ ensemble. These statistics generalize Wigner-Dyson level statistics from the discrete set of Dyson indices $beta = 1,2,4$ to the continuous range $0 < beta < infty$. The Gaussian $beta$ ensemble covers Poissonian level statistics for $beta to 0$, and provides a smooth interpolation between Poissonian and Wigner-Dyson level statistics. We establish the physical relevance of the level statistics of the Gaussian $beta$ ensemble by showing near-perfect agreement with the level statistics of a paradigmatic model in studies on many-body localization over the entire crossover range from the thermal to the many-body localized phase. In addition, we show similar agreement for a related Hamiltonian with broken time-reversal symmetry.
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