Bloch Electrons In Magnetic Field Research Articles

“More Is Different” and Sustainable Development Goals: Thermoelectricity

The thermal Green's function formalism bridging between macroscopic observables and microscopic processes via linear response theory was established in the early 1960s, when I started my research career. I recall stimulating experiences with the help of this technique in exploring transport and thermodynamic properties of Bloch electrons in magnetic fields, especially orbital magnetism and the Hall effect, and this technique is useful for understanding narrow gap systems like Dirac and Weyl electrons, which are of current interest. Recent ongoing challenges on thermoelectricity based on the Kubo–Luttinger formula are briefly introduced; it is an important scientific issue in view of sustainable development goals that awaits contributions from condensed matter physics, where the thermal Green's function is again a powerful tool.

Topologically universal spectral hierarchies of quasiperiodic systems

Topological properties of energy spectra of general one-dimensional quasiperiodic systems, describing also Bloch electrons in magnetic fields, are studied for an infinity of irrational modulation frequencies corresponding to irrational numbers of flux quanta per unit cell. These frequencies include well-known ones considered in works on Fibonacci quasicrystals. It is shown that the spectrum for any such frequency exhibits a self-similar hierarchy of clusters characterized by universal (system-independent) values of Chern topological integers which are exactly determined. The cluster hierarchy provides a simple and systematic organization of all the spectral gaps, labeled by universal topological numbers which are exactly determinable, thus avoiding their numerical evaluation using rational approximants of the irrational frequency. These numbers give both the quantum Hall conductance of the system and the winding number of the edge-state energy traversing a gap as a Bloch quasimomentum is varied.

Hofstadter butterflies in nonlinear Harper lattices, and their optical realizations

The ubiquitous Hofstadter butterfly describes a variety of systems characterized by incommensurable periodicities, ranging from Bloch electrons in magnetic fields and the quantum Hall effect to cold atoms in optical lattices and more. Here, we introduce nonlinearity into the underlying (Harper) model and study the nonlinear spectra and the corresponding extended eigenmodes of nonlinear quasiperiodic systems. We show that the spectra of the nonlinear eigenmodes form deformed versions of the Hofstadter butterfly and demonstrate that the modes can be classified into two families: nonlinear modes that are a ‘continuation’ of the linear modes of the system and new nonlinear modes that have no counterparts in the linear spectrum. Finally, we propose an optical realization of the linear and nonlinear Harper models in transversely modulated waveguide arrays, where these Hofstadter butterflies can be observed. This work is relevant to a variety of other branches of physics beyond optics, such as disorder-induced localization in ultracold bosonic gases, localization transition processes in disordered lattices, and more.

Quantum diffusion, fractal spectra, and chaos in semiconductor microstructures

Abstract We review how unbounded quantum mechanical diffusion is related to multifractal properties of the spectrum, its level statistics, and to the algebraic decay of correlations. This new field could be called “quantum chaology” of fractal spectra and should be contrasted with the dynamical localization of the kicked rotator and other previously studied quantum systems. These fascinating properties are found in systems described by a quasiperiodic Schrodinger equation, e.g. the Fibonacci chain for quasicrystals and the Harper model, a single-band description of Bloch electrons in magnetic fields. The semiclassical limit of Bloch electrons in magnetic fields is realized in recent experiments on lateral surface superlattices. There we show that classical chaos and nonlinear resonances are clearly reflected in the magnetotransport and thereby explain a series of magnetoresistance peaks observed in antidot arrays on semiconductor heterojunctions. We also find the counterintuitive result that electrons move in opposite direction to the free electron E × B - drift when subject to a two-dimensional periodic potential. This phenomenon arises from chaotic channeling trajectories and by a subtle mechanism leads to a negative value of the Hall resistivity for small magnetic fields.

Conductivity and thermopower of 2D Bloch electrons in magnetic fields with electron-phonon instabilities

The anomalies in the conductivity and thermopower associated with the charge-density-wave instabilities of 2D Bloch electrons in magnetic fields are studied. It is assumed that the electrons are scattered elastically by randomly distributed impurities. The minimum of the conductivity sigma xx and the maximum of the thermopower Qxx are split into several minima and maxima in the same way as the Van Hove singularities after the lattice distortion takes place. If the electron scattering is anisotropic (low angle) the results are essentially the same as in the case of isotropic scattering, with the only difference being that sigma xx is multiplied by a large factor.

Bethe-ansatz for the Bloch electron in magnetic field.

We present a new approach to the problem of Bloch electrons in magnetic field, by making explicit a natural relation between magnetic translations and the quantum group ${\mathit{U}}_{\mathit{q}}$(${\mathit{sl}}_{2}$). The approach allows us to express the spectrum and the Bloch function as solutions of the Bethe-ansatz equations typical for completely integrable quantum systems.

QUANTUM GROUP AND MAGNETIC TRANSLATIONS: BETHE ANSATZ SOLUTION FOR THE HARPER’S EQUATION

We present the Bethe ansatz solution for the problem of Bloch electrons in magnetic field described by the Harper’s equation [Formula: see text] We made explicit a natural relation between magnetic translations and the quantum group Uq(sl2) and express the midband spectrum and the Bloch function in terms of solutions of the Bethe ansatz equations. In this letter we present solution for the critical value λ=1.

Bloch electrons in magnetic fields: Classical chaos and Hofstadter's butterfly.

The single-band description of Bloch electrons in magnetic fields (e.g., by Harper's equation) leads to classically integrable Hamiltonians and thus fails for lateral surface superlattices (LSSLs) where chaotic trajectories prevail near the classical limit. We derive a new model [Eq. (18)], which is reminescent of Harper's equation, but is exact under the most general conditions and exhibits chaos in the classical limit. A matrix continued fraction expansion allows us to study the influence of classical chaos on the fractal spectrum known as Hofstadter's butterfly and to make predictions on its observability in LSSLs.

Bloch electrons in a magnetic field: Rigorous justification of the Peierls-Onsager effective Hamiltonian

We present a rigorous justification of the Peierls-Onsager one-band effective Hamiltonian, starting from the one-particle Hamiltonian for Bloch electrons in a homogeneous magnetic field.

Bloch Electrons in Magnetic Fields. Rationality, Irrationality, Degeneracy

AbstractThe “irrational” equations for determining the energy eigenvalues developed earlier are studied. It is shown that, for rational fields, they can be transformed into the “rational” equations which are generally preferred. The two energy spectra are therefore the same, and there exists no conflict. Further, the question of infinite degeneracy familiar from the Landau functions is taken up. It was established earlier that as long as the field is irrational, this degeneracy is also present for Bloch electrons; now a criterion is developed for the condition under which this degeneracy may be lifted. The criterion comes out to be a rationality condition on the field: it differs in general from the usual (Brown) rationality; it is called “green” rationality.

Energy Levels of Bloch Electrons in Magnetic Fields

Energy Levels of Bloch Electrons in Magnetic Fields

Energy Levels of Bloch Electrons in Magnetic Fields

From the calculation of the energy levels of conduction electrons in a simple model of a metal in a magnetic field three interesting results emerge. (a) Quantum corrections to Onsager's semiclassical quantization formula are found to be sufficiently small that the usual interpretation of the de Haas-van Alphen effect should provide an accurate measure of Fermi-surface cross sections in potassium. (b) The undetermined constant appearing in Onsager's formula is found to be $\frac{3}{4}$ for effects due to "lens" orbits in hexagonal metals when the magnetic field lies in the basal plane. (c) The quantization of energy levels in a variety of periodic open orbits is predicted to be observable in principle through cyclotron-resonance experiments in a suitable geometry in the intermediate magnetic-breakdown regime.

Magnetic Breakdown: Effective Hamiltonian and de Haas-van Alphen Effect

Magnetic breakdown on usual effective Hamiltonian theory for Bloch electrons in magnetic field, using simple two-dimensional rectangular model