Semiclassical analysis with vanishing magnetic fields

Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field

There are a number of excellent texts available on semi-classical analysis. The focus of the present monograph, however, is an aspect of the subject somewhat less systematically developed in other texts: In semi-classical analysis, many of the basic results involve asymptotic expansions in which the terms can be computed by symbolic techniques, and the focus of this monograph is the symbol created thus. In particular, the techniques involved in this symbolic calculus have their origins in symplectic geometry, and the first seven chapters of the present work are a discussion of this underlying symplectic geometry. Another feature which differentiates this monograph from other texts is an emphasis on the global aspects of the subject: A considerable amount of time is spent here showing that the objects studied are coordinate-invariant and hence make sense on manifolds. Wherever possible, intrinsic coordinate-free descriptions of these objects are given. Topics discussed include wave and heat trace formulas for globally defined semi-classical differential operators on manifolds, and equivariant versions of these results involving Lie group actions.

We consider the Harper model which describes two dimensional Bloch electrons in a magnetic field. For irrational flux through the unit-cell the corresponding energy spectrum is known to be a Cantor set with multifractal properties. In order to relate the maximal and minimal fractal dimension of the spectrum of Harper's equation to the irrational number involved, we combine a refined version of the Hofstadter rules with results from semiclassical analysis and tunneling in phase space. For quadratic irrationals $\omega$ with continued fraction expansion $\omega = [0;\overline{n}]$ the maximal fractal dimension exhibits oscillatory behavior as a function of $n$, which can be explained by the structure of the renormalization flow. The asymptotic behavior of the minimal fractal dimension is given by $\amin \sim {\rm const.} \ln n / n$. As the generalized dimensions can be related to the anomalous diffusion exponents of an initially localized wavepacket, our results imply that the time evolution of high order moments $< r^{q} >, q \to \infty$ is sensible to the parity of $n$.

[1] Using data from the high-energy neutral atom (HENA) imager onboard the IMAGE satellite, we examined the relation between the SYM-H index and the ring current energy during a storm main phase. The energy range of the energetic neutral atom (ENA) flux data used here is 16–120 keV for hydrogen and <180 keV for oxygen. From the data for the period 2000–2002, we selected 24 storm main phase events during which the IMAGE satellite was located at a geomagnetic latitude of ≥45° and a geocentric distance of ≥6 RE. According to the Dessler-Parker-Sckopke (DPS) equation, the ring current energy is expected to increase as the SYM-H index decreases. When the ENA energy flux is superimposed as a function of the SYM-H index for all 24 events, their overall correlation is negative; that is, the relation between the ENA energy flux and the SYM-H index is generally consistent with the DPS equation. However, an analysis of individual events showed only 10 events (42%) in which the ENA energy flux was negatively correlated with the SYM-H index (negative correlation events). There were 10 events showing no clear correlation between the ENA energy flux and the SYM-H index (no correlation events) and 4 events which contradicted the DPS equation (positive correlation events). In the superimposed plot, we noted that a smooth curve can be drawn for an upper limit of the data distribution, and data from the no correlation or positive correlation events create downward branches in the distribution. These observational results are not explained by the conventional DPS equation but by the “generalized” DPS equation, which includes a term representing energy stored in the stretched magnetic field. We can reasonably presume that the stretched magnetic field prevents energetic particles from being injected into the ring current. From the generalized DPS equation, we conclude that the total (kinetic and magnetic) energy stored in the stretched field and ring current loss mechanisms are important for understanding the relation between the ground magnetic field variation and ring current energy variation.

From constant to non-degenerately vanishing magnetic fields in superconductivity

We calculate the Landauer conductance through chaotic ballistic devices in the semiclassical limit, to all orders in the inverse number of scattering channels without and with a magnetic field. Families of pairs of entrance-to-exit trajectories contribute, similarly to the pairs of periodic orbits making up the small-time expansion of the spectral form factor of chaotic dynamics. As a clue to the exact result we find that close self-encounters slightly hinder the escape of trajectories into leads. Our result explains why the energy-averaged conductance of individual chaotic cavities, with disorder or "clean," agrees with predictions of random-matrix theory.

Magnetotransport and magneto-optics experiments offer a very powerful probe for studying the physical properties of materials. Here, we investigate the second-order nonlinear magnetoconductivity of the tilted type-I Weyl and multi-Weyl semimetals. In contrast to the existence of chiral charge pumping in the linear response regime, we reproduce the absence of chiral charge pumping in the nonlinear transport regime, using the Boltzmann transport framework with the Landau levels. We predict that an inversion symmetry broken and tilted Weyl semimetal can support finite longitudinal nonlinear magnetoconductivity, which is otherwise absent in untilted Weyl semimetals. The nonlinear magnetoconductivity vanishes in the ultraquantum limit, oscillates in the intermediate magnetic field regime, and saturates in the semiclassical limit. The nonlinear magnetoconductivity depends intricately on the tilt orientation, and it can be used to determine the tilt orientation in Weyl and multi-Weyl semimetals, via nonlinear magnetoresistivity or second harmonic generation experiments.

We study a Dirichlet boundary problem related to the fractional Laplacian in a manifold. Its variational formulation arises in the study of magnitude, an invariant of compact metric spaces given by the reciprocal of the ground state energy. Using recent techniques developed for pseudodifferential boundary problems we discuss the structure of the solution operator and resulting properties of the magnitude. In a semiclassical limit we obtain an asymptotic expansion of the magnitude in terms of curvature invariants of the manifold and the boundary, similar to the invariants arising in short-time expansions for heat kernels.

We give a survey of some results, mainly obtained by the authors and their collaborators, on spectral properties of the magnetic Schr\odinger operators in the semiclassical limit. We focus our discussion on asymptotic behavior of the individual eigenvalues for operators on closed manifolds and existence of gaps in intervals close to the bottom of the spectrum of periodic operators.

Environmental hazards (wildfires, floods, oil spills) are often modeled as “continuous objects” which evolve in space and time taking irregular shapes. Tracking their boundaries and accurately predicting their spatiotemporal spreading patterns is of paramount importance to combat their often catastrophic consequences. Wireless Sensor Networks (WSN) have been very instrumental for this purpose. However, current WSN-based methods require a prohibitively large density of deployed sensors to achieve a reasonable boundary reconstruction accuracy because they are based on the explicit identification of nodes close to the boundary. We present a novel approach that can track and predict the global boundary using only a relatively small number of distributed local front estimates. Our approach first filters and fuses the available sparse set of local front estimates (e.g. vectors of local orientation, direction and speed) and then uses the resulting information to reconstruct a smooth curve prediction of the evolving object's boundary at a future time. Moreover, since the uncertainty of the local front parameters is modeled, it can provide a heatmap representation of the evolving object, indicating the probability for each point in the area of interest to be reached at a future time by the spreading hazard. These predictive modeling capabilities when combined enable effective decision support for crisis management. We demonstrate that different types of continuous objects can be tracked with accuracy, even when only a relatively small number of noisy local front estimates is available. Our approach is practical since it can be applied in many situations where global boundary prediction updates are important to obtain by new sparsely distributed noisy local front estimates as soon as they reach a control center while the hazard is still progressing.

This article tackles the spectral analysis of the Robin Laplacian on a smooth bounded two-dimensional domain in the presence of a constant magnetic field. In the semiclassical limit, a uniform description of the spectrum located between the Landau levels is obtained. The corresponding eigenfunctions, called edge states, are exponentially localized near the boundary. By means of a microlocal dimensional reduction, our unifying approach allows on the one hand to derive a very precise Weyl law and a proof of quantum magnetic oscillations for excited states, and on the other hand to refine simultaneously old results about the low-lying eigenvalues in the Robin case and recent ones about edge states in the Dirichlet case.

Let {\check{X}_0} be a semi-flat Calabi–Yau manifold equipped with a Lagrangian torus fibration {\check{p}:\check{X}_0 \rightarrow B_0} . We investigate the asymptotic behavior of Maurer–Cartan solutions of the Kodaira–Spencer deformation theory on {\check{X}_0} by expanding them into Fourier series along fibres of {\check{p}} over a contractible open subset {U\subset B_0} , following a program set forth by Fukaya [Graphs and Patterns in Mathematics and Theoretical Physics (2005)] in 2005. We prove that semi-classical limits (i.e. leading order terms in asymptotic expansions) of the Fourier modes of a specific class of Maurer–Cartan solutions naturally give rise to consistent scattering diagrams, which are tropical combinatorial objects that have played a crucial role in works of Kontsevich and Soibelman [The Unity of Mathematics (2006)] and Gross and Siebert [Ann. of Math. (2) 174 (2011)] on the reconstruction problem in mirror symmetry.

Transposing the Berezin quantization into the setting of analytic microlocal analysis, we construct approximate semiclassical Bergman projections on weighted $L^{2}$ -spaces with analytic weights, and show that their kernel functions admit an asymptotic expansion in the class of analytic symbols. As a corollary, we obtain new estimates for asymptotic expansions of the Bergman kernel on $\\mathbb{C}^{n}$ and for high powers of ample holomorphic line bundles over compact complex manifolds.

We investigate the fine structure in the energy spectrum of bilayer graphene in the presence of various stacking defaults, such as a translational or rotational mismatch. This fine structure consists of four Dirac points that move away from their original positions as a consequence of the mismatch and eventually merge in various manners. The different types of merging are described in terms of topological invariants (winding numbers) that determine the Landau-level spectrum in the presence of a magnetic field as well as the degeneracy of the levels. The Landau-level spectrum is, within a wide parameter range, well described by a semiclassical treatment that makes use of topological winding numbers. However, the latter need to be redefined at zero energy in the high-magnetic-field limit as well as in the vicinity of saddle points in the zero-field dispersion relation.

A semiclassical analysis of a two-dimensional electron droplet in a high, nonuniform magnetic field predicts that the droplet will form "fingered" patterns upon increasing the number of electrons. We construct explicit examples of these patterns using methods first developed for the flow of two-dimensional viscous fluids. We complement our analytical results with Monte Carlo simulations of the droplet wave function, and find that at the point where the semiclassical analysis predicts a cusp on the interface, the droplet fissions-a type of "quantum breakup" phenomenon.