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Max Lein

Publications and source records attributed to Max Lein.

27 records · Page 2Linked to original sources
arXiv2013

Topological Polarization in Graphene-like Systems

In this article we investigate the possibility of generating piezoelectric orbital polarization in graphene-like systems which are deformed periodically. We start with discrete two-level models which depend on control parameters; in this setting, time-dependent model hamiltonians are described by loops in parameter space. Then, the gap structure at a given Fermi energy generates a non-trivial topology on parameter space which then leads to possibly non-trivial polarizations. More precisely, we show the polarization, as given by the *King-Smith--Vanderbilt formula*, depends only on the homotopy class of the loop; hence, a necessary condition for non-trivial piezo effects is that the fundamental group of the gapped parameter space must not be trivial. The use of the framework of non-commutative geometry implies our results extend to systems with weak disorder. We then apply this analysis to the uniaxial strain model for graphene which includes nearest-neighbor hopping and a stagger potential, and show that it supports non-trivial piezo effects; this is in agreement with recent physics literature.

math-ph↗
arXiv2013

Resonance phenomena in the interaction of a many-photon wave packet and a qubit

We demonstrate that a highly excited quantum electromagnetic mode strongly interacting with a single qubit exhibits several distinct resonances in addition to the Bloch-Siegert resonance condition that arises in the interaction with classical radiation. The resonance phenomena are associated with non-classical effects: Collapse of Rabi oscillations, splitting of the photon wave packet, and field-qubit entanglement. The analysis is based on a semiclassical phase-space flow of 2-by-2 matrices and becomes exact when the photon number tends to infinity. The flow equations are solved perturbatively and numerically, showing that the resonant detuning of the field and qubit frequencies lie on two branches in the leading order, that further split into distinct curves in higher orders.

quant-ph↗
arXiv2012

Semi- and Non-relativistic Limit of the Dirac Dynamics with External Fields

We show how to approximate Dirac dynamics for electronic initial states by semi- and non-relativistic dynamics. To leading order, these are generated by the semi- and non-relativistic Pauli hamiltonian where the kinetic energy is related to $\sqrt{m^2 + ξ^2}$ and $ξ^2 / 2m$, respectively. Higher-order corrections can in principle be computed to any order in the small parameter v/c which is the ratio of typical speeds to the speed of light. Our results imply the dynamics for electronic and positronic states decouple to any order in v/c << 1. To decide whether to get semi- or non-relativistic effective dynamics, one needs to choose a scaling for the kinetic momentum operator. Then the effective dynamics are derived using space-adiabatic perturbation theory by Panati et. al with the novel input of a magnetic pseudodifferential calculus adapted to either the semi- or non-relativistic scaling.

math-ph↗
arXiv2011

Exponentially Localized Wannier Functions in Periodic Zero Flux Magnetic Fields

In this work, we investigate conditions which ensure the existence of an exponentially localized Wannier basis for a given periodic hamiltonian. We extend previous results in [Pan07] to include periodic zero flux magnetic fields which is the setting also investigated in [Kuc09]. The new notion of magnetic symmetry plays a crucial role; to a large class of symmetries for a non-magnetic system, one can associate "magnetic" symmetries of the related magnetic system. Observing that the existence of an exponentially localized Wannier basis is equivalent to the triviality of the so-called Bloch bundle, a rank m hermitian vector bundle over the Brillouin zone, we prove that magnetic time-reversal symmetry is sufficient to ensure the triviality of the Bloch bundle in spatial dimension d=1,2,3. For d=4, an exponentially localized Wannier basis exists provided that the trace per unit volume of a suitable function of the Fermi projection vanishes. For d>4 and d \leq 2m (stable rank regime) only the exponential localization of a subset of Wannier functions is shown; this improves part of the analysis of [Kuc09]. Finally, for d>4 and d>2m (unstable rank regime) we show that the mere analysis of Chern classes does not suffice in order to prove trivility and thus exponential localization.

math-ph↗
arXiv2011

Applications of Magnetic PsiDO Techniques to Space-adiabatic Perturbation Theory

In this review, we show how advances in the theory of magnetic pseudodifferential operators (magnetic $Ψ$DO) can be put to good use in space-adiabatic perturbation theory (SAPT). As a particular example, we extend results of [PST03] to a more general class of magnetic fields: we consider a single particle moving in a periodic potential which is subjectd to a weak and slowly-varying electromagnetic field. In addition to the semiclassical parameter $\eps \ll 1$ which quantifies the separation of spatial scales, we explore the influence of additional parameters that allow us to selectively switch off the magnetic field. We find that even in the case of magnetic fields with components in $C_b^{\infty}(\R^d)$, e. g. for constant magnetic fields, the results of Panati, Spohn and Teufel hold, i.e. to each isolated family of Bloch bands, there exists an associated almost invariant subspace of $L^2(\R^d)$ and an effective hamiltonian which generates the dynamics within this almost invariant subspace. In case of an isolated non-degenerate Bloch band, the full quantum dynamics can be approximated by the hamiltonian flow associated to the semiclassical equations of motion found in [PST03].

math-ph↗
arXiv2010

Quantization and Semiclassics

This course is aimed at graduate students in physics in mathematics and designed to give a comprehensive introduction to Weyl quantization and semiclassics via Egorov's theorem. Chapter 2 gives a quick overview of classical and quantum mechanics on R^d. Some mathematical preliminaries concerning Hilbert space theory, operator theory and tempered distributions are detailed in Chapters 3-5. Weyl quantization and semiclassics are the content of Chapters 6 and 7. Finally, an application of Weyl calculus to Born-Oppenheimer systems is discussed in Chapter 8.

math-ph↗
arXiv2010

Two-parameter Asymptotics in Magnetic Weyl Calculus

This paper is concerned with small parameter asymptotics of magnetic quantum systems. In addition to a semiclassical parameter \eps, the case of small coupling $λ$ to the magnetic vector potential naturally occurs in this context. Magnetic Weyl calculus is adapted to incorporate both parameters, at least one of which needs to be small. Of particular interest is the expansion of the Weyl product which can be used to expand the product of operators in a small parameter, a technique which is prominent to obtain perturbation expansions. Three asymptotic expansions for the magnetic Weyl product of two Hörmander class symbols are proven: (i) \eps \ll 1 and λ\ll 1, (ii) \eps \ll 1 and λ= 1 as well as (iii) \eps = 1 and λ\ll 1. Expansions (i) and (iii) are impossible to obtain with ordinary Weyl calculus. Furthermore, I relate results derived by ordinary Weyl calculus with those obtained with magnetic Weyl calculus by one- and two-parameter expansions. To show the power and versatility of magnetic Weyl calculus, I derive the semirelativistic Pauli equation as a scaling limit from the Dirac equation up to errors of 4th order in 1/c.

math-ph↗
arXiv2009

Magnetic pseudodifferential operators with coefficients in C*-algebras

In previous articles, a magnetic pseudodifferential calculus and a family of C*-algebras associated with twisted dynamical systems were introduced and the connections between them have been established. We extend this formalism to symbol classes of Hörmander type with an x-behavior modelized by an abelian C*-algebra. Some of these classes generate C*-algebras associated with the twisted dynamical system. We show the relevance of these classes to the spectral analysis of pseudodifferential operators with anisotropic symbols and magnetic fields.

math-ph↗