arXiv subjects

Max Lein

Publications and source records attributed to Max Lein.

At least 19 recordsLinked to original sources

arXiv2024

A Proof of $\mathfrak{L}^2$-Boundedness for Magnetic Pseudodifferential Super Operators via Matrix Representations With Respect to Parseval Frames

A fundamental result in pseudodifferential theory is the Calder\'on-Vaillancourt theorem, which states that a pseudodifferential operator defined from a H\"ormander symbol of order $0$ defines a bounded operator on $L^2(\mathbb{R}^d)$. In this work we prove an analog for pseudodifferential \emph{super} operator, \ie operators acting on other operators, in the presence of magnetic fields. More precisely, we show that magnetic pseudodifferential super operators of order $0$ define bounded operators on the space of Hilbert-Schmidt operators $\mathfrak{L}^2 \bigl ( \mathcal{B} \bigl ( L^2(\mathbb{R}^d) \bigr ) \bigr )$. Our proof is inspired by the recent work of Cornean, Helffer and Purice and rests on a characterization of magnetic pseudodifferential super operators in terms of their "matrix element" computed with respect to a Parseval frame.

math-ph↗

arXiv2022

A Magnetic Pseudodifferential Calculus for Operator-Valued and Equivariant Operator-Valued Symbols

In this monograph we develop magnetic pseudodifferential theory for operator-valued and equivariant operator-valued functions and distributions from first principles. These have found plentiful applications in mathematical physics, including in rigorous perturbation theory for slow-fast systems and perturbed periodic operators. Yet, a systematic treatise was hitherto missing. While many of the results can be found piecemeal in appendices and as sketches in other articles, this article does contain new results. For instance, we have established Beals-type commutator criteria for both cases, which then imply the existence of Moyal resolvents for (equivariant) selfadjoint-operator-valued, elliptic H\"ormander symbols and allows one to construct functional calculi. What is more, we give criteria on the function under which a magnetic pseudodifferential operator is (locally) trace class. Our aims for this article are three-fold: (1) Create a single, solid work that colleagues can refer to. (2) Be pedagogical and precise. And (3) give a straightforward strategy for extending results from the operator-valued to the equivariant case, pointing out some caveats and pitfalls that need to be kept in mind.

math-ph↗

arXiv2022

A Calculus for Magnetic Pseudodifferential Super Operators

This work develops a magnetic pseudodifferential calculus for super operators OpA(F); these map operators onto operators (as opposed to Lp functions onto Lq functions). Here, F could be a tempered distribution or a H\"ormander symbol. An important example is Liouville super operators defined in terms of a magnetic pseudodifferential operator. Our work combines ideas from magnetic Weyl calculus developed in [MP04, IMP07, Lei11] and the pseudodifferential calculus on the non-commutative torus from [HLP18a, HLP18b]. Thus, our calculus is inherently gauge-covariant, which means all essential properties of OpA(F) are determined by properties of the magnetic field B = dA rather than the vector potential A. There are conceptual differences to ordinary pseudodifferential theory. For example, in addition to an analog of the (magnetic) Weyl product that emulates the composition of two magnetic pseudodifferential super operators on the level of functions, the so-called semi-super product describes the action of a pseudodifferential super operator on a pseudodifferential operator.

math-ph↗

arXiv2020

On Choosing a Physically Meaningful Topological Classification for Non-Hermitian Systems and the Issue of Diagonalizability

The topological classification of hermitian operators is solely determined by the presence or absence of certain discrete symmetries. For non-hermitian operators we in addition need to specify the type of spectral gap. They come in the flavor of a point gap or a line gap. Since the presence of a line gap implies the existence of a point gap, there is usually more than one mathematical classification applicable to a physical system. That raises the question: which of these gap-type classifications is physically meaningful? To decide this question, I propose a simple criterion, namely the choice of physically relevant states. This generalizes the notion of Fermi projection that plays a crucial role in the topological classification of fermionic condensed matter systems, and enters as an auxiliary quantity in the bulk classification of photonic and magnonic crystals. After that the classification is entirely algorithmic, the system's topology is encoded in (pairs of) projections with symmetries and constraints. A crucial point in my investigation is the relevance of diagonalizability. Even for existing topological classifications of non-hermitian systems diagonalizability needs to be assumed to ensure that continuous deformations of the hamiltonian lead to continuous deformations of the spectra, projections and unitaries.

cond-mat.mes-hall↗

arXiv2019

The Krein-Schr\"odinger Formalism of Bosonic BdG and Certain Classical Systems and Their Topological Classification

To understand recent works on classical and quantum spin equations and their topological classification, we develop a unified mathematical framework for bosonic BdG systems and associated classical wave equations; it applies not just to equations that describe quantized spin excitations in magnonic crystals but more broadly to other systems that are described by a BdG hamiltonian. Because here the generator of dynamics, the analog of the hamiltonian, is para-, aka Krein-, hermitian but not hermitian, the theory of Krein spaces plays a crucial role. For systems which are thermodynamically stable, the classical equations can be expressed as a Schr\"odinger equation with a hermitian hamiltonian. We then proceed to apply the Cartan-Altland-Zirnbauer classification scheme: to properly understand what topological class these equations belong to, we need to conceptually distinguish between symmetries and constraints. Complex conjugation enters as a particle-hole constraint (as opposed to a symmetry), since classical waves are necessarily real-valued. Because of this distinction only commuting symmetries enter in the topological classification. Our arguments show that the equations for spin waves in magnonic crystals are a system of class A, the same topological class as quantum hamiltonians describing the Integer Quantum Hall Effect. Consequently, the magnonic edge modes first predicted by Shindou et al. are indeed analogs of the Quantum Hall Effect, and their net number is topologically protected.

cond-mat.other↗

arXiv2019

Topological non-Hermitian origin of surface Maxwell waves

Maxwell electromagnetism, describing the wave properties of light, was formulated 150 years ago. More than 60 years ago it was shown that interfaces between optical media (including dielectrics, metals, negative-index materials) can support surface electromagnetic waves, which now play crucial roles in plasmonics, metamaterials, and nano-photonics. Here we show that surface Maxwell waves at interfaces between homogeneous, isotropic media described by real permittivities and permeabilities have a purely topological origin explained by the bulk-boundary correspondence. Importantly, the topological classification is determined by the helicity operator, which is generically non-Hermitian even in lossless optical media. The corresponding topological invariant, which determines the number of surface modes, is a Z4 number (or a pair of Z2 numbers) describing the winding of the complex helicity spectrum across the interface. Moreover, there is an additional pair of non-topological Z2 indices, which describe zones of the TE and TM polarizations at the phase diagram of surface modes. Our theory provides a new twist and insights for several areas of wave physics: Maxwell electromagnetism, topological quantum states, non-Hermitian wave physics, and metamaterials.

physics.optics↗

arXiv2018

Equivalence of Electric, Magnetic and Electromagnetic Chern Numbers for Topological Photonic Crystals

Haldane predicted an analog of the Integer Quantum Hall Effect in gyrotropic photonic crystals, where the net number of electromagnetic edge modes moving left-to-right is given by a bulk Chern number. His prediction --- topological effects are bona fide wave and not quantum phenomena --- has been confirmed in a number of experiments. However, theoretical physicists have tacitly used three different definitions for the bulk Chern numbers that enter the bulk-edge correspondence --- on the basis of electromagnetic Bloch functions, electric Bloch functions and magnetic Bloch functions. We use vector bundle theoretic arguments to prove that in media such as those considered by Haldane these three potentially different Chern numbers necessarily agree with one another, and consequently, any one of them can be used in Haldane's photonic bulk-edge correspondence.

math-ph↗

arXiv2018

Taking Inspiration from Quantum-Wave Analogies --- Recent Results for Photonic Crystals

Similarities between quantum systems and analogous systems for classical waves have been used to great effect in the physics community, be it to gain an intuition for quantum systems or to anticipate novel phenomena in classical waves. This proceeding reviews recent advances in putting these quantum-wave analogies on a mathematically rigorous foundation for classical electromagnetism. Not only has this Schr\"odinger formalism of electromagnetism led to new, interesting mathematical problems for so-called Maxwell-type operators, it has also improved the understanding of the physics of topological phenomena in electromagnetic media. For example, it enabled us to classify electromagnetic media by their material symmetries, and explained why "fermionic time-reversal symmetries" --- that were conjectured to exist in the physics literature --- are in fact forbidden.

math-ph↗

arXiv2017

Symmetry Classification of Topological Photonic Crystals

In a seminal paper Haldane conjectured that topological phenomena are not particular to quantum systems, and indeed experiments realized unidirectional, backscattering-free edge modes with electromagnetic waves. This raises two immediate questions: (1) Are there other topological effects in electromagnetic media? And (2) is Haldane's Quantum Hall Effect for light really analogous to the Quantum Hall Effect? We conclusively answer both of these questions by classifying topological photonic crystals according to material (as opposed to crystallographic) symmetries. It turns out there are four topologically distinct types of media, of which only one, gyrotropic media, is topologically non-trivial in $d = 2 , 3$. That means there are no as-of-yet undiscovered topological effects; in particular, there is no analog of the Quantum Spin Hall Effect in classical electromagnetism. Moreover, at least qualitatively, Haldane's Quantum Hall Effect for light is analogous to the Quantum Hall Effect from condensed matter physics as both systems as in the same topological class, class A. Our ideas are directly applicable to other classical waves.

physics.optics↗

arXiv2017

The Schr\"odinger Formalism of Electromagnetism and Other Classical Waves --- How to Make Quantum-Wave Analogies Rigorous

This paper systematically develops the Schr\"odinger formalism that is valid also for gyrotropic media where the material weights $W = \left ( \begin{smallmatrix} \varepsilon & \chi \chi^* & \mu \end{smallmatrix} \right ) \neq \overline{W}$ are complex. This is a non-trivial extension of the Schr\"odinger formalism for non-gyrotropic media (where $W = \overline{W}$) that has been known since at least the 1960s. Here, Maxwell's equations are rewritten in the form $\mathrm{i} \partial_t \Psi = M \Psi$ where the selfadjoint (hermitian) Maxwell operator $M = W^{-1} \, \mathrm{Rot} \, \big |_{\omega \geq 0} = M^*$ takes the place of the Hamiltonian and $\Psi$ is a complex wave representing the physical field $(\mathbf{E},\mathbf{H}) = 2 \mathrm{Re} \, \Psi$. Writing Maxwell's equations in Schr\"odinger form gives us access to the rich toolbox of techniques initially developed for quantum mechanics and allows us to apply them to classical waves. To show its utility, we explain how to identify conserved quantities in this formalism. Moreover, we sketch how to extend our ideas to other classical waves.

physics.optics↗

arXiv2016

Linear Response Theory: A Modern Analytic-Algebraic Approach

Linear response theory is a tool with which one can study systems that are driven out of equilibrium by external perturbations. This monograph presents a thoroughly modern framework to make linear response theory rigorous for a wide array of systems, that is suitable for novel applications such as periodic and random light conductors not yet covered in the literature. Our analytic-algebraic approach, based on von Neumann algebras and associated non-commutative $L^p$-spaces, can deal with discrete and continuous models alike, and include effects of disorder. First, we explain the mathematical setting, give a complete list of our hypotheses and state the main results, which include Kubo and Kubo-Streda formulas. To make our book accessible to a wide audience, we spend Chapters 3 and 4 explaining the mathematical underpinnings such as non-commutative $L^p$- and Sobolev spaces, and generalized commutators. Furthermore, we show how to construct a von Neumann algebras from a topological dynamical system and a 2-cocycle, a procedure which applies to discrete and continuous quantum systems. We dedicate Chapters 5 and 6 to the proofs of our main results. We close the book by sketching a novel application, linear response theory for periodic and random light conductors. This monograph is aimed at advanced students in mathematical physics and researchers.

math-ph↗

arXiv2015

Differential Equations of Mathematical Physics

These lecture notes for the course APM 351 at the University of Toronto are aimed at mathematicians and physicists alike. It is not meant as an introductory course to PDEs, but rather gives an overview of how to view and solve differential equations that are common in physics. Among others, I cover Hamilton's equations, variations of the Schr\"odinger equation, the heat equation, the wave equation and Maxwells equations.

math-ph↗

arXiv2015

Derivation of Ray Optics Equations in Photonic Crystals Via a Semiclassical Limit

In this work we present a novel approach to the ray optics limit: we rewrite the dynamical Maxwell equations in Schr\"odinger form and prove Egorov-type theorems, a robust semiclassical technique. We implement this scheme for periodic light conductors, photonic crystals, thereby making the quantum-light analogy between semiclassics for the Bloch electron and ray optics in photonic crystals rigorous. Our main results, Theorems 3.3 and 4.1, give a ray optics limit for quadratic observables and, among others, apply to local averages of energy density, the Poynting vector and the Maxwell stress tensor. Ours is the first rigorous derivation of ray optics equations which include all sub-leading order terms, some of which are also new to the physics literature. While the ray optics limit we prove initially (Theorem 3.3) applies to photonic crystals of any topological class, we also consider the ray optics limit for real electromagnetic fields propagating in non-gyrotropic photonic crystals. Such an extension is non-trivial, because the ray optic limit for real fields is necessarily a multiband problem.

math-ph↗

arXiv2014

On the Role of Symmetries in the Theory of Photonic Crystals

We discuss the role of the symmetries in photonic crystals and classify them according to the Cartan-Altland-Zirnbauer scheme. Of particular importance are complex conjugation C and time-reversal T, but we identify also other significant symmetries. Borrowing the jargon of the classification theory of topological insulators, we show that C is a particle-hole-type symmetry rather than a time-reversal symmetry if one consider the Maxwell operator in the first-order formalism where the dynamical Maxwell equations can be rewritten as a Schrödinger equation; The symmetry which implements physical time-reversal is a chiral-type symmetry. We justify by an analysis of the band structure why the first-order formalism seems to be more advantageous than the second-order formalism. Moreover, based on the Schrödinger formalism, we introduce a class of effective (tight-binding) models called Maxwell-Harper operators. Some considerations about the breaking of the particle-hole-type symmetry in the case of gyrotropic crystals are added at the end of this paper.

cond-mat.mes-hall↗

arXiv2013

The Perturbed Maxwell Operator as Pseudodifferential Operator

As a first step to deriving effective dynamics and ray optics, we prove that the perturbed periodic Maxwell operator in d = 3 can be seen as a pseudodifferential operator. This necessitates a better understanding of the periodic Maxwell operator M_0. In particular, we characterize the behavior of M_0 and the physical initial states at small crystal momenta $k$ and small frequencies |ω|. Among other things, we prove that generically the band spectrum is symmetric with respect to inversions at k = 0 and that there are exactly 4 ground state bands with approximately linear dispersion near k = 0.

math-ph↗

arXiv2013

Semiclassics for particles with spin via a Wigner-Weyl-type calculus

We show how to relate the full quantum dynamics of a spin-1/2 particle on R^d to a classical Hamiltonian dynamics on the enlarged phase space R^d x S^2 up to errors of second order in the semiclassical parameter. This is done via an Egorov-type theorem for normal Wigner-Weyl calculus for R^d [Lei10,Fol89] combined with the Stratonovich-Weyl calculus for SU(2) [VGB89]. For a specific class of Hamiltonians, including the Rabi- and Jaynes-Cummings model, we prove an Egorov theorem for times much longer than the semiclassical time scale. We illustrate the approach for a simple model of the Stern-Gerlach experiment.

math-ph↗

arXiv2013

Effective Light Dynamics in Perturbed Photonic Crystals

In this work, we rigorously derive effective dynamics for light from within a limited frequency range propagating in a photonic crystal that is modulated on the macroscopic level; the perturbation parameter $\lambda \ll 1$ quantifies the separation of spatial scales. We do that by rewriting the dynamical Maxwell equations as a Schr\"odinger-type equation and adapting space-adiabatic perturbation theory. Just like in the case of the Bloch electron, we obtain a simpler, effective Maxwell operator for states from within a relevant almost invariant subspace. A correct physical interpretation for the effective dynamics requires to establish two additional facts about the almost invariant subspace: (1) The source-free condition has to be verified and (2) it has to support real states. The second point also forces one to consider a multiband problem even in the simplest possible setting; This turns out to be a major difficulty for the extension of semiclassical methods to the domain of photonic crystals.

math-ph↗

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↗