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Giuseppe De Nittis

Publications and source records attributed to Giuseppe De Nittis.

At least 37 records · Page 2Linked to original sources
arXiv2021

Perturbation Theory for the Thermal Hamiltonian: 1D Case

This work continues the study of the thermal Hamiltonian, initially proposed by J. M. Luttinger in 1964 as a model for the conduction of thermal currents in solids. The previous work [DL] contains a complete study of the "free" model in one spatial dimension along with a preliminary scattering result for convolution-type perturbations. This work complements the results obtained in [DL] by providing a detailed analysis of the perturbation theory for the one-dimensional thermal Hamiltonian. In more detail the following result are established: the regularity and decay properties for elements in the domain of the unperturbed thermal Hamiltonian; the determination of a class of self-adjoint and relatively compact perturbations of the thermal Hamiltonian; the proof of the existence and completeness of wave operators for a subclass of such potentials.

math-ph↗
arXiv2020

The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

This work provides a first step towards the construction of a noncommutative geometry for the quantum Hall effect in the continuum. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the $C^*$-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.

math-ph↗
arXiv2020

Quantization of edge currents along magnetic interfaces: A K-theory approach

The purpose of this paper is to investigate the propagation of topological currents along magnetic interfaces (also known as magnetic walls) of a two-dimensional material. We consider tight-binding magnetic models associated to generic magnetic multi-interfaces and describe the K-theoretical setting in which a bulk-interface duality can be derived. Then, the (trivial) case of a localized magnetic field and the (non trivial) case of the Iwatsuka magnetic field are considered in full detail. This is a pedagogical preparatory work that aims to anticipate the study of more complicated multi-interface magnetic systems.

math-ph↗
arXiv2020

Spectral Theory of the Thermal Hamiltonian: 1D Case

In 1964 J. M. Luttinger introduced a model for the quantum thermal transport. In this paper we study the spectral theory of the Hamiltonian operator associated to the Luttinger's model, with a special focus at the one-dimensional case. It is shown that the (so called) thermal Hamiltonian has a one-parameter family of self-adjoint extensions and the spectrum, the time-propagator group and the Green function are explicitly computed. Moreover, the scattering by convolution-type potentials is analyzed. Finally, also the associated classical problem is completely solved, thus providing a comparison between classical and quantum behavior. This article aims to be a first contribution in the construction of a complete theory for the thermal Hamiltonian.

math-ph↗
arXiv2019

The Geometry of (non-Abelian) Landau levels

The purpose of this paper is threefold: First of all the topological aspects of the Landau Hamiltonian are reviewed in the light (and with the jargon) of theory of topological insulators. In particular it is shown that the Landau Hamiltonian has a generalized even time-reversal symmetry (TRS). Secondly, a new tool for the computation of the topological numbers associated with each Landau level is introduced. The latter is obtained by combining the Dixmier trace and the (resolvent of the) harmonic oscillator. Finally, these results are extended to models with non-Abelian magnetic fields. Two models are investigated in details: the Jaynes-Cummings model and the "Quaternionic" model.

math-ph↗
arXiv2019

Spectral Continuity for Aperiodic Quantum Systems I. General Theory

How does the spectrum of a Schrödinger operator vary if the corresponding geometry and dynamics change? Is it possible to define approximations of the spectrum of such operators by defining approximations of the underlying structures? In this work a positive answer is provided using the rather general setting of groupoid $C^\ast$-algebras. A characterization of the convergence of the spectra by the convergence of the underlying structures is proved. In order to do so, the concept of continuous field of groupoids is slightly extended by adding continuous fields of cocycles. With this at hand, magnetic Schrödinger operators on dynamical systems or Delone systems fall into this unified setting. Various approximations used in computational physics, like the periodic or the finite cluster approximations, are expressed through the tautological groupoid, which provides a universal model for fields of groupoids. The use of the Hausdorff topology turns out to be fundamental in understanding why and how these approximations work. Corrigendum: A correct statement of Theorem 4 is provided. The change does not affect the main results.

math.SP↗
arXiv2018

Differential geometric invariants for time-reversal symmetric Bloch-bundles II: The low dimensional "Quaternionic" case

This paper is devoted to the construction of differential geometric invariants for the classification of "Quaternionic" vector bundles. Provided that the base space is a smooth manifold of dimension two or three endowed with an involution that leaves fixed only a finite number of points, it is possible to prove that the Wess-Zumino term and the Chern-Simons invariant yield topological quantities able to distinguish between inequivalent realization of "Quaternionic" structures.

math-ph↗
arXiv2018

Computing the Strategy to Commit to in Polymatrix Games (Extended Version)

Leadership games provide a powerful paradigm to model many real-world settings. Most literature focuses on games with a single follower who acts optimistically, breaking ties in favour of the leader. Unfortunately, for real-world applications, this is unlikely. In this paper, we look for efficiently solvable games with multiple followers who play either optimistically or pessimistically, i.e., breaking ties in favour or against the leader. We study the computational complexity of finding or approximating an optimistic or pessimistic leader-follower equilibrium in specific classes of succinct games---polymatrix like---which are equivalent to 2-player Bayesian games with uncertainty over the follower, with interdependent or independent types. Furthermore, we provide an exact algorithm to find a pessimistic equilibrium for those game classes. Finally, we show that in general polymatrix games the computation is harder even when players are forced to play pure strategies.

cs.AI↗
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

Facing Multiple Attacks in Adversarial Patrolling Games with Alarmed Targets

We focus on adversarial patrolling games on arbitrary graphs, where the Defender can control a mobile resource, the targets are alarmed by an alarm system, and the Attacker can observe the actions of the mobile resource of the Defender and perform different attacks exploiting multiple resources. This scenario can be modeled as a zero-sum extensive-form game in which each player can play multiple times. The game tree is exponentially large both in the size of the graph and in the number of attacking resources. We show that when the number of the Attacker's resources is free, the problem of computing the equilibrium path is NP-hard, while when the number of resources is fixed, the equilibrium path can be computed in poly-time. We provide a dynamic-programming algorithm that, given the number of the Attacker's resources, computes the equilibrium path requiring poly-time in the size of the graph and exponential time in the number of the resources. Furthermore, since in real-world scenarios it is implausible that the Defender knows the number of attacking resources, we study the robustness of the Defender's strategy when she makes a wrong guess about that number. We show that even the error of just a single resource can lead to an arbitrary inefficiency, when the inefficiency is defined as the ratio of the Defender's utilities obtained with a wrong guess and a correct guess. However, a more suitable definition of inefficiency is given by the difference of the Defender's utilities: this way, we observe that the higher the error in the estimation, the higher the loss for the Defender. Then, we investigate the performance of online algorithms when no information about the Attacker's resources is available. Finally, we resort to randomized online algorithms showing that we can obtain a competitive factor that is twice better than the one that can be achieved by any deterministic online algorithm.

cs.AI↗
arXiv2018

How to Maximize the Spread of Social Influence: A Survey

This survey presents the main results achieved for the influence maximization problem in social networks. This problem is well studied in the literature and, thanks to its recent applications, some of which currently deployed on the field, it is receiving more and more attention in the scientific community. The problem can be formulated as follows: given a graph, with each node having a certain probability of influencing its neighbors, select a subset of vertices so that the number of nodes in the network that are influenced is maximized. Starting from this model, we introduce the main theoretical developments and computational results that have been achieved, taking into account different diffusion models describing how the information spreads throughout the network, various ways in which the sources of information could be placed, and how to tackle the problem in the presence of uncertainties affecting the network. Finally, we present one of the main application that has been developed and deployed exploiting tools and techniques previously discussed.

cs.SI↗
arXiv2018

The non-commutative topology of two-dimensional dirty superconductors

Non-commutative analysis tools have successfully been applied to the integer quantum Hall effect, in particular for a proof of the stability of the Hall conductance in an Anderson localization regime and of the bulk-boundary correspondence. In this work, these techniques are implemented to study two-dimensional dirty superconductors described by Bogoliubov-de Gennes Hamiltonians. After a thorough presentation of the basic framework and the topological invariants, Kubo formulas for the thermal, thermoelectric and spin Hall conductance are analyzed together with the corresponding edge currents.

math-ph↗
arXiv2018

Spectral Continuity for Aperiodic Quantum Systems II. Periodic Approximations in 1D

The existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schrödinger operators. In a forthcoming work [9] (arXiv:1709.00975) this task was boiled down to the existence and construction of periodic approximations of the underlying dynamical systems in the Hausdorff topology. As a result the one-dimensional systems admitting such approximations are completely classified in the present work. In addition explicit constructions are provided for dynamical systems defined by primitive substitutions covering all studied examples such as the Fibonacci sequence or the Golay-Rudin-Shapiro sequence. One main tool is the description of the Hausdorff topology by the local pattern topology on the dictionaries as well as the GAP-graphs describing the local structure. The connection of branching vertices in the GAP-graphs and defects is discussed.

math.SP↗
arXiv2018

Chiral vector bundles

This paper focuses on the study of a new category of vector bundles. The objects of this category, called chiral vector bundles, are pairs given by a complex vector bundle along with one of its automorphisms. We provide a classification for the homotopy equivalence classes of these objects based on the construction of a suitable classifying space. The computation of the cohomology of the latter allows us to introduce a proper set of characteristic cohomology classes: some of those just reproduce the ordinary Chern classes but there are also new odd-dimensional classes which take care of the extra topological information introduced by the chiral structure. Chiral vector bundles provide a geometric model for topological quantum systems in class AIII, namely for systems endowed with a (pseudo-)symmetry of chiral type. The classification of the chiral vector bundles over sphere and tori (explicitly computable up to dimension 4), recovers the commonly accepted classification for topological insulator of class AIII which is usually based on the K-group $K_1$. However, our classification turns out to be even richer since it takes care also for possible non-trivial Chern classes.

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 Schrödinger Formalism of Electromagnetism and Other Classical Waves --- How to Make Quantum-Wave Analogies Rigorous

This paper systematically develops the Schrödinger formalism that is valid also for gyrotropic media where the material weights $W = \left ( \begin{smallmatrix} \varepsilon & χχ^* & μ\end{smallmatrix} \right ) \neq \overline{W}$ are complex. This is a non-trivial extension of the Schrödinger 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 $\mathrm{i} \partial_t Ψ= M Ψ$ where the selfadjoint (hermitian) Maxwell operator $M = W^{-1} \, \mathrm{Rot} \, \big |_{ω\geq 0} = M^*$ takes the place of the Hamiltonian and $Ψ$ is a complex wave representing the physical field $(\mathbf{E},\mathbf{H}) = 2 \mathrm{Re} \, Ψ$. Writing Maxwell's equations in Schrödinger 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↗
arXiv2017

The cohomological nature of the Fu-Kane-Mele invariant

In this paper we generalize the definition of the FKMM-invariant introduced in [DG2] for the case of "Quaternionic" vector bundles over involutive base spaces endowed with free involution or with a non-finite fixed-point set. In [DG2] it has already be shown how the FKMM-invariant provides a cohomological description of the Fu-Kane-Mele index used to classify topological insulators in class AII. It follows that the FKMM-invariant described in this paper provides a cohomological generalization of the Fu-Kane-Mele index which is applicable to the classification of protected phases for other type of topological quantum systems (TQS) which are not necesarily related to models for topological insulators (e.g. the two-dimensional models of adiabatically perturbed systems discussed in [GR]). As a byproduct we provide the complete classification of "Quaternionic" vector bundles over a big class of low dimensional involutive spheres and tori.

math-ph↗