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Giuseppe De Nittis

Publications and source records attributed to Giuseppe De Nittis.

At least 19 recordsLinked to original sources
arXiv2026

Asymptotic quasi--local structure induced by Magnetic field

We introduce a magnetic coherent--state frame over the Landau levels of the two--dimensional Landau Hamiltonian, indexed by a discrete metric space combining spatial and energy degrees of freedom. Although overcomplete, the frame is almost--orthogonal, and we use it to endow the CAR $C^*$--algebra over the one--particle Hilbert space with an \emph{asymptotic quasi--local} (AQL) structure, in which graded commutators of observables localized on disjoint index sets decay exponentially rather than vanish exactly. By means of a family of conditional expectations onto the resulting local subalgebras, we construct a Fréchet $*$--subalgebra of almost--local observables, invariant under the discrete group of lattice translations, which act on it by continuous automorphisms. For the full continuous group of magnetic translations we establish a quantitative control on how local observables spread through this almost--local algebra, and we show that this control suffices to extend graded asymptotic abelianness from the lattice to the entire continuous translation group. The results provide a discrete--continuum dictionary intended as the starting point for extending Lieb--Robinson--type techniques from lattice systems to interacting continuum electrons in a magnetic field.

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arXiv2026

Higher Abelian Quantum Double Models

This paper develops a rigorous $C^*$-algebraic framework for higher abelian quantum double models, generalizing Kitaev's construction to simplicial complexes of arbitrary finite dimension and local regularity. We fully characterize the frustration-free ground state space through an associated algebra of logical operators: its state space is shown to be homeomorphic to the space of frustration-free ground states, and to obey generalized canonical commutation relations. When the relevant homology and cohomology groups are finite, the logical algebra decomposes into a commutative factor and a full matrix algebra, thereby separating the classical and quantum parts of the frustration-free ground state structure. We further prove that the vanishing of these groups is necessary and sufficient for the core algebra of the model to form a Cartan pair with the full algebra of observables, a property expected to pave the way toward classifying the model's equilibrium (KMS) states, extending recent results for the planar case.

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arXiv2026

Hilbert Grassmannians as classifying spaces

In this short work we prove that the Hilbert Grassmannians endowed with the weak topology are models for the classifying spaces of the unitary groups. As application of this result one can use Hilbert Grassmannians for the presentation of the $K$-theory of topological spaces by computing equivalences classes of homotopy equivalent maps.

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arXiv2026

A scheme for topological phases of the Weyl $C^*$-algebra

In this work, we introduce a classification scheme for topological phases of matter based on the topology of the space of pure states of a model $C^*$-algebra. Under it, topological phases are described by homotopy classes of sections of certain fiber bundles of (pure) states. Applying this classification procedure on states of the Weyl $C^*$-algebra that are invariant under translations by a lattice, we recover the $K$-theoretic classification of gapped spectral projectors for topological insulators of types A and AI, thus essentially generalizing this notion.

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arXiv2025

The Lieb-Robinson condition and the Fréchet topology

We define various notions of locality for *-automorphisms of the algebra of observables for an infinitely extended quantum spin system and study their relationship. In particular, we show that the ubiquitous characterization which arises from the Lieb-Robinson bound implies but is not equivalent to continuity with respect to the natural Fréchet topology of almost local observables, which is a non-commutative analog of the Schwartz space.

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arXiv2025

Topological description of pure invariant states of the Weyl $C^*$-algebra

In this work we study the topology of certain families of states of the Weyl $C^*$-algebra with finite degrees of freedom. We focus on families of pure states characterized by symmetries and a (semi-)regularity condition, and obtain precise topological descriptions through homeomorphisms with other explicit spaces. Of special importance are the families of pure, semi-regular states invariant under either continuous (plane-wave states) or discrete (Bloch-wave states) spatial translations, and the family of states invariant under discrete, mutually commuting spatial and momentum translations (Zak-wave states), all of which we completely characterize.

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arXiv2025

Topological phases of non-interacting systems: A general approach based on states

In this work we provide a classification scheme for topological phases of certain systems whose observable algebra is described by a trivial $C^*$-bundles. The classification is based on the study of the homotopy classes of \emph{configurations}, which are maps from a \emph{quantum parameter space} to the space of pure states of a reference \emph{fiber} $C^*$-algebra. Both the quantum parameter space and the fiber algebra are naturally associated with the observable algebra. A list of various examples described in the last section shows that the common classification scheme of non-interacting topological insulators of type A is recovered inside this new formalism.

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arXiv2024

On the K-theory of magnetic algebras: Iwatsuka case

In the tight-binding approximation, an Iwatsuka magnetic field is modeled by a function on $\mathbb{Z}^2$ with constant, but distinct values in the two parts of the lattice separated by a straight line of slope $α\in [-\infty,\infty]$. In this paper, the $K$-theory of the magnetic $C^*$-algebras generated by an Iwatsuka magnetic field for any possible $α$ is computed. One interesting aspect concerns the analysis of the behavior of the system in the transition from rational to irrational $α$. It turns out that when $α$ is irrational, the magnetic hull associated with the flux operator forms a Cantor set. On the other hand, for rational $α$ this set coincides with the two-point compactification of $\mathbb{Z}$. This characterization, along with the use of the Pimsner-Voiculescu exact sequence, is the main ingredient for the computation of the $K$-theory. Once the $K$-theory is known, with the use of the index theory one can deduce the bulk-interface correspondence for tight-binding Hamiltonians subjected to an Iwatsuka magnetic field. Notably, it occurs that the topological quantization of the interface currents remains independent of the slope $α$.

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arXiv2024

Lieb-Robinson bounds in the continuum via localized frames

We study the dynamics of interacting fermions in the continuum. Our approach uses the concept of lattice-localized frames, which we introduce here. We first prove a Lieb-Robinson bound that is valid for a general class of local interactions, which implies the existence of the dynamics at the level of the CAR algebra. We then turn to the physical situation relevant to the (fractional) quantum Hall effect, namely the quasi-free second quantized Landau Hamiltonian to which electron-electron interactions can be added.

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arXiv2023

Dixmier trace and the DOS of perturbed magnetic operators

The main goal of this work is to provide a description of the {trace per unit volume} in terms of the {Dixmier trace} (regularized by the resolvent of the harmonic oscillator) for a large class of two-dimensional \emph{magnetic operators} perturbed by (homogeneous) {potentials}. One of the payoffs of this result is the possibility of reinterpreting the {density of states} (DOS) of these perturbed magnetic systems via the {Dixmier trace}, and taking advantage of the fact that this quantity can be conveniently calculated on the basis of the Laguerrre functions that diagonalize the harmonic oscillator.

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arXiv2022

A new light on the FKMM invariant and its consequences

"Quaternionic" vector bundles are the objects which describe the topological phases of quantum systems subjected to an odd time-reversal symmetry (class AII). In this work we prove that the FKMM invariant provides the correct fundamental characteristic class for the classification of "Quaternionic" vector bundles in dimension less than, or equal to three (low dimension). The new insight is provided by the interpretation of the FKMM invariant from the viewpoint of the Bredon equivariant cohomology. This fact, along with basic results in equivariant homotopy theory, allows us to achieve the expected result.

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arXiv2022

Topological polarization in disordered systems

Deformations in piezoelectric materials lead to conduction effects, which are due to two contributions: the relative displacements of the ionic cores, and the so-called orbital polarization. This work is devoted to the rigorous derivation of the celebrated King-Smith and Vanderbilt formula for orbital polarization in a generalized setting that includes continuous random systems among others.

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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 Hörmander 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.

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arXiv2021

The noncommutative geometry of the Landau Hamiltonian: Differential aspects

In this work we study the differential aspects of the noncommutative geometry for the magnetic $C^*$-algebra which is a 2-cocycle deformation of the group $C^*$-algebra of $\mathbb{R}^2$. This algebra is intimately related to the study of the Quantum Hall Effect in the continuous, and our results aim to provide a new geometric interpretation of the related Kubo's formula. Taking inspiration from the ideas developed by Bellissard during the 80's, we build an appropriate Fredholm module for the magnetic $C^*$-algebra based on the magnetic Dirac operator which is the square root (à la Dirac) of the quantum harmonic oscillator. Our main result consist of establishing an important piece of Bellissard's theory, the so-called second Connes' formula. In order to do so, we establish the equality of three cyclic 2-cocycles defined on a dense subalgebra of the magnetic $C^*$-algebra. Two of these 2-cocycles are new in the literature and are defined by Connes' quantized differential calculus, with the use of the Dixmier trace and the magnetic Dirac operator.

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arXiv2021

Dixmier trace and the DOS of magnetic operators

The main goal of this work is to provide two new formulas for the computation of the trace per unit volume, and consequently the integrated density of states (IDOS), for magnetic operators. These formulas also permit the use of the Dixmier trace in the spectral analysis of magnetic operators. The second of these formulas, named energy shell formula, permits to approximate the IDOS by a finite sums of averaged expectation values of the spectral projections.

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arXiv2021

Spectral and scattering theory of one-dimensional coupled photonic crystals

We study the spectral and scattering theory of light transmission in a system consisting of two asymptotically periodic waveguides, also known as one-dimensional photonic crystals, coupled by a junction. Using analyticity techniques and commutator methods in a two-Hilbert spaces setting, we determine the nature of the spectrum and prove the existence and completeness of the wave operators of the system.

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arXiv2021

The cohomology invariant for class DIII topological insulators

This work concerns with the description of the topological phases of band insulators of class DIII by using the equivariant cohomology. The main result is the definition of a cohomology class for general systems of class DIII which generalizes the well known $\mathbb{Z}_2$-invariant given by the Teo-Kane formula in the one-dimension case. In the two-dimensional case this cohomology invariant allows a complete description of the strong and weak phases. The relation with the KR-theory, the Noether-Fredholm index and the classification of "Real" gerbes are also discussed.

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