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Giuseppe De Nittis

Publications and source records attributed to Giuseppe De Nittis.

62 records · Page 4Linked to original sources
arXiv2012

The topological Bloch-Floquet transform and some applications

We investigate the relation between the symmetries of a Schrödinger operator and the related topological quantum numbers. We show that, under suitable assumptions on the symmetry algebra, a generalization of the Bloch-Floquet transform induces a direct integral decomposition of the algebra of observables. More relevantly, we prove that the generalized transform selects uniquely the set of "continuous sections" in the direct integral decomposition, thus yielding a Hilbert bundle. The proof is constructive and provides an explicit description of the fibers. The emerging geometric structure is a rigorous framework for a subsequent analysis of some topological invariants of the operator, to be developed elsewhere. Two running examples provide an Ariadne's thread through the paper. For the sake of completeness, we begin by reviewing two related classical theorems by von Neumann and Maurin.

math-ph↗
arXiv2012

Topological aspects of generalized Harper operators

A generalized version of the TKNN-equations computing Hall conductances for generalized Dirac-like Harper operators is derived. Geometrically these equations relate Chern numbers of suitable (dual) bundles naturally associated to spectral projections of the operators.

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

Thermodynamic phase transitions and shock singularities

We show that under rather general assumptions on the form of the entropy function, the energy balance equation for a system in thermodynamic equilibrium is equivalent to a set of nonlinear equations of hydrodynamic type. This set of equations is integrable via the method of the characteristics and it provides the equation of state for the gas. The shock wave catastrophe set identifies the phase transition. A family of explicitly solvable models of non-hydrodynamic type such as the classical plasma and the ideal Bose gas are also discussed.

nlin.SI↗
arXiv2011

Generalized TKNN-equations

We derive generalized TKNN-equations via bundle representations of the noncommutative torus with rational deformation parameter, the bundle coming from spectral projections in the torus algebra. These equations relate Chern numbers of dual bundles which we interpret as Hall conductances for Dirac-like Hamiltonians describing magnetic Bloch electrons in a strong magnetic field. We also present their generalizations for irrational values of the deformation parameter.

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

Effective models for conductance in magnetic fields: derivation of Harper and Hofstadter models

Some relevant transport properties of solids do not depend only on the spectrum of the electronic Hamiltonian, but on finer properties preserved only by unitary equivalence, the most striking example being the conductance. When interested in such properties, and aiming to a simpler model, it is mandatory to check that the simpler effective Hamiltonian is approximately unitarily equivalent to the original one, in the appropriate asymptotic regime. In this paper, we consider the Hamiltonian of an electron in a 2-dimensional periodic potential (e.g. generated by the ionic cores of a crystalline solid) under the influence of a uniform transverse magnetic field. We prove that such Hamiltonian is approximately unitarily equivalent to a Hofstadter-like (resp. Harper-like) Hamiltonian, in the limit of weak (resp. strong) magnetic field. The result concerning the case of weak magnetic field holds true in any dimension. Finally, in the limit of strong uniform magnetic field, we show that an additional periodic magnetic potential induces a non-trivial coupling of the Landau bands.

math-ph↗