##### Research Statement (eng) | (fra)

The present research statement briefly describes the research activity in mathematical physics I have carried out in last years and develops prospects for future research. *Solid state physics* has always been a source of great inspiration for me. The principal research fields of my activity (ordered by personal interest) are:

- Topological effects (e.g. Topological Insulators, Quantum Hall Effect, Piezoelectricity, etc.),
- Electronic Properties of Periodic and Aperiodic Structures,
- Thermodynamics.

The wealth of mathematical knowledge that I accumulated during my research includes:

- Functional Analysis,
- Operator Algebras,
- Vector bundle theory and K-theory,
- Non-commutative Geometry,
- Differential Geometry,
- Partial Differential Equations.

### Past Achievements

**+ Magnetic Hamiltonians and TKNN-equations
**During my PhD, I have been working mostly on problems related to

*Quantum Hall Effect*(QHE), with emphasis on its geometric (commutative and non-commutative) aspects. My thesis entitled

*“Hunting colored (quantum) butterflies: a geometric derivation of the TKNN-equations”*, focused on the geometric analysis of the duality proposed by D. J. Thouless, M. Kohmoto, M. P. Nightingale and M. Nijs in the seminal paper of 1982. This duality relates the Hall conductances associated to a system of non-interacting 2D electrons in a periodic potential (crystal) under the influence of a uniform orthogonal magnetic field (with rational flux) in the opposite limits of a very strong (

*Harper regime*) and a very weak (

*Hofstadter regime*) magnetic field. The quantized conductances are associated (via Kubo formula) with the Chern numbers of suitable vector bundles labeled by the spectral projections of the effective Hamiltonians which describe the system in the two limit regimes. In this paper the authors argued that the values of the conductance exhibited by the system in the two limits are pairwise coupled by means of a diophantine relations called

*TKNN-equation*.

The main goal achieved in my thesis was the rigorous derivation of the TKNN-equation based on a complete understanding of the deep geometric aspects of the duality proposed by Thouless & coworkers. The proof is based on three steps:

- First of all I derived the effective models for the description of the QHE in the limit regimes of strong and weak magnetic field. It turns out that these effective models live in two isomorphic representations of the non-commutative torus. I used the Space Adiabatic Perturbation Theory (developed by G. Panati, H. Spohn and S. Teufel in 2003) in order to prove the (asymptotic) unitary equivalence between the effective models and the true physical Hamiltonian. Afterwards, these results have been extended to a larger class of slowly varying unbounded magnetic fields in a later work in collaboration with M. Lein. This generalization was made possible by the combined use of the Space Adiabatic Perturbation Theory and the
*Magnetic Weyl quantization*. - The second step concerns the analysis of the connection between geometry and symmetry. I showed how the existence of an Abelian symmetry for a quantum system leads, in general, to a unique vector bundle structure which encodes relevant physical information in terms of topological invariants. Such an analysis is based on a suitable algebraic generalized of the
*Bloch-Floquet-Wannier transform*. - The discussion of the geometric duality between vector bundles which is at the origin of the TKNN-equation is the last step. The geometric approach developed in my thesis led to fruitful generalizations. In fact, in a later work in collaboration with G. Landi, I proved that the TKNN-equation can be extended to a larger family of representations of the non-commutative torus. Each of these representations is labeled by an integer which corresponds to the Chern number of an associated vector bundle. Possible applications of such a generalization are the determination of the Hall conductance (in the strong magnetic field regime) when a periodic magnetic field is turned on or in the case of an hexagonal lattice (G
*raphene*).

**+ Localizzation of Wannier functions**

Any realistic model for conduction in crystals should consider also the effect due to a *periodic* magnetic field. The results obtained in my thesis are still valid in the presence of a periodic magnetic field provided that certain topological obstructions are removed (this is necessary for the application of the Space Adiabatic Perturbation Theory). I discussed this problem in collaboration with M. Lein. The main result of this work is the proof of the existence of a basis of *exponentially localized Wannier functions* in the multi-band case even when the time-reversal symmetry is broken, provided certain conditions are met.

**+ Piezoelectricity in Graphene**

In collaboration with M. Lein I investigated the possibility of generating *piezoelectric orbital polarization* in graphene-like systems which are deformed periodically. In this work we looked at discrete two-level models which depend on control parameters; in this setting, time-dependent model Hamiltonians are described by loops in parameter space. 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 showed the polarization, as given by the (topological) *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 this analysis extend to systems with weak disorder. This analysis has been applied to the *uniaxial strain model* for graphene (which includes nearest-neighbor hopping and a stagger potential) showing that this model supports non-trivial piezo effects. This result seems to be in agreement with recent physics literature.

**+ Perturbed Maxwell Dynamics in Photonic Crystals**

*Photonic crystals* are periodic optical nanostructures that affect the motion of photons in much the same way that semiconductors affect electrons. Mathematically, the dynamics of the light in photonic crystals is described by a first order differential Hamiltonian, called *Maxwell operator*. If the periodic structure of the photonic crystal is perturbed by an external, slow varying, field an adiabatic expansion (in the spirit of the Spatial Adiabatic Perturbation Theory) seems to be possible. As a first step to deriving effective dynamics (more properly called *ray optics*), I proved in a preliminar work in collaboration with M. Lein that the perturbed periodic Maxwell operator can be seen as a *pseudodifferential operator*. This necessitates a better characterization of the behavior of the physical initial states at small crystal momenta and small frequencies.

**+ Thermodynamics and Integrable Systems**

A secondary line of research, which has always fascinated me since the time of my bachelor degree, concerns the thermodynamics of *phase transitions*. In a work in collaboration with A. Moro I investigated the relations between thermodynamic phase transitions and *shock waves* of hydrodynamic type. The application of techniques proper of the theory of Integrable Systems to thermodynamic equations seems to be very promising in order to study phase transitions of composite and non-additive thermodynamic systems.

### Future Plans

The research activity that I would like to carry out in the next years can be divided into two main lines: the first concerns the study of *aperiodic solids* while the latter focusses on *topological effects* in condensed matter.

**+ Spectral Properties of Aperiodic Solids
**The aim of this project is to develop a new mathematical method capable of analysing the spectrum of the Schrödinger operators with potentials created by the atomic nuclei of an aperiodic solid with long range order. This method is based on the Aperiodic Wannier Transform which is the most natural generalization of the usual Wannier Transform able to take care of the loss of translation symmetry. The Aperiodic Wannier Transform has been conceived by J. Bellissard and has been intensely studied by J. Bellissard, V. Milani and myself during the 2010 at Georgia Institute of Technology. Many properties of this transform like unitarity, covariance, boundary condition, etc., have been definitively understood during this period. The first paper on this project, concerning the one-dimensional case, is expected to be ready in the next weeks.

The construction of the Aperiodic Wannier Transform is based on the following generalizations of notions typical of the periodic case:

- The periodic arrangement of atoms is replaced by a
*Delone set*(denoted by L), i.e. a discrete set with a finite local density of points (absence of collapse) which is nowhere zero (absence of holes); - The Abelian group of the d-dimensional translations is replaced by the (generally) higher dimensional Abelian group generated by L-L (the set of vectors joining each pair of points in L), called
*Lagarias group*; - The
*Brillouin zone*is replaced by a torus which has the same dimension of the Lagarias group (more precisely by the Pontryagin dual of the Lagarias group); - The notion of fundamental cell is replaced by the construction of the Voronoi tiling; the latter leads to the definition of the
*Anderson-Putnam complex*which is a CW-complex which encodes in its topology some properties of the Delone set L.

The Aperiodic Wannier Transform is used to define a Bloch-like decomposition for the quasicrystal-Hamiltonians, namely the Schrödinger operators with a potential energy given by the sum of short range atomic potentials with atoms located at the points of the set L. More precisely, it is possible to show that the quasicrystal-Hamiltonian can be represented as a continuous family of operators which live on the Anderson-Putnam complex. Any representative of the family is parametrized by a point of the “enlarged” Brillouin zone through suitable boundary conditions. The latter are characterized in terms of the action a cohomological operator.

The analysis of spectral quantities for quasycristals in dimension greater than one is a tremendously challenging problem. For example, numerical computations are made by means of periodic approximant models. However, depending upon the type of approximation, defects may occur in the aperiodic limit. It is expected that the method based on the application of the Aperiodic Wannier Transform could be immune to such problems. The problem of computing the spectrum of quasicrystal-Hamiltonians through a Bloch-like decomposition is still not achieved. A more deep analysis is required to overcome some technical complications related to a loss of information due to the underlying aperiodicity. The long term idea is to employ a renormalization-like method based on the self-similarity of certain Delone sets (substitution tiling) in order to rebuild in the thermodynamic limit the full spectrum of the quasicrystal-Hamiltonian. The success of such a program would provide a powerful toll capable to address some intriguing open questions, such as: *in dimension greater than then 2 is the spectrum of the quasicrystal-Hamiltonian purely absolutely continuous for large enough energy? Are gaps closed at high-energy regimes (Bethe-Sommerfeld conjecture)?*

+ Physics of Topological Phases in Insulators and Photonic Crystals

*– Graphene and the fate of Dirac points –*

Graphene is a configuration of carbon consisting of a mononuclear layer of carbon atoms located at the sites of a two-dimensional honeycomb lattice. The investigation of this new material is a very interesting and fascinating topic in physics. The 2010 Nobel Prize in physics was awarded to Geim and Novoselov for groundbreaking experiments regarding this new two-dimensional material. My main interest concerns the peculiar electronic properties of graphene. The most simple model is a discrete Laplaican on a honeycomb lattice. Because the lattice is bipartite with two triangular sublattices, this Laplacian is a block operator. Its most interesting feature is the appearance of so-called Dirac points at which the dispersion relation is linear and thus similar to that of a two-dimensional Dirac operator without mass. At these points the density of states vanish. There are now various mechanisms to open a gap at the Dirac points, in particular, by adding a staggered potential or a spin-orbit coupling. According to the results of Kane and Mele, these two ways of opening the gap differ considerably, namely the first one leads to a classical insulator and the second one to a particular example of a so-called topological insulator, a new class of condensed matter phases classified by their symmetry classes. One of the main feature of these phases is the expected stability of non-trivial edge modes. In the case of the Kane-Mele model these are spin edge currents. The proposed research aims at an analysis of the effects of disorder on all the different classes of topological insulators:

- What happens with density of states at the Dirac points if a small disorder is added? What is the scaling law in the coupling constant of the disorder? This should be analyzed without adding a gap-opening term and could alternatively also be studied effectively for a two-dimensional continuous Dirac operator.
- How does the density of states behave under a disorder perturbation when the gap at the Dirac point is open? Are there Lifshitz tails in this situation or not? How does the behavior depend on the phase of the topological insulator?
- What is the asymptotic formula for the low frequency conductivity? For random Schrödinger operators this has been proved to be close to the form proposed by Mott.
- Can one prove Anderson localization at the band edges in any of the above situations? Are the spin edge currents stable under perturbation by disorder when one is in the non-trivial topological insulator phase? This would be a result similar to the stability of edge currents in the Quantum Hall Effect. An invariant describing the non-trivial topology of the bulk system also in presence of disorder has been put forward by Prodan, but the edge effects still have to be analyzed for a half-plane model.

*– Structural analysis of models of topological insulators –*

The study of the concrete models on a honeycomb lattice described above naturally leads to the question whether a similar analysis can be carried out for other topological insulators, in particular, also for three-dimensional models and some quantum spin systems. Furthermore, and this is rather the focus of this point of the proposal, there remains a lot to understand about the classification and stability analysis of topological insulators (even though this is considered a finished story in the physicists community). While physicists like Kitaev provided a general classification scheme, it has yet to be applied to many concrete models, actually containing disorder not only in a way typical for random matrix theory. A mathematical contribution to a very concrete model is by Hasting and Loring. This work uses real K-theory in order to determine the relevant invariants. This is a very promising point of view that should be further developed (albeit in a somewhat other direction than). In particular, the various Z2-invariants appearing in the theory of topological insulators should be identified as the index of some skew-adjoint Fredholm operator to be determined. A prime example where this is achieved also on a mathematical satisfactory level is the case of Quantum Hall systems. For these systems, also the edge physics has been analyzed rigorously (note that this edge physics is used as a guiding principle to determine the classification in vorious theoretical physics works). As shows already for the example of the spin Hall effect in Kane-Mele models described above, the analysis of this edge physics is not as simple as in quantum Hall systems. Nevertheless, as already pointed out, the analysis of graphene should lead the way to a more conceptual understanding.

*– Photonic Quantum Hall Effect –*

The Photonic Hall Effect (PHE) is a topological effect which is nowadays under the attention of physicists. This is the equivalent of the well known Quantum Hall Effect (QHE) but for electromagnetic waves traveling through a photonic crystal (i.e. a periodic arrangement of dielectrics within an environment dielectric like the air). This effect was predicted on a theoretical level by F. D. Haldane and S. Raghu, but the proposed derivation is based only on an analogy with the well-known electronic cases. In collaboration with M. Lein we started a research program in order to obtain a rigorous description of the PHE. The first goal of this project is to obtain an effective model for the semiclassical propagation of the light associated with a given “photonic conduction band”. This can be done by means of the Space Adiabatic Perturbation Theory. Such a model should play for the PHE the same role that the Harper operator plays in the case of standard QHE. In a second stage, we want also to consider the the effect of random perturbations. This should be done using the non-commutative geometry in the same spirit proposed by J. Bellissard for the case of the usual QHE.