Conical intersections in Mathematical Physics

Paris - May 29-31, 2013

"Variational and Spectral Methods in Quantum Mechanics"

The goal of the mini-workshop is to get together leading experts from different communities in an interdisciplinary atmosphere, to stimulate a fruitful exchange of ideas and methods.

Among the topics covered:

### conical intersections in Born-Oppenheimer molecular dynamics

### Dirac cones and pseudospin winding number in (multilayer) graphene

### eigenvalue intersections in ultracold Fermion gases

### applications to Quantum Control Theory

Title: Asymptotic stability for the Schroedinger equation in dimension three with a pointwise nonlinearity

Abstract: Point interactions for the Schroedinger equation are defined in the linear case by means of the Von Neumann-Krein's theory of self-adjoint extensions of symmetric operators. It turns out that the family of such extensions can be parametrized by a real number, that represents the scattering length of the chosen interaction. Imposing a dependence of such parameter on the wave function, one obtains a nonlinear equation, whose nonlinear term is concentrated at a point. Specializing the shape of the nonlinearity, it is possible to produce stationary states, and then to study the problem of their orbital and asymptotic stability. We give some results on one of those models. This is a joint work with D. Noja and C. Ortoleva (Milan)

Andrei Agrachev (SISSA, Trieste, Italy)

Title: Homological invariants for families of quadratic forms.

Abstract: We introduce "Quadratic cohomology" that reflects the structure of the parameters sets of multiple eigenvalues for families of real symmetric or Hermitian matrices. The talk is based on the papers: "Quadratic cohomology" arXiv:1301.2059 and "On the space of symmetric operators with multiple ground states" arXiv:1107.3010

Francesca Chittaro (LSIS, Toulon)

Title: Geometry of conical intersections: constructive methods for adiabatic control.

Abstract: In this talk we discuss in more details the results on controllability of the bilinear Schroedinger equation presented by M. Sigalotti. In particular, we focus on the geometry of conical intersections: we introduce the notions of ``conicity matrix'', that describes the behaviour of the eigenstates in a neighborhood of the intersection, and of ``non-mixing curves'', that is special curves, in the space of controls, that improve the precision of the adiabatic passage through the conical intersection.

Clotilde Fermanian (Universite Paris EST)

Title: A nonlinear Landau Zener formula.

Abstract: In this talk, we describe the results obtained in collaboration with R. Carles about the large time behaviour of the solutions of some non linear extension of the celebrated Landau-Zener system : existence of scattering states, of wave operator and comparison of the scattering operator with its linear counterpart. This system may appear from different physical models, for example as an envelope equation in Bose-Einstein condensation.

George Hagedorn (Virginia Tech, USA)

Title: Two Molecular Problems Involving Electron Energy Level Crossings

Abstract: We discuss two mathematical problems that involve level crossings in molecular quantum mechanics.

1. We present a summary of analysis by former graduate student Mark Herman on Born-Oppenheimer approximations near Renner-Teller Crossings. These crossings can occur for triatomic linear molecules, and they produce two different bending vibrational frequencies.

2. We present a mathematical problem that is motivated by the H

Alain Joye (Grenoble, France)

Title: Semiclassical Determination of Intermode Transitions

Abstract: We consider the semiclassical limit of systems of autonomous PDEs in 1 + 1 spacetime dimensions, with analytic matrix-valued coefficients and non-degenerate real-valued dispersion relations only. Considering time-dependent solutions to the PDE that are carried asymptotically in the past and as x → -∞ along one mode only, we determine the piece of the solution that is carried for x → +∞ along some other mode in the future. Because of the nondegeneracy of the modes, such transitions are exponentially small in the semiclassical parameter by the Landau-Zener mechanism. We elucidate the spacetime properties of the leading term of this exponentially small wave, when the semiclassical parameter is small, for large values of x and t, when some avoided crossing takes place between the involved modes. Joint work with Magali Marx.

Mathieu Lewin (Universite de Cergy-Pontoise, France)

Title: Ground state properties of graphene in Hartree-Fock theory

Abstract: In this talk I will discuss the Hartree-Fock approximation of graphene in infinite volume, with instantaneous Coulomb interactions. First I will explain how to construct the translation-invariant ground state of the system. The zero-mass Dirac dispersion relation of graphene gets renormalized and the effective Fermi velocity becomes logarithmically divergent at zero momentum. Then I will discuss the existence of ground states in the presence of local defects and some properties of the associated nonlinear response. Joint work with C. Hainzl and C. Sparber.

Carolin Lasser (Munich, Germany)

Title: Microlocal surface hopping through conical intersections

Abstract: We review a microlocal approximation of the dynamics associated with vibrational Schroedinger equations. Its key property are effective non-adiabatic transitions at phase space points, where the eigenvalue gap has a local minimum. The talk links the analytical approximation with a surface hopping algorithm and presents numerical simulations for the internal conversion of pyrazine through a conical intersection.

Domenico Monaco (SISSA, Trieste, Italy)

Title: Topological invariants of eigenvalue intersections and decrease of Wannier functions in graphene

Abstract: Motivated by the study of localisation of electrons in mono- and multilayer graphene, we introduce a topological invariant for the family of Bloch eigenspaces, baptised eigenspace vorticity. This invariant characterises the behaviour of such eigenspaces around an eigenvalue intersection. After a comparison with the pseudospin winding number, we exhibit a canonical model for the local topology of the eigenspaces, one for each value n in Z of the eigenspace vorticity. A suitable universality theorem for these models is then proved. This allows us to extract the asymptotic decrease of the Wannier functions for the valence and conduction band of graphene, both in the monolayer and the bilayer case. We show that that the single band Wannier function w satisfies w(x) ~ |x|

Marcello Porta (Bonn, Germany)

Title: The bulk-edge duality for topological insulators

Abstract: Topological insulators are newly discovered materials, which behave as usual insulators in the bulk but carry spin currents on their edges. The presence of the currents is stable against perturbations which respect time-reversal symmetry. In this talk I will discuss a duality between the bulk and edge descriptions of two-dimensional topological insulators, namely the relation between the presence/absence of edge currents and the value of a certain bulk Z_2 topological invariant. This is joint work with G. M. Graf.

Dario Prandi (Ecole Polytechnique, Paris)

Title: Dynamics of a quantum particle on a conical-like surface

Abstract: We consider the evolution of a free particle on a two-dimensional manifold endowed with the degenerate Riemannian metric $ds^2=dx^2+|x|^{2\alpha}d\theta^2$, where $x\in R$, $\theta\in S^1$ and the parameter $\alpha R$. For $\alpha$ smaller or equal to $-1$ this metric describes cone-like manifolds (for $\alpha=-1$ it is a flat cone). For $\alpha=0$ it is a cylinder. For $\alpha$ bigger or equal to $1$ it is a Grushin-like metric. In particular we discuss whether the free particle can cross the singular set ${x=0}$ or not. This will be achieved through a study of the self-adjoint extensions of the Laplace-Beltrami operator. In the second part of the talk we focus on the same problem for the evolution of the heat. We study the Markovianity of the self-adjoint extensions of the Laplace-Beltrami operator and their stochastic completeness. In particular we describes in which cases the singularity absorbs the heat. This is a joint work with U. Boscain.

Mario Sigalotti (INRIA, Paris, France)

Title: Control of the Schroedinger equation via adiabatic paths through conical intersections

Abstract: We present a constructive method to control the bilinear Schroedinger equation via two controls. The method is based on adiabatic techniques and works if the spectrum of the Hamiltonian admits conical eigenvalue intersections. We discuss in particular how to spread on several levels connected by conical intersections a state initially concentrated in a single energy level. We provide sharp estimates on the dependence of the error with respect to the controllability time, based on the construction of some special curves in the space of controls that improve the precision of the adiabatic approximation with respect to classical adiabatic theory. We also discuss the relation of the proposed sufficient condition for controllability with the classical Lie algebraic controllability criterion.

Andrea Trombettoni (SISSA, Trieste, Italy)

Title: Simulations of Dirac fermions with ultracold atoms in optical lattices

Abstract: In the first part of the talk I discuss the developing area of quantum simulations with ultracold atoms, especially in connection with the use of optical lattices. A major example is given by the simulation of Dirac fermions, which has been recently experimentally implemented in a two-dimensional tunable-geometry optical lattice. In the second part I discuss a proposal for the experimental realization of (3+1) relativistic Dirac fermions using ultracold atoms in a rotating optical lattice or, alternatively, in a synthetic magnetic field. This approach has the advantage to give mass to the Dirac fermions by coupling the ultracold atoms to a Bragg pulse. A dimensional crossover from (3+1) to (2+1) Dirac fermions can be obtained by varying the anisotropy of the lattice. The effect of attractive interactions, and the consequent semimetal-superfluid transition, will be also analyzed.

Luigi Barletti (Florence)

Ariane Garon (Munich)

Lysianne Hari (Cergy Pontoise)

Paolo Mason (CNRS, LSS, Orsay)

If you are interested to give a scientific communication in the framework of the workshop, please include title and abstract of a possible talk.