Workshop
Conical
intersections in Mathematical Physics
SCHEDULE
Aim and topics:
In the last
decade, conical eigenvalue intersections appeared to be ubiquitous in
Mathematical Physics, with applications ranging from molecular
dynamics, to solid-state physics (e.g. the well-known Dirac-points
in graphene), to systems of ultracold Fermi atoms in optical
lattices, to the somehow surprising applications in Quantum
Control Theory .
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
Scientific
and Organizing Committee:
U. Boscain
(CNRS, Paris), G.
Panati (Roma I).
Speakers:
Riccardo
Adami (Turin, Italy)
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 H3+ ion
absorbing an electron and emitting a photon to produce H3
in its electronic ground state, but with the nuclei in an equilateral
triangle configuration. This equilateral triangle configuration is
precisely at a conical intersection of electron energy levels, and
the system is unstable. Typically H3 --->
H2 +H very quickly, but we do not understand the dynamics
precisely.
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|-3/2 as
|x| → ∞. In
particular, the expectation value of the modulus of the position
operator is infinite. Joint work with G. Panati.
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.
Other
participants
Luigi Barletti (Florence)
Ariane Garon (Munich)
Lysianne Hari (Cergy Pontoise)
Paolo
Mason (CNRS, LSS, Orsay)
Registration:
Please send,
before May 15th , an e-mail to the
address
boscain@cmap.polytechnique.fr
specifying
your name and affiliation. No conference fee is required.
If
you are interested to give a scientific
communication in the framework of the workshop, please include
title and abstract of a possible talk.
Sponsors:
ERC project
“GeCoMethods”, ANR GCM, AST project "Wannier functions".
We are grateful to Insistut Henri Poincaré for the kind hospitality.