ANR-Blan-0238

VoLQuan

(Vortex Lattice and Quantum Hall Effect)

Rencontres:

* 28 novembre 2008

* 23 Mai 2008


Buts du projet

The aim of this project is to tackle mathematical issues that are relevant for problems
at the core of international interest in physics, and for which several Nobel prizes have been
awarded : the Quantum Hall effect in Bose Einstein condensates. For that purpose, our group
is made up of two teams, one of mathematicians and one of physicists.

International context in physics : Superfluidity and superconductivity are two spectacular
manifestations of quantum mechanics at the macroscopic scale. Among their striking charac-
teristics is the existence of vortices with quantized circulation. The physics of such vortices
is of tremendous importance in the field of quantum fluids and extends beyond condensed
matter physics. Indeed, rearrangements of a vortex lattice in the interior of neutron stars
have been proposed as an explanation of the so-called "glitches" in the rotation frequency
of pulsars. The cosmic strings are topological line defects of cosmological extent akin to the
vortices of liquid Helium and their role in galaxy formation is the subject of present research
in cosmology. Ultracold gaseous Bose-Einstein condensates allow tests in the laboratory to
study various aspects of macroscopic quantum physics. In particular, the quest for vortices
has been immediate. The possibility of tuning many parameters of the system through atomic
physics techniques turns these gaseous systems into extremely attractive objects.
Motivation of the project : In 2000, the ENS team of Jean Dalibard at Laboratoire Kastler-
Brossel succeeded in observing vortices in a single component Bose Einstein condensate, fol-
lowed by teams at the MIT and Oxford. This has triggered of many mathematical works on
vortices and the Abrikosov vortex lattice.1 All these experimental situations are described by
a mean-field approach, where the system is characterized by a macroscopic wave function sol-
ving the Gross-Pitaevskii or nonlinear Schrodinger equation. The mathematical tools involve
PDE's, energy estimates, homogeneisation, analysis of the spectrum of some operator, and
more recently semi-classical analysis and the introduction of Bargmann spaces. Our interest
in this project is still in the vortex lattice, but in a situation where the mean field model is
no longer valid because the states are highly correlated, similarly to the fractional quantum
Hall states of electrons in two-dimensional structures. Thus, one has to consider the quantum
N body hamiltonian.
Current Interest and objectives : To nucleate vortices, one has to add a certain amount
of angular momentum to the gas. This can be done by rotation. Current physical interest
is now for a rapid rotation regime : the rotation frequency becomes close to the transverse
trapping frequency. The centrifugal and trapping forces then nearly compensate each other
and the spatial extent of the condensate becomes very large. This is a most interesting regime
since it is equivalent to the situation leading to the quantum Hall effect in two-dimensional
electron gases. Depending on the ratio between the number of vortices and the number of
particles, the ground state of the system will be well approximated by a macroscopic wave
function or it will be a strongly correlated state. The former case is the one that we have
started to study but that still leads to very interesting open questions, in particular on the
study of the Abrikosov lattice, as well as the behaviour of the lattice (melting) as a function
of temperature. The case of strongly correlated states is even more fascinating as it is directly
connected to the physics of fractional Quantum Hall phenomena. This regime has not yet
been reached experimentally. It is the core of our project and requires the analysis of the
quantum N body hamiltonian for bosons. The issues that we want to understand are new
and open mathematically. Indeed, all the mathematical works on the thermodynamical limit
lead to situations where the asymptotic problem is de-correlated and thus the methods cannot
be used as such. Our problems display links with very di±cult questions in random matrix
theory, in particular for estimating the norm of the so-called Laughlin wave functions. We
will need to use as much as possible tools coming from statistical physics for Coulomb gases
and problems on electrons. The team of physicists is expert for this type of problems. In
previous collaborations with physicists, the mathematical team have proved its ability and
dynamism for interdisciplinary aspects. It involves well known specialists of the tools that
will be needed. Our team has the talent to match the challenges raised by this ambitious and
innovative research program. We hope to gain knowledge of a fundamental nature which may
then be related to current experiments and applied to more conventional condensed matter
systems.





CMAP UMR 7641 École Polytechnique CNRS, Route de Saclay, 91128 Palaiseau Cedex France, Tél: +33 1 69 33 46 00 Fax: +33 1 69 33 46 46