Lectures

ABSTRACT

The 1983 discovery of the fractional quantum Hall effect marks a milestone in condensed matter physics: systems of “ordinary particles at ordinary energies” displayed highly exotic effects, most notably fractional quantum numbers. It was later recognized that this was due to emergent quasi-particles carrying a fraction of the charge of an electron. It was also conjectured (and, very recently, experimentally confirmed) that these quasi-particles had fractional statistics, i.e. a behavior interpolating between that of bosons and fermions, the only two types of fundamental particles.

These lectures will be an introduction to the basic physics of the fractional quantum Hall effect, with an emphasis on the challenges to rigorous many-body quantum mechanics emerging thereof.

The cornerstone of our theoretical understanding is the Laughlin state, a well-educated ansatz for the ground state of 2D particles subjected to large magnetic fields and strong interactions. The two latter effects conspire to generate strong and very specific correlations between particles.

We shall discuss mathematically the rigidity these correlations display in their response to perturbations. The main message is that trapping and disorder potentials can be taken into account by generating uncorrelated quasi-particles (Laughlin quasi-holes) on top of the Laughlin state.

SCHEDULE

REFERENCES

• N. Rougerie, "The classical Jellium and the Laughlin phase", arXiv:2203.06952 [math.AP];

• N. Rougerie, "On the Laughlin function and its perturbations", arXiv:1906.11656 [math-ph];

• N. Rougerie, J Yngvason, "Holomorphic quantum Hall states in higher Landau levels", J. Math. Phys. 61 (2020), 041101;

• E.H. Lieb, N. Rougerie, J. Yngvason, "Local incompressibility estimates for the Laughlin phase", arXiv:1701.09064 [math-ph].