MATHEMATICAL PHYSICS SEMINAR

Fall 2006 Lectures

Regular meeting time: Tuesdays 4:30--5:30
Place: Jadwin 343

Date Speaker Title
Sept.. 19
Stanislav Smirnov,
Univ. Geneva

Conformal invariance in the Ising model
We will discuss how to show that cluster interfaces in the Ising model (as well as its random cluster representation) have conformally invariant scaling limits, which can be identified with Schramm's SLE curves.  The proof is based on the construction of covariant discrete holomorphic observables (fermions), and much of it can be generalized to other O(n) and random cluster models, with the appropriate values of the spin.

Note Special Date, Time
and Location

Friday
Sept.. 22
3:00 p.m.
Jadwin 303

 Lincoln Chayes, UCLA

On the Absence of Ferromagnetism in Typical 2D Ferromagnets
The Ising ferromagnet in d “ 2 is the simplest model of interacting magnetic constituents that exhibits a cooperative phase transition.  The Ising systems serve not only as models of magnetic materials, there are other phenomenological interpretations related to binary alloys, adsorbed gasses etc. not to mention a host of uses in other areas of science: political distributions, dueling protagonists and so forth. In all cases, the well known phase transition is presumed to teach us something about the collective behaviors in these systems. But for the primary interpretation, other forces must be taken into account, in particular the long-range dipolar interactions. As it turns out, these forces have significant impact on the phase structure of 2D Ising ferromagnets and, as the title indicates, exclude the possibility of ferromagnetism in these systems.
Sept. 26 Heinz Siedentop, University of Munich

The ground state energy of heavy atoms: Absence of relativistic effects
We will show that the ground state energy of heavy atoms is, to leading order, given by the non-relativistic Thomas-Fermi energy. The proof is based on the relativistic Hamiltonian of Brown and Ravenhall which is derived from quantum electrodynamics yielding energy levels correctly up to order $\alpha2$Ry.
Oct. 3 Rowan Killip,
UCLA		 Circular beta Ensembles
I will describe two results concerning Dyson's circular ensembles (i.e., the Coulomb gas on the unit circle) at arbitrary temperature: (i) their appearance as the eigenvalue distribution of a Schroedinger-like model with random decaying potential; and (ii) Gaussian fluctuations for the number of particles in an interval.
Oct. 10 David Brydges,
Univ of British Columbia, IAS		 Self-avoiding loop correlations and loop erasure
Oct.17 Alessandro Giuliani,
Princeton Univ.		 Ising models with long range competing interactions: striped nature of the ground states
I will consider a d-dimensional Ising model with nearest neighbor ferromagnetic interaction and a long range antiferromagnetic interaction, decaying as the distance to the power -(d+1). In 1D I will show that the ground state is periodic, consisting of ferromagnetic blocks of alternating magnetization. I will then discuss partial results for the multidimensional case, for which a similar phenomenon of spontaneous stripe formation in the ground state has been conjectured. The talk is based on joint work with Joel Lebowitz and Elliott Lieb.
Oct. 24 Robert Seiringer
Princeton Univ.		 Correlation Estimates for Quantum Many-Body Systems at Positive Temperature
We present a method for obtaining bounds on the expectation value of certain two-body interaction potentials in a general state on Fock space in terms of the corresponding expectation value for thermal equilibrium states of non-interacting systems. The difference can be dominated by the difference in the free energies. This method can be viewed as a rigorous form of first order perturbation theory for many-body systems at positive temperature. One of the key ingredients is the strong subadditivity of the von-Neumann entropy. As an application, we give a proof of the first two terms in a high density (and high temperature) expansion of the free energy of jellium with Coulomb interactions, both in the fermionic and bosonic case. For bosons, our method works above the non-interacting gas transition temperature for Bose-Einstein condensation.
Oct. 31    
Nov. 7
Frederic Klopp,
Universite Paris-Nord
 On the multifractal structure of the generalized eigenfunctions of certain sparse Schrödinger operators
The talk is devoted to the study of the generalized eigenfunctions for certain Schrödinger operators with sparse potentials on the half-line. This study reduces to that of an ergodic matrix cocycle for which we develop an exact renormalization analysis based upon the monodromization procedure. The analysis results in a multi-fractal description of the solutions to the Schrödinger equation. As a consequence, we obtain a complete description of the set of ergodic parameters for which the Lyapunov exponent of the cocycle, hence, the modified Lyapunov exponent of the Schrödinger equation, exists and does not exist. It also gives a multi-fractal description of the generalized eigenfunctions of the operator. The talk is based on a joint work, which is in progress, with A. Fedotov (St Petersburg).
Nov. 14
CANCELLED		 E. Trubowitz, ETH, Zurich ************CANCELLED*********
Many Bosons
Nov. 21 Sandro Graffi, 
Univ. of Bologna
 Mean-Field and Classical Limit of Many-body Schroedinger Dynamics for Bosons
A new proof of the convergence of the N-particle Schroedinger dynamics for bosons towards the dynamics generated by the Hartree equation in the mean-field limit. For a restricted class of two-body interactions, we obtain convergence estimates uniform in h- bar, up to an exponentially small remainder. For h-bar = 0, the classical dynamics in the mean-field limit is given by the Vlasov equation. (Joint work with J.Froehlich and S.Schwarz)
Nov. 28 Emil Prodan, 
Princeton University 		The analytic structure of Bloch functions
In 1959, Walter Kohn descovered that the band energies of periodic Schroedinger operators in 1 dimension have a beautiful structure when one lets the k-wavevector be complex. He found that the energies of different bands are nothing but the same function evaluated on different sheets of a certain Riemann surface. This Riemann surface is generic in 1 dimension, in the sense that its shape does not depend on the particular form of the periodic potential. The exact asymptotic behavior of most of the correlation functions can be easily computed if the Riemann surface is known. In this talk I will discuss recent results that generalize all the above to linear molecular chains in 3D. The new approach is quite different from the original one and it relies on topological arguments (plus elementary functional analysis). I will discuss the generic structure of the Riemann surface for periodic molecular chains and, if time allows, I will present some explicitly calculated surfaces and go over several applications.

Please note special date and time
Fri., Dec. 15
2:30 p.m.

Y. Avron, Technion, Israel Entangled photons from biexciton decay in a quantum dot