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ICMP 2003 > Sessions > Condensed matter physics |
Session organized by J. Yngvason (Wien), Y. Avron (Haïfa).
| A. Elgart | New York |
| J. Feldman | Vancouver |
| G. M. Graf | Zürich |
| M. Salmhofer | Leipzig |
| R. Seiringer | Princeton |
| S. Teufel | TU München, Germany |
Adiabatic transport, Kubo formula and Anderson localization in some lattice and continuum models
The different explanations of Quantum Hall Effect rely on the validity of the linear response theory for a system that has infinite extent. We will present recent results on the adiabatic charge transport in this context for two dimensional lattice (joint work with M. Aizenman and J. Schenker) and continuum (joint work with B. Schlein) models of a non interacting electron gas. It is proved that if the Fermi energy falls in the localization regime then Hall transport is correctly described by the linear response Kubo formula. The localization condition is set forth by fractional moment method, which is by now extended also to continuum models (joint work with M. Aizenman, S. Naboko, J. Schenker and G. Stoltz). In the present talk, besides of the localization criteria, we will discuss some ideas -- Nenciu's asymptotic expansion, generalized space-momentum inequalities, and finite speed of propagation estimates which enter the proof.
Construction of a 2-d Fermi liquid: the results
This is one of two self-contained talks at ICMP 2003 concerning the proof (of Horst Knoerrer, Eugene Trubowitz and J. F.) that the temperature zero renormalized perturbation expansions of a class of interacting many-fermion models in two space dimensions have nonzero radius of convergence. The models have "asymmetric" Fermi surfaces and short range interactions. One consequence of the convergence is the existence of a discontinuity in the particle number density at the Fermi surface. The detailed proof is now available on the net at http://www.math.ubc.ca/~feldman/. This talk will concentrate on the statement of results. The other, which is in the session on Equilibrium Statistical Mechanics, will highlight the strategy of the construction.
Adiabatic charge pumping in open quantum systems
Transport between coupled reservoirs at equilibrium occurs if the junction between them is externally driven. We consider such pumps in the case where they are connected to reservoirs of independent electrons, and the driving is slow compared to the dwell time of particles in the pump. The charge transport associated with a given change of pump parameters is characterized in terms of S-matrices pertaining to time-independent junctions. In fact, several transport properties (charge, dissipation and, at positive temperature, noise and entropy production) may be expressed in terms of the matrix of energy shift which, like Wigner's time delay to which it is dual, is determined by the S-matrix.
This is joint work with J. Avron, A. Elgart, L. Sadun, K. Schnee.
Fermi systems in two dimensions and fermi surface flows
A proof of Fermi liquid behaviour of weakly coupled Hubbard--like models in two spatial dimensions at positive temperature, in the sense of finiteness of the quasiparticle interactions and regularity of the selfenergy, is discussed. The proof is by a renormalization group flow in which the Fermi surface gets adjusted dynamically during the flow, so that no counterterms are needed. To show the required regularity properties of the selfenergy and the Fermi surface, the technique of improving power counting by single and double overlaps is implemented in a nonperturbative setting.
This is joint work with Walter de Siqueira Pedra (University and MPI--MIS, Leipzig).
One-dimensional behavior of dilute, trapped Bose gases
Recent experimental and theoretical work has shown that under certain conditions trapped, low-density Bose gases behave like one-dimensional systems, which are typically described by an (exactly solvable) model of a 1D Bose gas with delta-function interaction. This one-dimensional behavior is an intrinsically quantum-mechanical phenomenon because it is not necessary to have a trap width that is of the order of the size of an atom - as might have been supposed - but it suffices merely that the trap width is small compared to the "healing length". We give a rigorous proof of the one-dimensional behavior, at least as far as the ground state is concerned, and determine the relevant parameter regimes. In particular, we derive the 1D Bose gas with delta-function interaction as a limit of a 3D Bose gas of hard spheres. We also investigate the phenomenon of Bose-Einstein condensation for these effectively one-dimensional systems.
(Joint work with E.H. Lieb and J. Yngvason.)
Effective dynamics for Bloch Electrons: Peierls substitution and beyond
The semiclassical equations of motion of a quantum particle in a slowly perturbed periodic potential are discussed. Based on recent results from a joint work with G. Panati and H. Spohn I explain how the well known "semiclassical model" of solid state physics is related to the Schrödinger equation. I also present a rigorous result concerning the first order corrections to the "semiclassical model" recently discovered by Niu et al. These corrections involve, in particular, a contribution from the curvature of the Berry connection defined by the Bloch eigenfunctions. They provide a semiclassical explanation for the quantization of the Hall conductivity in quantum Hall systems.
(Joint work with G. Panati and H. Spohn)
Lower bound for the segregation energy in the Falicov-Kimball model
A class of exactly solvable models of 2D correlated fermions
The Meyer property for quasicrystals
Effective dynamics for Bloch electrons: Peierls substitution and beyond
Fractal properties of the spectrum of the three-dimensional periodic Landau operators
Coincidence site lattices for cubic lattices
Disorder and vortices in unconventional superconductors
Do bosons condense in a homogeneous magnetic field?
Bose condensation of trapped bosons
A condition for Bose-Einstein condensation in a trap
Quantum field theory of degenerate systems
Mathematical simulation of photoacoustic effect in layered solid structures
Algebraic thermodynamic properties of anharmonic molecules and linear chains
On the theory of electric DC-conductivity: linear and non-linear microscopic evolution vs. macroscopic behaviour