Shear viscosity and spin sum rules in strongly interacting Fermi gases

Fermi gases with short-range interactions are ubiquitous in ultracold atomic systems. In the absence of spin-flipping processes the number of atoms in each spin species is conserved separately, and we discuss the associated Ward identities. For contact interactions the spin conductivity spectral function sigma_s(omega) has universal power-law tails at high frequency. We derive the spin f-sum rule and show that it is not affected by these tails in d<4 dimensions. Likewise the shear viscosity spectral function eta(omega) has universal tails; in contrast they modify the viscosity sum rule in a characteristic way.Comment: 7 pages, published versio

A New Look at the Multidimensional Inverse Scattering Problem

As a prototype of an evolution equation we consider the Schr\"odinger equation i (d/dt) \Psi(t) = H \Psi(t), H = H_0 + V(x) for the Hilbert space valued function \Psi(.) which describes the state of the system at time t in space dimension at least 2. The kinetic energy operator H_0 may be propotional to the Laplacian (nonrelativistic quantum mechanics), H_0 = \sqrt{-\Delta + m^2} (relativistic kinematics, Klein-Gordon equation), the Dirac operator, or ..., while the potential V(x) tends to 0 suitably as |x| to infinity. We present a geometrical approach to the inverse scattering problem. For given scattering operator S we show uniqueness of the potential, we give explicit limits of the high-energy behavior of the scattering operator, and we give reconstruction formulas for the potential. Our mathematical proofs closely follow physical intuition. A key observation is that at high energies translation of wave packets dominates over spreading during the interaction time. Extensions of the method cover e.g. Schr\"odinger operators with magnetic fields, multiparticle systems, and wave equations.Comment: LaTeX2e, 16 pages, to be published in: Understanding Physics, A.K. Richter ed., Copernicus Gesellschaft, Katlenburg-Lindau, 1998, pp. 31-; ISBN 3-9804862-2-2 (Proceedings Bonn 1996). For other formats see http://www.iram.rwth-aachen.de/~enss/ or ftp://work1.iram.rwth-aachen.de/pub/papers/enss

Lightcone renormalization and quantum quenches in one-dimensional Hubbard models

The Lieb-Robinson bound implies that the unitary time evolution of an operator can be restricted to an effective light cone for any Hamiltonian with short-range interactions. Here we present a very efficient renormalization group algorithm based on this light cone structure to study the time evolution of prepared initial states in the thermodynamic limit in one-dimensional quantum systems. The algorithm does not require translational invariance and allows for an easy implementation of local conservation laws. We use the algorithm to investigate the relaxation dynamics of double occupancies in fermionic Hubbard models as well as a possible thermalization. For the integrable Hubbard model we find a pure power-law decay of the number of doubly occupied sites towards the value in the long-time limit while the decay becomes exponential when adding a nearest neighbor interaction. In accordance with the eigenstate thermalization hypothesis, the long-time limit is reasonably well described by a thermal average. We point out though that such a description naturally requires the use of negative temperatures. Finally, we study a doublon impurity in a N\'eel background and find that the excess charge and spin spread at different velocities, providing an example of spin-charge separation in a highly excited state.Comment: published versio

High-Velocity Estimates and Inverse Scattering for Quantum N-Body Systems with Stark Effect

In an N-body quantum system with a constant electric field, by inverse scattering, we uniquely reconstruct pair potentials, belonging to the optimal class of short-range potentials and long-range potentials, from the high-velocity limit of the Dollard scattering operator. We give a reconstruction formula with an error term.Comment: In this published version we have added remarks and we have edited the pape

Perturbation Theory for the Quantum Time-Evolution in Rotating Potentials

The quantum mechanical time-evolution is studied for a particle under the influence of an explicitly time-dependent rotating potential. We discuss the existence of the propagator and we show that in the limit of rapid rotation it converges strongly to the solution operator of the Schr\"odinger equation with the averaged rotational invariant potential.Comment: To appear in Proceedings of the Conference QMath-8 "Mathematical Results in Quantum Mechanics" Taxco, Mexico, December 200