On Strongly Correlated Eelectrons in Metals

A procedure, dedicated to superconductivity, is extended to study the properties of interacting electrons in normal metals in the thermodynamic limit. Each independent-electron band is shown to split into two correlated-electron bands. Excellent agreement is achieved with Bethe's wave-function for the one-dimensional Hubbard model. The groundstate energy, reckoned for the two-dimensional Hubbard Hamiltonian, is found to be lower than values, obtained thanks to the numerical methods. This analysis applies for any spatial dimension and temperature.

THE 1D HEISENBERG ANTIFERROMAGNET MODEL BY THE VARIATION AFTER PROJECTION METHOD

The four site and eight site 1D anti-ferromagnetic Heisenberg chains in the Jordan–Wigner representation are investigated within the standard Hartree–Fock and random phase approximation (RPA) approaches, both in the symmetry unbroken and in the symmetry broken phases. A translation invariant groundstate, obtained by the projection method as a linear combination of a symmetry-broken HF state and its image under reflection, is also considered, for each chain type. It is found that the projection method considerably improves the HF treatment for instance as far as the groundstate energy is concerned, but also with respect to the RPA energies. The results are furthermore confronted with the ones obtained within so-called SCRPA scheme.

On the groundstate energy of tight knots

New results on the groundstate energy of tight, magnetic knots are presented. Magnetic knots are defined as tubular embeddings of the magnetic field in an ideal, perfectly conducting, incompressible fluid. An orthogonal, curvilinear coordinate system is introduced and the magnetic energy is determined by the poloidal and toroidal components of the magnetic field. Standard minimization of the magnetic energy is carried out under the usual assumptions of volume- and flux-preserving flow, with the additional constraints that the tube cross section remains circular and that the knot length (ropelength) is independent from internal field twist (framing). Under these constraints the minimum energy is determined analytically by a new, exact expression, function of ropelength and framing. Groundstate energy levels of tight knots are determined from ropelength data obtained by the SONO tightening algorithm. Results for torus knots are compared with previous work, and the groundstate energy spectrum of the first prime knots — up to 10 crossings — is presented and analysed in detail. These results demonstrate that ropelength and framing determine the spectrum of magnetic knots in tight configuration.

FROM DEFECTS TO BOUNDARIES

In this paper we describe how relativistic field theories containing defects are equivalent to a class of boundary field theories. As a consequence previously derived results for boundaries can be directly applied to defects, these results include reduction formulas, the Coleman–Thun mechanism and Cutcosky rules. For integrable theories the defect crossing unitarity equation can be derived and defect operator found. For a generic purely transmitting impurity we use the boundary bootstrap method to obtain solutions of the defect Yang–Baxter equation. The groundstate energy on the strip with defects is also calculated.

GROUND STATES FOR LARGE SAMPLES OF TWO-DIMENSIONAL ISING SPIN GLASSES

We have developed a combinatoric matching method to find the exact groundstate energy for the 2-D Ising spin glass with the ± J distribution and equal numbers of positive and negative bonds. For the largest size (1800×1800 plaquettes of spins), we averaged results from 278 samples and for the smaller ones up to 374, 375 samples. We also studied the behavior of the distributions of computer time (CPU) and memory as functions of sample size. We present finite size scaling leading to a groundstate energy estimate of E∞=-1.40193±2 for the infinite system. We found that the memory scales as the square of sample length and that for a given size, the CPU time appears to have a skewed and high-tailed distribution.

Variationsmethode zur Berechnung der Grundzustandsenergie

Variational Method for Calculations of the Groundstate Energy For Hamiltonians without any symmetry properties, a variational method for the calculation of the groundstate energy is proposed. The numerical computation for the case of the few-electron-problem is based on the evaluation of functional integrals.

Zur exakten Behandlung von kollektiven Rotationen Teil A: Theorie / On the Exact Treatment of Collective Rotations Part A: Theory

Basic in the treatment of collective rotations is the definition of a body-fixed coordinate system. A kinematical method is derived to obtain the Hamiltonian of a n-body problem for a given definition of the body-fixed system. From this exact Hamiltonian, a consequent perturbation expansion in terms of the total angular momentum leads to two exact expressions: one for the collective rotational energy which has to be added to the groundstate energy in this order of perturbation and a second one for the effective inertia tensor in the groundstate. The discussion of these results leads to two criteria how to define the best body-fixed coordinate system, namely a differential equation and a variational principle. The equivalence of both is shown.