Introduction to the Bethe ansatz I

The Bethe ansatz for the one-dimensional s=1/2 Heisenberg ferromagnet is introduced at an elementary level. The presentation follows Bethe's original work very closely. A detailed description and a complete classification of all two-magnon scattering states and two-magnon bound states are given for finite and infinite chains. The paper is designed as a tutorial for beginning graduate students. It includes 10 problems for further study.Comment: 8 pages, 4 figure

Dynamics of the Boxy Elliptical Galaxy NGC 1600

We use three--integral models to infer the distribution function (DF) of the boxy E3-E4 galaxy NGC 1600 from surface brightness and line profile data on the minor and major axes. We assume axisymmetry and that the mass-to-light ratio is constant in the central ~1 R_e. Stars in the resulting gravitational potential move mainly on regular orbits. We use an approximate third integral K from perturbation theory, and write the DF as a sum of basis functions in the three integrals E, L_z and K. We then fit the projected moments of these basis functions to the kinematic observables and deprojected density, using a non-parametric algorithm. The deduced dynamical structure is radially anisotropic, with sigma_theta/sigma_r ~ sigma_phi/sigma_r ~ 0.7 on the major axis. Both on the minor axis and near the centre the velocity distribution is more isotropic; thus the model is flattened by equatorial radial orbits. The kinematic data is fit without need for a central black hole; the central mass determined previously from ground-based data therefore overestimates the actual black hole mass. The mass-to-light ratio of the stars is M/L_V = 6 h_50. The anisotropy structure of NGC 1600 with a radially anisotropic main body and more nearly isotropic centre is similar to that found recently in NGC 1399, NGC 2434, NGC 3379 and NGC 6703, suggesting that this pattern may be common amongst massive elliptical galaxies. We discuss a possible merger origin of NGC 1600 in the light of these results.Comment: 14 pages, 9 figures, re-submitted to Monthly Notice

Computational probes of collective excitations in low-dimensional magnetism

The investigation of the dynamics of quantum many-body systems is a concerted effort involving computational studies of mathematical models and experimental studies of material samples. Some commonalities of the two tracks of investigation are discussed in the context of the quantum spin dynamics of low-dimensional magnetic systems, in particular spin chains. The study of quantum fluctuations in such systems at equilibrium amounts to exploring the spectrum of collective excitations and the rate at which they are excited from the ground state by dynamical variables of interest. The exact results obtained via Bethe ansatz or algebraic analysis (quantum groups) for a select class of completely integrable models can be used as benchmarks for numerical studies of nonintegrable models, for which computational access to the spectrum of collective excitations is limited.Comment: 22 pages. Talk given at the 7th Summer School on Neutron Scattering, Zuoz Switzerland, August 199

The strong Centre Conjecture: an invariant theory approach

The aim of this paper is to describe an approach to a a strengthened form of J. Tits' Centre Conjecture for spherical buildings. This is accomplished by generalizing a fundamental result of G. R. Kempf from Geometric Invariant Theory and interpreting this generalization in the context of spherical buildings. We are able to recapture the conjecture entirely in terms of our generalization of Kempf's notion of a state. We demonstrate the utility of this approach by proving the Centre Conjecture in some special cases.Comment: 30 pages, minor changes, new subsection on rationality; v3 updated bibliography and affiliation of second autho

Complete reducibility and separable field extensions

Let G be a connected reductive linear algebraic group. The aim of this note is to settle a question of J-P. Serre concerning the behaviour of his notion of G-complete reducibility under separable field extensions. Part of our proof relies on the recently established Tits Centre Conjecture for the spherical building of the reductive group G.Comment: 5 pages; to appear in Comptes rendus Mathematiqu

Quasiparticles in the XXZ model

The coordinate Bethe ansatz solutions of the XXZ model for a one-dimensional spin-1/2 chain are analyzed with focus on the statistical properties of the constituent quasiparticles. Emphasis is given to the special cases known as XX, XXX, and Ising models, where considerable simplifications occur. The XXZ spectrum can be generated from separate pseudovacua as configurations of sets of quasiparticles with different exclusion statistics. These sets are complementary in the sense that the pseudovacuum of one set contains the maximum number of particles from the other set. The Bethe ansatz string solutions of the XXX model evolve differently in the planar and axial regimes. In the Ising limit they become ferromagnetic domains with integer-valued exclusion statistics. In the XX limit they brake apart into hard-core bosons with (effectively) fermionic statistics. Two sets of quasiparticles with spin 1/2 and fractional statistics are distinguished, where one set (spinons) generates the XXZ spectrum from the unique, critical ground state realized in the planar regime, and the other set (solitons) generates the same spectrum from the twofold, antiferromagnetically ordered ground state realized in the axial regime. In the Ising limit, the solitons become antiferromagnetic domain walls.Comment: 6 figure