Exact dynamics in the inhomogeneous central-spin model

We study the dynamics of a single spin-1/2 coupled to a bath of spins-1/2 by inhomogeneous Heisenberg couplings including a central magnetic field. This central-spin model describes decoherence in quantum bit systems. An exact formula for the dynamics of the central spin is presented, based on the Bethe ansatz. This formula is evaluated explicitly for initial conditions such that the bath spins are completely polarized at the beginning. For this case we find, after an initial decay, a persistent oscillatory behaviour of the central spin. For a large number of bath spins $N_b$, the oscillation frequency is proportional to $N_b$, whereas the amplitude behaves like $1/N_b$, to leading order. No asymptotic decay due to the non-uniform couplings is observed, in contrast to some recent studies.Comment: 7 pages, 3 figure

Correlation functions of integrable models: a description of the ABACUS algorithm

Recent developments in the theory of integrable models have provided the means of calculating dynamical correlation functions of some important observables in systems such as Heisenberg spin chains and one-dimensional atomic gases. This article explicitly describes how such calculations are generally implemented in the ABACUS C++ library, emphasizing the universality in treatment of different cases coming as a consequence of unifying features within the Bethe Ansatz.Comment: 30 pages, 8 figures, Proceedings of the CRM (Montreal) workshop on Integrable Quantum Systems and Solvable Statistical Mechanics Model

Yang-Baxter equation in spin chains with long range interactions

We consider the $su(n)$ spin chains with long range interactions and the spin generalization of the Calogero-Sutherland models. We show that their properties derive from a transfer matrix obeying the Yang-Baxter equation. We obtain the expression of the conserved quantities and we diagonalize them.Comment: Saclay-t93/00

Families of quasi-exactly solvable extensions of the quantum oscillator in curved spaces

We introduce two new families of quasi-exactly solvable (QES) extensions of the oscillator in a $d$-dimensional constant-curvature space. For the first three members of each family, we obtain closed-form expressions of the energies and wavefunctions for some allowed values of the potential parameters using the Bethe ansatz method. We prove that the first member of each family has a hidden sl(2,$\mathbb{R}$) symmetry and is connected with a QES equation of the first or second type, respectively. One-dimensional results are also derived from the $d$-dimensional ones with $d \ge 2$, thereby getting QES extensions of the Mathews-Lakshmanan nonlinear oscillator.Comment: 30 pages, 8 figures, published versio

Laughlin's wave functions, Coulomb gases and expansions of the discriminant

In the context of the fractional quantum Hall effect, we investigate Laughlin's celebrated ansatz for the groud state wave function at fractional filling of the lowest Landau level. Interpreting its normalization in terms of a one component plasma, we find the effect of an additional quadrupolar field on the free energy, and derive estimates for the thermodynamically equivalent spherical plasma. In a second part, we present various methods for expanding the wave function in terms of Slater determinants, and obtain sum rules for the coefficients. We also address the apparently simpler question of counting the number of such Slater states using the theory of integral polytopes.Comment: 97 pages, using harvmac (with big option recommended) and epsf, 7 figures available upon request, Saclay preprint Spht 93/12

Spectra and Symmetry in Nuclear Pairing

We apply the algebraic Bethe ansatz technique to the nuclear pairing problem with orbit dependent coupling constants and degenerate single particle energy levels. We find the exact energies and eigenstates. We show that for a given shell, there are degeneracies between the states corresponding to less and more than half full shell. We also provide a technique to solve the equations of Bethe ansatz.Comment: 15 pages of REVTEX with 2 eps figure

The spin 1/2 Calogero-Gaudin System and its q-Deformation

The spin 1/2 Calogero-Gaudin system and its q-deformation are exactly solved: a complete set of commuting observables is diagonalized, and the corresponding eigenvectors and eigenvalues are explicitly calculated. The method of solution is purely algebraic and relies on the co-algebra simmetry of the model.Comment: 15 page

Phenomenology of chiral damping in noncentrosymmetric magnets

A phenomenology of magnetic chiral damping is proposed in the context of magnetic materials lacking inversion symmetry breaking. We show that the magnetic damping tensor adopts a general form that accounts for a component linear in magnetization gradient in the form of Lifshitz invariants. We propose different microscopic mechanisms that can produce such a damping in ferromagnetic metals, among which spin pumping in the presence of anomalous Hall effect and an effective "$s$-$d$" Dzyaloshinskii-Moriya antisymmetric exchange. The implication of this chiral damping in terms of domain wall motion is investigated in the flow and creep regimes. These predictions have major importance in the context of field- and current-driven texture motion in noncentrosymmetric (ferro-, ferri-, antiferro-)magnets, not limited to metals.Comment: 5 pages, 2 figure

Fermi-Bose transformation for the time-dependent Lieb-Liniger gas

Exact solutions of the Schrodinger equation describing a freely expanding Lieb-Liniger (LL) gas of delta-interacting bosons in one spatial dimension are constructed. The many-body wave function is obtained by transforming a fully antisymmetric (fermionic) time-dependent wave function which obeys the Schrodinger equation for a free gas. This transformation employs a differential Fermi-Bose mapping operator which depends on the strength of the interaction and the number of particles.Comment: 4+ pages, 1 figure; added reference

Diagonalization of an Integrable Discretization of the Repulsive Delta Bose Gas on the Circle

We introduce an integrable lattice discretization of the quantum system of n bosonic particles on a ring interacting pairwise via repulsive delta potentials. The corresponding (finite-dimensional) spectral problem of the integrable lattice model is solved by means of the Bethe Ansatz method. The resulting eigenfunctions turn out to be given by specializations of the Hall-Littlewood polynomials. In the continuum limit the solution of the repulsive delta Bose gas due to Lieb and Liniger is recovered, including the orthogonality of the Bethe wave functions first proved by Dorlas (extending previous work of C.N. Yang and C.P. Yang).Comment: 25 pages, LaTe