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

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

Momentum distribution of a freely expanding Lieb-Liniger gas

We numerically study free expansion of a few Lieb-Liniger bosons, which are initially in the ground state of an infinitely deep hard-wall trap. Numerical calculation is carried out by employing a standard Fourier transform, as follows from the Fermi-Bose transformation for a time-dependent Lieb-Liniger gas. We study the evolution of the momentum distribution, the real-space single-particle density, and the occupancies of natural orbitals. Our numerical calculation allows us to explore the behavior of these observables in the transient regime of the expansion, where they are non-trivially affected by the particle interactions. We derive analytically (by using the stationary phase approximation) the formula which connects the asymptotic shape of the momentum distribution and the initial state. For sufficiently large times the momentum distribution coincides (up to a simple scaling transformation) with the shape of the real-space single-particle density (the expansion is asymptotically ballistic). Our analytical and numerical results are in good agreement.Comment: small changes; references correcte

Plasmon channels in the electronic relaxation of diamond under high-order harmonics femtosecond irradiation

We used high order harmonics of a femtosecond titanium-doped sapphire system (pulse duration 25 fs) to realise Ultraviolet Photoelectron Spectroscopy (UPS) measurements on diamond. The UPS spectra were measured for harmonics in the range 13 to 27. We also made ab initio calculations of the electronic lifetime of conduction electrons in the energy range produced in the UPS experiment. Such calculations show that the lifetime suddenly diminishes when the conduction electron energy reaches the plasmon energy, whereas the UPS spectra show evidence in this range of a strong relaxation mechanism with an increased production of low energy secondary electrons. We propose that in this case the electronic relaxation proceeds in two steps : excitation of a plasmon by the high energy electron, the latter decaying into individual electron-hole pairs, as in the case of metals. This process is observed for the first time in an insulator and, on account of its high efficiency, should be introduced in the models of laser breakdown under high intensity

Exact Results for Three-Body Correlations in a Degenerate One-Dimensional Bose Gas

Motivated by recent experiments we derive an exact expression for the correlation function entering the three-body recombination rate for a one-dimensional gas of interacting bosons. The answer, given in terms of two thermodynamic parameters of the Lieb-Liniger model, is valid for all values of the dimensionless coupling $\gamma$ and contains the previously known results for the Bogoliubov and Tonks-Girardeau regimes as limiting cases. We also investigate finite-size effects by calculating the correlation function for small systems of 3, 4, 5 and 6 particles.Comment: 4 pages, 2 figure

Geometry of quantum observables and thermodynamics of small systems

The concept of ergodicity---the convergence of the temporal averages of observables to their ensemble averages---is the cornerstone of thermodynamics. The transition from a predictable, integrable behavior to ergodicity is one of the most difficult physical phenomena to treat; the celebrated KAM theorem is the prime example. This Letter is founded on the observation that for many classical and quantum observables, the sum of the ensemble variance of the temporal average and the ensemble average of temporal variance remains constant across the integrability-ergodicity transition. We show that this property induces a particular geometry of quantum observables---Frobenius (also known as Hilbert-Schmidt) one---that naturally encodes all the phenomena associated with the emergence of ergodicity: the Eigenstate Thermalization effect, the decrease in the inverse participation ratio, and the disappearance of the integrals of motion. As an application, we use this geometry to solve a known problem of optimization of the set of conserved quantities---regardless of whether it comes from symmetries or from finite-size effects---to be incorporated in an extended thermodynamical theory of integrable, near-integrable, or mesoscopic systems

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

Discretized vs. continuous models of p-wave interacting fermions in 1D

We present a general mapping between continuous and lattice models of Bose- and Fermi-gases in one dimension, interacting via local two-body interactions. For s-wave interacting bosons we arrive at the Bose-Hubbard model in the weakly interacting, low density regime. The dual problem of p-wave interacting fermions is mapped to the spin-1/2 XXZ model close to the critical point in the highly polarized regime. The mappings are shown to be optimal in the sense that they produce the least error possible for a given discretization length. As an application we examine the ground state of a interacting Fermi gas in a harmonic trap, calculating numerically real-space and momentum-space distributions as well as two-particle correlations. In the analytically known limits the convergence of the results of the lattice model to the continuous one is shown.Comment: 7 pages, 5 figure

Extended trigonometric Cherednik algebras and nonstationary Schr\"odinger equations with delta-potentials

We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schr\"odinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schr\"odinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations in their spectral parameter. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with pairwise delta-function interactions is indicated.Comment: 23 pages; Version 2: expanded introduction and misprints correcte

Diagonalization of infinite transfer matrix of boundary Uq,p(AN−1(1))U_{q,p}(A_{N-1}^{(1)}) face model

We study infinitely many commuting operators $T_B(z)$, which we call infinite transfer matrix of boundary $U_{q,p}(A_{N-1}^{(1)})$ face model. We diagonalize infinite transfer matrix $T_B(z)$ by using free field realizations of the vertex operators of the elliptic quantum group $U_{q,p}(A_{N-1}^{(1)})$.Comment: 36 pages, Dedicated to Professor Etsuro Date on the occassion of the 60th birthda