Classical ground states in one dimension

We give conditions on an interaction sufficient to guarantee that in one dimension it yield a periodic ground state with one or two particles per unit cell.Research supported in part by NSF Grant MCS 81-01596-A01.     

J-matrix method of scattering in one dimension: The nonrelativistic theory

We formulate a theory of nonrelativistic scattering in one dimension based on the J-matrix method. The scattering potential is assumed to have a finite range such that it is well represented by its matrix elements in a finite subset of a basis that supports a tridiagonal matrix representation for the reference wave operator. Contrary to our expectation, the 1D formulation reveals a rich and highly nontrivial structure compared to the 3D formulation. Examples are given to demonstrate the utility and accuracy of the method. It is hoped that this formulation constitutes a viable alternative to the classical treatment of 1D scattering problem and that it will help unveil new and interesting applications.     

Large orders in the 1/N perturbation theory by inverse scattering in one dimension

When one tries to compute large orders in the 1/N series à la Lipatov a complicated non-linear equation for the instanton is found in ø⁴ or non-linear sigma models.We solve here this equation in the one-dimensional case (quantum mechanics) by inverse scattering techniques. From the instanton solutions we obtain theK ^th order of the 1/N perturbation theory up to 0(K ^–1) for the 0(N) symmetric anharmonic oscillator and up to a factor 0(K ⁰) for a non-symmetric model. In the symmetric case we agree with results recently obtained in quantum mechanics by Hikami and Brézin following a different procedure. For the non-symmetric anharmonic oscillator we believe our formulae are new.     

Dynamical theory of superfluidity in one dimension

A theory accounting for the dynamical aspects of the superfluid response of one dimensional (1D) quantum fluids is reported. In long 1D systems, the onset of superfluidity is related to the dynamical suppression of quantum phase slips at low temperatures. The effect of this suppression as a function of frequency and temperature is discussed within the framework of the experimentally relevant momentum response function. Applications of these results to the understanding of the superfluid properties of helium confined in 1D pores with nanometer diameter, dislocations in solid 4He, and ultracold atomic gases are also briefly discussed.     

Classical simulation of infinite-size quantum lattice systems in one spatial dimension

Invariance under translation is exploited to efficiently simulate one-dimensional quantum lattice systems in the limit of an infinite lattice. Both the computation of the ground state and the simulation of time evolution are considered.     

Entanglement perturbation theory for the elementary excitation in one dimension

The entanglement perturbation theory is developed to calculate the excitation spectrum in one dimension. Applied to the spin- antiferromagnetic Heisenberg model, it reproduces the des Cloiseaux-Pearson Bethe ansatz result. As for spin-1, the spin-triplet magnon spectrum has been determined for the first time for the entire Brillouin zone, including the Haldane gap at k=π.     

Classical dynamical theory of heavy ion fusion and scattering

A classical dynamical model which includes dissipative forces is suggested for heavy ion reactions high above the Coulomb barrier. Internal degrees of freedom corresponding to rotation of the ions are included. The reactions divide into three parts: (1) quasi-elastic scattering, with relatively small energy loss, associated with higher angular momenta, (2) deep inelastic scattering, with larger energy loss and considerable transfer of mass, associated with intermediate angular momenta, and (3) complete fusion where a highly excited compound state is formed, generally associated with the lowest angular momenta. One can predict a fusion cross section for two values of the friction coefficient, “weak” and “strong” friction cases. Reasonable values for fusion can be obtained in the weak friction case, but scattering angular distributions are not consistent with available experimental data. In the strong friction case it is more difficult to fit all the fusion cross sections with a single friction parameter. But the predicted angular distributions and energy losses are in better agreement with experiment than for the weak friction case.     

Bridge function and fundamental measure theory: a test in dimension one

A formal and general expression for the bridge function is obtained from the Fundamental Measure Theory. This expression involves solely the knowledge of an analytical form of the free-energy, the weight functions and the pair correlation function, and comes out in terms of combinations of convolution products of the two latter quantities, thus lifting the irreducibility of the concerned diagrams. It is applicable to mixtures and inhomogeneous fluids. The validity of this expression is then tested in the case of a one-dimensional hard rods fluid, for which exact expression of the correlations are known, and the Fundamental Measure Theory is also exact. The formalism allows one to sum the infinite series and produce a compact analytical expression for the exact bridge function. However, the numerical term by term evaluation of the series shows that severe convergence problems arise in the high density regime, questioning the pertinence of usual such attempts. The present approach opens interesting perspectives for the Theory of Liquids.     

Bose-Fermi mixtures in one dimension

We analyze the phase stability and the response of a mixture of bosons and spin-polarized fermions in one dimension (1D). Unlike in 3D, phase separation happens for low fermion densities. The dynamics of the mixture at low energy is independent of the spin-statistics of the components, and the modes are essentially undamped.     

Delocalization transition in one dimension

Singular extended states among random system localized states are proved to be a general rule. When a Fermi energy coincides with the extended state energy, a residual resistance becomes proportional to the square of the system length.