For one-dimensional many-boson systems there exists a considerable body of analytical treatments of model systems. The existence of some exactly known properties of these systems and the fact that some of these properties differ from those of ${}^{4}\mathrm{He}$ in higher dimensions are a strong impetus to apply microscopic many-body methods to quasi-one-dimensional systems with realistic interactions. The results on the model systems can help as a guide to understanding those systems, where due to the nature of the interaction, exact results are not available. In this paper, we examine three model systems: hard rods, the Morse potential, and the Lennard-Jones 6-12 potential using the nonperturbative variational approach based on Feenberg's method of correlated basis functions. In the hard rod system we can compare our equation of state with the exact result and discuss the low-lying excitations. We examine both finite-size rods and the point rod limit. For the Morse system, the existence of a two-body bound state, dimerization, can be determined analytically. We examine the effect of dimerization on the many-body equation of state and argue that dimerization is manifested in the many-boson system as the appearance of a many-body bound state (i.e., a zero pressure, negative energy system with nonzero density). For the Lennard-Jones system we numerically determine the onset of dimerization and, as with the Morse system, we show that in one dimension the many-body bound state only exists for those systems which dimerize. For each of these systems we track high-density instabilities in the equations of state which we tentatively identify as zero-temperature liquid-solid transitions.
For one-dimensional many-boson systems there exists a considerable body of analytical treatments of model systems. The existence of some exactly known properties of these systems and the fact that some of these properties differ from those of 4He in higher dimensions are a strong impetus to apply microscopic many-body methods to quasi-one-dimensional systems with realistic interactions. The results on the model systems can help as a guide to understanding those systems, where due to the nature of the interaction, exact results are not available. In this paper, we examine three model systems: hard rods, the Morse potential, and the Lennard-Jones 6-12 potential using the nonperturbative variational approach based on Feenberg's method of correlated basis functions. In the hard rod system we can compare our equation of state with the exact result and discuss the low-lying excitations. We examine both finite-size rods and the point rod limit. For the Morse system, the existence of a two-body bound state, dimerization, can be determined analytically. We examine the effect of dimerization on the many-body equation of state and argue that dimerization is manifested in the many-boson system as the appearance of a many-body bound state (i.e., a zero pressure, negative energy system with nonzero density). For the Lennard-Jones system we numerically determine the onset of dimerization and, as with the Morse system, we show that in one dimension the many-body bound state only exists for those systems which dimerize. For each of these systems we track high-density instabilities in the equations of state which we tentatively identify as zero-temperature liquid-solid transitions.
A quantum Monte Carlo simulation of a system of bosonic hard rods in one dimension is presented and discussed. The calculation is exact since the analytical form of the wave function is known and is in excellent agreement with predictions obtained from asymptotic expansions valid at large distances. The analysis of the static structure factor and the pair distribution function indicates that a solidlike and a gaslike phases exist at high and low densities, respectively. The one-body density matrix decays following a power law at large distances and produces a divergence in the low density momentum distribution at k=0 which can be identified as a quasicondensate.
It has been suggested recently that 4He can be prepared and studied as a quasi-one-dimensional quantum fluid. In this paper we calculate the static and dynamic properties of one-dimensional 4He using variational methods based upon the Jastrow-Feenberg wave function and its extension to dynamic systems with time-dependent, correlated wave functions. We calculate the zero temperature equation of state and show that in one dimension 4He is just barely self-bound with a binding energy of 0.002 K at a density of 0.036 ${\mathrm{\AA{}}}^{\ensuremath{-}1}.$ We calculate the Feynman excitation spectrum and corrections that contain multiphonon processes and study the density dependence of the roton feature as well as the static response function. In addition we demonstrate the presence of strong anomalous dispersion in the phonon regime. Finally, we introduce a 3He impurity and calculate the zero concentration chemical potential as a function of 4He linear density. We also compute the 3He-3He effective interaction in the 4He background and compute the energy of dimerization.
In this thesis, we study the physical properties of several ultracold few-boson systems depending on the interactions between their constituents. Nowadays, experimentally, it is possible to have great control with high precision over the geometry and the interactions between the particles, making them an excellent setup to test directly the principles of quantum mechanics. A very interesting point is to study the evolution of their properties with the number of particles. The theoretical study of these systems pretends to microscopically understand the current experimental results and give support to new experimental developments.\n\n\t\t\t\t The method that will be used is the exact diagonalization of the Hamiltonian of the system. As we will see, in spite of the attempts to improve it, the method is limited by the fact that, in practice, it is only useful to study few-particle systems. The method has several advantages. First of all, one has access to both the ground and the excited states. In second place, the method is variational and converges to the exact solution as long as the Hilbert space in which we diagonalize is enlarged.\n\t\t\t\t Moreover, since we have access to the states of the system, it is possible to calculate any observable quantity of interest.\n\n\t\t\t\t First, we will study a system of spinless bosons trapped in a two-dimensional harmonic potential. The effect of the trap is to keep the system bound. It will be seen how the presence of a repulsive interaction changes the energy spectrum and other properties of the system. For instance, the density profile, which is usually measurable, and also the two-body distribution function, which is intimately related to the existence of correlations.\n\n\t\t\t\t Afterwards, the focus will be on the particular case of having only two bosons in the system interacting through a strong repulsive force. Inspired by the one-dimensional case where the fermionization phenomenon takes place in the strongly-interacting limit, we will study whether in two dimensions there is a resembling reminiscent effect. In other words, we will analyze if there are properties of the two strongly-interacting bosons in two dimensions that are like the ones of two noninteracting fermions.\n\n\t\t\t\t After that, we will tackle the localization phenomenon in a one-dimensional system that is caused by an external speckle potential that introduces disorder in the system. We will show that the localization is a robust phenomenon against repulsive contact interactions.\n\n\t\t\t\t Finally, we will study the influence of the spin-orbit coupling in a system of bosons with two possible pseudospin components, associated, for instance, to two hyperfine levels, confined in a two-dimensional harmonic trap. We will present an exhaustive analysis of the combined effects of the interaction and the spin-orbit coupling in the spectrum and the properties of the system. In particular we show the existence of a crossover in the ground state of the system susceptible to be experimentally identified.
Variational calculations of the ground-state energy of hard-sphere, helium-4, Coulomb, and nuclear Yukawa fluids are carried out. The long-range part v(r>d) and a constant part $\ensuremath{\lambda}$ at r<d of the bare interaction are assumed to contribute to the average field. The correlation function f(r) is obtained by minimizing the two-body cluster contribution due to the remainder interaction, and the distance d is determined by minimizing the energy as calculated with the hierarchy of hypernetted-chain-type equations. The convergence of two- and three-particle distribution functions given by these equations is discussed. The method is shown to be exact in the low-density limit, and the comparison of the present results with the available exact (standard) results in the high-density limit (region) verifies its accuracy over the entire density range. It is shown that the variationally obtained value of d in the high-density Coulomb fluid can be understood with conventional meanfield theory. A solidification criterion is obtained by studying the energies in liquid and solid phases with the present f(r). This criterion explains the solidification of the Coulomb, hard-sphere, and helium systems, and the inability of the nuclear Yukawa system to solidify. The variational method in the form presented here could be very useful for studying complicated quantum systems.
A variational Monte Carlo calculation of the one-body density matrix and momentum distribution of a system of Fermi hard rods (HRs) is presented and compared with the same quantities for its bosonic counterpart. The calculation is exact within statistical errors since we sample the exact ground-state wave function, whose analytical expression is known. The numerical results are in good agreement with known asymptotic expansions valid for Luttinger liquids. We find that the difference between the absolute value of the bosonic and fermionic density matrices becomes marginally small as the density increases. In this same regime, the corresponding momentum distributions merge into a common profile that is independent of the statistics. Nonanalytical contributions to the one-body density matrix are also discussed and found to be less relevant with increasing density.
We extend to the N-body case previous techniques for solving three-body problems with repulsive interparticle potentials and periodic boundary conditions on each particle. For clarity, we begin with one-dimensional problems, although the techniques are not peculiar to them and can be generalized to three dimensions. We decompose the wavefunction ψ into N(N − 1)/2 parts, according to Ψ= ∑ i<j=1Nψij;from the Schrödinger equation we find a basic equation for a typical ψij. As a test of this equation, we apply it for N = 4 and the case of δ-function interactions, and solve it numerically to find close agreement between the energy per particle of a four-body system and a system of the same density in which N → ∞. The numerical results are consistent with the analytic one that Eg(∞), which is the ground-state energy per particle of the system with an infinite number of particles, is related to Eg(N), which is that energy for the N-body system of the same density, by Eg(N) = Eg(∞)[1 − (1/N)] for weak repulsion and by Eg(N) = Eg(∞)[1 − (1/N2)] for very strong repulsion. We begin a similar comparison for the three-dimensional hard-sphere case by working out the problem of two hard spheres with periodic boundary conditions over a length L2 such that the density is the same as for an N-body problem in volume V=LN3:2/L23=N/LN3. We find that we get the analog of the one-dimensional result, for which case Eg(2) = Eg(∞)/2 for N = 2. That is, we find Eg(2) = πa2h/mv, which is just one-half of the many-body theory result. We also calculate the two-body correlation function and find good agreement with the many-body one, as calculated by Lee, Huang, and Yang.