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arXiv:cond-mat/9406044 · Physical review. B, Condensed matter

Pseudoparticle-operator description of an interacting bosonic gas

A. H. Castro Neto, H. Q. Lin, Y.-H. Chen, J. M. P. Carmelo

Published Nov 15, 1994• 26 citations
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Abstract

We write the Hamiltonian of the Bose gas with two-body repulsive \ensuremath{\delta}-function potential in a pseudoparticle-operator basis which diagonalizes the problem via the Bethe ansatz. In this operator basis the original bosonic interactions are represented by zero-momentum forward-scattering interactions between Landau-liquid pseudoparticles. We find that this pseudoparticle-operator algebra is complete: all the Hamiltonian eigenstates are generated by acting pseudoparticle operators on the system vacuum. It is shown that one boson of vanishing momentum and energy is a composite of a one-pseudoparticle excitation and a collective pseudoparticle excitation. These excitations have finite opposite momenta and cannot be decomposed. Our formalism enables us to calculate the various quantities that characterize the static and dynamic behavior of the system at low energies.

DOI
10.1103/physrevb.50.14032

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We write the Hamiltonian of the Bose gas with two-body repulsive \ensuremath{\delta}-function potential in a pseudoparticle-operator basis which diagonalizes the problem via the Bethe ansatz. In this operator basis the original bosonic interactions are represented by zero-momentum forward-scattering interactions between Landau-liquid pseudoparticles. We find that this pseudoparticle-operator algebra is complete: all the Hamiltonian eigenstates are generated by acting pseudoparticle operators on the system vacuum. It is shown that one boson of vanishing momentum and energy is a composite of a one-pseudoparticle excitation and a collective pseudoparticle excitation. These excitations have finite opposite momenta and cannot be decomposed. Our formalism enables us to calculate the various quantities that characterize the static and dynamic behavior of the system at low energies.
We introduce an operator algebra for the description of the low-energy physics of one-dimensional, integrable, multicomponent quantum liquids. Considering the particular case of the Hubbard chain in a magnetic field and chemical potential, we show that at low energy its Bethe-ansatz solution can be interpreted in terms of a pseudoparticle-operator algebra. Our algebraic approach provides a concise interpretation of, and justification for, several recent studies of low-energy excitations and trasnport which have been based on detailed analyses of specific Bethe-ansatz eigenfunctions and eigenenergies. A central point is that the exact ground state of the interacting many-electron problem is the noninteracting pseudoparticle ground state. Furthermore, in the pseudoparticle basis, the quantum problem becomes perturbative, i.e., the two-pseudoparticle forward-scattering vertices and amplitudes do not diverge, and one can define a many-pseudoparticle perturbation theory. We write the general quantum-liquid Hamiltonian in the pseudoparticle basis and show that the pseudoparticle-perturbation theory leads, in a natural way, to the generalized Landau-liquid approach.
We introduce the pseudohole and heavy-pseudoparticle operator algebra that generates all Hubbard-chain eigenstates from a single reference vacuum. In addition to the pseudoholes already introduced for the description of the low-energy physics, this involves the heavy pseudoparticles associated with Hamiltonian eigenstates whose energy spectrum has a gap relative to the many-electron ground state. We introduce a generalized pseudoparticle perturbation theory that describes the relevant finite-energy ground-state transitions. In the present basis these excitations refer to a small density of excited pseudoparticles. Our operator basis goes beyond the Bethe-ansatz solution and it is the suitable and correct starting point for the study of the finite-frequency properties, which are of great relevance for the understanding of the unusual spectral properties detected in low-dimensional novel materials.
We use the Landau-liquid character of the low-energy eigenstate spectrum of the Hubbard chain in a magnetic field to investigate its two-particle spectral properties. The picture that emerges is that of a gas of two coupled fluids of pseudoparticles. The kinetic equations that govern the flow of pseudoparticles are derived. The macroscopic currents are expressed in terms of the elementary pseudoparticle currents. As a central result we evaluate in the limit of zero momentum the wave function for the electronic excitations obtained by applying the up-spin and down-spin density-fluctuation operators on the ground state. This wave function is calculated on the basis of the pseudoparticle single-pair eigenstates. It is defined by the microscopic two-particle matrix elements of the corresponding fluctuations taken between the ground state and the elementary excitations. The on-site interaction U, electronic density n, and magnetic field H dependence of these matrix elements, which play a major role in the two-particle spectral functions, provide relevant information on the microscopic foundation of the one-dimensional Landau-liquid theory. This leads to the introduction of an adiabatic continuity principle, which is valid in the Hilbert subspace of the pseudoparticle single-pair eigenstates of lowest energy and momentum. The non-Landau-liquid character of the zero-magnetic-field L\"uttinger phase, where such an adiabatic continuity principle is not valid, is explained.Exact expressions for the two-pseudoparticle forward-scattering amplitudes and for the charge-charge and spin-spin correlation functions at small momentum and low frequency are obtained. The dependence of the charge and spin stiffnesses of the conductivity spectra on U, n, and H, which is determined by the pseudoparticle transport masses, is investigated in detail. The study of these masses and of the effective charge and spin projection of the pseudoparticles gives information on their structure in the electronic representation. It is shown that the decoupling that characterizes the low-energy excitation spectrum does not correspond in the case of finite magnetic fields to a real charge-spin separation. Our results provide a method for the study of the low-frequency dynamics of interacting systems solvable with the Bethe ansatz, and we identify the spectral parameters that control the nonclassical critical exponents with microscopic two-electron matrix elements.
We examine and explain the Luttinger-liquid character of models solvable by the Bethe ansatz by introducing a suitable bosonic operator algebra. In the case of the Hubbard chain, this involves two bosonic algebras which apply to all values of U, electronic density, and magnetization. Only at zero magnetization does not lead to the usual charge-spin separation. We show that our ``pseudoparticle'' operator approach clarifies, unifies, and extends several recent results, including the existence of independent right and left equations of motion and the concept of ``pseudoparticle'' (also known as ``Bethe quasiparticle'').
Introduction The one-dimensional Bose gas with point-like interaction of the particles (the quantum variant of the nonlinear Schrödinger equation) is one of the main and most important models which can be Solved by means of the Bethe Ansatz. This model has been thoroughly investigated (and. We shall start with the construction of eigenfunctions of the Hamiltonian in a finite volume. Quantities interesting from the physical point of view (in the thermodynamic limit at zero temperature) are considered; the thermodynamics at finite temperatures is investigated in detail as well. A number of essential ideas which will be applied to other models are introduced. The construction of eigenfunctions of the Hamiltonian is explained in section 1. Their explicit form and, in particular, the two-particle reducibility, are common features of the models solvable by means of the Bethe Ansatz method. Periodic boundary conditions are imposed on the wave function in section 2; the Bethe equations for the particle' momenta are introduced and analyzed. Taken in the logarithmic form, these equations realize the extremum condition of a certain functional, the corresponding action being called the Yang-Yang action. The transition to the thermodynamic limit is considered in section 3. In that same section the ground state of the gas is constructed. The density of the particle distribution in momentum space and the energy of the ground state are calculated. The method of the transition to the thermodynamic limit described in this section is rather general and may be applied to any model solvable by means of the Bethe Ansatz. In section 4, excitations above the ground state are constructed and their main characteristics (energy, momentum and scattering matrix) are determined by means of the dressing equations. The ground state of the model is the Dirac sea (also called the Fermi sphere).