The low energy properties of different one-dimensional fermionic lattice models are investigated using the bosonization technique. We attach much importance to a proper consideration of the Klein factors which are neglected or inaccurately treated in the majority of the literature. In order to deal with the nonlinear terms, arising from the dimerization or similar scattering processes, we use the self-consistent harmonic approximation (SCHA), a variational method where the nonlinearities are replaced by a quadratic potential. We construct an extension of the SCHA in order to account for the existence of Klein factors in bosonized Hamiltonians. As prototypical models we consider one-dimensional dimerized spinless fermions and Hubbard models with periodic modulation of the hopping and of the chemical potential, respectively. In an attempt to describe more realistic systems we also consider a model where one-dimensional electrons are coupled to a three-dimensional conduction band via a local interaction. In order to see how the one-dimensional electron system is modified in the presence of the three-dimensional environment, we calculate the charge and the spin susceptibility, and analyze the collective excitations within RPA. We find that the spin excitations remain unaffected while in the charge channel, the collective modes of the one-dimensional and three-dimensional subsystems are coupled.
We use our recently developed functional bosonization approach to bosonize interacting fermions in arbitrary dimension d beyond the Gaussian approximation. Even in d=1 the finite curvature of the energy dispersion at the Fermi surface gives rise to interactions between the bosons. In higher dimensions scattering processes describing momentum transfer between different patches on the Fermi surface (around-the-corner processes) are an additional source for corrections to the Gaussian approximation. We derive an explicit expression for the leading correction to the bosonized Hamiltonian and the irreducible self-energy of the bosonic propagator that takes the finite curvature as well as around-the-corner proceses into account. In the special case that around-the-corner scattering is negligible, we show that the self-energy correction to the Gaussian propagator is negligible if the dimensionless quantities (
qc/
kF)dF0 [1+
F0${]}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\mu}/${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\Vert}\ensuremath{\partial}${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$/\ensuremath{\partial}\ensuremath{\mu}\ensuremath{\Vert} are small compared with unity for all patches \ensuremath{\alpha}. Here
qc is the cutoff of the interaction in wave-vector space,
kF is the Fermi wave vector, \ensuremath{\mu} is the chemical potential,
F0 is the usual dimensionless Landau interaction parameter, and ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$ is the local density of states associated with patch \ensuremath{\alpha}. We also show that the well-known cancellation between vertex and self-energy corrections in one-dimensional systems, which is responsible for the fact that the random-phase approximation (RPA) for the density-density correlation function is exact in d=1, exists also in d>1, provided (1) the interaction cutoff
qc is small compared with
kF, and (2) the energy dispersion is locally linearized at the Fermi surface. Finally, we suggest a systematic method to calculate corrections to the RPA, which is based on the perturbative calculation of the irreducible bosonic self-energy arising from the non-Gaussian terms of the bosonized Hamiltonian.
The method of self-consistent-field approximation is applied for the study of the ground-state properties of a one-dimensional system of bosons, of density $\ensuremath{\rho}$, first interacting through finite rigid rods, of length
l, and again via a repulsive $\ensuremath{\delta}$-function potential. We obtain the energy spectrum and the sound velocity. The results are close to the exact one for $\ensuremath{\rho}l<0.5$. The energy spectrum obtained shows a rotonlike region which is qualitatively similar to the two- and three-dimensional versions. For a Bose gas interacting through a repulsive $\ensuremath{\epsilon}$ potential of strength
2c, we derive the energy spectrum, the sound velocity, and the ground-state energy. The results are compared with Girardeau and Lieb's exact expressions. In the small-coupling limit our calculations show the same results as obtained by Bogoliubov's perturbation theory. The ground-state energy obtained for small $\ensuremath{\gamma}=\frac{c}{\ensuremath{\rho}}$ is found to agree closely with several different works found in the literature. Unfortunately our theory diverges for the ground-state energy in the limit $\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\infty}$. The sound velocity in this limit is finite but twice as small as from Girardeau's exact results.
In this work, we extend the analysis of interacting bosons at 2D-1D dimensional crossover for finite size and temperature by using the field theory approach ( b o s o n i z a t i o n ) and quantum Monte Carlo simulations. Stemming from the fact that finite size low-dimensional systems are allowed only to have quasiordered phase, we consider the self-consistent harmonic approximation to compute the fraction of quasicondensate and its scaling with the system size. It allows us to understand the important role played by finite size and temperature across the dimensional crossover for deciding the condensate nature. Furthermore, we consider a mean-field approximation to compute the finite size effect on the crossover temperature for both weak and strong interactions. All the physical quantities we discuss here provide essential information for quantum gas at dimensional crossover and are directly detectable in cold atom experiments.