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Journal de physique

Coulomb screening in quasi one-dimensional conductors. II. Bosonization method

Srećko Botrić, S. Barišić

Published Jan 1, 1984• 14 citations
Publisher

Abstract

The Tomonaga model for a system of parallel chains with the long range Coulomb interactions is investigated by the bosonization technique. The respective rôles of the screened long range and the bare intermediate range interactions are examined in the 2 k F CDW/SDW correlation function. The accurate expression for this latter function is derived in the weak-coupling limit.

DOI
10.1051/jphys:01984004502018500

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•Jan 1, 2010•0 citations
The Tomonaga model for a system of parallel chains with the long range Coulomb interactions is investigated by the bosonization technique. The respective rôles of the screened long range and the bare intermediate range interactions are examined in the 2 k F CDW/SDW correlation function. The accurate expression for this latter function is derived in the weak-coupling limit.
Variational wave function is proposed to describe electronic properties of an array of one-dimensional conductors coupled by transverse hopping and interaction. For weak or intermediate in-chain interaction the wave function has the following structure: Tomonaga-Luttinger bosons with momentum higher than some variational quantity $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Lambda}}$ are in their ground state while other bosons (with $|k|<\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Lambda}})$ form kinks --- fermionlike excitations of the Tomonaga-Luttinger boson field. The nature of the ground state for these quasiparticles can be determined by solving a three-dimensional effective Hamiltonian. Since the anisotropy of the effective Hamiltonian is small the use of mean-field theory is justified. For repulsive interaction possible phases are density wave and p-wave superconductivity. Our method allows us to calculate the low-energy part of different electronic Green's functions. In order to do this it is sufficient to apply standard perturbation theory technique to the effective Hamiltonian. When the in-chain interaction is strong $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Lambda}}$ vanishes and no fermionic excitation is present in the system. In this regime the dynamics is described by transversally coupled Tomonaga-Luttinger bosons.
Experiments on quasi-one-dimensional systems such as quantum wires and metallic chains on surfaces suggest the existence of electron-electron interactions of substantial range and hence physics beyond the Hubbard model. We therefore investigate one-dimensional, quarter-filled chains with a Coulomb potential with variable screening length by quantum Monte Carlo methods and exact diagonalization. The Luttinger liquid interaction parameter ${K}_{\ensuremath{\rho}}$ decreases with increasing interaction strength and range. Experimentally observed values close to 1/4 require strong interactions and/or large screening lengths. As predicted by bosonization, we find a metal-insulator transition at ${K}_{\ensuremath{\rho}}=1/4$. Upon increasing the screening length, the charge and spin correlation functions reveal the crossover from dominant 2kF2{k}_{\mathrm{F}}2kF​ spin correlations to dominant 4kF4{k}_{\mathrm{F}}4kF​ charge correlations, and a strong enhancement of the charge velocity. In the metallic phase, the signatures of spin-charge separation in the single-particle spectrum, spinon and holon bands, remain robust even for rather long-ranged interactions. The charge-density-wave state exhibits backfolded shadow bands.
This dissertation is devoted to theoretical studies of strongly interacting one-dimensional and quasi one-dimensional electron systems. The properties of one-dimensional electron systems can be studied within the bosonization technique, which presents fermions as collective bosonic density excitations. The power of this approach is the ability to treat electron-electron interaction exactly in the low-energy limit. The approach predicts the failure of Fermi liquid and an absence of long-range order in one-dimensions. The low-energy description of one-dimensional interacting systems is called the Tomonaga-Luttinger liquid theory. For example, the edges of quantum Hall systems are one-dimensional and described by a chiral Tomonaga-Luttinger liquid. Another example is a quantum spin Hall system with helical edge states, which are also described by a Tomonaga-Luttinger liquid. In our first work, a study of magnetized edge states of quantum spin-Hall system is presented. A magnetic field dependent signature of such edges is obtained, which can be verified in a Coulomb drag experiment. The second part of the dissertation is devoted to quasi-one dimensional antiferromagnetic lattices. A spatially anisotropic lattice antiferromagnet can be viewed as an array of one dimensional spin chains coupled in a way to match the lattice symmetry. This allows to use the non-Abelian bosonization technique to describe the low-energy physics of spin chains and study the inter-chain interactions perturbatively. The work presented in the dissertation studies the effect of Dzyaloshinskii-Moriya interaction on the magnetic phase diagram of the spatially anisotropic kagome antiferromagnet. We predict a Dzyaloshinskii-Moriya interaction driven phase transition from Neel to Neel+dimer state. In the third work, a novel model of the fractional quantum Hall effect is given. Wave functions of two-dimensional electrons in strong and quantizing magnetic field are essentially one-dimensional. That invites one to use the one-dimensional phenomenological bosonization to describe the density fluctuations of the two-dimensional interacting electrons in magnetic field. Remarkably, the constructed trial bosonized fermion operator describing the electron states with a fixed Landau gauge momentum is effectively two-dimensional.
We study a one-dimensional system of spinless electrons in the presence of a long-range Coulomb interaction (LRCI) and a random chemical potential at each site. We present a Tomonaga-Luttinger liquid description of the system. We use the bosonization technique followed by the replica trick to average over the quenched randomness. An expression for the localization length of the system is obtained using the renormalization-group method and also a physical argument. We then find the density of states for different values of the energy; we get different expressions depending on whether the scale of energy is larger than or smaller than the inverse of the localization length. We work in the limit of weak disorder where the localization length is very large, at that length scale, the LRCI has the effect of reducing the interaction parameter K to a value much smaller than the noninteracting value of unity.
By applying the renormalization group method to two-coupled chains in the Tomonaga model, the role of interchain hopping has been studied in the entire energy region. The energy for a crossover from the perturbational regime to the relevant regime becomes smaller than that of the interchain hopping due to one-dimensional fluctuations of the mutual interaction. From the calculation of response functions for charge density waves and superconducting states, the phase diagram of dominant and subdominant states has been obtained in the plane of mutual interactions with fixed energy. Organic conductors, 1) which exhibit low dimensional fluctuations, have been studied by using the model of quasi-one-dimensional systems, where the role of the interchain hopping plays an important role. As a basic model for such a system, two-coupled chains of interacting electrons
For nearly a decade, quasi-one-dimensional conductors have been the subject of intensive study. Theoretically, much attention has been devoted to the development of one-dimensional Fermi gas models, some which may be solved exactly, and to the calculation of their response functions. After a review of this theory, a different approach is adopted in the investigation of two models. The dielectric response theory of the three-dimensional Coulomb gas has been applied to an anisotropic system in which the particles interact with an effective one-dimensional long-range potential. Within the framework of the approximation of Singwi, Tosi, Land, and Sjolander, the dielectric properties of the model are examined in order to determine the conditions under which it is unstable with respect to formation of a charge density wave state. It is found that the positive neutralizing background must be polarizable in order for such an instability to occur. The same approximation method, when applied to a one-dimensional fermion gas with a ʃ-function interaction may be compared with the exact solution of Yang. This solution, which exists in the form of coupled integral equations, has been calculated numerically, and, as predicted by the Lieb-Mattis theorem, the ground state is found to be non-magnetic. The approximation of Singwi et al. proves to give better correlation energies than other inexact methods, particularly at higher densities.