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Progress of Theoretical Physics

A Theory of a One-Dimensional System of Interacting Electrons in a 2kF-Periodic Potential

T. Sugiyama, J. Kumagai, Kazuki Okamoto

Published Apr 1, 1984• 5 citations
Publisher

Abstract

An effect of mutual interaction is studied in a one-dimensional system of electrons with spins scattered by a 2kF-periodic potential. The phase representation based on the boson method is used. It is shown that the system becomes metallic at T=0 depending on the strength of the interaction between electrons. Response functions are calculated and the phase diagram of the ground state is described. Further, the behavior of the system of electron in the 2kF-periodic potential is compared with that in a random potential.

DOI
10.1143/ptp.71.677

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Physical review. B, Condensed matter•Apr 15, 1997•17 citations
An effect of mutual interaction is studied in a one-dimensional system of electrons with spins scattered by a 2kF-periodic potential. The phase representation based on the boson method is used. It is shown that the system becomes metallic at T=0 depending on the strength of the interaction between electrons. Response functions are calculated and the phase diagram of the ground state is described. Further, the behavior of the system of electron in the 2kF-periodic potential is compared with that in a random potential.
An effect of mutual interaction is studied in a one-dimensional system of electrons with spins in a 2kF-periodic potential. The phase representation based on the boson method is used. It is shown that the system is metallic at T=0 in some regions of the g1H-g2 plane, where the g’s are the coupling constants of the interaction between electrons. The metal-insulator boundary depends on the sign of g3⊥. A formulation of the self-consistent harmonic approximation with two kinds of variables is founded from a stand-point of the variational method.
A one-dimensional electron system at quarter-filling has been examined by applying the renormalization group method to a bosonized model with on-site (U) and nearest-neighbor (V) repulsive interactions. By evaluating both normal scattering and Umklapp scattering perturbatively, we obtain a phase diagram in which a metallic state with a 2k_F spin density wave (k_F is the Fermi wave number) moves into an insulating state with charge disproportionation of a 4k_F charge density wave with an increase in both U and V. The effect of the next-nearest-neighbor repulsion is also discussed.
A one-dimensional interacting electron gas in a random potential exhibits a localized-delocalized transition for increasingly attractive interactions. We develop here a renormalization-group approach to study this transition. Our treatment allows us to obtain the phase diagram and the exponents of the correlation functions in the delocalized regime. The boundary between the two regimes is found to depend both on disorder and the strength of the interactions. For (nearly) spin-isotropic interactions the delocalized phase is dominated by superconducting fluctuations of either singlet or triplet type. The temperature dependence of the conductivity in the delocalized phase is also obtained and a nonuniversal power-law behavior is found. A description of the crossover towards the localized phase is given and the localization length is computed. An analogous description is developed for the localized-superfluid transition of a one-dimensional boson gas. In this case the transition to the localized regime occurs for increasingly repulsive interactions. We suggest a phase diagram with two different localized phases. Finally, we discuss some possible implications of our model for real quasi-one-dimensional metals.
The model Hamiltonian of the one-dimensional electron gas with different types of interaction (forward and backward scattering, Umklapp processes) is studied by assuming a sharp bandwidth cut-off. Exact equalities among different response functions are obtained for specific values of coupling constants from simple symmetry considerations. The ground-state energy derived in mean-field approximation and the asymptotic behaviour of response functions as calculated by various authors are discussed with respect to these exact relations. The agreement of phase diagrams as obtained with different methods is traced back to the underlying symmetry. Giving up this symmetry, e.g. by assuming different cut-offs above and below the Fermi energy, changes the phase diagram. Furthermore this allows one to construct a mean-field ground state where superconductivity and density waves coexist in a finite region of coupling space. If the electron-phonon interaction is also taken into account it follows that the charge-density wave instability (together with a periodic lattice distortion) is favoured.
The effects of a long range electronic potential on a one dimensional chain of spinless fermions are investigated by numerical techniques and analytic calculations. Due to a competition between the 2kF oscillations in the density and the log divergence of the long wavelength part of the Coulomb potential, the metallic character of the system is enhanced at intermediate (and up to quite large) lengthscales. We found that this quasi-metallic regime is not of the Luttinger Liquid type, as evidenced by strong deviations from predictions of conformal field theory and in agreement with the picture of a (very weakly pinned) Wigner crystal. When the strength of the Coulomb interaction is substantially increased, we observe a smooth cross-over to a strongly localized charge density wave.
We discuss interaction effects for the one-dimensional electron gas with a repulsive \ensuremath{\delta}-function interaction potential by using the random-phase approximation and a local-field correction. Analytical results for the local-field correction of charge-density fluctuations and spin-density fluctuations are obtained. The ground-state energy is found to be in better agreement with the exact result than the ground-state energy calculated within the random-phase approximation. We calculate the pair-correlation functions for electrons with parallel and antiparallel spins and the paramagnetic susceptibility. The energies of the collective density modes ${\mathrm{\ensuremath{\omega}}}_{\mathrm{d}}$ (q) and the collective spin modes ${\mathrm{\ensuremath{\omega}}}_{\mathrm{s}}$ (q) are calculated and compared with the energy of the electron-hole excitations ${\mathrm{\ensuremath{\omega}}}_{\mathrm{eh}\ifmmode\pm\else\textpm\fi{}}$ (q): ${\mathrm{\ensuremath{\omega}}}_{\mathrm{d}}$ (q\ensuremath{\rightarrow}0)>${\mathrm{\ensuremath{\omega}}}_{\mathrm{eh}\ifmmode\pm\else\textpm\fi{}}$ (q\ensuremath{\rightarrow}0)>${\mathrm{\ensuremath{\omega}}}_{\mathrm{s}}$ (q\ensuremath{\rightarrow}0). The critical exponents for the long-distance behavior of correlation functions are found to be described by local-field corrections. We compare our results for the critical exponents with recent results obtained by bosonization techniques and the conformal field theory.