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Physical review. B, Condensed matter

Symmetry properties of one-dimensional Fermi systems: Some nonperturbative results

Edward H. Rezayi, Josef Sak, Jenő Sólyom

Published Feb 1, 1981• 6 citations
Publisher

Abstract

Several exact results are proved for two models of one-dimensional conductors, a model with transfer cutoff and spin- and charge-carrying interactions and the usual backscattering model with a bandwidth cutoff. A set of Ward identities is derived which connects the three-point charge vertex parts with single-particle Green's functions. These are combined with a modified renormalization-group scheme to prove the scale invariance of the couplings between charge densities in the absence of umklapp processes. The Ward identities are also used for the computation of the correlation functions in the charge sector of the system. Then some canonical transformations are introduced which allow one to relate various response functions among themselves. From these relations the phase boundaries of the ground state can be established. It is shown that the transfer cutoff and the backscattering models have the same phase diagram.

DOI
10.1103/physrevb.23.1342

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Several exact results are proved for two models of one-dimensional conductors, a model with transfer cutoff and spin- and charge-carrying interactions and the usual backscattering model with a bandwidth cutoff. A set of Ward identities is derived which connects the three-point charge vertex parts with single-particle Green's functions. These are combined with a modified renormalization-group scheme to prove the scale invariance of the couplings between charge densities in the absence of umklapp processes. The Ward identities are also used for the computation of the correlation functions in the charge sector of the system. Then some canonical transformations are introduced which allow one to relate various response functions among themselves. From these relations the phase boundaries of the ground state can be established. It is shown that the transfer cutoff and the backscattering models have the same phase diagram.
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 one-dimensional Fermi system with Tomonaga-type (g2{g}_{2}g2​) and back-scattering (g1{g}_{1}g1​) interactions is studied by a weak-coupling renormalization-group method at the three-loop level. When the transfer cutoff procedure is used, the Gell-Mann-Low function for g1{g}_{1}g1​ depends on g1{g}_{1}g1​ alone, $\frac{d{g}_{1}}{d\ensuremath{\xi}}={{g}_{1}}^{2}+\frac{1}{2}{{g}_{1}}^{3}+\frac{1}{8}{{g}_{1}}^{4}$. Further, in the absence of umklapp processes the combination ${g}_{1}\ensuremath{-}2{g}_{2}$ is a scale invariant. These conclusions agree with the results of Luther and Emery derived by the boson representation method. Some novel calculational techniques for the derivation of the scaling equations are described and compared to the conventional scheme.
We investigate the Fermi gas model for one-dimensional (1D) conductors as a function of chemical potential and of the applied magnetic field. The bosonization method allows to prove the equivalence between these parameters, the first acting on charge degrees of freedom, the second on spin degrees of freedom. A number of results available in the literature can thus be connected. We also use the mapping to the 2D classical Coulomb gas and a two-cutoff renormalization-group method, the two energy scales being the bandwidth and the Zeeman energy (or the chemical potential). We obtain a coherent description of the Fermi gas phase diagram as a function of these two ``fields.''
One-dimensional metals have a particular symmetry associated with the discrete structure of the Fermi surface: separate charge conservation in low-energy-scattering processes for particles near the left and right Fermi points, respectively. The field-theoretical renormalization group allows for an efficient exploitation of the Ward identities following from this symmetry. As a first application we prove that the \ensuremath{\beta} function of the Luttinger model vanishes identically. The same symmetry ensures the finiteness of the compressibility, thus making possible the existence of stable metallic phases with anomalous dimensions in d=1.
Asymptotic properties of nearly half-filled one-dimensional conductors coupled with phonons are studied through a renormalization-group method. Due to spin-charge coupling via an electron-phonon interaction, the spin correlation varies with filling as well as the charge correlation. Depending on the relation between cutoff energy scales of the umklapp process and of the electron-phonon interaction, various phases appear. We find a metallic phase with a spin gap and a dominant charge-density-wave correlation near half filling between a gapless density-wave phase (like in the doped repulsive Hubbard model) and a superconductor phase with a spin gap. The spin gap is produced by phonon-assisted backward scatterings that interfere with the umklapp process constructively or destructively depending on the character of electron-phonon coupling. \textcopyright{} 1996 The American Physical Society.