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Molecular crystals and liquid crystals science technology. Section A, Molecular crystals and liquid crystals/Molecular crystals and liquid crystals science and technology. Section A, Molecular crystals and liquid crystals

Spin Gap in One-Dimensional Electron-Phonon Systems

Kenji Yonemitsu, Masatoshi Imada

Published Jul 1, 1996• 2 citations
Publisher

Abstract

The exponents of spin and charge correlations of one-dimensional electron systems are calculated by a renormalization-group method. We study combined effects of electron-phonon interaction and the Umklapp process on renormalization flows. As scattering processes are successively integrated out, these two processes are interfered with each other depending on the type of electron-phonon coupling, phonon frequency and filling. We demonstrate how these factors control a spin gap formation and density-wave/superconductor correlations.

DOI
10.1080/10587259608030805

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The exponents of spin and charge correlations of one-dimensional electron systems are calculated by a renormalization-group method. We study combined effects of electron-phonon interaction and the Umklapp process on renormalization flows. As scattering processes are successively integrated out, these two processes are interfered with each other depending on the type of electron-phonon coupling, phonon frequency and filling. We demonstrate how these factors control a spin gap formation and density-wave/superconductor correlations.
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.
On the basis of the extended Hubbard model we consider phonon mediated forward scattering in one-dimensional systems. Exact results for the two-particle spectral functions are derived for the case when Umklapp and backward scattering due to electron–electron interaction are neglected. Due to the electron–phonon interaction the phase diagram of the interacting electron system changes remarkably. In dependence on the phonon frequency and on the electron–phonon coupling constant a new divergence may appear in a two-particle spectral function or an existing one may be removed. The inclusion of Umklapp and backward scattering leads to the appearance of an energy gap and a polaron energy shift. The electron–phonon coupling constant is strongly renormalized and vanishes faster than the charge velocity as half-filling is approached. Beyond half-filling the polaron shift may diverge at certain values of the coupling parameters. This divergency is related to the Wentzel-Bardeen singularity.
An exact solution of the one-dimensional electron gas on a lattice is given for the special case in which two of the four coupling constants have particular values. It is shown that umklapp scattering has the same effect on the charge-density waves as backward scattering across the Fermi "surface" has on the spin-density waves and that the method of Luther and Emery can be used to solve this more general problem. For repulsive electron-electron interactions the umklapp scattering produces a gap in the charge-density wave spectrum and this appears for low-lying excitations when there is a half-filled band. The use of renormalization-group scaling to solve for general values of the coupling constants is discussed.
The influence of umklapp scattering on the resistivity of an interacting-one-dimensional-fermion system (Luttinger liquid) is studied. By using a renormalization-group calculation and a memory-function approximation for the conductivity, it is possible to obtain its frequency and temperature dependence at arbitrary filling. At high temperature the conductivity behaves as a power law of the temperature with an exponent depending on the interactions. Away from half filling there is a crossover between this behavior and an exponential increase of the conductivity. At half filling, the low-temperature conductivity behaves as ${\mathit{e}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\Delta}}/\mathit{T}}$, where \ensuremath{\Delta} is the gap in the charge spectrum. It is argued that to get such behavior other scattering processes or phase-breaking processes are needed since in the presence of only electron-electron scattering the conductivity should, strictly speaking, be infinite at every finite temperature, even at half filling. Finally some results on the exponents of correlation functions and on the weight of the Drude peak obtained previously for the Hubbard model are shown to be generic features of any Luttinger liquid.
Using a multistep renormalization group method, we study the low-temperature phases of the interacting one-dimensional (1D) electron gas coupled to phonons. We obtain analytic expressions for the weak-coupling quantum phase boundaries of the 1D extended Holstein-Hubbard model and the 1D extended Peierls-Hubbard model for general band-filling and phonon frequency. Away from half-filling, the phase diagrams are characterized by a delicate competition between spin density wave, charge density wave, and superconducting orders. We study the dependence of the ground state on the electron-phonon (el-ph) and electron-electron (el-el) coupling strengths, the screening length, electron bandwidth, phonon frequency, doping, and type of phonon. Unlike the case in Fermi liquids, in 1D the el-ph coupling is strongly renormalized, often to stronger values. Even when the bare phonon-induced attraction is weak compared to the bare el-el repulsion, a small amount of retardation can cause the renormalized el-ph interaction to dominate the problem. We find cases in which a repulsive el-el interaction enhances the superconducting susceptibility in the presence of a retarded el-ph interaction. The spin gap and superconducting susceptibility are found to be strongly dependent on the deviation from half-filling (doping). In some cases, the superconducting susceptibility varies nonmonotonically with doping and exhibits a maximum at a particular doping. For a quasi-1D array of weakly coupled, fluctuating 1D chains, the superconducting transition temperature Tc{T}_{c}Tc​ also exhibits a maximum as a function of doping. The effect of changing the ion mass (isotope effect) on Tc{T}_{c}Tc​ is found to be largest near half-filling and to decrease rapidly with doping.
The one-dimensional Mott metal-insulator transition is a typical strong correlation effect triggered by the Umklapp scattering. However, in a physical system, the Umklapp scattering coexists with the normal scattering, including both forward and backward scattering, which conserves the total momentum of scattered electrons. Therefore, it is not easy to quantify the contribution of the Umklapp scattering in a Mott metal-insulator transition. To resolve this difficulty, we propose to explore these scattering processes separately. We study the contribution of each scattering process in the one-dimensional Hubbard model using the momentum-space density-matrix renormalization group (kDMRG) and bosonization methods. Our kDMRG calculation confirms that the Mott charge gap results from the Umklapp scattering, but the normal scattering processes strongly renormalize its value. Furthermore, we present a scaling analysis of the Mott charge gap in the bosonization theory and show that the interplay between the Umklapp and forward scattering dictates the charge dynamics in the half-filled Hubbard model.