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Journal of the Physical Society of Japan

Many-Polaron Theory for CDW, SDW and Superconducting Phases in One-Dimensional N-Sites N-Elctrons System –Quantum-Classic Crossover–

Keiichirō Nasu

Published May 15, 1985• 41 citations
Publisher

Abstract

A one-dimensional many-electron system with on-site electron-electron(e-e) and electron-phonon(e-p) interactions is studied in the context of polaron theory, so as to clarify the quantum effect of phonon on the phase diagram. Lattice distortions moving with electrons as well as a frozen one are taken into account by a variational method combined with mean field theory. The phase diagram is given in a tetrahedral coordinate, spanned by four parameters : inter-site transfer of electron T , e-e repulsion U , e-p coupling S and phonon energy ω. This “T-U-S-ω tetrahedron” is shown to be divided into CVW and SDW regions by the boundary S = U . In the CDW region, the polaron radius increases as the retardation effect increases, and in its long limit the superconducting state(SCS) becomes far unstable than CDW, while in its short limit, SCS is degenerate with CDW, indicating the importance of this radius in quantum-classic crossover.

DOI
10.1143/jpsj.54.1933

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Journal of the Physical Society of Japan•Oct 15, 1982•23 citations
A one-dimensional many-electron system with on-site electron-electron(e-e) and electron-phonon(e-p) interactions is studied in the context of polaron theory, so as to clarify the quantum effect of phonon on the phase diagram. Lattice distortions moving with electrons as well as a frozen one are taken into account by a variational method combined with mean field theory. The phase diagram is given in a tetrahedral coordinate, spanned by four parameters : inter-site transfer of electron T , e-e repulsion U , e-p coupling S and phonon energy ω. This “T-U-S-ω tetrahedron” is shown to be divided into CVW and SDW regions by the boundary S = U . In the CDW region, the polaron radius increases as the retardation effect increases, and in its long limit the superconducting state(SCS) becomes far unstable than CDW, while in its short limit, SCS is degenerate with CDW, indicating the importance of this radius in quantum-classic crossover.
The phase diagram of a two-dimensional N-site N-electron system (N\ensuremath{\gg}1) with site-diagonal electron-phonon (e-ph) coupling is studied in the context of polaron theory, so as to clarify the competition between the superconducting (SC) state and the charge-density wave (CDW) state. The Fermi surface of noninteracting electrons is assumed to be a complete circle with no nesting-type instability in the case of weak e-ph coupling, so as to focus on such a strong coupling that even the standard ``strong-coupling theory'' for superconductivity breaks down. Phonon clouds moving with electrons as well as a frozen phonon are taken into account by a variational method, combined with a mean-field theory. It covers the whole region of three basic parameters characterizing the system: the intersite transfer energy of electron T, the e-ph coupling energy S, and the phonon energy \ensuremath{\omega}. The resultant phase diagram is given in a triangular coordinate space spanned by T, S, and \ensuremath{\omega}. In the adiabatic region \ensuremath{\omega}\ensuremath{\ll}(T,S) near the T-S line of the triangle, each electron becomes a large polaron with a thin phonon cloud, and the system changes discontinuously from the SC state to the CDW state with a frozen phonon as S/T increases. In the inverse-adiabatic limit \ensuremath{\omega}\ensuremath{\gg}(T,S) near the \ensuremath{\omega} vertex of the triangle, on the other hand, each electron becomes a small polaron, and the SC state is always more stable than the CDW state, because the retardation effect is absent. Thus, the polaron radius decreases and the SC region expands in the triangle as T/\ensuremath{\omega} decreases. It is found, for the first time, that the energy gap of the SC state for a given T and S becomes maximum at the intermediate region \ensuremath{\omega}\ensuremath{\sim}T, indicating the importance of the polaron effect. The collective excitation within the gap of the SC state is also studied by the random-phase approximation, and is found to change its nature continuously from the pair-breaking type to the superfluid type as S/T increases.
Stable and metastable phases of a one-dimensional many-electron system with short range electron-electron (e-e) repulsion and site-diagonal electron-phonon (e-p) interaction are studied within the mean field theory and in the adiabatic limit. The total energy of the system is calculated as a function of the three parameters: the transfer energy of the electron between neighbouring two sites T , the on-site e-e repulsive energy U and the e-p coupling energy S . The resultant phase diagram is such that the “ T - U - S triangle ” is bisected into the SDW and the CDW regions by the border-line with U = S . The metastable SDW (CDW) phase exists in addition to the stable CDW (SDW) phase in a narrow region around the bisector. The border-lines for the metastable phases and the bisector between the two stable phases flow together into the T - vertex of the triangle, indicating the intriguing instability of one-dimensional metal against U and S .
A one-dimensional many-electron system with site-diagonal electron-phonon coupling, intra- and inter-site electron-electron (e-e) repulsions is studied within the RPA for electrons and in the adiabatic limit for phonons, so as to clarify effects of fluctuation on the phase diagram of the ground state within the mean field theory (MFT), and also on various excitations therefrom. Energies of bound electron-hole (e-h) pairs in the MFT energy-gap such as magnon and charge-transfer (CT) excitons are calculated together with the ground-state energy-gain due to their virtual excitations. The resultant new phase diagram is such that the SDW phase erodes the CDW region given by the MFT, because of the magnon excitation. It is shown that the inter-site e-e repulsion always gives a stable singlet CT exciton that dominates the optical property of the CDW.
The Holstein-Hubbard model serves as a useful framework to investigate this interplay between the phonon-induced electron-electron attractive interaction and the direct Coulomb repulsion and can afford interesting phase diagrams due to competition among charge-density wave (CDW), spin-density wave (SDW), and superconductivity. However the detailed nature of the CDW-SDW transition is still not very well known. It is generally believed that the system undergoes a direct insulator to insulator transition from CDW to SDW with the increase of the on-site Coulomb repulsion for a given strength of the electron-phonon coupling and this is the main bottleneck for the polaronic/bipolaronic mechanism of high-temperature superconductivity. We have recently made an investigation to study the nature of the transition from SDW phase to CDW phase within the framework of a one-dimensional Holstein-Hubbard model at half-filling using a variational method. We find that an intervening metallic phase may exist at the crossover region of the CDW-SDW transition. We have also observed that if the anharmonicity of the phonons is taken into account, this metallic phase widens and the polarons become more mobile, which is a more favorable situation from the point of view of superconductivity. We shall finally show that an improved variational calculation widens the metallic phase and makes the polarons more mobile, which reconfirms the existence of the intermediate metallic phase at the SDW-CDW crossover region.
The phase diagram for one-dimensional many-electron system with the electron-electron and electron-phonon interactions is studied within the RPA and in the adiabatic limit, so as to clarify the effect of the fluctuation on the phase diagram given by the mean field theory (MFT). The ground-state energy gain due to the virtual excitation of the bound and unbound electron-hole pairs is calculated. The newly obtained phase diagram is such that the SDW region erodes the CDW region given by the MFT. In the CDW, the lowest allowed excitation is a pair of charge transfer type excitons with the energy nearly equal to the gap, while in the SDW it is a pair of magnons just above the ground state. This difference results in the erosion, indicating that the comprehensive study of all excited states is essential in obtaining the ground state itself.