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Crossover between High and Low Energy-States in Two-Coupled Chains of Tomonaga Model

Masahisa Tsuchiizu, Hideo Yoshioka, Yoshikazu Suzumura

Published Jan 1, 1997• 2 citations
OpenAlex

Abstract

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

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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
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.
Two chains of spinless Tomonaga model coupled by a transverse hopping are investigated in terms of phase Hamiltonian where separation of Fermi momentum in the presence of the hopping is taken into account explicitly. It is shown that there appears a gap in one of the two excitations by using a renormalization group method. As a result, correlation of charge density wave with interchain ordering and out of phase between the chains (superconductivity with intrachain ordering and in phase between the chains) becomes the most dominant one in the case of the repulsive (attractive) intrachain interaction.
Two chains of interacting spinless fermions in the presence of interchain hopping are investigated by use of phase Hamiltonian based on bosonization method. Excitation spectrum and phase diagram at absolute zero temperature are obtained by means of renormalization group method. Momentum distribution functions are also calculated. It is shown that the interchain hopping gives rise to drastic effects on properties of ground state compared to interchain interaction and is crucial to appearance of ordered states in higher dimensions.
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.
Phase transitions of a quasi-one-dimensional Fermi gas are studied by the use of the renormalization group method for the fermion system without the bosonization. Intrachain backward and forward scatterings, pair electron hopping between different chains and exchange scattering of electrons in different chains are taken account of. Pair electron hopping gives rise to the singlet and triplet superconductor phase transitions. Electron exchange scattering gives rise to the spin density wave and charge density wave transitions. Calculation is made by the use of the one-dimensional dispersion. Following Menyhárd's method (Ref. 1) for the model with interchain backward scattering only, the change of the fluctuation from the one-dimensional region to the three-dimensional region is described. The transition temperatures are given by the bare couplings at high temperatures.
The one-particle Green's function for the two-chain Tomonaga–Luttinger (TL) model coupled by interchain hopping including spin degrees of freedom is calculated as a function of distance and time by the higher-dimensional bosonization method. In the short distance and time region, the one-dimensional TL liquid behavior is reproduced, while the long distance and time behavior is different from that for either TL or Fermi liquid. As the interaction becomes stronger, the behavior of Green's function becomes closer to that for the one-dimensional TL liquid, showing that the one-dimensionality is enhanced for stronger electron interaction, and is interpreted as the precursor of the “confinement”.