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arXiv:cond-mat/0310471 · Physical review. B, Condensed matter

Variational description of the dimensional crossover in an array of coupled one-dimensional conductors

A. V. Rozhkov

Published Sep 22, 2003• 9 citations
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Abstract

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.

DOI
10.1103/physrevb.68.115108

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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 \ensuremathΛ\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Lambda}}\ensuremathΛ\ifmmode~\else˜\fi​ are in their ground state while other bosons (with ∣k∣<\ensuremathΛ\ifmmode~\else˜\fi)|k|<\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Lambda}})∣k∣<\ensuremathΛ\ifmmode~\else˜\fi​) 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 \ensuremathΛ\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Lambda}}\ensuremathΛ\ifmmode~\else˜\fi​ vanishes and no fermionic excitation is present in the system. In this regime the dynamics is described by transversally coupled Tomonaga-Luttinger bosons.
We use higher-dimensional bosonization to study the normal state of electrons in weakly coupled metallic chains interacting with long-range Coulomb forces. Particular attention is paid to the crossover between Luttinger and Fermi-liquid behavior as the interchain hopping ${t}_{\ensuremath{\perp}}$ is varied. Although in the physically interesting case of finite but small ${t}_{\ensuremath{\perp}}$ the quasiparticle residue does not vanish, the single-particle Green's function exhibits the signature of Luttinger-liquid behavior (i.e., anomalous scaling and spin-charge separation) in a large intermediate parameter regime. Using realistic parameters, we find that the scaling behavior in this regime is characterized by an anomalous dimension of the order of unity, as suggested by recent experiments on quasi-one-dimensional conductors. Our calculation also gives insights into the approximations inherent in higher-dimensional bosonization; in particular, we show that the replacement of a curved Fermi surface by a finite number MMM of flat patches can give rise to unphysical nesting singularities in the single-particle Green's function, which disappear only in the limit $M\ensuremath{\rightarrow}\ensuremath{\infty}$ or if curvature effects are included. We also compare our approach with other methods. This work extends our recent letter [Phys. Rev. Lett. 74, 2997 (1995)].
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
The validity of the quasiparticle picture for electrons in weakly coupled linear chains is investigated using for a single chain the Tomonaga-Luttinger model of interacting one-dimensional electrons. For interchain coupling without jumps between the chains it is proved exactly that no stable quasiparticles exist. The excitation spectrum is exhausted by density oscillations and the single-particle occupation number shows a power-law behavior. For finite-interchain hopping, lowest-order-perturbation theory yields a quasiparticle peak in the spectral density. Its weight is $z={[1\ensuremath{-}2\ensuremath{\alpha}\mathrm{ln}(\frac{{W}_{\ensuremath{\perp}}}{{W}_{\ensuremath{\parallel}}})]}^{\ensuremath{-}1}$, where $\ensuremath{\alpha}$ is the interaction parameter and ${W}_{\ensuremath{\parallel}}$ and ${W}_{\ensuremath{\perp}}$ are the widths of the conduction band in momentum direction parallel and perpendicular to the chains. The relevance of our results for real quasi-one-dimensional conductors will be discussed.
We provide a quasi-one-dimensional model which can support a pair-density-wave (PDW) state, in which the superconducting (SC) order parameter modulates periodically in space, with gapless Bogoliubov quasiparticle excitations. The model consists of an array of strongly interacting one-dimensional systems, where the one-dimensional systems are coupled to each other by local interactions and tunneling of the electrons and Cooper pairs between them. Within the interchain mean-field theory, we find several SC states from the model, including a conventional uniform SC state, PDW SC state, and a coexisting phase of the uniform SC and PDW states. In this quasi-1D regime we can treat the strong correlation physics essentially exactly using bosonization methods and the crossover to the 2D system by means of interchain mean-field theory. The resulting critical temperatures of the SC phases generically exhibit a power-law scaling with the coupling constants of the array, instead of the essential singularity found in weak-coupling BCS-type theories. Electronic excitations with an open Fermi surface, which emerge from the electronic Luttinger liquid systems below their crossover temperature to the Fermi liquid, are then coupled to the SC order parameters via the proximity effect. From the Fermi surface thus coupled to the SC order parameters, we calculate the quasiparticle spectrum in detail. We show that the quasiparticle spectrum can be fully gapped or nodal in the uniform SC phase and also in the coexisting phase of the uniform SC and PDW parameters. In the pure PDW state, the excitation spectrum has a reconstructed Fermi surface in the form of Fermi pockets of Bogoliubov quasiparticles.
This dissertation is devoted to theoretical studies of strongly interacting one-dimensional and quasi one-dimensional electron systems. The properties of one-dimensional electron systems can be studied within the bosonization technique, which presents fermions as collective bosonic density excitations. The power of this approach is the ability to treat electron-electron interaction exactly in the low-energy limit. The approach predicts the failure of Fermi liquid and an absence of long-range order in one-dimensions. The low-energy description of one-dimensional interacting systems is called the Tomonaga-Luttinger liquid theory. For example, the edges of quantum Hall systems are one-dimensional and described by a chiral Tomonaga-Luttinger liquid. Another example is a quantum spin Hall system with helical edge states, which are also described by a Tomonaga-Luttinger liquid. In our first work, a study of magnetized edge states of quantum spin-Hall system is presented. A magnetic field dependent signature of such edges is obtained, which can be verified in a Coulomb drag experiment. The second part of the dissertation is devoted to quasi-one dimensional antiferromagnetic lattices. A spatially anisotropic lattice antiferromagnet can be viewed as an array of one dimensional spin chains coupled in a way to match the lattice symmetry. This allows to use the non-Abelian bosonization technique to describe the low-energy physics of spin chains and study the inter-chain interactions perturbatively. The work presented in the dissertation studies the effect of Dzyaloshinskii-Moriya interaction on the magnetic phase diagram of the spatially anisotropic kagome antiferromagnet. We predict a Dzyaloshinskii-Moriya interaction driven phase transition from Neel to Neel+dimer state. In the third work, a novel model of the fractional quantum Hall effect is given. Wave functions of two-dimensional electrons in strong and quantizing magnetic field are essentially one-dimensional. That invites one to use the one-dimensional phenomenological bosonization to describe the density fluctuations of the two-dimensional interacting electrons in magnetic field. Remarkably, the constructed trial bosonized fermion operator describing the electron states with a fixed Landau gauge momentum is effectively two-dimensional.
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”.