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arXiv:cond-mat/9709146 · Physical review. B, Condensed matter

Theory of hole propagation in one-dimensional insulators and superconductors

Sandro Sorella, Alberto Parola

Published Mar 15, 1998• 40 citations
arXivHTMLPDF

Abstract

The dynamical properties of hole motion in an antiferromagnetic background are determined in one-dimensional models in zero magnetic field, where spin isotropy holds, as well as in an external magnetic field. The latter case is also relevant, via particle-hole transformation, to the problem of hole propagation in one-dimensional ``superconductors.'' The singularities in the spectral function are investigated by means of bosonization techniques and perturbation theories. Results are then compared with Bethe ansatz solutions and Lanczos diagonalizations. The formalism also leads to interesting connections to the single impurity problem in Luttinger liquids. A rich structure is found in the spectral function whenever spin isotropy is broken, suggesting the presence of exotic momentum dependence in photoemission spectra of (quasi) one dimensional materials.

DOI
10.1103/physrevb.57.6444

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The dynamical properties of hole motion in an antiferromagnetic background are determined in one-dimensional models in zero magnetic field, where spin isotropy holds, as well as in an external magnetic field. The latter case is also relevant, via particle-hole transformation, to the problem of hole propagation in one-dimensional ``superconductors.'' The singularities in the spectral function are investigated by means of bosonization techniques and perturbation theories. Results are then compared with Bethe ansatz solutions and Lanczos diagonalizations. The formalism also leads to interesting connections to the single impurity problem in Luttinger liquids. A rich structure is found in the spectral function whenever spin isotropy is broken, suggesting the presence of exotic momentum dependence in photoemission spectra of (quasi) one dimensional materials.
Conformal field theory and Bethe ansatz are used to investigate the low energy features of the spectral function in one dimensional models which exhibit a gap in the spin or in the charge excitation spectrum. Exotic behavior is found in the superconducting case, where the Green function displays momentum dependent Tomonaga-Luttinger liquid exponents. The predictions of the formalism are confirmed by Lanczos diagonalizations in the $t\ensuremath{-}J$ model up to 32 sites. These results may be relevant in connection to photoemission experiments in quasi-one-dimensional insulators or superconductors.
A formal theory of one-electron states in insulators and semiconductors is developed from a many-particle point of view. The techniques of second quantization are utilized for this purpose in a manner analogous to that introduced recently for the study of Fermi liquids, i.e., by the study of matrix elements of the electron field operator which describe the propagation of particles or holes. Both types of motion are described symmetrically by means of the one-particle Green's function or propagator. The utility of these constructs for the present study derives from the existence of a gap against single-particle excitation.The basic result of this paper is that the motion of electrons near the bottom of the conduction band in the presence of external electric and magnetic fields whose spatial variation over one lattice spacing is small and which contain no frequencies comparable with the gap frequency is governed by a simple Schr\"odinger equation. The latter contains as parameters only the effective mass (as measured in a cyclotron resonance experiment) and the static dielectric constant and magnetic permeability of the solid. Our proof, within the restriction of a fixed, perfect lattice, takes into account all many-body effects. A similar theorem obtains for the motion of holes.
Bariev's model for correlated hopping in one dimension is integrable and soluble within the framework of two nested Bethe Ansätze. For repulsive interaction and finite magnetization the low energy excitation spectrum is a two-component Luttinger liquid with conformal charges c=1. In zero magnetic field the spectrum of spin excitations is gapped. For higher energy excitations the single-hole (single-particle) spectral functions display edge singularities that deviate from those expected from a Luttinger liquid. The system can be described as a mobile impurity coupled to the Luttinger liquid yielding an effective x-ray edge type model. Using the Bethe Ansatz solution for repulsive interaction we obtain expressions for the critical exponents for the single-particle and single-hole Green's functions in zero magnetic field.
We explore with exact diagonalization the propagation of a single hole in four magnetic phases of the $t\text{\ensuremath{-}}J$-like Kitaev-Heisenberg model on a honeycomb lattice: the N\'eel antiferromagnetic, stripe, zigzag, and Kitaev spin-liquid phases. We find coherent propagation of spin-polaron quasiparticles in the antiferromagnetic phase by a mechanism similar to that in the $t\text{\ensuremath{-}}J$ model for high-Tc{T}_{c}Tc​ cuprates. In the stripe and zigzag phases clear quasiparticles features appear in spectral functions of those propagators where holes are created and annihilated on one sublattice, while they remain largely hidden in those spectral functions that correspond to photoemission experiments. As the most surprising result, we find a totally incoherent spectral weight distribution for the spectral function of a hole moving in the Kitaev spin-liquid phase in the strong-coupling regime $t\ensuremath{\gg}J$ relevant for iridates. At intermediate coupling the finite systems calculation reveals a well-defined quasiparticle at the $\ensuremath{\Gamma}$ point; however, we find that the gapless spin excitations wipe out quasiparticles at finite momenta. Also for this more subtle case we conclude that in the thermodynamic limit the lightly doped Kitaev liquid phase does not support quasiparticle states in the neighborhood of $\ensuremath{\Gamma}$, and therefore yields a non-Fermi liquid, contrary to earlier suggestions based on slave-boson studies. These observations are supported by the presented study of the dynamic spin-structure factor in the Kitaev spin liquid regime.
We compute the spectral function of a spin S hole injected into a two-dimensional antiferromagnet or superconductor in the vicinity of a magnetic quantum critical point. We show that, near Van Hove singularities, the problem maps onto that of a static vacancy carrying excess spin S. The hole creation operator is characterized by a new boundary anomalous dimension and a vanishing quasiparticle residue at the critical point. We discuss possible relevance to photoemission spectra of cuprate superconductors near the antinodal points.
An exact expression for the Green's function G(k,\ensuremath{\omega}) of one hole in the U\ensuremath{\rightarrow}\ensuremath{\infty} one-dimensional Hubbard model is derived from the known Bethe-ansatz solution of the model. Due to the spin-charge decoupling of the one-hole excitation spectrum and the Fermi-like statistics of the spin excitations, G shows branch-cut singularities at \ensuremath{\omega}=\ifmmode\pm\else\textpm\fi{}2 sink and a nontrivial dependence on the momentum of the hole. As a result, the hole can propagate although the quasiparticle weight vanishes.