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Physical review. B, Condensed matter

Bariev model for correlated hopping with crystalline field splitting

Rui‐Hong Yue, P. Schlottmann

Published Jan 16, 2003• 0 citations
Publisher

Abstract

The Bethe Ansatz solution of the one-dimensional Bariev model for correlated hopping for two orbital and two spin degrees of freedom per site is used to study the evolution of the electron bands as a function of crystalline field splitting $\ensuremath{\Delta}.$ The formation of charge bound states (nonlocal Cooper pairs) leads to a charge sector with Fermi surface, while all other excitations are gapped in the absence of fields. The gap for the interband transitions is gradually closed with increasing $\ensuremath{\Delta}.$ Two critical fields ${\ensuremath{\Delta}}_{1}$ and ${\ensuremath{\Delta}}_{2}$ $({\ensuremath{\Delta}}_{1}<{\ensuremath{\Delta}}_{2})$ have to be distinguished. For two electrons per site the system is a one-component Luttinger liquid if $\ensuremath{\Delta}<{\ensuremath{\Delta}}_{1},$ a two-component Luttinger liquid for ${\ensuremath{\Delta}}_{1}<\ensuremath{\Delta}<{\ensuremath{\Delta}}_{2},$ and an insulator for $\ensuremath{\Delta}>{\ensuremath{\Delta}}_{2}.$ The critical behavior of superconducting fluctuations is also discussed.

DOI
10.1103/physrevb.67.035107

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The Bethe Ansatz solution of the one-dimensional Bariev model for correlated hopping for two orbital and two spin degrees of freedom per site is used to study the evolution of the electron bands as a function of crystalline field splitting \ensuremathΔ.\ensuremath{\Delta}.\ensuremathΔ. The formation of charge bound states (nonlocal Cooper pairs) leads to a charge sector with Fermi surface, while all other excitations are gapped in the absence of fields. The gap for the interband transitions is gradually closed with increasing \ensuremathΔ.\ensuremath{\Delta}.\ensuremathΔ. Two critical fields \ensuremathΔ1{\ensuremath{\Delta}}_{1}\ensuremathΔ1​ and \ensuremathΔ2{\ensuremath{\Delta}}_{2}\ensuremathΔ2​ (\ensuremathΔ1<\ensuremathΔ2)({\ensuremath{\Delta}}_{1}<{\ensuremath{\Delta}}_{2})(\ensuremathΔ1​<\ensuremathΔ2​) have to be distinguished. For two electrons per site the system is a one-component Luttinger liquid if \ensuremathΔ<\ensuremathΔ1,\ensuremath{\Delta}<{\ensuremath{\Delta}}_{1},\ensuremathΔ<\ensuremathΔ1​, a two-component Luttinger liquid for \ensuremathΔ1<\ensuremathΔ<\ensuremathΔ2,{\ensuremath{\Delta}}_{1}<\ensuremath{\Delta}<{\ensuremath{\Delta}}_{2},\ensuremathΔ1​<\ensuremathΔ<\ensuremathΔ2​, and an insulator for \ensuremathΔ>\ensuremathΔ2.\ensuremath{\Delta}>{\ensuremath{\Delta}}_{2}.\ensuremathΔ>\ensuremathΔ2​. The critical behavior of superconducting fluctuations is also discussed.
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 demonstrate that for general spin-charge separated Luttinger liquids there exists a critical value of the inter-Luttinger-liquid single particle hopping, ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}^{\mathit{c}}$, below which there is no coherent single particle hopping between the liquids. In the absence of coherent single particle hopping, two Luttinger liquids coupled by ${\mathit{t}}_{\mathrm{\ensuremath{\perp}}}$ will not exhibit split Fermi surfaces. For many Luttinger liquids, no band dipsersing between the liquids will form, and thus the system will retain a one-dimensional Fermi surface. This will have dramatic implications for the physical properties of such a system.
One-dimensional conductors are a long-standing topic of research with direct applications to organic conductors and mesoscopic rings. The discovery of the ceramic high-temperature superconductors has revitalized the interest in low-dimensional charge and spin fluctuations of highly correlated electron systems. Several mechanisms proposed to explain the high-T c superconductors invoke properties of the two-dimensional Hubbard model, but probably also some one-dimensional aspects are relevant. Numerous one-dimensional models for correlated electrons have been studied with various approximate, asymptotically exact and exact methods. These results lead to the concept of Luttinger liquid for interacting electron gases without excitation gaps (metallic systems). Characteristic of Luttinger liquids are the charge and spin separation, marginal Fermi liquid properties, e.g. the absence of quasiparticles in the vicinity of the Fermi surface, nonuniversal power-law singularities in the one-particle spectral function and the related absence of a discontinuity in the momentum distribution at the Fermi level, the power-law decay of correlation functions for long times and large distances, persistent currents in finite rings, etc. Due to the peculiarities of the phase space in one dimension some of the models have sufficient conserved currents to be completely integrable. We review exact results derived within the framework of Bethe's ansatz for integrable one-dimensional models of correlated electrons. The Bethe-ansatz method is presented by explicitly showing the steps leading to the solution of the N-component electron gas interacting via a δ-function potential (repulsive and attractive interaction), which is probably the simplest model of correlated electrons. Emphasis is given to the procedure to extract the groundstate properties, the classification of states, the excitation spectrum, the thermodynamics and finite size effects, such as critical exponents of correlation functions and persistent currents. The method is then applied to numerous other models, e.g. (i) a two-band model involving attractive and repulsive potentials and crystalline fields splitting the bands, (ii) the traditional Hubbard chain with attractive and repulsive U, (iii) the degenerate Hubbard model with repulsive U, which displays a metal–insulator transition at a finite U, (iv) a two-band Hubbard model with repulsive U, (v) the traditional supersymmetric t–J model (vi) a two-band supersymmetric t–J model with band-splitting and (vii) the N-component supersymmetric t–J model. Finally, results for models with long-range interactions, in particular r-2 and sinh -2(r) potentials, are briefly reviewed.
The ground-state phase diagram of the one-dimensional Hubbard chain with pair-hopping interaction is studied. The analysis of the model is performed using the continuum-limit field theory approach and exact diagonalization studies. At half-filling the phase diagram is shown to consist of two superconducting states with Cooper-pair center-of-mass momentum $Q=0 (\mathrm{BCS}\ensuremath{-}{\ensuremath{\eta}}_{0}$ phase) and $Q=\ensuremath{\pi} ({\ensuremath{\eta}}_{\ensuremath{\pi}}$ phase) and four insulating phases corresponding to the Mott antiferromagnet, the Peierls dimerized phase, the charge-density-wave (CDW) insulator, and an unconventional insulating phase characterized by the coexistence of a CDW and a bond-located staggered magnetization. Away from half-filling the phase diagram consists of the superconducting $\mathrm{BCS}\ensuremath{-}{\ensuremath{\eta}}_{0}$ and ${\ensuremath{\eta}}_{\ensuremath{\pi}}$ phases and the metallic Luttinger-liquid phase. The $\mathrm{BCS}\ensuremath{-}{\ensuremath{\eta}}_{0}$ phase exhibits a smooth crossover from a weak-coupling BCS type to a strong-coupling local-pair regime. The ${\ensuremath{\eta}}_{\ensuremath{\pi}}$ phase shows the properties of the doublon (zero-size Cooper-pair) superconductor with Cooper-pair center-of-mass momentum $Q=\ensuremath{\pi}.$ The transition into the ${\ensuremath{\eta}}_{\ensuremath{\pi}}$-paired state corresponds to an abrupt change in the ground-state structure. After the transition the conduction band is completely destroyed and a new ${\ensuremath{\eta}}_{\ensuremath{\pi}}$-pair band, corresponding to the strongly correlated doublon motion, is created.
A model of two interacting one-dimensional fermion systems (``Luttinger liquids'') coupled by single-particle hopping is investigated. Bosonization allows a number of exact statements to be made. In particular, for forward scattering only, the model contains two massless boson sectors and an Ising-type critical sector. For general repulsive interactions, there is a spin excitation gap and either d-type pairing or orbital antiferromagnetic fluctuations dominate. It is argued that the same behavior is also found for strong interactions. A possible scenario for the crossover to a Fermi liquid in a many-chain system is discussed. \textcopyright{} 1996 The American Physical Society.