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arXiv:cond-mat/9908296 · Physical review. B, Condensed matter

Correlation functions for a two-dimensional electron system with bosonic interactions and a square Fermi surface

John Ove Fjærestad, Asle Sudbø, Alan H. Luther

Published Nov 15, 1999• 14 citations
arXivHTMLPDF

Abstract

We calculate zero-temperature correlation functions for a model of two-dimensional interacting electrons with short-range interactions and a square Fermi surface. The model was arrived at by mapping electronic states near a square Fermi surface with Hubbard-like interactions onto one-dimensional quantum chains, retaining terms that can be written in terms of bosonic density operators. Interactions between orthogonal chains, corresponding to orthogonal faces of the square Fermi surface, are neglected. The correlation functions become sums of Luttinger-type correlation functions due to the bosonic model. However, the correlation function exponents differ in form from those of the Luttinger model. As a consequence, the simple scaling relations found to exist between the Luttinger model exponents, do not carry over to the leading exponents of our model. We find that for repulsive effective interactions, charge-density wave/spin-density wave instabilities are dominant. We do not consider d-wave instabilities here.

DOI
10.1103/physrevb.60.13361

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We calculate zero-temperature correlation functions for a model of two-dimensional interacting electrons with short-range interactions and a square Fermi surface. The model was arrived at by mapping electronic states near a square Fermi surface with Hubbard-like interactions onto one-dimensional quantum chains, retaining terms that can be written in terms of bosonic density operators. Interactions between orthogonal chains, corresponding to orthogonal faces of the square Fermi surface, are neglected. The correlation functions become sums of Luttinger-type correlation functions due to the bosonic model. However, the correlation function exponents differ in form from those of the Luttinger model. As a consequence, the simple scaling relations found to exist between the Luttinger model exponents, do not carry over to the leading exponents of our model. We find that for repulsive effective interactions, charge-density wave/spin-density wave instabilities are dominant. We do not consider d-wave instabilities here.
Electronic states near a square Fermi surface are mapped onto quantum chains. Using boson-fermion duality on the chains, the bosonic part of the interaction is isolated and diagonalized. These interactions destroy Fermi-liquid behavior. Nonboson interactions are also generated by this mapping, and give rise to an alternate perturbation theory about the boson problem. A case with strong repulsions between parallel faces is studied and solved. This solution discards irrelevant operators in a new interaction Hamiltonian. There is spin-charge separation and the square Fermi surface remains square under doping. At half-filling, there is a charge gap and insulating behavior together with gapless spin excitations. This mapping appears to be a general tool for understanding the properties of interacting electrons on a square Fermi surface.
Experiments on quasi-one-dimensional systems such as quantum wires and metallic chains on surfaces suggest the existence of electron-electron interactions of substantial range and hence physics beyond the Hubbard model. We therefore investigate one-dimensional, quarter-filled chains with a Coulomb potential with variable screening length by quantum Monte Carlo methods and exact diagonalization. The Luttinger liquid interaction parameter ${K}_{\ensuremath{\rho}}$ decreases with increasing interaction strength and range. Experimentally observed values close to 1/4 require strong interactions and/or large screening lengths. As predicted by bosonization, we find a metal-insulator transition at ${K}_{\ensuremath{\rho}}=1/4$. Upon increasing the screening length, the charge and spin correlation functions reveal the crossover from dominant 2kF2{k}_{\mathrm{F}}2kF​ spin correlations to dominant 4kF4{k}_{\mathrm{F}}4kF​ charge correlations, and a strong enhancement of the charge velocity. In the metallic phase, the signatures of spin-charge separation in the single-particle spectrum, spinon and holon bands, remain robust even for rather long-ranged interactions. The charge-density-wave state exhibits backfolded shadow bands.
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 dynamic density response function, structure factor, and spectral function of a Luttinger liquid with Coulomb interaction are studied with the emphasis on the short-range electron correlations. The Coulomb interaction changes dramatically the density response function as compared to the case of the short-ranged interaction. The coordinate dependence of the density response is smoothing with time, and the oscillatory structure appears. However, the spectral functions remain qualitatively the same. The dynamic structure factor contains the $\ensuremath{\delta}$ peak in the long-wave region, corresponding to one-boson excitations. In addition, the multi-boson-excitations band exists in the wave-number region near to 2kF,{2k}_{F},2kF​, where kF{k}_{F}kF​ is the Fermi wave number. We develop a method to analyze the asymptotic behavior of the spectral functions near to the edges of the multi-boson-excitations band at zero temperature. The dynamic structure factor diverges at the edges of this band, while the dielectric function goes to zero there. Hence, a new collective charge mode appears in the one-dimensional electron liquid as a consequence of short-range dynamic correlations. The mode energy goes to zero at 2kF,{2k}_{F},2kF​, which means that the 2kF{2k}_{F}2kF​ mode is soft.
By removing one empty site between two occupied sites, we map the ground states of chains of hardcore bosons and spinless fermions with infinite nearest-neighbor repulsion to ground states of chains of hardcore bosons and spinless fermions without nearest-neighbor repulsion, respectively, and ultimately in terms of the one-dimensional Fermi sea. We then introduce the intervening-particle expansion, where we write correlation functions in such ground states as a systematic sum over conditional expectations, each of which can be ultimately mapped to a one-dimensional Fermi-sea expectation. Various ground-state correlation functions are calculated for the bosonic and fermionic chains with infinite nearest-neighbor repulsion, as well as for a ladder model of spinless fermions with infinite nearest-neighbor repulsion and correlated hopping in three limiting cases. We find that the decays of these correlation functions are governed by surprising power-law exponents.
We summarize results on the asymptotics of the two-particle Green functions of interacting electrons in one dimension.Below a critical value of the chemical potential the Fermi surface vanishes, and the system can no longer be described as a Luttinger liquid.Instead, the non-relativistic Fermi gas with infinite point-like repulsion becomes the universal model for the long-wavelength, low temperature physics of the one-dimensional electrons.This model, which we call the impenetrable electron gas, allows for a rigorous mathematical treatment.In particular, a so-called determinant representation for the two particle Green function could be derived.This determinant representation is related to an integrable classical evolution equation and to a Riemann-Hilbert problem, that enable the exact calculation of the asymptotics of the two-particle Green functions.