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International Journal of Modern Physics B

STRONG COUPLING APPROACH TO ONE-DIMENSIONAL HUBBARD MODEL AND RELATED LATTICE MODELS

Hiroyuki Shiba, Karlo Penc, Frédéric Mila

Published Mar 10, 1999• 1 citation
Publisher

Abstract

The strong coupling limit of one-dimensional correlated systems is nontrivial, but simple in some sense. Here we demonstrate it, discussing the wave functions for the 1D U→ +∞ Hubbard model, which are factorized as a direct product of the charge part and the spin part and determining the one-particle spectral functions explicitly.
DOI
10.1142/s021797929900062x

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We give the details of the calculation of the spectral functions of the one-dimensional Hubbard model using the spin-charge factorized wave function for several versions of the U\ensuremath{\rightarrow}+\ensuremath{\infty} limit. The spectral functions are expressed as a convolution of charge and spin dynamical correlation functions. A procedure to evaluate these correlation functions very accurately for large systems is developed, and analytical results are presented for the low-energy region. These results are fully consistent with the conformal field theory. We also propose a direct method of extracting the exponents from the matrix elements in more general cases.
A recent study of the large-U limit of the one-dimensional (1D) Hubbard model is presented and discussed. It is pointed out first that the wave function has a simple structure in this limit. Namely, it is a product of Slater determinant of noninteracting spinless fermions and the wave function of 1D S=1/2 Heisenberg antiferromagnet. Secondly, by using this property, a direct calculation of momentum distribution and spin correlation function is carried out for the ground state at zero-field and finite-field cases. The results show various singularities exactly at the same position where one expects for the small-U case in both zero-field and finite-field cases. The critical exponents estimated from the size dependence are in reasonable agreement with those predicted by the Tomonaga-Luttinger-liquid and the conformal field theory.
Employing the equation of motion method for the Green's function, we have obtained the self-energy for the Hubbard model in the weak correlation limit up to the 2nd order in U, the on-site interaction strength. The quasi-particle spectral density of states is determined numerically, using the local approximation which is known to be exact for the infinite dimensional Hubbard model. Further, assuming a linear dispersion for the bare electronic spectra for the 1D Hubbard model, we have evaluated analytically the imaginary part of the self-energy, Im Sigma, near the Fermi energy (E(F)) and the renormalization coefficient at E(F), Z(F). It is found that the 1D Hubbard model system exhibits a non-Fermi liquid behavior within the 2nd order perturbational treatment.
Exact relations are derived for the Fermi Hubbard spectral weight function for infinite U at zero temperature in the thermodynamic limit for any dimension,any lattice structure and general hopping matrix. These relations involve moments of the spectral weight function but differ from similar work by (a) restricting the moments over the interesting low energy (lower Hubbard band) spectrum and (b) without any of the usual approximations (e.g. decoupling) for the requisite higher order correlation functions. The latter are determined exactly as explicit functions of the single particle correlation function in the infinite U limit. The methods behind these results have the additional virtue of simplicity - cumbersome, but entirely straightforward. In a companion paper, the relations offer rich insights into the model behavior and offer severe tests - a computer lab - of any proposed solution to the Hubbard model in the strong correlation (large finite U) region. As the electron density approaches one electron per site, then the correlation functions become local, so are trivial to compute exactly. In that limit and for the paramagnetic case, the second central moment of the spectral weight function is not zero, but quite large.
In the limit of dominant nearest-neighbour hopping and infinitesimal secondand third-nearest-neighbour hopping, the Hubbard model is solvable in the sense that it separates into two models which have previously been analysed in the literature. The solutions are spin-charge separated and involve highly correlated motion of the electrons. Although in general the solutions are too complicated to analyse, there are two quite specific states which are amenable to direct analytical attack: the phase boundary between the quantum paramagnet and the Nagaoka ferromagnet, and the 'dimer' phase where all spins pair up into nearest-neighbour singlets. We have been able to evaluate both (n(k)) and (S -k .S k ) accurately enough to deduce the qualitative behaviour for the two types of state in question. The second of these correlation functions is directly relevant to neutron-scattering studies. Although the solutions are highly correlated and spin-charge separated, the correlations are reminiscent of the non-interacting 'spinfull' free-electron gas solution, the spin response exhibiting a divergence where the noninteracting susceptibility does, for example.
We show that the one-particle spectral functions of the one-dimensional Hubbard model diverge at the Fermi energy like $|\ensuremath{\omega}\ensuremath{-}{\ensuremath{\varepsilon}}_{F}{|}^{\ensuremath{-}3/8}$ in the $U\ensuremath{\rightarrow}+\ensuremath{\infty}$ limit. The Luttinger liquid behavior $|\ensuremath{\omega}\ensuremath{-}{\ensuremath{\varepsilon}}_{F}{|}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}\ensuremath{\rightarrow}1/8$ as $U\ensuremath{\rightarrow}+\ensuremath{\infty}$, should be limited to $|\ensuremath{\omega}\ensuremath{-}{\ensuremath{\varepsilon}}_{F}|\ensuremath{\sim}{t}^{2}/U$ (for UUU large but finite), which shrinks to a single point, $\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\varepsilon}}_{F}$, in that limit. The consequences for the observation of the Luttinger liquid behavior in photoemission and inverse photoemission experiments are discussed.