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Physical Review Letters

Asymptotic spin-spin correlations of theU→∞ one-dimensional Hubbard model

Alberto Parola, Sandro Sorella

Published Apr 9, 1990• 117 citations
Publisher

Abstract

The large-distance behavior of the spin-spin correlation function of the one-dimensional repulsive Hubbard model is evaluated analytically in the strong-coupling regime at quarter filling. In this case, its power-law decay is characterized by an exponent \ensuremath{\gamma}=3/2. We have found that this behavior is generally valid at any nonzero doping, although our argument is not mathematically rigorous away from quarter filling. These results strongly suggest that the renormalization-group scaling to the Tomonaga-Luttinger model is exact in the U\ensuremath{\rightarrow}\ensuremath{\infty} Hubbard model.

DOI
10.1103/physrevlett.64.1831

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The large-distance behavior of the spin-spin correlation function of the one-dimensional repulsive Hubbard model is evaluated analytically in the strong-coupling regime at quarter filling. In this case, its power-law decay is characterized by an exponent \ensuremath{\gamma}=3/2. We have found that this behavior is generally valid at any nonzero doping, although our argument is not mathematically rigorous away from quarter filling. These results strongly suggest that the renormalization-group scaling to the Tomonaga-Luttinger model is exact in the U\ensuremath{\rightarrow}\ensuremath{\infty} Hubbard model.
The long-distance decay of correlation functions in the one-dimensional Hubbard model is determined for arbitrary band filling and correlation strength, using the exact solution of Lieb and Wu. In particular, for either infinitely strong on-site repulsion U, or in the close proximity of half filling for any U, spin-spin correlations decay like cos(2kF{\mathit{k}}_{\mathit{F}}kF​x)${\mathit{x}}^{\mathrm{\ensuremath{-}}3/2}$ ln1/2{\mathrm{ln}}^{1/2}ln1/2(x). For infinite U the results are generalized to the case of nonzero nearest-neighbor interaction. The behavior of the frequency-dependent conductivity is also discussed, in particular in the proximity of the metal-insulator transitions occurring for half and quarter filling.
We examine the density-density correlation function in the Tomonaga-Luttinger liquid state for the one-dimensional extended Hubbard model with the on-site Coulomb repulsion UUU and the intersite repulsion VVV at quarter filling. By taking into account the effect of the marginally irrelevant umklapp scattering operator based on the bosonization and renormalization-group methods, we obtain the generalized analytical form of the correlation function. We show that, in the proximity to the gapped charge-ordered phase, the correlation function exhibits anomalous crossover between the pure power-law behavior and the power-law behavior with logarithmic corrections, depending on the length scale. Such a crossover is also confirmed by the highly accurate numerical density-matrix renormalization-group method.
We have studied correlation functions of the one-dimensional Hubbard model using the density-matrix numerical renormalization-group approach. The singularity exponents of the momentum distribution at k=kF{\mathit{k}}_{\mathit{F}}kF​ and 3kF{\mathit{k}}_{\mathit{F}}kF​, as well as the power-law singularity of the spin-correlation function, are computed for large U (U/t=103{10}^{3}103). The momentum-distribution exponent at k=kF{\mathit{k}}_{\mathit{F}}kF​ and the spin-correlation exponent are in agreement with the analytic results, while the calculated exponent for momentum distribution at k=3kF{\mathit{k}}_{\mathit{F}}kF​ is about 3/4, which disagrees with the analytically predicted value 9/8. We also discuss several techniques for an accurate calculation of the correlation functions.
The one-dimensional Hubbard model in a magnetic field is equivalent under renormalization-group transformation to a multicomponent Tomonaga-Luttinger model. The mapping between the Coulomb repulsion U of the Hubbard model and the couplings of the Tomonaga-Luttinger model is obtained for aribitrary magnetic field h and band filling n from a comparison of the correlation function exponents. These quantities are calculated for the Hubbard model from the finite-size corrections to the excitation energies, since the assumption of conformal invariance relates them to the critical exponents. On the other hand, the correlation functions of the multicomponent Tomonaga-Luttinger model are determined by solving the exact equation of motion derived by the use of generalized Ward identities.
We study the one-dimensional Anderson-Hubbard model using the density-matrix renormalization-group method. The influence of disorder on the Tomonaga- Luttinger liquid behavior is quantitatively discussed. Based on the finite-size scaling analysis of density-density correlation functions, we find the following results: (i) the charge exponent is significantly reduced by disorder at low filling and near half filling, (ii) the localization length decays as $\ensuremath{\xi}\ensuremath{\sim}{\ensuremath{\Delta}}^{\ensuremath{-}2}$, where $\ensuremath{\Delta}$ is the disorder strength, independently of the on-site Coulomb interaction as well as band filling, and (iii) the localization length is strongly suppressed by the on-site Coulomb interaction near half filling in association with the formation of the Mott plateaus.