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Physical Review Letters

Correlation exponents and the metal-insulator transition in the one-dimensional Hubbard model

Heinz J. Schulz

Published Jun 4, 1990• 652 citations
Publisher

Abstract

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(2${\mathit{k}}_{\mathit{F}}$x)${\mathit{x}}^{\mathrm{\ensuremath{-}}3/2}$ ${\mathrm{ln}}^{1/2}$(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.

DOI
10.1103/physrevlett.64.2831

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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)x\ensuremath−3/2{\mathit{x}}^{\mathrm{\ensuremath{-}}3/2}x\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.
A brief introduction to the bosonization method for interacting one-dimensional fermion systems is given. Using these results, the long-distance decay of correlation functions in the one-dimensional Hubbard model is determined exactly for arbitrary bandfilling and correlation strength, using the exact solution of Lieb and Wu. For infinite U the results are generalized to the case of nonzero nearest-neighbour interaction. The behaviour of thermodynamic quantities, of the frequency-dependent conductivity, and of the thermopower is also discussed, in particular in the proximity of the metal-insulator transitions occurring for half- and quarter-filling. The one-dimensional Luttinger liquid is shown to be unstable in the presence of interchain hopping. The results for the metal-insulator transition are compared with other scenarios developed in higher dimensions.
A brief introduction to the bosonization method for interacting one-dimensional fermion systems is given. Using these results, the long-distance decay of correlation functions in the one-dimensional Hubbard model is determined exactly for arbitrary bandfilling and correlation strength, using the exact solution of Lieb and Wu. For infinite U the results are generalized to the case of nonzero nearest-neighbour interaction. The behaviour of thermodynamic quantities, of the frequency-dependent conductivity, and of the thermopower is also discussed, in particular in the proximity of the metal-insulator transitions occurring for half- and quarter-filling. The one-dimensional Luttinger liquid is shown to be unstable in the presence of interchain hopping. The results for the metal-insulator transition are compared with other scenarios developed in higher dimensions.
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.
Correlation effects in the one-dimensional Hubbard model are studied by application of the L-resolvent method. The calculations are well beyond one-particle mean field approximation. All correlations involving two neighboring sites are included exactly and higher order correlations are approximated. The effects of varying temperature, band filling, Coulomb repulsion and external magnetic field are investigated. Two-site correlations for non-half-filled band fillings are calculated for the first time. Some exact relations between correlation functions are derived by electron-hole symmetry. Results indicate that short-range spin correlations always favor antiferromagnetic order. Large Coulomb repulsion induces a gap in the excitation spectrum and a metal-insulator transition is predicted as the Coulomb repulsion parameter is changed.
We examine the critical behavior near the metal-insulator transition (MIT) in the one-dimensional extended Hubbard model with the on-site and the nearest-neighbor interactions UUU and VVV at quarter filling using a combined method of the numerical diagonalization and the renormalization group (RG). The Luttinger-liquid parameter ${K}_{\ensuremath{\rho}}$ is calculated with the exact diagonalization for finite size systems and is substituted into the RG equation as an initial condition to obtain ${K}_{\ensuremath{\rho}}$ in the infinite size system. This approach also yields the charge gap $\ensuremath{\Delta}$ in the insulating state near the MIT. The results agree very well with the available exact results for $U=\ensuremath{\infty}$ even in the critical regime of the MIT where the characteristic energy becomes exponentially small and the usual finite size scaling is not applicable. When the system approaches the MIT critical point $V\ensuremath{\rightarrow}{V}_{c}$ for a fixed UUU, ${K}_{\ensuremath{\rho}}$ and $\ensuremath{\Delta}$ behave as ${\ensuremath{\mid}\mathrm{l}\mathrm{n}\ensuremath{\Delta}\ensuremath{\mid}}^{\ensuremath{-}2}={c}_{\ensuremath{\Delta}}(V∕{V}_{c}\ensuremath{-}1)$ and ${({K}_{\ensuremath{\rho}}\ensuremath{-}1∕4)}^{2}={c}_{K}(1\ensuremath{-}V∕{V}_{c})$, where the critical value Vc{V}_{c}Vc​ and the coefficients ${c}_{\ensuremath{\Delta}}$ and cK{c}_{K}cK​ are functions of UUU. These critical properties, which are known to be exact for $U=\ensuremath{\infty}$, are observed also for finite UUU case. We also observe the same critical behavior in the limit of the MIT critical point $U\ensuremath{\rightarrow}{U}_{c}$ when UUU is varied for a fixed VVV.
The critical exponents for the charge, spin, electron, and superconducting correlation functions of the one-dimensional t-J model at t=J are obtained for arbitrary band filling. Our method is based on the Bethe-ansatz solution and the finite-size scaling analysis in conformal field theory. The long-distance behavior of the model belongs to the same universality class as the repulsive Hubbard model. Near half-filling the exponents take the values expected in the strong-correlation limit of the Hubbard model. In the low-density limit the exponents are those of the noninteracting system.