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Europhysics Letters (EPL)

Non–Fermi-Liquid Exponents of the One-Dimensional Hubbard Model

Sandro Sorella, Alberto Parola, Michele Parrinello, Erio Tosatti

Published Aug 15, 1990• 70 citations
Publisher

Abstract

We have applied newly numerical schemes to study the ground-state properties of the 1D Hubbard model. Through finite-size scaling, we determine the singularity of the momentum distribution at k F and the power law decay of the spin and charge correlations in real space. Our results confirm the non-Fermi-liquid nature of the system. The model is shown to scale to the Tomonaga-Luttinger model even in the large- U regime where perturbative results are invalid.

DOI
10.1209/0295-5075/12/8/009

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Europhysics Letters (EPL)•Jun 15, 1990•15 citations
We have applied newly numerical schemes to study the ground-state properties of the 1D Hubbard model. Through finite-size scaling, we determine the singularity of the momentum distribution at k F and the power law decay of the spin and charge correlations in real space. Our results confirm the non-Fermi-liquid nature of the system. The model is shown to scale to the Tomonaga-Luttinger model even in the large- U regime where perturbative results are invalid.
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.
In contrast to normal Fermi liquids, where the momentum distribution n(k) has a discontinuous jump at k=kF{\mathit{k}}_{\mathit{F}}kF​, in Luttinger liquids a power-law singularity appears in n(k). It was argued that this singularity shows up not only at kF{\mathit{k}}_{\mathit{F}}kF​ but at 3kF{\mathit{k}}_{\mathit{F}}kF​,5kF{\mathit{k}}_{\mathit{F}}kF​, . . . as well, and the anomaly exponent has been detemined for the one-dimensional Hubbard model using numerical methods or assuming conformal invariance. We present an exact calculation of the anomaly exponent for the Tomonaga-Luttinger model and compare it with the results for the Hubbard model.
We perform a rigorous computation of the spin and charge density correlations of the 1d repulsive Hubbard model at weak coupling, focusing on the properties of the Fourier transform at momentum 0 and ±2 p F , if p F is the Fermi momentum. We prove that the interaction changes the singularity at ±2 p F (the discontinuity in the derivative becomes a power law singularity) while the singularity at 0 is essentially unchanged. Our results show that the logarithmic divergences at zero momentum recently found in [7], which would be in contrast with Luttinger liquid behaviour, are indeed spurious.
The electronic system of the 1D Hubbard model is not stable due to Peierls instability; the correlations are strong even for the weak Coulomb interaction. The resulting strongly correlated state without Landau quisi--particle excitations is known as the Luttinger liquid. Critical exponents of a power--law dependence of correlation functions at low energies differ substantially for the Luttinger and Fermi liquids. In this paper we evaluate two critical exponents that define non--trivial behavior of the density of electronic states at low frequences and the momentum distribution of the occupation number nearby kFk_FkF​.
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.
We calculate the behaviour of the momentum distribution function n ( k ) for the ground state of the one-dimensional Hubbard model with arbitrary band filling in the weak-coupling regime. For the incommensurate case and U > 0 we derive the variation of n ( k ) in the vicinity of the Fermi wave vector k F and find a power law behaviour with a maximum exponent of one instead of a discontinuous jump. We also make a comparison with the n ( k ) obtained from recent Monte Carlo simulations and conclude that these numerical results are still seriously affected by finite-size effects. We finally consider the half-filled band and find a linear variation of n ( k ) close to k F , the scale being set by the charge gap of the Hubbard model.