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Journal of Physics A Mathematical and General

Finite size scaling approach to the 1D Hubbard model

Katarina Uzelac

Published Feb 1, 1984• 6 citations
Publisher

Abstract

Finite size scaling is applied to the one-dimensional Hubbard model with the half filled energy band at zero temperature. It is shown that, even for the scaling between small blocks of four, six and eight atoms, the essential singularity of the weak coupling limit is reproduced with remarkable accuracy, the error being less than 1% for the exponent and a few percent for the multiplicative constant. The results are discussed in comparison with the quantum renormalisation group approach, which fails to give the right exponent, and the advantages of the present method are pointed out.

DOI
10.1088/0305-4470/17/2/010

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Finite size scaling is applied to the one-dimensional Hubbard model with the half filled energy band at zero temperature. It is shown that, even for the scaling between small blocks of four, six and eight atoms, the essential singularity of the weak coupling limit is reproduced with remarkable accuracy, the error being less than 1% for the exponent and a few percent for the multiplicative constant. The results are discussed in comparison with the quantum renormalisation group approach, which fails to give the right exponent, and the advantages of the present method are pointed out.
The finite-cell-scaling method is applied to the one-dimensional Hubbard model. In the half-filled-band case and at zero temperature, the numerical results obtained for the ground-state energy differ from the exact solution by approximately 1%. For the range 1<Ut<51<\frac{U}{t}<51<tU​<5, where UUU is the Coulomb interaction and ttt is the transfer integral, the value of the normalized gap $\frac{\ensuremath{\Delta}}{t}$ agrees well with the exact result while it differs significantly for 0.2<Ut<0.90.2<\frac{U}{t}<0.90.2<tU​<0.9. For the latter range, we show that the gap varies as ${(\frac{U}{t})}^{\frac{1}{2}}\mathrm{exp}[{(a+\frac{\mathrm{bU}}{t})}^{\ensuremath{-}1}]$, where aaa and bbb are negative constants which depend on the size of the system, instead of ${(\frac{U}{t})}^{\frac{1}{2}}\mathrm{exp}(\frac{\ensuremath{-}2\ensuremath{\pi}t}{U})$ as in the exact solution.
The one-dimensional Hubbard model is studied using a real-space renormalisation-group block method previously applied to other quantum systems. The authors primarily consider the half-filled band for which the ground state is found to be insulating for all non-zero values of the Hubbard interaction, in agreement with the exact solution. When the method is applied to first order in perturbation, they find that the insulating gap for small U/t behaves like exp (-At 2 /U 2 ) where U is the Hubbard interaction and t is the nearest-neighbour transfer integral. When the method is extended to second order, the authors find that for small U/t the gap behaves like (U/t) alpha with alpha approximately=11.2. The exact asymptotic behaviour of the gap ( approximately (U/t) 1/2 exp(-At/U)) is not recovered, but a good semiquantative agreement with the exact solution is obtained for both the gap and the ground state energy. Extensions to non-half-filled bands and to higher dimensions are discussed. The validity of the renormalisation group method is also discussed.
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.
A recently proposed ground-state renormalisation-group approach to the Hubbard model is generalised to finite temperatures. This new method provides a global and consistent description of the thermodynamics of the model at all temperatures. Application to the Hubbard chain with a half-filled band gives values for various thermodynamic functions which compare remarkably well with existing exact or numerical data, over all temperature ranges. The difficulties posed by Fermi-Dirac statistics in the set-up of the transformation are discussed to some extent and it is shown that these do not prevent one, at least in principle, from extending the approach to higher dimensions.
Recent computer simulations performed on the one-dimensional Hubbard model with a new algorithm developed by Sorella et al. confirm the absence of a Fermi surface in this model at half filling. However, these same simulations suggest the existence of a Fermi surface at zero temperature away from half filling, contrary to expectations based on renormalization-group analysis. The results found here, using a more standard algorithm and a simple low-temperature extrapolation, appear consistent with the above-mentioned numerical results in the half-filled and quarter-filled band cases. However, it is argued that the discontinuity at the Fermi level found in all these simulations is likely to come from the zero-temperature finite-size dependence of the quasiparticle weight Z, which is also discussed here.