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Physical review. B, Condensed matter

Application of the finite-cell-scaling method to the one-dimensional Hubbard model

B. Fourcade, G. Spronken

Published May 1, 1984• 13 citations
Publisher

Abstract

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 $1<\frac{U}{t}<5$, where $U$ is the Coulomb interaction and $t$ 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 $0.2<\frac{U}{t}<0.9$. 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 $a$ and $b$ 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.

DOI
10.1103/physrevb.29.5012

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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 \ensuremathΔt\frac{\ensuremath{\Delta}}{t}t\ensuremathΔ​ 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 (Ut)12exp[(a+bUt)\ensuremath−1]{(\frac{U}{t})}^{\frac{1}{2}}\mathrm{exp}[{(a+\frac{\mathrm{bU}}{t})}^{\ensuremath{-}1}](tU​)21​exp[(a+tbU​)\ensuremath−1], where aaa and bbb are negative constants which depend on the size of the system, instead of (Ut)12exp(\ensuremath−2\ensuremathπtU){(\frac{U}{t})}^{\frac{1}{2}}\mathrm{exp}(\frac{\ensuremath{-}2\ensuremath{\pi}t}{U})(tU​)21​exp(U\ensuremath−2\ensuremathπt​) 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.
The ground state of the half-filled Hubbard model is studied using a real-space renormalization-group technique. We first study in detail the one-dimensional case. We find the ground state to be insulating for all finite coupling, in agreement with the exact solution. We compute the ground-state energy, localization length, energy gap, magnitude of the local moment, and spin-density autocorrelation function. For those quantities that are exactly known we find good agreement with the exact results. Using a simple extension of our one-dimensional calculation, we are able to study approximately two- and three-dimensional lattices. We find a Mott transition at finite interaction for these cases. The critical exponents for these transitions are found to satisfy an approximate interdimensional scaling law.
Exact results for the one-hole excitation spectra of the one-dimensional half-filled Hubbard Hamiltonian are presented. We consider short chains of N=2, 4, 6 atoms and introduce modified periodic boundary conditions which allow us to make a comparison between different N and to extrapolate the results up to infinite chains. By the increasing weight of the scattering states in the one-hole spectrum, we demonstrate a gradual disappearance of the free-electron (band) picture with increasing local electron-electron interaction U. The exact spectra of the N=6 chain verify the results of second-order perturbation theory for U/W1, where W is the bandwidth. The extrapolation to the infinite system, made with the finite-cell-scaling method, gives the gap which agrees rather well with the exact result for U/W>0.2. The average energies and weights of the band and scattering states obtained with this method for an infinite chain do not differ significantly from those of the N=6 chain.
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 Hubbard model is investigated in the framework of lattice density functional theory (LDFT). The single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ with respect to the lattice sites is considered as the basic variable of the many-body problem. A new approximation to the interaction-energy functional $W[\ensuremath{\gamma}]$ is proposed which is based on its scaling properties and which recovers exactly the limit of strong electron correlations at half-band filling. In this way, a more accurate description of W is obtained throughout the domain of representability of ${\ensuremath{\gamma}}_{\mathrm{ij}},$ including the crossover from weak to strong correlations. As examples of applications results are given for the ground-state energy, charge-excitation gap, and charge susceptibility of the Hubbard model in one-, two-, and three-dimensional lattices. The performance of the method is demonstrated by comparison with available exact solutions, with numerical calculations, and with LDFT using a simpler dimer ansatz for W. Goals and limitations of the different approximations are discussed.