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Journal of Physics C Solid State Physics

Hubbard chain with reflecting ends

Hermann Schulz

Published Jan 30, 1985• 108 citations
Publisher

Abstract

The one-dimensional Hubbard model bounded by infinitely high potential walls is exactly diagonalised by Bethe-Yang ansatz techniques. The 'surface' energy is studied for the half-filled band and in the Fermi gas limit. The analysis starts with a solution of a more general model in which the particles also interact with their mirror-image sites. In the linear band limit the states of this system become the Bethe eigenstates of the Kondo model at zero coupling.

DOI
10.1088/0022-3719/18/3/010

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The one-dimensional Hubbard model bounded by infinitely high potential walls is exactly diagonalised by Bethe-Yang ansatz techniques. The 'surface' energy is studied for the half-filled band and in the Fermi gas limit. The analysis starts with a solution of a more general model in which the particles also interact with their mirror-image sites. In the linear band limit the states of this system become the Bethe eigenstates of the Kondo model at zero coupling.
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The one-dimensional Hubbard model at half-filling is studied in the framework of the Composite Operator Method using a static approximation. A solution characterized by strong antiferromagnetic correlations and a gap for any non-zero on-site interaction U is found. The corresponding ground-state energy, double occupancy and specific heat are in excellent agreement with those obtained within the Bethe Ansatz. These results show that the Composite Operator Method is an appropriate framework for the half-filled Hubbard chain and can be applied to evaluate properties, like the correlation functions, which cannot be obtained by means of the Bethe Ansatz, except for some limiting cases.
The authors have calculated the electronic self-energy of the two-dimensional Hubbard model as a function of frequency and momentum for various band fillings. They present this up to second order in the interaction parameter and also up to infinite order for ladder diagrams in the two-particle scattering channel. The authors have carried out direct k-space integrations using a triangle method. For a half-filled perfectly nested band, the imaginary part, lm Sigma (k f , omega ), has a linear- omega dependence for nested Fermi surfaces which is consistent with the marginal Fermi liquid phenomenology for high-T c superconductors. At, and near, half-filling there is a strong low-energy peak in lm Sigma (2) (k, omega ) when k is off the Fermi surface. The physical origin of this peak is discussed. The peak is enhanced in the ladder approximation. The corresponding structure in Re Sigma (2) (k, omega ) leads to multiple solutions of the Dyson equation for a wide range of values of the parameters. This leads to an appealing explanation of observed structure in some experimental angle-resolved photoemission spectra of high-T c materials. The authors also find anti-bound states split off from the band in the two-particle scattering ladders consistent with the suggestion of Anderson.
Numerical exact diagonalisation (ED) of the Hubbard model is traditionally constrained by the exponential growth of the Hilbert space. In this study, we implemented a symmetry-based ED framework to investigate the one-dimensional Hubbard model at half-filling for system sizes up to L=16. By exploiting particle number conservation, spin projection, and translation invariance, we significantly reduced the Hamiltonian dimension, enabling the resolution of ground state energies and dispersion relations without additional approximations. Our results confirmed the transition from a metallic to a correlated insulating regime with increasing on-site repulsion U. Furthermore, extending calculations from ten to sixteen sites revealed scaling trends that asymptotically approached the thermodynamic limit predicted by the Bethe Ansatz (𝐿= ∞). These results served as precise, reproducible benchmarks for validating approximate many-body techniques such as the Density Matrix Renormalisation Group (DMRG).
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.
A new analytical method is proposed for solving the Hubbard model in d-dimensions. The method permits, in principle, the study of the Hubbard model at any temperature and any band filling. The accuracy of the method is tested against known Bethe ansatz results of Lieb and Wu (1968), and against known weak- and strong-coupling expansion results. The one-dimensional path integral solution obtained for the ground-state (half-filled band case) is in excellent qualitative agreement with Bethe ansatz results for all couplings and exactly reproduces weak- and strong-coupling expansion results.