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arXiv:cond-mat/9302011 · Physical review. B, Condensed matter

Hole-hole correlations in theU=∞ limit of the Hubbard model and the stability of the Nagaoka state

M W Long, X. Zotos

Published Jul 1, 1993• 16 citations
arXivHTMLPDF

Abstract

We use exact diagonalization in order to study the infinite-U limit of the two-dimensional Hubbard model. As well as looking at single-particle correlations, such as ${\mathit{n}}_{\mathit{k}\mathrm{\ensuremath{\sigma}}}$=〈${\mathit{c}}_{\mathit{k}\mathrm{\ensuremath{\sigma}}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{c}}_{\mathit{k}\mathrm{\ensuremath{\sigma}}}$〉, we also study N-particle correlation functions, which compare the relative positions of all the particles in different models. In particular we study 16- and 18-site clusters and compare the charge correlations in the Hubbard model with those of spinless fermions and hard-core bosons. We find that although low densities of holes favor a ``locally ferromagnetic'' fermionic description, the correlations at larger densities resemble those of pure hard-core bosons surprisingly well.

DOI
10.1103/physrevb.48.317

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We use exact diagonalization in order to study the infinite-U limit of the two-dimensional Hubbard model. As well as looking at single-particle correlations, such as nk\ensuremathσ{\mathit{n}}_{\mathit{k}\mathrm{\ensuremath{\sigma}}}nk\ensuremathσ​=〈{\mathit{c}}_{\mathit{k}\mathrm{\ensuremath{\sigma}}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{c}}_{\mathit{k}\mathrm{\ensuremath{\sigma}}}〉, we also study N-particle correlation functions, which compare the relative positions of all the particles in different models. In particular we study 16- and 18-site clusters and compare the charge correlations in the Hubbard model with those of spinless fermions and hard-core bosons. We find that although low densities of holes favor a ``locally ferromagnetic'' fermionic description, the correlations at larger densities resemble those of pure hard-core bosons surprisingly well.
The hole-hole correlation function in the 2D Hubbard model is investigated in the low doping regime both via Monte Carlo simulations for U=4 and Lanczos diagonalization on a 16 site cluster for very large U. The results are compared with available theoretical predictions for the strong coupling limit. There is a clear short range repulsion, but no signature of hole-hole attraction is found in either case. At U=4 correlations are still remarkably weak-coupling-like. For large U, they become rather similar to those of hard core bosons.
Ferromagnetic domains in the large-UUU Hubbard model with a few holes: an FCIQMC study
Two-dimensional Hubbard lattices with two or three holes are investigated as a function of UUU in the large-UUU limit. In the so-called Nagaoka limit (one-hole system at infinite UUU), it is known that the Hubbard model exhibits a ferromagnetic ground-state. Here, by means of exact FCIQMC simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing UUU. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centred on individual holes. The overall total spin of the wavefunctions remain the lowest possible (0, or 12\frac{1}{2}21​, depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than one) number of holes.
Two-dimensional Hubbard lattices with two or three holes are investigated as a function of UUU in the large-UUU limit. In the so-called Nagaoka limit (one-hole system at infinite UUU), it is known that the Hubbard model exhibits a ferromagnetic ground state. Here, by means of exact full configuration interaction quantum Monte Carlo simulations applied to periodic lattices up to 24 sites, we compute spin-spin correlation functions as a function of increasing UUU. The correlation functions clearly demonstrate the onset of ferromagnetic domains, centered on individual holes. The overall total spin of the wave functions remains the lowest possible (0 or 12\frac{1}{2}21​, depending on the number of holes). The ferromagnetic domains appear at interaction strengths comparable to the critical interaction strengths of the Nagaoka transition in finite systems with strictly one hole. The existence of such ferromagnetic domains is the signature of Nagaoka physics in Hubbard systems with a small (but greater than 1) number of holes.
The need for suitable many- or infinite-fermion correlation functions to describe strongly correlated systems is discussed, and the question linked to the need for a correlated basis, in which the ground state may be positive definite for certain low-dimensional geometries. In seeking a positive-definite basis a particular trial basis is proposed, based on that for hard-core bosons in pure one-dimensional systems. Single-particle correlations in this basis are evaluated for the case of the ground state of a quasi-1D Hubbard model in the limit of extreme correlation. The model is a strip of the 2D square lattice wrapped around a cylinder, and is related to a ladder geometry with periodic boundary conditions along its edges. This is done using both a novel mean-field theory and exact diagonalization, and the basis is indeed found to be well suited for examining (quasi-) order in the model. The model has a paramagnetic region and a Nagaoka ferromagnetic region. In the numerical calculation the correlation function in the paramagnetic phase has a power-law decay and the charge motion is qualitatively hard-core bosonic. The mean field leads to an example of a BCS-type model with single-particle bosonic long-range order.
The infinite-U Hubbard model on finite two-dimensional square lattices is investigated by the use of exact numerical diagonalization. Configurations with one spin flip away from the ferromagnetic state and up to three holes are studied for lattice sizes ranging from 4*4 to 12*12. For all these situations, the state with one spin flip is found to be more stable than the Nagaoka ferromagnetic state. The crystal momentum of the ground state in this spin sector depends on the system size. Systematic comparison with the Gutzwiller wavefunction shows that it reproduces qualitatively the crystal momentum dependence of the energy, but not the exact correlation between the spin flip and the holes.