General remarks on the Hubbard model in the strong coupling limit

Question

Some results are known for the Fermi-Hubbard model under certain assumptions. For instance, it is known that at half-filling and in the strong coupling limit, the Hubbard model reduces to a Heisenberg antiferromagnet. For reference, the Hubbard model is: \begin{equation} H = -t\sum_{\langle ij \rangle\sigma}c^\dagger_{i\sigma}c_{j\sigma}+U\sum_i n_{i\uparrow}n_{i\downarrow} \end{equation}

I'm wondering: is it possible to make general remarks on how the ground state of the system behaves by increasing the on-site interaction $U$? What I'm looking for is some basic predictions on, say, how the densities on a given lattice might evolve if $U/t$ gets higher regardless of the filling or the specific lattice.

In general, $U$ favours having a non-zero spin on each site and thus one might expect that for any lattice and any filling a sort of antiferromagnetic configuration emerges as the ground state at high $U$. Is it fair to say so? Can we say anything else? I hope the question isn't too generic.

Answer 1

Your question is indeed rather generic. The most general results are theorems like those of Lieb, Lieb-Schultz-Mattis-Oshikawa-Hastings and Thouless-Nagaoka-Tasaki, which provide results for total spin of the ground state. The latter implies that the ground state for $U\rightarrow \infty$, $t\geq 0$ and $N=L-1$ electrons (i.e. 1 electron shy of half-filling) is actually ferromagnetic, serving as a counterexample of your statement above. In general, the ground state depends strongly on the filling.

There were two excellent reviews published in Annual Review of Condensed Matter Physics earlier this year that provide a good overview for 2- and 3-dimensional lattices, so I recommend having a look at

• D. P. Arovas, E. Berg, S. A. Kivelson, and S. Raghu: The Hubbard Model, Annu. Rev. Condens. Matter 13, 239-274 (2022) (arXiv link).
• M. Qin, T. Schäfer, S. Andergassen, P. Corboz, and E. Gull: The Hubbard Model: A Computational Perspective, Annu. Rev. Condens. Matter 13, 275-302 (2022) (arXiv link).