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I was always curios what real material are described by the fermionic Hubbard model.

$$H = \sum_{\left< i, j\right> \sigma} t_{ij} c^{\dagger}_{i, \sigma} c_{j, \sigma} + \sum_i U_i n_{i\uparrow} n_{\downarrow} + \sum_{i\sigma} \mu_{i \sigma} n_{i \sigma}$$

It has been popular for years, but, I guess, because it is simplest quantum model that has insulating and conducting properties at different limits. ($t = 0$ for insulating, $U = 0$ for conducting) with non trivial interplay in some intermediate state. Plus it predicts Mott insulating phase.

It is nice that it can be solved in 1 dimension, in the $t=0$ and $U=0$ limits and serve in solid state physics as Ising model in statistical physics.

But how is it related to the real life?

I know only about square lattice describing $CuO_2$ plane in cuprates and cold atom experiments.

Are there any other materials that can be described with it?

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V2O3 may be the material, which can be described by the pure Hubbard model. For CuO2, t-J model or even the Emery's three band model may be a more appropriate starting point as can be seen from optical experiments.

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The Hubbard Model is a parametric model - that is U and t are adjustable parameters which can be tuned with experimental data. However, there are so many uses for this, that the effective value of these parameters depend not only on the material, but also the type of information that you want to get out of the model. So, from doping to long range interactions and magnetism or high Tc superC. there are so many ways to use it. Also, corrections to include spectral weight transfer etc., all add to a massive state of confusion. To use it purely as a phenomenological model, is the safest application, as for example, just going to U=0 and U>0. But, the ratio of U/t can be used to normalize the energy scales, and a large set of other pertubative approaches can be used. To me, the Hubbard I model - see original Hubbard paper - is the most elegant and simple demonstration of the phenomenology. The rest is all adaptations to particular materials.

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The model has sort of a revival due to realistic applications of density functional theory + dynamical mean-field theory (https://www.cond-mat.de/events/correl14/manuscripts/). In particular multi-orbital Hubbard models have interesting properties due to the intra-atomic Hund's rule exchange that also apply to realistic materials, cf. 'Hund's metals' (https://arxiv.org/abs/1707.03282).

As mentioned by others, the Hubbard model is a successful 'effective model', in particular when there are some narrow bands near the Fermi level and the high energy bands can be projected out, giving rise to an effective t and U.

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