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I am now learning the many-body physics and having some questions about the insulator behavior of large $U$ limit for the Hubbard model :

\begin{equation} H = -t\sum_{\left\langle {i,j} \right\rangle, \sigma}c^{\dagger}_{i,\sigma}c^{}_{j,\sigma} + U\sum_{i}n_{i,\uparrow}n_{i,\downarrow}. \end{equation}

I already know that by assuming $\frac{U}{t} >> 1$ and the half-filling limit the energy spectrum of the system will have a charge gap with order $\approx U$.

But if we only consider the behavior for the group of lowest energy states (namely below the charge gap), we can perturbatively obtain the effective low energy Hamiltonian as

\begin{equation} H \approx \frac{4t^2}{U} \sum_{\left\langle {i,j} \right\rangle} {\bf S}_{i} \cdot {\bf S}_{j} \end{equation} which is the so called Heisenberg antiferromagnet.

Also, the effective Hamiltonian describing the charge distribution gives one electron per site.

My question is as follow :

1.

The large $U$ limit of the Hubbard model is also called the Mott insulator because the electrons are now localized in each site of the lattice. Although it might seem trivial that the localization of the electron give rise to insulating behavior, I am wondering since there will be some low energy spin excitation corresponds to spin-1 magnon, is the system really a insulator? Given the fact that there are gapless excitations, I can not understand why it is a insulator. (although there is actually a large charge gap, we can still achieve a excitation by magnon)

2.

It is widely accepted that this diagram characterize the metal-Mott insulator transition : enter image description here

However I can not understand why the density of state will be completely filled in the lower energy part of the splitting due to $e^{-}-e^{-}$ interaction. Is there any rigorous derivation to prove that these lower energy states (below the charge gap $U$) is completely filled to give an insulating state?

If I make some mistakes in the above reasoning please let me know :)

I would be extremely grateful for any suggestions on my questions!

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    $\begingroup$ In my limited understanding, your point of view on the first question is correct. Here the Mott insulator should be literally an insulator whose current-current correlator decays exponentially whereas spin-spin correlator does not. Thus this concept of an insulator is simply having zero electric conductivity, which does not exclude gaplessness. $\endgroup$ – Smart Yao Mar 18 at 10:12

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