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Consider the 3D Fermi-Hubbard model in a cold-atom setting (harmonic confinement, $\epsilon_i$):

$ H = - t \sum_{\langle i, j\rangle, \sigma} c^{\dagger}_{i, \sigma}c_{j,\sigma} + U\sum_{i}n_{i,\sigma} n_{i, \bar{\sigma}} + \sum_i \epsilon_i (n_{i, \sigma} + n_{i,\bar{\sigma}})$

I am interested in the temperature change if I adiabatically change $t$ (tunnelling) but keep all other terms ($U$, $\epsilon$, and particle number $N$ fixed, entropy $S$ is fixed due to adiabaticity). Any ideas?

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It seems there have been some DMFT calculations which show that for low U/t (our case) there is almost no temperature change when changing adiabatically the tunnelling:

https://arxiv.org/pdf/cond-mat/0504003.pdf

I don't think there is a back-of-the-envelope calculation that can explain this.

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