Lowest entropy per site reachable in a doped homogeneous Hubbard sample
In plain words
Pairing in the Hubbard model needs very cold atoms, and the best experiments reached their coldest temperatures only at half filling, one atom per site. The question is how cold a doped sample with holes can be made, and what stops further cooling.
Precise statement
Homogeneous (flat-bottom) 2D square-lattice Fermi-Hubbard sample with at least 100 sites, $U/t = 8$, hole doping $\delta = 0.1\ \text{to}\ 0.2$. Determine the minimum entropy per site $s_{\mathrm{min}}$ (units of $k_{B}$) and temperature $T/t$ reachable by entropy redistribution into reservoirs and adiabatic loading, with independent thermometry, and identify the limiting heating processes (lattice-light scattering, technical noise, reservoir capacity). Answer: $s_{\mathrm{min}}$ and $T/t$ in the doped region with an error budget, and whether $T/t < 0.05$ is reachable at $\delta = 0.125$.
What would settle it
A doped homogeneous sample with thermometry validated against finite-temperature numerics, plus a measured heating budget.
Status in the literature
Unverified note
A 2025 experiment reached $T/t = 0.05 (+0.06/-0.05)$ at half filling by entropy redistribution (Xu et al., Nature 642, 909, arXiv:2502.00095).