Triangular lattice Hubbard model physics at intermediate temperatures

Moire systems offer an exciting playground to study many-body effects of strongly correlated electrons in regimes that are not easily accessible in conventional material settings. Motivated by a recent experiment on $\text{WSe}_2/\text{WS}_2$ moire bilayers [Y. Tang et al., Nature 579, 353-358 (2020)], which realizes a triangular superlattice with a small hopping (of approximately 10 Kelvin), with tunable density of holes, we explore the Hubbard model on the triangular lattice for intermediate temperatures $t \lesssim T < U$. Employing finite temperature Lanczos calculations, and closely following the fitting protocols used in the experiment, we recover the observed trends in the reported Curie-Weiss temperature $\Theta$ with filling, using the reported interaction strength $U/t=20$. We focus on the large increase of $|\Theta|$ on decreasing the density below half filling and the sign change of $\Theta$ at higher fillings, which signals the onset of ferromagnetism. The increase in $|\Theta|$ is also seen in the $t$-$J$ model (the low energy limit of the Hubbard model) in the intermediate temperature range, which we clarify is opposite to the trend in its high temperature limit. Differences between the low, intermediate and high temperature regimes are discussed. Our numerical calculations also capture the crossover between short-range antiferromagnetic to ferromagnetic order in the intermediate temperature regime, a result broadly consistent with the experimental findings. We find that this behavior is a finite-temperature remnant of the underlying zero temperature phase transition, which we explore with ground state density matrix renormalization group calculations. We provide evidence of ferromagnetism characterized by weak (but robust) correlations that explain the small $\Theta$ seen in the experiment.