Fermions in 2D Optical Lattices: Temperature and Entropy Scales for Observing Antiferromagnetism and Superfluidity

One of the major challenges in realizing antiferromagnetic and superfluid phases in optical lattices is the ability to cool fermions. We determine constraints on the entropy for observing these phases in two-dimensional Hubbard models using determinantal quantum Monte Carlo simulations. We find that an entropy per particle $\simeq \ln 2$ is sufficient to observe the insulating gap in the repulsive Hubbard model at half-filling, or the pairing pseudogap in the attractive case. Observing antiferromagnetic correlations or superfluidity in 2D systems requires a further reduction in entropy by a factor of 3 or more. In contrast with higher dimensions, we find that adiabatic cooling is not useful to achieve the required low temperatures. We also show that double-occupancy measurements are useful for thermometry for temperatures greater than the nearest-neighbor hopping energy.

Figure 1

(a) Phase diagram of the repulsive ($U>0$) 2D Hubbard Model at half-filling in the ($U/t$, $T/t$) plane obtained from QMC simulations on $N={10}^2$ lattices. Two important crossover scales are denoted by filled symbols: the insulating gap ${\Delta }_{\mathrm{ch}}$ (obtained from the compressibility) and $T_{\mathrm{spin}}$ (obtained from the peak in the uniform magnetic susceptibility). We also show curves of constant entropy per particle $s=S/N{k}_B$. (b) Phase diagram in the ($s$, $U/t$) plane showing the two crossover scales. In the regime I, the system is effectively gapless; in regime II, the insulating gap develops and local moments form; and in regime III, the spins get correlated over ever increasing distances. (c) Temperature dependence of the double-occupancy $D(T)$ for various $U/t$. Dynamical mean field theory (DMFT) [20] shows a much more pronounced anomalous region with $dD/dT<0$ compared to our QMC results. The inset shows that $D$ is insensitive to linear system size $L$.

Figure 2

(a) Phase diagram of the attractive ($U<0$) 2D Hubbard Model at $\rho =0.7$ (away from half-filling) obtained on $N\ge {10}^2$ lattices. We show the crossover scale $T^{*}$ corresponding to the pairing pseudogap (obtained from the susceptibility) and the critical temperature $T_c$ [estimated from the superfluid density ${\rho }_s(T)$]. We also show curves of constant entropy per site $s=S/N{k}_B$, Inset: $|U|$-dependence of the superfluid $T_c$. (b) Phase diagram in the ($|U|/t$, $s$) plane showing the three regions: normal, pseudogap, and superfluid. (c) Temperature dependence of the double-occupancy $D(T)$ for various $|U|/t$.