Thermodynamics of the 3D Hubbard Model on Approaching the Néel Transition

We study the thermodynamic properties of the 3D Hubbard model for temperatures down to the Néel temperature by using cluster dynamical mean-field theory. In particular, we calculate the energy, entropy, density, double occupancy, and nearest-neighbor spin correlations as a function of chemical potential, temperature, and repulsion strength. To make contact with cold-gas experiments, we also compute properties of the system subject to an external trap in the local density approximation. We find that an entropy per particle $S/N\approx 0.65(6)$ at $U/t=8$ is sufficient to achieve a Néel state in the center of the trap, substantially higher than the entropy required in a homogeneous system. Precursors to antiferromagnetism can clearly be observed in nearest-neighbor spin correlators.

Figure 1

Entropy per lattice site $s$ of the Hubbard model as a function of temperature $T/t$, for $U/t=8$, at half filling. Dashed vertical lines (black): $T_N$ from Ref. 20; dotted lines (blue): according to DMC simulations. Dashed horizontal lines (black): entropy per lattice site $s$ at $T_N$ [20]; dotted lines (blue): according to DMC simulations; $\log (2)$ is shown as a solid (green) horizontal line. Inset: Energy $E/t$ per lattice site. The DMC data were extrapolated linearly in $1/L$ from the data at $L=6,8,10$.

Figure 2

Double occupancy of the Hubbard model as a function of $T/t$, at half filling. Extrapolated DCA results are shown as solid lines and DMFT values as dashed lines. Vertical lines: See Fig. 1.

Figure 3

Nearest-neighbor spin-spin correlation of the Hubbard model as a function of $T/t$, at half filling. Inset: Density dependence for $U/t=8$ and selected temperatures. Vertical lines: See Fig. 1.

Figure 4

Entropy per lattice site $s$ and the entropy per particle $S/N$ (inset) of the Hubbard model at the temperature $T=0.35t\approx T_N$, as a function of density $n$, for $U/t=8$.

Figure 5

Entropy profiles (entropy per lattice site) plotted over the trap in the LDA approximation for different temperatures with an interaction strength $U/t=8$. Error bars, shown every 5 lattice spacings, are smaller than symbol size.

Figure 6

Entropy per particle averaged over the trap as a function of temperature relative to $T_N$ for different $U$. The Néel temperature is reached at $T/t=0.333(7)$ for $U/t=8$, when the average entropy is $S/N=0.65(6)$. Errors (not shown) in $S(T_N)$ are estimated to be in the 10% range, with the largest contribution caused by the uncertainty in $T_N$.