Universal probes for antiferromagnetic correlations and entropy in cold fermions on optical lattices

We determine antiferromagnetic (AF) signatures in the half-filled Hubbard model at strong coupling on a cubic lattice and in lower dimensions. Upon cooling, the transition from the charge-excitation regime to the AF Heisenberg regime is signaled by a universal minimum of the double occupancy at entropy $s\equiv S/\left(N{k}_{\text{B}}\right)=s^{*}\approx \ln \left(2\right)$ per particle and a linear increase of the next-nearest-neighbor (NNN) spin correlation function for $s<s^{*}$. This crossover, driven by a gain in kinetic exchange energy, appears as the essential AF physics relevant for current cold-atom experiments. The onset of long-range AF order (at low $s$ on cubic lattices) is hardly visible in nearest-neighbor spin correlations versus $s$, but could be detected in spin correlations at or beyond NNN distances.

Figure 1

Hypercubic lattice ($1\le d\le 3$) at strong coupling $U/\left(\sqrt{Z}t\right)\approx 6$: (a) Double occupancy $D\left(T\right)$ as estimated from DMFT ($d=3$, circles), DQMC ($d=3$: diamonds; $d=2$: squares), and BA ($d=1$, dotted line). (b) Corresponding estimates of entropy per particle $s=S/\left(N{k}_{\text{B}}\right)$.

Figure 2

Hypercubic lattice at strong coupling: Double occupancy vs entropy. In all cases, the minimum of the double occupancy corresponds to $s^{*}\approx \ln \left(2\right)$ (dotted line). The shaded area indicates the nonmagnetic contribution to $D$. Inset: $D\left(s\right)$ in $d=1$ for various interactions.

Figure 3

Hypercubic lattice at strong coupling: (a) Rescaled total and kinetic energy vs entropy. (b) Spin correlations $\langle {\widehat{\mathit{\sigma }}}_i·{\widehat{\mathit{\sigma }}}_{i+\delta }\rangle$ between nearest [${\delta }_{d=1}=1$, ${\delta }_{d=2}=(1,0)$, ${\delta }_{d=3}=(1,0,0)$] and next-nearest [${\delta }_{d=1}=2$, ${\delta }_{d=2}=(1,1)$, ${\delta }_{d=3}=(1,1,0)$] neighbors. Inset: NNN and infinite-range correlations for Heisenberg model ($d=1$ from Ref. [38]).

Figure 4

Derivatives of NN (squares) and NNN (circles) spin correlation functions of the Heisenberg model with respect to entropy (main panel) and temperature (inset) in $d=3$. Thin solid and dashed lines: Heisenberg results in $d=2$.