Numerical study of antiferromagnetic nearest-neighbor spin correlations of an SU(6) Fermi gas in an optical lattice

Experimental progress with ultracold alkaline-earth-like atoms allows quantum simulation of novel phenomena and phases of matter. Many interesting SU($N$) phases are characterized by their magnetic correlations. Recently, the experiment at Kyoto has detected nearest-neighbor antiferromagnetic (AFM) spin-correlations in an SU(6) $^{173}$Yb Fermi gas loaded in 1D, 2D and 3D optical lattices. Such systems are effectively described by the SU($N$) Fermi Hubbard Model. We have developed and applied the Exact Diagonalization and Determinant Quantum Monte Carlo methods to solve such model in all of the relevant geometries. With these tools we calculate the density, double occupancy, entropy and magnetic correlations in a trapped gas under the local density approximation. We compare the calculated and measured nearest-neighbor AFM correlations, and we find they agree quantitatively. In 2D and 3D the experiments reach lower temperatures than is possible to accurately compute with theory, but in 1D, the temperature estimated from theory comparison is $k_BT/t = 0.08(2)$, being the lowest temperature ever reported for a Fermi gas in an optical lattice.