Discerning Incompressible and Compressible Phases of Cold Atoms in Optical Lattices

Experiments with cold atoms trapped in optical lattices offer the potential to realize a variety of novel phases but suffer from severe spatial inhomogeneity that can obscure signatures of new phases of matter and phase boundaries. We use a high temperature series expansion to show that compressibility in the core of a trapped Fermi-Hubbard system is related to measurements of changes in double occupancy. This core compressibility filters out edge effects, offering a direct probe of compressibility independent of inhomogeneity. A comparison with experiments is made.

Figure 1

Total compressibility (top panel), system radius (middle panel), and the core compressibility (bottom panel) plotted as a function of the chemical potential in the trap center for $t=0.054\mathrm{kHz}$ and $U=5\mathrm{kHz}$. The arrows in the bottom panel indicate the Mott regime at $T=0.2\mathrm{kHz}$.

Figure 2

Compressibility (dotted line) and core compressibility (solid line) versus chemical potential in a uniform system, $\gamma =0$, with $U=7\mathrm{kHz}$ and $t=0.054\mathrm{kHz}$. The top (bottom) panel sets $T=1\mathrm{kHz}$ ($T=0.5\mathrm{kHz}$). The three incompressible regions correspond to a pinning of the density at 0, 1, and 2 for chemical potentials near $\mu =-U/2$, $U/2$, and $3U/2$, respectively. For $\mu \lesssim U/2$, ${\kappa }^{\mathrm{C}}$ vanishes because the system shows very little double occupancy.

Figure 3

The core compressibility ratio versus on-site interaction strength for $\mu =4\mathrm{kHz}$. The top panel fixes $t=0.054\mathrm{kHz}$ for several temperatures, $T=0.07\mathrm{kHz}$ (dot-dashed line), 0.2 (dashed line), 0.5 (solid line), and 1.0 (dotted line). The bottom panel fixes $T=0.2\mathrm{kHz}$ for $t=0.01\mathrm{kHz}$ (dashed line) and 0.1 (dotted line). The insets plot the density along a cross section in the cubic lattice for the same parameters as the bottom panel of Fig. 2 but in a trap and with $\mu =6.5\mathrm{kHz}$ (left) and $\mu =3.5\mathrm{kHz}$ (right). The circles are computed in a full second order expansion for the density while the solid line is computed in the second order LDA. The Mott gap pins the density near $⟨n_i⟩=1$ at the trap center (right panel). For $\mu >U/2$ (left panel), a central compressible region arises with $n_i>1$ at the trap center.

Figure 4

The core compressibility ratio (solid line) and the experimentally observed quantity $\delta d/\delta N$ (dashed line), plotted as a function of the on-site interaction strength for $\mu =4\mathrm{kHz}$, $t=0.054\mathrm{kHz}$ and $T=0.3\mathrm{kHz}$. The central density is unity for $U\gtrsim 6\mathrm{kHz}$. The two lines merge when there is little double occupancy. The inset plots the same, but the values are obtained by matching the entropy of $N=80000$ atoms in the dipole trap (at $T_D/T_F=0.15$) to those in the lattice with $t=0.054$.