Quantitative Determination of the Hubbard Model Phase Diagram from Optical Lattice Experiments by Two-Parameter Scaling

We propose an experiment to obtain the phase diagram of the fermionic Hubbard model, for any dimensionality, using cold atoms in optical lattices. It is based on measuring the total energy for a sequence of trap profiles. It combines finite-size scaling with an additional “finite-curvature scaling” necessary to reach the homogeneous limit. We illustrate its viability in the 1D case, simulating experimental data in the Bethe-ansatz local-density approximation. Including experimental errors, the filling corresponding to the Mott transition can be determined with better than 3% accuracy.

Figure 1

(a) The trap potential $V(x)$ for $d=1$ and various values of $\alpha$. (b) Dependence of total energy on system size for a $d=1$ optical lattice loaded with noninteracting, spin-$1/2$ fermions. The filling is fixed at $f=1$. The trap exponents are $\alpha =2$ ($+$), 4 (*), 6 (□), 8 (▪), and $\infty$ ($\times$). Inset: the same quantity in the thermodynamic limit plotted against $1/\alpha$. The dash-dotted lines mark the thermodynamic limit result for $\alpha =\infty$: $Ea/2Lt=-(4/\pi )\sin (\pi f/2)$.

Figure 2

(a) Same as Fig. 1b, but with interaction strength $U/t=4$, and with $L/a=400$ data used for the inset. In this case the thermodynamic limit result for $\alpha =\infty$ is $E=-0.57372937$. (b) Solid data points and line: the $\alpha =6$ data from (a) (upper curve) and Fig. 1b (lower curve). Open data points: the same, but with random errors of $\pm 5\%$ added to the values of $E$ and $f$. Dashed line: least-squares fit to these “noisy” data. The resulting error in the thermodynamic limit of the energy is $\lesssim 0.6\%$.

Figure 3

Energy per site (a) and numerical estimate of the chemical potential (b) vs $f$ for $U/t=4$, $L/a=400$, $\alpha =2$, 4, 6, 8, and $\infty$ (lines with no symbols). In the inset in panel (b), the filling $f_{\max }$ where $\mu (f)$ has maximum slope is plotted as a function of ${\alpha }^{-1}$ (circles). The parabola interpolating the points coming from $\alpha =4$, 6, and 8 (dashed line) extrapolates to $f_{\max }(0)=1.025$, indicating that even those relatively small trap exponents provide a good estimate of the critical filling.