Local Quantum Criticality in Confined Fermions on Optical Lattices

Using quantum Monte Carlo simulations, we show that the one-dimensional fermionic Hubbard model in a harmonic potential displays quantum critical behavior at the boundaries of a Mott-insulating region. A local compressibility defined to characterize the Mott-insulating phase has a nontrivial critical exponent. Both the local compressibility and the variance of the local density show universality with respect to the confining potential. We determine a generic phase diagram, which allows the prediction of the phases to be observed in experiments with ultracold fermionic atoms trapped on optical lattices.

Figure 1

Density profiles along the trap for different fillings. Flat terraces are the Mott-insulating regions.

Figure 2

(a) Density profiles for (△) $N_f=30$, $U=6t$, and $V_2=15t$; (○) $N_f=70$, $U=6t$, and $V_2=6.25t$; and (▽) $N_f=74$, $U=6t$, and $V_2=7t$. (b) Variance of the local density. (c) Local compressibility ${\kappa }^l$ as defined in Eq. (2).

Figure 3

The local compressibility ${\kappa }^l$ vs $\delta =1-n$ at $\delta \to 0$ for (△) $N_f=70$, $U=8t$, and $V_2=6.25t$; (▽) $N_f=70$, $U=6t$, and $V_2=6.25t$; (○) $N_f=72$, $U=6t$, and a quartic potential with $V_4=6.25t$; (◇) unconfined periodic system with $U=6t$. Inset: dependence of the critical exponent $\varpi$ on $\Delta {\tau }^2$.

Figure 4

Variance $\Delta$ vs $n$ for (○) harmonic potential $V_2=6.25t$ with $N=100$, (△) quartic potential $V_4=15.82t$ with $N=150$, (▽) harmonic potential $V_2=10t+\mathrm{\text{cubic}}$ $V_3=2.5t+\mathrm{\text{quartic}}$ $V_4=7.5t$ with $N=50$, and (full line) unconfined periodic potential with $N=102$ sites. The curves correspond from top to bottom to $U/t=0,2,4,8$. For a discussion of the insets, see the text.

Figure 5

Phase diagram for a system with $N=100$ (▽) and $N=150$ (○) sites. The phases are explained in the text.