Thermodynamic properties of two-component fermionic atoms trapped in a two-dimensional optical lattice

We study the finite-temperature properties of two-component fermionic atoms trapped in a two-dimensional (2D) optical lattice. We apply the self-energy functional approach to the 2D Hubbard model with a harmonic-trapping potential, and systematically investigate the thermodynamic properties of this system. We find that entropy and grand potential provide evidence of a crossover between the Mott-insulating and the metallic phases at certain temperatures. In addition, we find that entropy exhibits a cusplike anomaly at lower temperatures, suggesting a second- or higher-order antiferromagnetic transition. We estimate the antiferromagnetic transition temperatures, and clarify how the trapping potential affects this magnetic transition.

Figure 1

A schematic of the 2D lattice, where the cross is the center of the harmonic-trapping potential, $L$ is the total number of sites, $a$ is the lattice distance, and $r_i$ is the distance between the center of the trap and the $i$th site. The numbers written in the top right of the lattice sites correspond to the site indices.

Figure 2

Rescaled cloud size $R_{\mathrm{SC}}$ as a function of $E_t/W$ for different $U/W$ at $T/W=0.1$.

Figure 3

Density profiles of atomic cloud [i.e., the number of atoms $N_i$ as a function of the rescaled distance $r_{\mathrm{SC}}$ for several fixed values of $E_t/W$: (a) 0.6, (b) 0.8, (c) 1.0, and (d) 2.0]. The interaction strength is varied from $U/W=0.5$ to $U/W=2.0$. The temperature is fixed at $T/W=0.1$.

Figure 4

(a) Entropy per atom $S/N_{\mathrm{tot}}$ and (b) grand potential per atom $\Omega /N_{\mathrm{tot}}$ for several values of $U/W$ at $T/W=0.1$. Thin (light blue) lines in the panel (b) mean $\Omega =-E_t{N}_{\mathrm{tot}}/2$ and $\Omega =-E_t{N}_{\mathrm{tot}}$ from top to bottom.

Figure 5

Rescaled cloud size $R_{\mathrm{SC}}$ vs $E_t/W$ for different $T/W$ values. The interaction strength is fixed at $U/W=1.5$.

Figure 6

Density profiles of atomic clouds for two characteristic trap energies: (a) $E_t/W~0.6$ and (b) $E_t/W~0.8$. The temperature is varied from $T/W=0.05$ to $T/W=0.3$. The interaction strength is fixed at $U/W=1.5$.

Figure 7

Entropy per atom $S/N_{\mathrm{tot}}$ vs $E_t/W$ for several $T/W$ values. The interaction strength is fixed at $U/W=1.0$.

Figure 8

(Top) the number of atoms $N_i$ and (bottom) the magnetization $M_i$ as functions of the rescaled distance $r_{\mathrm{SC}}$ for two characteristic trap energies: (a) $E_t/W~0.6$ and (b) $E_t/W~0.8$. The interaction strength is varied from $U/W=0.5$ to $U/W=2.0$, and the temperature is fixed at $T/W=0.025$.

Figure 9

Real-space distribution of the magnetization at $U/W=1.5$ and $T/W=0.025$ for (a) $E_t/W~0.6$ and (b) $E_t/W~0.8$.

Figure 10

Absolute value of the magnetization $\left|M_i\right|$ as a function of $r_{\mathrm{SC}}$ and $T/W$ for fixed values of (a) $E_t/W~0.6$ and (b) $E_t/W~0.8$.

Figure 11

Entropy per atom $S/N_{\mathrm{tot}}$ as a function of $T/W$ for several values of $E_t/W$ at $U/W=1.5$. Inset is an enlarged view, and arrows indicate the magnetic transition temperatures.

Figure 12

AF transition temperatures $T_c/W$ vs $E_t/W$ for different $U/W$ values. The lines are guides for the eye. The arrows indicate AF transition temperatures for the uniform Hubbard model at half-filling for $U/W=1.0$, 1.5, and 2.0 from top to bottom.

Figure 13

The optimized hybridization $V_i^{*}$ as a function of $r_{\mathrm{SC}}$ for two choices of the interaction strength: (a) $U/W=0.5$ and (b) $U/W=1.5$. Other parameters are fixed at $T/W=0.05$ and $E_t/W~0.8$. For convenience, we also plot the number of atoms $N_i$.