Thermodynamics of the three-dimensional Hubbard model: Implications for cooling cold atomic gases in optical lattices

We present a comprehensive study of the thermodynamic properties of the three-dimensional fermionic Hubbard model, with application to cold fermionic atoms subject to an optical lattice and a trapping potential. Our study is focused on the temperature range of current experimental interest. We employ two theoretical methods—dynamical mean-field theory and high-temperature series—and perform comparative benchmarks to delimit their respective range of validity. Special attention is devoted to understand the implications that thermodynamic properties of this system have on cooling. Considering the distribution function of local occupancies in the inhomogeneous lattice, we show that, under adiabatic evolution, the variation of any observable (e.g., temperature) can be conveniently disentangled into two distinct contributions. The first contribution is due to the redistribution of atoms in the trap during the evolution, while the second one comes from the intrinsic change of the observable. Finally, we provide a simplified picture of a recently proposed cooling procedure, based on spatial entropy separation, by applying this method to an idealized model.

Figure 1

Density $n$ versus chemical potential $\mu$ for different interaction strength $U$ and at fixed inverse temperature $\beta 6J=10$ (DMFT). For large interaction strength a clear Mott plateau of filling $n=1$ develops.

Figure 2

Density $n$ versus chemical potential $\mu$ for increasing interaction $U$ and fixed inverse temperature $\beta 6J=1$ (DMFT). Only a reminiscent behavior of the Mott plateau is left for large interaction strength.

Figure 3

Double occupancy $d$ versus chemical potential $\mu$ at fixed inverse temperature $\beta 6J=10$ and different values of the interaction strength $U$ (DMFT).

Figure 4

Double occupancy $d$ versus density $n$ at fixed inverse temperature $\beta 6J=10$ and different values of the interaction strength $U$ (DMFT). (Inset) $d(n)/n$ at the inverse temperature $\beta 6J=10$ and different values of the interaction strength $U$ (DMFT).

Figure 5

Entropy per site $s$ versus chemical potential $\mu$ at the weak interaction strength $U/6J=0.5$ for different values of the inverse temperature $\beta$ (DMFT).

Figure 6

Entropy per site $s$ versus chemical potential $\mu$ at the strong interaction strength $U/6J=5$ for different values of the inverse temperature $\beta$ (DMFT). The curves legend is the same as in Fig. 5.

Figure 7

Entropy per site $s$ versus density $n$ at different interaction strengths $U$ and inverse temperatures $\beta$ (DMFT). The curves legend is the same as in Fig. 5.

Figure 8

Entropy per particle $s(n)/n$ at different interaction strengths $U$ and inverse temperatures $\beta$ (DMFT). The curves legend is the same as in Fig. 5.

Figure 9

(Left column) Double occupancy $d$ as a function of temperature $T$ for different interactions strength $U$ (DMFT) (from top to bottom: $n=1$, $n=0.8$, and $n=0.6$). (Right column) Entropy $s$ as a function of $U$ for different temperatures (DMFT) (from top to bottom: $n=1$, $n=0.8$, and $n=0.6$). The inverse temperature $\beta$ is measured in units of $1/6J$.

Figure 10

Comparison of the results for temperature dependence of the entropy $s$ and double occupancy $d$ at fixed chemical potential $\mu /6J=4$ and intermediate interaction strength $U/6J=2.5$ obtained by high-temperature series and DMFT.

Figure 11

Comparison of the results for temperature dependence of the entropy $s$ and double occupancy $d$ at fixed chemical potential $\mu /6J=2.45$ and intermediate interaction strength $U/6J=2.5$ obtained by high-temperature series and DMFT method.

Figure 12

Comparison of the results for the entropy $s$ versus the chemical potential at intermediate interaction strength $U/6J=2.5$ obtained by high-temperature series and DMFT method.

Figure 13

Contour plot of the absolute value of the difference between the results obtained with tenth- and sixth-order series for the entropy versus chemical potential and inverse temperature at $U/6J=2.5$.

Figure 14

Entropy per particle in a system at half filling and intermediate interaction strength $U/6J=2.5$ obtained by series expansion, DMFT, and QMC (for the Heisenberg model) [26].

Figure 15

State diagram of the gas in a three-dimensional optical lattice with parabolic trapping, for different temperatures (DMFT). The four characteristic regimes (see text) are labeled as follows: B (band insulator in the center of the trap), Mc (Mott insulator in the center of the trap, shaded areas), Ms (shell of Mott insulator away from the center), and L (liquid state). For each temperature the (crossover) lines indicate, from bottom to top, the $\rho$ values at which the central density takes the values 0.995, 1.005, and 1.995. The gray dashed line marks the crossover from the liquid to the Mott state with increasing interaction. The crosses indicate the points for which the density profiles are plotted in Fig. 16.

Figure 16

Density profiles (left column) and occupancy distribution $q$($n$) (right column) for four typical points in the state diagram (obtained with DMFT): a and b (in regime “L”), $U/6J=1$, $\rho =10$; c and d (in “B” for low $T$), $U/6J=1$, $\rho =20$; e and f (in “Mc”), $U/6J=3$, $\rho =10$; g and h (in “Ms”), $U/6J=3$, $\rho =20$.

Figure 17

Schematic evolution of the temperature as a function of a parameter $x$ along isentropic lines, in a case where cooling occurs as $x$ is increased. The continuous line corresponds to a lower entropy per particle than the dashed line, $s_1<s_2$. The figure illustrates that for this situation, increasing $x$ while keeping $T$ constant will increase the entropy [Eq. (14)].

Figure 18

Temperature dependence of the number of doubly occupied sites $D(T)$ for the interaction strength $U/6J=1$ and at different scaled particle numbers $\rho =5,10,15$ (DMFT).

Figure 19

Temperature dependence of the number of doubly occupied sites $D(T)$ for the interaction strength $U/6J=2$ and at different scaled particle numbers $\rho =5,10,15$ (DMFT).

Figure 20

Temperature dependence of the number of doubly occupied sites $D(T)$ for the interaction strength $U/6J=3$ and at different scaled particle numbers $\rho =5,10,15$ (DMFT).

Figure 21

The two contributions to $\delta D/\delta T$ for $U/6J=1$ (DMFT). Continuous line (left side, top to bottom: $\rho =5,10,15,20$) shows redistribution of atoms in the trap [reshaping, second term in Eq. (17)]. Dotted line (right side, top to bottom: $\rho =20,15,10,5$) shows intrinsic change of the double occupancy with temperature [first term in Eq. (17)].

Figure 22

Different contributions of the intrinsic and the reshaping term to $\delta D/\delta T$ for $U/6J=3$ (DMFT). Continuous line (left side, top to bottom: $\rho =5,10,15,20$) is the reshaping; dotted (left side, top to bottom: $\rho =20,15,10,5$) is the intrinsic double occupancy change with temperature.

Figure 23

Occupation number (dashed line, left top panel), entropy per particle (solid line, left top panel), and potential profile (solid line, bottom panel) for the idealized trapping potential. $Q(n)$ [solid line (full system), dashed line (core region), right top panel] as a function of the density. The core region is taken to be cylindrical with $r_0=51a$; the storage region is a rectangular box of size $300\times 300\times 100a^3$. Each region has a homogeneous density $n_{\mathrm{core}}=2$, $n_{\mathrm{storage}}=0.31$. The average entropy per particle for the total system is $\frac{S_T}{N_T}=1.65$ and $\frac{S_C}{N_C}=0$ for the core region. The ratio of particles in the core region versus the total particle number is $\frac{N_C}{N_T}=0.39$.

Figure 24

Occupation number (dashed line, left top panel), entropy per particle (solid line, left top panel), and potential profile (solid line, bottom panel) in the presence of the dimple and barriers, as a function of the transverse coordinate. The potential is offset such that $V=0$ is at the bottom of the dimple. $Q(n)$ [solid line (full system), dashed line (core region), right top panel] as a function of the density. We chose the following experimentally realistic parameters: $\frac{U}{6J}=0.5$, $\frac{V_h}{6J}=1.8104$, ${\gamma }^2=50$, $\frac{V_b}{6J}=6$, $r_b=15a$, ${\omega }_b=5a$, $\frac{V_d}{6J}=15$, ${\omega }_d=15a$, and $12\times {10}^4$ atoms. The average entropy per particle in the total system is $\frac{S_T}{N_T}=1.95$ and in the core region $\frac{S_C}{N_C}=0.198$. The ratio of particles in the core region versus the total particle number is $\frac{N_C}{N_T}=0.404$. Obtained with DMFT. See also Fig.2)of Ref. [43].

Figure 25

Modelization of the spectral function in the metallic (top) and insulating (bottom) phases. In the metallic phase $\Delta =0$, while in the insulating phase $\Delta >0$.

Figure 26

The gap parameter $\Delta (\mu ,T)$ as a function of $T$ for different interaction strength $U$.

Figure 27

Comparison of the state diagram obtained by DMFT (circles) and the simple analytic approximation (solid lines) (cf. Ref. [23]).