Generalized gradient expansion for inhomogeneous dynamical mean-field theory: Application to ultracold atoms in a harmonic trap

We develop a generalized gradient expansion of the inhomogeneous dynamical mean-field theory method for determining properties of ultracold atoms in a trap. This approach goes beyond the well-known local density approximation and at higher temperatures, in the normal phase, it shows why the local density approximation works so well, since the local density and generalized gradient approximations are essentially indistinguishable from each other (and from the exact solution within full inhomogeneous dynamical mean-field theory). But because the generalized gradient expansion only involves nearest-neighbor corrections, it does not work as well at low temperatures, when the systems enter into ordered phases. This is primarily due to the problem that ordered phases often satisfy some global constraints, which determine the spatial ordering pattern, and the local density and generalized gradient approximations are not able to impose those kinds of constraints; they also overestimate the tendency to order. The theory is applied to phase separation of different mass fermionic mixtures represented by the Falicov-Kimball model and to determining the entropy per particle of a fermionic system represented by the Hubbard model. The generalized gradient approximation is a useful diagnostic for the accuracy of the local density approximation—when both methods agree, they are likely accurate, when they disagree, neither is likely to be correct.

Figure 1

(Top row) Radial density for $T=0.5$. Blue is for heavy particles, red is for light particles. From left to right, we have the (a) LDA, (b) GGA, and (c) IDMFT results. (Bottom row) Radial density for $T=0.15$. Blue is for heavy particles, red is for light particles. From left to right, we have the (d) LDA, (e) GGA, and (f) IDMFT results.

Figure 2

(Top row) Radial density for $T=0.1$. Blue is for heavy particles, red is for light particles. From left to right, we have the (a) LDA, (b) GGA, and (c) IDMFT results. (Middle row) Light particle density for $T=0.1$ (red). The size of the symbol is proportional to the density of the light particles at that site. From left to right, we have the (d) LDA, (e) GGA, and (f) IDMFT results. (Bottom row) Heavy particle density for $T=0.1$ (blue). The size of the symbol is proportional to the density of the heavy particles at that site. From left to right, we have the (g) LDA, (h) GGA, and (i) IDMFT results.

Figure 3

(Top row) Radial density for $T=0.05$. Blue is for heavy particles, red is for light particles. From left to right, we have the (a) LDA, (b) GGA, and (c) IDMFT results. (Middle row) Light particle density for $T=0.05$ (red). The size of the symbol is proportional to the density of the light particles at that site. From left to right, we have the (d) LDA, (e) GGA, and (f) IDMFT results. (Bottom row) Heavy particle density for $T=0.05$ (blue). The size of the symbol is proportional to the density of the heavy particles at that site. From left to right, we have the (g) LDA, (h) GGA, and (i) IDMFT results.

Figure 4

(Top row) Radial density for $T=0.02$. Blue is for heavy particles, red is for light particles. From left to right, we have the (a) LDA, (b) GGA, and (c) IDMFT results. (Middle row) Light particle density for $T=0.02$ (red). The size of the symbol is proportional to the density of the light particles at that site. From left to right, we have the (d) LDA, (e) GGA, and (f) IDMFT results. (Bottom row) Heavy particle density for $T=0.02$ (blue). The size of the symbol is proportional to the density of the heavy particles at that site. From left to right, we have the (g) LDA, (h) GGA, and (i) IDMFT results.

Figure 5

Density of particles for $N=61455$ (a) LDA and (b) GGA for four different temperatures as labeled in the figure. Note how the system must be as large as it is to enclose all of the particles at high temperature and how at lower $T$ we start to see the kink in the distribution due to strong-coupling physics. The horizontal axis is an effective radius derived from the potential energy and using the geometric mean of the trap lengths. This allows all of the data for the 3D system to be plotted on one plot.

Figure 6

Density of particles for $N=332455$ (a) LDA and (b) GGA for four different temperatures as labeled in the figure. Note how the system must be as large as it is to enclose all of the particles at high temperature and how at lower $T$ we start to see the kink in the distribution due to strong-coupling physics. The horizontal axis is an effective radius derived from the potential energy and using the geometric mean of the trap lengths. This allows all of the data for the 3D system to be plotted on one plot.

Figure 7

Double occupancy versus particle number for experiment (symbols with error bars) and isoentropic lines with the entropy per particle equal to the most-likely fit interval 2.14, 2.24, or 2.34 $k_B$ and the extreme values at 1.3 and 2.5 $k_B$ for (a) LDA and (b) GGA. In both cases, we have $U/t=8.4$.

Figure 8

Posterior PDF for the LDA (black, solid) and GGA (orange, dashed) approaches, compared to the prior PDF (magenta, dashed). Note how it becomes strongly peaked around 2.24 $k_B$.