Self-consistent Hartree-Fock approach for interacting bosons in optical lattices

A theoretical study of interacting bosons in a periodic optical lattice is presented. Instead of the commonly used tight-binding approach (applicable near the Mott-insulating regime of the phase diagram), the present work starts from the exact single-particle states of bosons in a cubic optical lattice, satisfying the Mathieu equation, an approach that can be particularly useful at large boson fillings. The effects of short-range interactions are incorporated using a self-consistent Hartree-Fock approximation, and predictions for experimental observables such as the superfluid transition temperature, condensate fraction, and boson momentum distribution are presented.

Figure 1

We show in this plot the transition temperature $T_{\mathrm{c}}$, normalized to the recoil energy $E_{\mathrm{r}}$, as a function of the normalized optical lattice depth $q=V_0/4E_{\mathrm{r}}$, for bosons at unit filling in a periodic optical lattice potential. The blue points (and dashed curve) show the noninteracting case, the red circles show our interacting Hartree-Fock calculation, and the triangles show the experimental data from Ref. [10] (indicated as “Trotzky et al.”). The latter shows a clear suppression for larger $q$ as the Mott-insulating quantum critical point (at $q\simeq 3$ in this figure [20]) is approached.

Figure 2

The top panel shows $T_{\mathrm{c}}$ vs normalized optical lattice depth $q$ for parameters consistent with the Trotzky et al. experiment, but including larger filling values ($f=N/N_{\mathrm{sites}}$ with $N$ the particle number and $N_{\mathrm{sites}}$ the number of lattice sites). For each case, the solid curve is the interacting case and the dashed curve is the noninteracting case. Although these curves show a slight separation of the noninteracting and interacting curves with increasing filling $f$, the difference is quite small even for the largest filling. In the bottom panel we plot $T_{\mathrm{c}}$ vs $q$ for larger $a_s$ ($a_s/a=0.1$), which shows a significant enhancement of $T_{\mathrm{c}}$ due to mean-field interactions.

Figure 3

The Mathieu characteristic function for the even Mathieu function gives the dispersion, i.e., $a_{\nu }={\varepsilon }_{\nu }/E_{\mathrm{r}}$. Here $q=V_0/4E_{\mathrm{r}}=1,0.5,0$. The ground-state energy (the lowest of the curves) is lowered as the optical lattice potential $V_0$ increases. The characteristic function is reduced to a parabola when $V_0=0$.

Figure 4

Two calculated noninteracting boson densities are compared in this plot. Here the dashed blue and solid red curves are the modulus squared of the condensate wave function $n={\left|me(0,q,u)\right|}^2$ for $q=2$ and $q=0.6$, respectively. When $q$ is small, the density resembles a cosine shape plus a constant. However, the $q=2$ curves show deviations from this.

Figure 5

In this plot, we show the condensate fraction $N_0/N$ for both interacting and noninteracting gases, as a function of the lattice depth $q=V_0/4E_{\mathrm{r}}$ at temperature $T=0.1E_{\mathrm{r}}$. The condensate fraction decreases with increasing lattice depth $q$, approaching the phase transition. We also observe that the condensation fraction for the interacting gas is larger than that of the noninteracting gas. Here the system is at unit filling, with system parameters consistent with those of Ref. [10].

Figure 6

In this plot, we show the trend of the transition temperature $T_{\mathrm{c}}$ with varying lattice depth $q=V_0/4E_{\mathrm{r}}$ for the case of filling $f=5$ and other system parameters consistent with Ref. [10].

Figure 7

Plot of the condensate fraction for both noninteracting and interacting gases, as a function of temperature, over the temperature range from $T=0$ to $T_{\mathrm{c}}$ (where the fraction becomes zero), with filling $f=5$ and normalized optical lattice depth $q=1.75$, and other system parameters consistent with Ref. [10].

Figure 8

In this plot, we show the condensate fraction $N_0/N$ for both interacting and noninteracting gases, as a function of the lattice depth $q=V_0/4E_{\mathrm{r}}$ at temperature $T=0.1E_{\mathrm{r}}$. Here the system is at unit filling, with system parameters consistent with those of Ref. [10], except with a large scattering length $a_s=0.1a$.

Figure 9

The top panel shows the local density in a unit cell for $kz=\pi /2$ in the presence of repulsive interactions. Here, the system parameters are the same as those of Ref. [10] but with a larger scattering length $a_s\simeq 0.1a$ and filling $f=2.83$. For comparison, the bottom panel shows the noninteracting case.

Figure 10

This plot shows the interacting boson density (solid curve) as a function of the spatial variable $x$ in a unit cell at $ky=\pi /2$, $kz=\pi /2$ (crossing the unit cell center), showing a suppression of the central boson density relative to the noninteracting case (dashed curve).

Figure 11

This plot shows the interacting boson density (solid curve) as a function of the spatial variable $x$ in a unit cell at $ky=\pi /8$, $kz=\pi /8$ (near the unit cell edge), showing an increase of the central boson density relative to the noninteracting case (dashed curve).

Figure 12

The boson momentum distribution as a function of $k_x$ for $k_y=0$ and $k_z=0$, with system parameters given in the main text. Both plots are normalized so that $n\left(0\right)=1$. The top panel shows the interacting case, and the bottom panel shows the noninteracting case, with the height of the side peaks being smaller in the interacting case reflecting a more spatially uniform Bose gas.