Bose-Einstein-condensed cold atoms in a lattice system with $k$-space Berry curvature

In this paper we study the properties of cold bosons in two-dimensional optical lattices where Bose condensation occurs at a momentum point $\mathbf{k}$ with nonzero $k$-space Berry curvature. By combining results from both analytic and numerical approaches, the bulk and edge excitation spectra of the system are derived. We show that the boson system carries nonuniversal, nontopological temperature-dependent equilibrium angular momentum and edge current at low temperatures. The differences between boson systems with real- and $k$-space Berry curvature are pointed out. In particular, using a simple variational wave-function approach, we show that for bosons in a harmonic trap, the stability of the Bose-condensed state towards vortex formation is rather different in the two cases because of the different angular momentum scalings in the ground states.

Figure 1

The distribution of current density in units of $\frac{{\hslash }^2{\omega }_0^2\mathrm{\Omega }}{4U{R}_c}$ from our analytic result in a harmonic trap. Here the dot-dashed curve (I) is the harmonic trapping potential, and the dashed curve represents the chemical potential $\mu$ which determines the boundary of the system $R_c$ in the LEA. Along with the trap potential, the current amplitude as a function the position $J\left(R\right)$ is plotted as the solid curves II and III. Note that only regions with $|R-R_c|>\delta$ are plotted since our analytic result fails at a region of size $\delta$ around the boundary.

Figure 2

The density configuration (left panel) and the current flow (right panel) in the self-consistent calculation. We take $t=2$ eV, $\alpha =0.3$ eV, $B_z=1$ eV, $U=0.4$ eV, and $N=45$ in our calculation. The distribution of boson density is consistent with the LEA except near the edge. The current is chiral and localized around the edge.

Figure 3

(a) The charge current along the $x$ direction as $j_x\left(y\right)$ in our model Eq. (14) for several values of interaction strength $U$. The current is measured at a fixed $x=23$ line as a function of $y$. The values of other parameters are the same as those in Fig. 2. The current is mainly localized at the edge and the localization is strengthened by increasing $U$. (b) The scaling relation between $L$ and the size dimension $N\sim R_c$ for different interacting strengths. We find that the total angular momentum scales as $L\sim a+b{R}_c^2$, where $b$ is a function of $U$.

Figure 4

The density distribution of the minor component in momentum space $n_{\downarrow }\left(\mathbf{k}\right)$. Without SOC the density of the minor component at the ground state is almost zero due to the Zeeman field splitting. With SOC $\alpha =0.3$ eV and interaction $U=0.4$ eV, the density distribution of $n_{\downarrow }\left(\mathbf{k}\right)$ spreads out to higher momentum, with four maxima at finite $k$ points corresponding to the edge mode. The other parameters are just set as in Fig. 2.

Figure 5

The critical field $B_c$ as a function of interaction strength $\lambda =\frac{mUN}{2\pi {\hslash }^2}$. The three curves in the figure represent different trapping depths ${\omega }_0$.

Figure 6

The critical Berry curvature as a function of $\lambda =\frac{mUN}{2\pi {\hslash }^2}$ for different harmonic oscillator frequencies ${\omega }_0$.

Figure 7

The numerical simulation density for the GP equation with Berry curvature with different parameter settings. The parameter settings for the four subfigures are as follows: (a) $\lambda =1.5,\mathrm{\Omega }={10}^{26}$ s/kg; (b) $\lambda =2\times {10}^5,\mathrm{\Omega }={10}^{22}$ s/kg; (c) $\lambda =2\times {10}^5,\mathrm{\Omega }=3\times {10}^{22}$ s/kg; and (d) $\lambda =2\times {10}^5,\mathrm{\Omega }=5\times {10}^{22}$ s/kg. And the common parameters are $m={10}^{-26}$ kg and ${\omega }_0=100$ Hz.