Berry curvature of interacting bosons in a honeycomb lattice

We consider soft-core bosons with on-site interaction loaded in the honeycomb lattice with different site energies for the two sublattices. Using both a mean-field approach and quantum Monte Carlo simulations, we show that the topology of the honeycomb lattice results in a nonvanishing Berry curvature for the band structure of the single-particle excitations of the system. This Berry curvature induces an anomalous Hall effect when driving the system out of equilibrium. It is seen by studying the time evolution of a wave packet, namely, a superfluid ground state in a harmonic trap, subjected either to a constant force (Bloch oscillations) or to a sudden shift of the trap center.

Figure 1

Structure of a bipartite lattice with four nearest neighbors. Sites belonging to sublattice $A$ are represented by dark (blue) points in the figure, whereas $B$ sites are represented by light-gray (green) points. The hopping amplitudes are all equal in (a) while in (b) the hopping term $t_4$ along links $c_4$ is different from the other three near neighbor hopping terms.

Figure 2

Berry curvature in the first Brillouin zone for the brick wall lattice calculated from Eq. (9). The parameters are chosen as follows: $t_{1,2,3}=t,t_4=0,\mathrm{\Delta }=5t$.

Figure 3

Mean-field phase diagram of the Bose-Hubbard Hamiltonian equation (10) on the honeycomb lattice. (a) Density per site; (b) the superfluid order parameter. The parameters used in the calculation are $t_{1,2,3}=t,t_4=0,\mathrm{\Delta }/U=0.15,N=10$. The white lines correspond to $t/U=0.03$.

Figure 4

Comparison of QMC and mean-field results for the density per site as a function of total chemical potential $\mu =({\mu }_A+{\mu }_B)/2$. Parameters are $t_{1,2,3}=t,t_4=0,\mathrm{\Delta }/U=0.15$, as in Fig. 3, and the hopping amplitude is fixed to $t/U=0.03$. The blue solid curve is obtained from the mean-field approach (corresponding to the white line in Fig. 3), and the red dots are obtained from QMC simulations. The lattice size for QMC simulations is $16\times 16$, with a maximum atom number per lattice site $n^{(max)}=4$. The error bars (not shown) are smaller than the symbols.

Figure 5

Bogoliubov excitation spectrum $\omega \left(\mathbf{k}\right)$. Parameters are the same as in Fig. 4. The top panel corresponds to the Mott Phase $\mu /U=0$ and the bottom one to the superfluid phase $\mu /U=0.15$. As expected, the excitations are gapped in the Mott phase, whereas in the superfluid phase, they are gapless and exhibit a linear behavior at small momenta.

Figure 6

Berry curvature of the Bogoliubov excitations displayed in Fig. 5. One can clearly observe $\mathrm{\Omega }\left(\mathbf{k}\right)=-\mathrm{\Omega }(-\mathbf{k})$ (time-reversal symmetry), the antisymmetry with respect to the diagonal $\left(k_y\to -k_y\right)$, and the symmetry with respect to the antidiagonal $\left(k_x\to -k_x\right)$.

Figure 7

Pseudo Berry curvature, Eq. (20), obtained with the QMC equal-time Green functions for the brick wall and square lattices. The upper panel corresponds to the $n=1/2$ Mott phase for the brick wall lattice, i.e., $t_{1,2,3}=t,t_4=0,\mathrm{\Delta }/U=0.15,\mu /U=0$, and $t/U=0.03$. The lower panel corresponds to the $n=1/2$ Mott phase for the square lattice, i.e., $t_{1,2,3,4}=t$. For the square lattice, one obtains only a random pattern with an amplitude two orders of magnitude lower than the brick wall lattice results; the level of the fluctuations simply reflects the numerical accuracy of the QMC results. This emphasizes that the interacting system must really depict a nonvanishing Berry curvature in the excitation spectrum. In addition, one can see that the pseudo curvature has the same properties as the noninteracting system: it attains the largest (absolute) values around the location of the conical intersections and fulfills ${\mathrm{\Omega }}_g\left(\mathbf{k}\right)=-{\mathrm{\Omega }}_g(-\mathbf{k})$, due to the time-reversal symmetry. It is also odd under $k_y\to -k_y$ and even under $k_x\to -k_x$.

Figure 8

Initial wave packet in a harmonic trap. The atomic density in the $A$ sublattice (respectively, $B$ sublattice) is displayed in the top left (respectively, top right) plot. The harmonic trap is such that the chemical potential at the center corresponds to the superfluid phase described in the previous section. Since the chemical potential ${\mu }_A$ is larger than ${\mu }_B$, i.e., $\mathrm{\Delta }/U=-0.15$, the $A$ sites are in a Mott-like state with unit filling. On the contrary, the filling of the $B$ sublattice is low, such that the system is in a superfluid phase. Densities in the $\mathbf{k}$ space for both the sublattices are displayed in the bottom row and are found to be peaked around $\mathbf{k}=0$.

Figure 9

Evolution of the wave packet in momentum space. The upper panel displays the wave packet after a time $\tau =25/U$, and the lower one after $\tau =50/U$. The center of the wave packet moves at a constant velocity along the $k_y$ axis, i.e., ${\alpha }_1+{\alpha }_2=0$. The broadening of the wave packet is weak.

Figure 10

Bloch oscillations and AHE for a wave packet. The top plot shows the average value of $k_y$ (within the $B$ sublattice) as a function of time. After the adiabatic transition, the linear increase of $k_y$ with time is a clear signature of the Bloch oscillation. The two curves, i.e., the continuous (black) line and the dashed (red) line with the circles, correspond to opposite directions of the effective electric field. The AHE is demonstrated by the bottom plot, where one plots the displacement from the center of the trap of the wave packet along the $x$ axis, i.e., perpendicular to the applied force: the continuous (black) line with the squares and the dashed (red) line with the circles correspond to opposite directions of the effective electric field. For comparison, the corresponding displacement on a square lattice is also shown by the continuous (green) line with the diamonds.

Figure 11

Evolution of the average center of the wave packet, i.e., $\langle \mathbf{r}\rangle$, after a shift of the trap center by 20 lattice spacings in the $y$ direction [continuous (black) line with the squares]. The plot corresponds to a total duration $\tau U=100$. The wave packet clearly exhibits motion along the transverse direction, i.e., the $x$ axis. Similarly to Bloch oscillations, one has the following results: (i) nothing happens on the square lattice [see the (green) line with the diamonds]; (ii) nothing happens when the shift is along the $x$ axis where the Berry curvature always vanishes [see the (blue) line with the triangles]; and (iii) the displacement along $x$ is in the same direction whether the trap center is shifted towards the $+y$ or the $-y$ direction [see the dashed (red) line with the circles].