Bose-Hubbard models with staggered flux: Quantum phases, collective excitation, and tricriticality

We study the quantum phases of a Bose-Hubbard model with staggered magnetic flux in two dimensions, as was realized recently [M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro, B. Paredes, and I. Bloch, Phys. Rev. Lett. 107, 255301 (2011)]. Within mean-field theory, we show how the structure of the condensates evolves from the weak- to the strong-coupling limit, exhibiting a tricritical point at the Mott-superfluid transition. Nontrivial topological structures (Dirac points) in the quasiparticle (hole) excitations in the Mott state are found within random phase approximation and we discuss how interaction modifies their structures. The excitation gap in the Mott state closes at different $\mathbf{k}$ points when approaching the superfluid states, which is consistent with the findings of mean-field theory.

Figure 1

(a) Schematics of the two-dimensional square optical lattice with synthetic magnetic flux of magnitude $\frac{\pi }{2}$. The flux is periodic along the $\widehat{y}$ direction, but staggered along the $x$ direction. This gives rise to a unit cell with two nonequivalent lattice sites $A$ (solid circle) and $B$ (open circle). The hopping amplitudes from $A$ and $B$ sites to their nearest neighbors are shown schematically. (b) Phase diagram based on the inhomogeneous mean-field theory for average filling factor $n_A+n_B=2$. Three phases are found within the inhomogeneous mean-field theory with cluster of dimension $8\times 8$. Besides the Mott-insulating state, two superfluid states [plane-wave phase (PP) and stripe phase (SP)] are found and they terminate at a tricritical point. The dashed red line gives the boundary between PP and SP for a noninteracting gas. We have set $U$ as the energy unit.

Figure 2

(a) Quasihole excitation for the two bands closest to zero energy. There are two Dirac points in the spectrum at ${\mathbf{k}}_1=(\frac{\pi }{4},\frac{3\pi }{4})$ and ${\mathbf{k}}_2=(\frac{\pi }{4},-\frac{\pi }{4})$. The shaded region shows the Dirac cone at ${\mathbf{k}}_2$. (b) The velocity of the quasihole along the $k_x$ and $k_y$ directions for the Dirac point ${\mathbf{k}}_2$, as a function of $U$ in the Mott regime. The dashed line corresponds to the noninteracting value and has a ratio $\sqrt{2}$. (c) Quasiparticle and quasihole bands closest to zero energy in units of $U$. In the Mott regime, a finite gap exists between the quasiparticle and quasihole bands. The gap decreases as one approaches the superfluid phases. There is no particle-hole symmetry in the excitation spectrum. For (a–c), $K=J=0.03$. The average number of bosons per site is set to one.

Figure 3

Softening of the excitation spectrums in Mott-insulating state as a function of $k_y$, for $k_x=-\pi /4$, along two values of $J/K$, making the transition to PP and SP. (a) $J/K=1$, and the excitation spectrum goes to zero at $k_x=-\pi /4,k_y=-\pi /4$, consistent with the structure of the strong-coupling superfluid state. (b) $J/K=2$, and the excitation spectrum goes to zero at two values of $k_y$, indicating the emergence of condensate with two momentum components. (c, d) The phases (directions of the arrows) of superfluid order parameters at each site for PP and SP, respectively. Overall $\pi /4$ modulation along $y$ has been removed to make the comparison between (c) and (d) clear. We have set $U$ as the energy unit.