Spontaneous Quantum Hall Effect in an Atomic Spinor Bose-Fermi Mixture

We study a mixture of spin-1 bosonic and spin-$1/2$ fermionic cold atoms, e.g., ${}^{87}\mathrm{Rb}$ and ${}^6\mathrm{Li}$, confined in a triangular optical lattice. With fermions at $3/4$ filling, Fermi surface nesting leads to spontaneous formation of various spin textures of bosons in the ground state, such as collinear, coplanar, and even noncoplanar spin orders. The phase diagram is mapped out with varying boson tunneling and Bose-Fermi interactions. Most significantly, in one noncoplanar state the mixture is found to exhibit a spontaneous quantum Hall effect in fermions and crystalline superfluidity in bosons, both driven by interaction.

Figure 1

(a) Schematic picture of a triangular optical lattice and the four-sublattice magnetic ordering. (b) The Brillouin zone of the triangular lattice (solid hexagon) and the Fermi surface (dashed hexagon) at $3/4$ filling. Three nesting wave vectors are ${\mathbf{Q}}_1=(-2\pi /a,0)$, ${\mathbf{Q}}_2=(\pi /a,\sqrt{3}\pi /a)$, and ${\mathbf{Q}}_3=(\pi /a,-\sqrt{3}\pi /a)$, where $a$ is the lattice constant. (c) The spin configuration of various magnetic orderings of the spin-1 Bose-Einstein condensate, where arrows denote the direction of four spin vectors $⟨{\widehat{\mathbf{F}}}_i⟩$ ($i=1,2,3,4$): (I) all-out structure chiral spin order, (II) “umbrella” structure spin order, (III) stripe-cant spin order, (IV) ferromagnetic order, (V) coexistence of DW and 90° coplanar spin order, (VI) coexistence of DW and collinear spin order, and (VII) coexistence of DW and ferromagnetic order.

Figure 2

Zero temperature phase diagram of spinor Bose-Fermi mixtures in a triangular lattice at $3/4$ filling for fermions as functions of $t_b$ and $U_{bf}$. Here, $U_b+J_b=3.6t_f$ ($U_b>0>J_b$) and $J_{bf}=0.4t_f$; ${\overline{n}}_b=2$ is the averaged number of bosons per site.

Figure 3

The spectrum of the Hamiltonian ${\widehat{H}}_f^{\text{eff}}$ describing a single fermion moving in a fixed bosonic spin texture background in a strip geometry for different phases at $t_b=0.2\times {10}^{-3}{t}_f$ and (a) phase I, $U_{bf}=1.5t_f$, (b) phase V, $U_{bf}=2.35t_f$, and (c) phase VI, $U_{bf}=2.75t_f$. Here, red dash-dot lines denote the edge states.

Figure 4

(a) The lowest energy band of the Hamiltonian ${\widehat{H}}_b^{\text{sp}}$ with eigenenergy denoted by ${ϵ}_{\mathbf{k}}$. (b) Illustration of the spatially varying phase ${\xi }_i$ of the condensate wave function for the state I. The symbols squares, circles, triangles, inverted triangles, diamonds, right pointing triangles represent the different phase angles 0, $\pi /3$, $2\pi /3$, $\pi$, $4\pi /3$, and $5\pi /3$, respectively. One unit cell of ${\xi }_i$ is denoted by the dashed rhombus. (c) Time-of-fight pictures for three spin states $M_F=1$ (left), 0 (middle), and $-1$ (right) expected from the experiment for the state shown in (b) as a signal of the finite-momentum-condensate phase. Here, the outer hexagon indicates the first Brillouin zone of the triangular lattice and the inner hexagon denotes the first Brillouin zone for the ordered state with an enlarged unit cell with four sublattices.