Emergent nontrivial lattices for topological insulators

Materials with nontrivial lattice geometries allow for the creation of exotic states of matter like topologically insulating states. Therefore searching for such materials is an important aspect of current research in solid-state physics. In the field of ultracold gases there are ongoing studies aiming to create nontrivial lattices using optical means. In this paper we study two species of fermions trapped in a square optical lattice and show how nontrivial lattices can emerge due to strong interaction between atoms. We theoretically investigate regimes of tunable parameters in which such self-assembly may take place and describe the necessary experimental conditions. Moreover, we discuss the possibility of such emergent lattices hosting topologically insulating states.

Figure 1

One-dimensional schematic representation of the system considered. A larger (red) sphere refers to a $\uparrow$-fermion and a smaller (blue) sphere represents a $\downarrow$-fermion. (a) Top: One creates a band insulator for $\downarrow$-fermions and a half-filled system for $\uparrow$-fermions in the presence of weak interactions. Bottom: Increasing the attractive scattering length $a_s$ leads to the emergence of composites that form a checkerboard structure and the remaining $\downarrow$-fermions move between $s$ and $p$ orbitals (blue spheres and dumbbells). (b) The interaction-induced s-p band hybridization tunneling element corresponding to Hamiltonian (4). (c) Effective tunneling in the $p$ band when two composites occupy neighboring sites.

Figure 2

Comparison of different energy scales present in the system as a function of the effective interaction strength. Color lines represent different hopping parameters. The black line with circles represents the energy difference between the state of a system with half-filling of a $\uparrow$-fermion ($\mathbf{E}$) and a state with one $\uparrow$-fermion more than half-filled (${\mathbf{E}}_{+\mathbf{1}}$). Energies are in units of $E_R$, and lattice depths are $V_0=4E_R$ and $V_1=20E_R$.

Figure 3

(a) Phase diagram for the configurations of the composites. The largest (red) region denotes the period 1 checkerboard (CH1) configuration for the composites. The black region denotes the phase-separated state; the yellow region, the mixed phase. The shallow lattice parameter is $V_0=4E_R$. (b) Distribution of the composites in the CH1 phase. Filled (open) circles denote the presence (absence) of a composite. (c) Orbital degree of freedom for the excess $\downarrow$-fermions in the CH1 lattice. Colored circles denote $s$ orbitals. Horizontal (vertical) dumbbell shapes denote $p_x\left(p_y\right)$ orbitals. Basis states for the blue (red) lattice are denoted $\mathbf{A},\mathbf{B}$, and $\mathbf{C}$ (${\mathbf{A}}^{\prime },\mathbf{B}$, and $\mathbf{C}$). The corresponding sites in the CH1 structure are shown in Fig. 2. (d) Ground-state phase diagram for excess $\downarrow$-fermions corresponding to the composite CH1 phase. Phases are shown as a function of the dipolar strength $D$ and contact interaction strength $\alpha$ for $V_0=4E_R$. The upper (blue) region denotes the spin-nematic (SN) phase, whereas the lower (green) region denotes quantum anomalous Hall and quantum spin Hall (QAH/QSH) phases.

Figure 4

Dispersion relations of the Lieb lattice, as expressed in (7), for three values of $\Delta$ with lattice depths $V_0=4E_R$ and $V_1=20E_R$.

Figure 5

Important parameters in this paper: the energy cost $\left|\Delta \right|/E_R$ (dashed black line) and the relative strength of the s-p-band tunneling and the effective tunneling in the $p$ band, $J_{01}/\left|J_p\right|$ [solid (blue) line], as a function of the effective interaction strength $\alpha =a_s/a$. We fix the $\downarrow$-fermion lattice depth $V_0=4E_R$ and the $\uparrow$-fermion lattice depth $V_1=30E_R$, for which we see from Fig. 2 that the CH1 state is stable. For low $\alpha$, $\Delta$ is positive. As $\alpha$ becomes more negative, $\Delta$ decreases. For $\alpha ⪅-0.56$, $\Delta$ becomes negative and its absolute value increases. From the solid (blue) curve we also see that with an increase in $\left|\alpha \right|$, the effective $p$-band tunneling decreases and s-p tunneling increases, resulting in an increase in the ratio $J_{01}/J_p$. Around $\alpha \sim -0.6$ the contributions of all tunneling processes become of the same order of magnitude, facilitating the stability of the CH1 phase in the paper.