Interaction-Driven Topological Insulator in Fermionic Cold Atoms on an Optical Lattice: A Design with a Density Functional Formalism

We design an interaction-driven topological insulator for fermionic cold atoms in an optical lattice; that is, we pose the question of whether we can realize in a continuous space a spontaneous symmetry breaking induced by the interatom interaction into a topological Chern insulator. Such a state, sometimes called a “topological Mott insulator,” has yet to be realized in solid-state systems, since this requires, in the tight-binding model, large off-site interactions on top of a small on-site interaction. Here, we overcome the difficulty by introducing a spin-dependent potential, where a spin-selective occupation of fermions in $A$ and $B$ sublattices makes the on-site interaction Pauli forbidden, while a sizeable intersite interaction is achieved by a shallow optical potential with a large overlap between neighboring Wannier orbitals. This puts the system away from the tight-binding model, so that we adopt density functional theory for cold atoms, here extended to accommodate noncollinear spin structures emerging in the topological regime, to quantitatively demonstrate the phase transition to the topological Mott insulator.

Figure 1

(a)–(c) Spatial patterns of the optical lattice potentials $V_{\uparrow \uparrow }(\mathit{r})$ (a), $V_{\downarrow \downarrow }(\mathit{r})$ (b), and $V_{\uparrow \downarrow }(\mathit{r})$ (c), given, respectively, in Eqs. (2)–(4), here shown on the $x=0$ plane for $(V_A,V_B)=(0.8,1.2)$ in units of $E_R={\hslash }^2{\pi }^2/(4M{d}^2)$. $\circ (+)$ indicates positions of minima in $V_{\uparrow \uparrow }(V_{\downarrow \downarrow })$. (d) Spatial pattern of the hopping in the tight-binding limit. Red (blue) circles represent $A$ ($B$) sites, while red (blue) lines represent the hopping with positive (negative) amplitudes. (e) A transformed model.

Figure 2

Structure of the lowest four bands, in the noninteracting case (a) and for an interacting system (b) with $a_s=0.25d$. (c) The topological gap of the system against $a_s$. The parameters, all in units of $E_R$, are $V_A-V_B=0.4$ for case (A), $-0.4$ for case (B), and $-0.8$ for case (C) with $(V_A+V_B,V_C)=(2,0.25)$ for all the cases. A round off of the gap for stronger $a_s$ for case (C) signifies an emergence of a site-nematic order (see text). Case (D) displays the case of (C) with a magnetic field applied along $x$ with $V_M=0.2$. (d) The Chern density for the lowest two bands in the system is depicted in (b).

Figure 3

(a),(b) Spatial texture of the magnetization $\mathit{m}$ (arrows) for the noninteracting case (a) and for an interacting system with $a_s=0.25d$ (b) with the color representing the $z$-component magnetization, up (red) or down (blue). The axes of $\mathit{m}$ are taken as indicated for clearer viewing, and the loci of spins as we sweep the unit cell are depicted on Bloch spheres in the top insets. (c) The $y$ component of the magnetization (order parameter for the topological phase) for the state depicted in (b). (d) Spatial pattern of the atomic area density for the case of Fig. 2, case (C), with $a_s=0.2d$. (e) The same as (d) for a larger $a_s=0.27d$, for which a site-nematic order coexists with a topological gap.