Realization of fractional Chern insulators in the thin-torus limit with ultracold bosons

Topological states of interacting many-body systems are at the focus of current research due to the exotic properties of their elementary excitations. In this paper we suggest a realistic experimental setup for the realization of a simple version of such a phase. We show how $\delta$-interacting bosons hopping on the links of a one-dimensional ladder can be used to simulate the thin-torus limit of the two-dimensional (2D) Hofstadter-Hubbard model at one-quarter magnetic flux per plaquette. Bosons can be confined to ladders by optical superlattices, and synthetic magnetic fields can be realized by laser-assisted tunneling. We show that twisted boundary conditions can be implemented, enabling the realization of a fractionally quantized Thouless pump. Using numerical density-matrix-renormalization-group calculations, we show that the ground state of our model is an incompressible symmetry-protected topological charge density wave phase at average filling $\rho =1/8$ per lattice site, related to the $1/2$ Laughlin-type state of the corresponding 2D model.

Figure 1

Using commensurate optical lattice potentials, interacting bosons in 1D ladder systems (a) can be realized. When the hopping elements across the ladder alternate between values $t_1$ and $t_2$ [see Eq. (3)] and every second hopping on one leg along the ladder has a phase $\pi$, the thin-torus limit of the 2D Hofstadter-Hubbard model (at flux per plaquette $\alpha =1/4)$ can be realized (b). Which of the hoppings on the right leg have a nontrivial phase $\pi$ depends on the value of $\sigma =\pm 1$. The boundary conditions are periodic across the ladder, with a tunable twist angle ${\theta }_x$, corresponding to magnetic flux ${\theta }_x/2\pi$ threading the smaller perimeter of the torus. The ground state is an incompressible CDW with average occupation $\rho =1/8$ per lattice site, related to the $1/2$ Laughlin state of the 2D model. The density distribution $\langle {\widehat{n}}_{j,\mathrm{L}}\rangle =\langle {\widehat{n}}_{j,\mathrm{R}}\rangle$ is shown in two unit cells, chosen from a larger system with open boundary conditions, for different values of ${\theta }_x$ in (c). This corresponds to a quarter cycle of the fractionally quantized Thouless pump.

Figure 2

(a) For the 2D Hofstadter-Hubbard model at flux per plaquette $\alpha =1/4$ we make a gauge choice leading to a two-by-two magnetic unit cell. Supplemented by twisted boundary conditions in the $x$ direction (twist angle ${\theta }_x)$, an effective 1D ladder model is obtained when the thin-torus limit is considered (b). For notational simplicity, the imaginary unit $i=e^{i\pi /2}$ is used to express the complex hopping elements in (b). When an additional gauge transformation is applied, leaving invariant the magnetic flux ${\Phi }_{1,2}$ in every plaquette, the ladder model described by Eq. (1) is obtained (c).

Figure 3

(a) Possible realization of the ladder system with half a magnetic flux-quantum piercing every second plaquette; cf. [30]. By interference of standing waves from a red (r) and a blue (b) detuned sideband of the long-lattice laser in the $x$ direction, with corresponding slightly detuned running waves along the $y$ direction, two independent lattice modulations [upper (blue) and lower (red) plot in (b)] are created. Each acts on a single leg of the ladder and they move in opposite directions along $y$, as shown by the amplitude of the modulations for different times in (b).

Figure 4

(a) The particle-hole gap ${\Delta }_{\mathrm{CDW}}$ of the incompressible phase at filling $\rho =1/8$ for varying interaction strength $U/J=\infty ,5,2$ extrapolated from finite system size calculations at $L_y=18,24,32$. This gap corresponds to the plateaus in the $\rho \left(\mu \right)$ diagrams shown in (b) for $U/J=\infty$ at system size $L_y=48$. Note that for ${\theta }_x=\pi /2$ the plateau has a kink in its middle where $\rho \left(\mu \right)$ changes, corresponding to the addition of a single particle. This is not a bulk effect, however, because the additional particle is localized at the edge of the system.

Figure 5

Local density of hard-core bosons with $U/J=\infty$ at ${\theta }_x=0$ in a harmonic trap centered around $j=64.5$ with $V=8.4\times {10}^{-5}J$ and $\mu =-2.4J$ and $L_y=128$. While the density along the left leg (blue squares) demonstrates the density wave character, the averaged density (black solid line) reveals incompressible phases at fillings $\rho =1/4$ and $\rho =1/8$. The vertical red lines indicate the phase boundaries between compressible and incompressible (blue shading) phases in local density approximation.

Figure 6

Approximate Wannier orbitals (blue shaded) at the points ${\theta }_x$ of highest symmetry.

Figure 7

The U(2) Wilson-loop phase ${\phi }_{\mathrm{W}}\left({\theta }_x\right)=\mathrm{Im}ln\det \widehat{W}\left({\theta }_x\right)$ is shown for the $\rho =1/8$ CDW, the winding of which gives the total Chern number. We used exact diagonalization for a system of size $L_x=2,L_y=12$ with periodic boundary conditions and $N=3$ particles for $N_{\varphi }=6$ flux quanta.