Fractional Quantum Hall Phases of Bosons with Tunable Interactions: From the Laughlin Liquid to a Fractional Wigner Crystal

Highly tunable platforms for realizing topological phases of matter are emerging from atomic and photonic systems and offer the prospect of designing interactions between particles. The shape of the potential, besides playing an important role in the competition between different fractional quantum Hall phases, can also trigger the transition to symmetry-broken phases, or even to phases where topological and symmetry-breaking order coexist. Here, we explore the phase diagram of an interacting bosonic model in the lowest Landau level at half filling as two-body interactions are tuned. Apart from the well-known Laughlin liquid, Wigner crystal, stripe, and bubble phases, we also find evidence of a phase that exhibits crystalline order at fractional filling per crystal site. The Laughlin liquid transits into this phase when pairs of bosons strongly repel each other at relative angular momentum $4\hslash$. We show that such interactions can be achieved by dressing ground-state cold atoms with multiple different-parity Rydberg states.

Figure 1

(a) Energy gap $\mathrm{\Delta }E$ at $\mathbf{K}=0$ as a function of the pseudopotentials. The inferred phase diagram is indicated by white dashed lines. The Laughlin phase ($L$) is surrounded by a stripe phase ($S$), bubble phases ($B1$ and $B2$), an integer Wigner crystal (WC) phase, and a fractional Wigner crystal (FWC) phase. The star indicates the point for which the experimental realization is discussed in detail. (b) The absolute energy gap versus $u_2$ and $u_4$: Small gaps indicate a symmetry-broken phase, in contrast to the larger gaps seen in the Laughlin liquid phase. (c) The overlap with the Laughlin wave function versus $u_2$ and $u_4$: Nonzero overlap persists in the FWC phase. All data were obtained for eight bosons on a torus with ratio $a/b=0.9$.

Figure 2

The ground-state pair-correlation function $g_2(x,y)$ for $N=8$ and $a/b=0.9$ for different $u_2$ and $u_4$ corresponding to the different phases: (a) triangular lattice with $1/2$ boson per lattice site (fractional Wigner crystal), (b) square lattice with 1 boson per site (Wigner crystal), (c) triangular arrangement of “bubbles” (2 bosons per bubble), (d) square arrangement of “bubbles” (4 bosons per bubble), (e) clustering along the $x$ axis (stripe), and (f) homogeneous Laughlin liquid.

Figure 3

(a) Energy of the lowest two eigenstates at each $\mathbf{K}$, for different $u_4$ (with $u_2=0$). Increasing $u_4$ softens the magnetoroton branch around $\mathbf{K}=(4,0)$ and $\mathbf{K}=(2,4)$. (b) Direct energy gaps ${\mathrm{\Delta }}_{1\mathrm{EX}}$ and ${\mathrm{\Delta }}_{2\mathrm{EX}}$ of the first and second excited states at $\mathbf{K}=0$ and $\mathbf{K}=(4,0)$, as a function of $u_4$ (with $u_2=0$). (c) Overlap between the Laughlin state and the three lowest eigenstates ($\mathbf{K}=0$) of $H$ as a function of $u_4$, with $u_2=0$. The transition into the FWC phase ($u_4\approx 0.5$) occurs without a sudden drop of the overlap, in contrast to the transition into a bubble phase shown in the inset (overlap vs $u_2$, with $u_4=0$). (b) Overlap between the Laughlin magnetoroton state at $\mathbf{K}=(4,0)$ and the three lowest eigenstates. All data in (a–d) was obtained for $N=8$ and $a/b=0.9$.

Figure 4

We plot the folded magnetic Brillouin zone for different $N$ and mark with filled circles the symmetry sectors belonging to the ground-state manifold in the FWC phase. Doubly degenerate sectors are filled with two colors. The ground states form two stripes (red and blue solid lines), related to each other via reflection, winding twice around the zone.

Figure 5

Experimental realization. (a) The $s$ and $sp$ dressing lead to a standard soft-core potential (green and blue dashed lines, respectively), whereas $p$ dressing leads to the potential with a sharp dip (orange solid line). Together they lead to a hump-dip-hump potential (brown dot-dashed line). The outer hump, relevant for ensuring that $U_4>0$, is shown in the inset. (b) Level scheme for the dressing of the ground state $|g⟩$ with Rydberg states $|s⟩$ and $|+⟩$.