Multiparticle Interactions for Ultracold Atoms in Optical Tweezers: Cyclic Ring-Exchange Terms

Dominant multiparticle interactions can give rise to exotic physical phases with anyonic excitations and phase transitions without local order parameters. In spin systems with a global $\mathrm{SU}(N)$ symmetry, cyclic ring-exchange couplings constitute the first higher-order interaction in this class. In this Letter, we propose a protocol showing how $\mathrm{SU}(N)$-invariant multibody interactions can be implemented in optical tweezer arrays. We utilize the flexibility to rearrange the tweezer configuration on short timescales compared to the typical lifetimes, in combination with strong nonlocal Rydberg interactions. As a specific example, we demonstrate how a chiral cyclic ring-exchange Hamiltonian can be implemented in a two-leg ladder geometry. We study its phase diagram using density-matrix renormalization group simulations and identify phases with dominant vector chirality, a ferromagnet, and an emergent spin-1 Haldane phase. We also discuss how the proposed protocol can be utilized to implement the strongly frustrated $J–Q$ model, a candidate for hosting a deconfined quantum critical point.

Figure 1

Proposed setup: SU(2)-invariant chiral cyclic ring-exchange interactions can be realized by combining state-dependent lattices generated by optical tweezer arrays and strong Rydberg interactions with a central Rydberg-dressed control qubit ($C$). The auxiliary states $|\tau =1⟩|\sigma ⟩$ with $\sigma =\uparrow ,\downarrow$ (orange) of the atoms on the sites of the plaquette are subject to a state-dependent tweezer potential, which allows us to permute them coherently around the center. Our protocol makes use of stroboscopic $\pi$ pulses between the physical states $\tau =0$ (green) and the auxiliary states $\tau =1$, which only take place collectively on all sites and are conditioned on the absence of a Rydberg excitation in the control atom.

Figure 2

Proposed protocol: The sequence in (a) is repeated periodically with period $T=2(t_c+t_{\mathrm{rx}})$. When $t_c\ll 2\pi /{\mathrm{\Delta }}_c$, $1/{\mathrm{\Omega }}_c$ it implements a Trotterized time evolution of the effective Hamiltonian (8), which realizes CCR couplings when ${\mathrm{\Delta }}_c\gg {\mathrm{\Omega }}_c$. The individual time steps are illustrated in (b).

Figure 3

Phase diagram of the CCR Hamiltonian on a ladder, obtained from DMRG in a system with 64 sites: different observables are evaluated in the ground state of the Hamiltonian (1) to characterize the phases. Upon varying $\varphi$, three different phases can be identified: (a) a topological Haldane phase featuring a vanishing gap in the entanglement spectrum, (b) edge states with a nonzero local magnetization for $S_{\mathrm{tot}}^z=1$, (c) a symmetry-broken phase around $\varphi =\pi$ with long-range ferromagnetic correlations, and (d) a symmetric phase for small $\varphi$, where the staggered vector chirality remains nonvanishing over long distances.