Doped resonating-valence-bond states: Robustness of the spin-ice phases in three-dimensional Rydberg arrays

TL;DR

The paper investigates the robustness of three-dimensional quantum spin ice realized in Rydberg arrays by constructing an extended Rokhsar–Kivelson quantum dimer model with tunable monomer doping on diamond and cubic lattices. The ground state at the RK point is an equal-weight superposition over dimer configurations, enabling Monte Carlo sampling, while the single-mode approximation analyzes dimer and monomer excitations. It finds that infinitesimal monomer doping destabilizes the U(1) Coulomb spin-ice phase in the thermodynamic limit, triggering off-diagonal long-range order and Higgsing of the gauge field, though finite-size systems exhibit a crossover region where QSI features persist. The study establishes a doped QDM phase diagram, highlights the presence of Goldstone modes associated with monomer condensates, and demonstrates that the framework remains sign-free away from the RK point, offering a practical route to explore charged matter coupled to emergent gauge fields in 3D with potential experimental relevance for Rydberg arrays.

Abstract

Rydberg blockade effect provides a convenient platform for simulating locally constrained many-body systems, such as quantum dimer models and quantum loop models, especially their novel phases like topological orders and gapless quantum spin ice (QSI) phases. To discuss the possible phase diagram containing different QSIs in three-dimensional (3D) Rydberg arrays, we have constructed an extended Rokhsar-Kivelson (RK) Hamiltonian with equal weight superposition ground state in different fillings at the RK point. Therefore, the perfect QSIs with fixed local dimer filling and their monomer-doped states can be simulated directly by Monte Carlo sampling. Using single-mode approximation, the excitations of dimers and monomers have also been explored in different fillings. We find that, in the thermodynamical limit, even doping a small amount of monomers can disrupt the topological structure and lead to the existence of off-diagonal long-range order. However, in a finite size (as in cold-atom experiment), the property of QSI will be kept in a certain region like a crossover after doping. The phase diagram containing different QSIs and off-diagonal order phases is proposed.

Figures (16)

• Figure 1: The mapping between spin, dimer and flux. (a) Rydberg array can be mapped to the spin model. The atom at the Rydberg excited state is mapped to the spin-up (up arrows), while the atom at the atomic ground state is mapped to the spin-down (down arrows).(b) Spin can be mapped onto dimer (orange line). A dimer, corresponds to a spin pointing upward, $S_z=1/2$, while the absence of a dimer (blue line) corresponds to a spin pointing downward, $S_z=-1/2$. (c) They can be interpreted as fluxes, we identify a spin pointing up, with a flux vector $B$ from sublattice a to b, and vice versa.
• Figure 2: Pyrochlore lattice. (a) Each sites in the pyrochlore lattice is shared by an up-pointing tetrahedron and a down-pointing tetrahedron. Different colored dots indicate that the atoms are in ground states or Rydberg states. The configuration satisfies the ice rule. (b) The pink dashed line connects the centers of the nearest-neighbor tetrahedron to form a hexagonal cell. We label the atoms at different positions from 1 to 6 respectively. The atoms in the Rydberg state are mapped to dimers, while the atoms in the ground state are mapped to empty links. Therefore, the Rydberg atomic array on the pyrochlore lattice can be mapped to a QDM on diamond lattice. (c) The pyrochlore lattice is a composite lattice composed of four face-centered cubic (FCC) lattices. Each vertex of a tetrahedron belongs to a different sublattice. The sites in different sublattices are labeled as 1, 2, 3, and 4. (d) For one sublattice, the direct lattice vectors are $\overrightarrow{a_1}, \overrightarrow{a_2}, \overrightarrow{a_3}$. (e) The reciprocal lattice is body-centered cubic (BCC) with vectors $\overrightarrow{b_1}, \overrightarrow{b_2}, \overrightarrow{b_3}$. We measure the dispersion along the $\overrightarrow{b_1}+\overrightarrow{b_3}$ and $\overrightarrow{b_3}$.
• Figure 3: Dispersion of dimer density correlation for the original RK-QDM obtained by SMA for the diamond lattice ($24\times 24\times 24$) with strict one dimer per site (star) and two dimers per site (circle) along the Brillouin zone path $\Gamma (0, 0, 0) \rightarrow (0,4\pi,0)$ and $\Gamma \rightarrow (2\pi,2\pi,-2\pi)$.
• Figure 4: The dispersion of dimer density operator of the doped QDM at the RK point $t=t'=V=V'=1$. We take the Brillouin zone path $\Gamma~(0,0,0) \rightarrow (0,4\pi,0)$ and $\Gamma \rightarrow (2\pi,2\pi,-2\pi)$ on a diamond lattice with system size $24\times 24\times 24$. Different colors correspond to (a) one dimer per site with varying doping densities ($0.904\%$, $25.0\%$, $49.9\%$, $70.0\%$ and $95.8\%$) and (b) two dimers per site with varying doping densities ($0.904\%$, $25.0\%$, and $49.9\%$) .
• Figure 5: Off-diagonal dimer correlation of the doped QDM at the RK point $t=t'=V=V'=1$. The circles represent the off-diagonal correlations along the $\overrightarrow{a_1}$ direction as a function of the distance $|i-j|$, Different colors correspond to (a) one dimer per site with the varying doping density ($0\%, 4.17\%, 25.0\%, 49.9\%$ and $83.3\%$) and (b) two dimers per site with the varying doping density ($0\%, 4.17\%, 25.0\%,$ and $49.9\%$) , where $0\%$ corresponds to applying a strong constraint to the system that prohibits the presence of monomers. Here, the distance between nearest-neighbor links in the sublattice is set to 1. The values of the off-diagonal dimer correlation are significantly larger than their corresponding errors, the error bars are obscured by the circular data markers.