Optimizing the dynamical preparation of quantum spin lakes on the ruby lattice

Abstract

Quantum spin liquids are elusive long-range entangled states. Motivated by experiments in Rydberg quantum simulators, recent excitement has centered on the possibility of dynamically preparing a state with quantum spin liquid correlation even when the ground state phase diagram does not exhibit such a topological phase. Understanding the microscopic nature of such quantum spin "lake" states and their relationship to equilibrium spin liquid order remains an essential question. Here, we extend the use of approximately symmetric neural quantum states for real-time evolution and directly simulate the dynamical preparation in systems of up to $N=384$ atoms. We analyze a variety of spin liquid diagnostics as a function of the preparation protocol and optimize the extent of the quantum spin lake thus obtained. In the optimal case, the prepared state shows spin-liquid properties extending over half the system size, with a topological entanglement entropy plateauing close to $γ= \ln 2$. We extract two physical length scales $λ$ and $ξ$ which constrain the extent of the quantum spin lake $\ell$ from above and below.

Figures (11)

• Figure 1: (a) The approximately gauge-symmetric neural network architecture, comprising an exactly symmetric path (upper) and an unconstrained path (lower). The symmetric path takes as input the parity of the number of Rydberg states on atoms adjacent to each vertex of the kagome lattice. (b) Dynamical preparation of a QSL by ramping the Hamiltonian to excite more atoms from the ground state (purple) to the Rydberg state (orange) subjected to Rydberg blockade.
• Figure 2: (a) Infidelity of the approximately-symmetric NQS (orange dots) and Jastrow ansatz in Ref. mauron2025predicting (blue squares) on $N=24$ system under the ramping profile of Ref. semeghini2021probing (inset) as compared with exact diagonalization. (b) Comparison of the TEE ($\gamma$) from approximately-symmetric NQS (orange) and Jastrow ansatz (blue) on an experimentally studied $N=219$ system with open boundary conditions. The inset shows the geometry of the lattice and the partitions (red borders) used to compute the TEE at scale $L_A$.
• Figure 3: Optimizing preparation protocol for spin lake. (a) Topological entanglement entropy with ramping rate $\Gamma\approx 0.096~\Omega$ varying the final $\delta$. The inset shows the evolution of stabilizers $A_v$ and $B_p$ with final $\delta$ over a wider range. (b) Topological entanglement entropy over different subsystems (see Fig. \ref{['fig:partition']}(b) in suppinfo) fixing the final $\delta/\Omega = 4.5$ and varying the ramp rate $\Gamma$. The hatched region indicates a proxy for finite-size spin-liquid-like behavior over the range $\Gamma \sim 0.03$ to $\sim 0.2$ with a TEE close to $\ln 2$. The subsystem size $L_A$ is depicted in the inset of Fig. \ref{['fig:dynamic_benchmark']}(b).
• Figure 4: (a) The area-law scaling of $W_A^{\mathcal{Z}}/ W_A^{\mathcal{Z}, gs}$ at different ramp rates. (b) The perimeter-law scaling of $W_A^\mathcal{X}$. (c) Pictorial depiction of TEE with respect to length scale $L_A$ of the subsystem $A$. TEE approaches $\ln 2$ for $\lambda > L_A > \xi$. The inset shows the free $e$-gas with average distance $\lambda$ and tightly bound $m$-pairs with pair length $\xi$. (d) The upper limit $\lambda$ for the dynamical spin liquid size defined by the average distance between unbound leftover $e$-anyons, the lower limit $\xi$ defined by the $m$-anyon pair length. For each length scale, the adiabatic limit is set by the energy scale of the respective excitation species.
• Figure 5: Ground state phase diagram as a function $\delta/\Omega$. (a) Excitation pattern in the three phases: symmetric trivial, symmetry-breaking valence bond solid (VBS), and symmetry-breaking stripe solid (SS). The unit vectors in each phase are shown with arrows. The VBS state doubles the unit cell as the lattice in both unit vector directions, while the SS does so in one direction. (b) Ground state energy with VBS and SS initializations offset by $E_{gs}$ computed with random initialization. (c) Correlations [as defined by Eqn. \ref{['eq:order_params']}] detecting the symmetry-breaking pattern of the VBS (blue solid) and SS (red dashed) phases.