Stabilization of bulk quantum orders in finite Rydberg atom arrays

1. Problem Statement

Rydberg atom arrays are a premier platform for quantum simulation, capable of realizing complex many-body phenomena such as quantum spin models, topological phases, and quantum scars. However, a persistent discrepancy exists between theoretical predictions (derived for infinite or periodic systems) and experimental realizations on finite-sized arrays.

• The Core Issue: Strong boundary effects in finite systems distort the bulk physics. In Rydberg systems, atoms at the edges have reduced interaction energy compared to interior atoms. This leads to the "pinning" of Rydberg excitations ($|r\rangle$) at the boundaries.
• Consequences:
  • In 1D: The "floating phase" (a gapless, incommensurate phase between crystalline orders) is destroyed. Instead of a continuous variation of the density-wave wavevector $k$, finite systems exhibit strictly quantized, discrete wavevectors due to edge pinning.
  • In 2D: Boundary pinning favors competing low-energy orders (e.g., a "square" phase) over the thermodynamically stable bulk order (e.g., a "star" phase), even in large lattices ($\sim$200 atoms).
• Limitations of Current Solutions: Simply increasing system size is insufficient to mitigate these effects, and modifying global interactions is often experimentally unfeasible or undesirable.

2. Methodology

The authors propose a general, experimentally feasible strategy to suppress boundary effects by engineering a spatially non-uniform Hamiltonian.

• The Strategy:
  • Bulk Region: The center of the array retains the target Hamiltonian parameters ($\delta_{bulk}$) corresponding to the desired quantum phase.
  • Boundary Region: The detuning parameter $\delta_i$ is smoothly tuned (e.g., via a linear gradient) from the bulk value down to a value deep within the disordered (paramagnetic) phase ($\delta_{boundary} \ll \Omega$) at the physical edges.
• Physical Mechanism:
  • The disordered phase in Rydberg systems ($\delta \approx \Omega$) is not a simple featureless state but a highly correlated superposition of configurations.
  • Crucially, this disordered ground state contains a broad superposition of local structures that have significant overlap with proximal ordered phases (e.g., $Z_3$, $Z_4$, or star phases).
  • By placing this "flexible" disordered subsystem at the boundary, the strong interactions between the bulk and the boundary allow the boundary atoms to naturally adopt configurations compatible with the bulk order, rather than being pinned to a specific, incompatible edge state.
• Simulation Tools:
  • The authors utilize large-scale Density Matrix Renormalization Group (DMRG) simulations.
  • They employ the BLOCK2 package to handle long-range interactions ($1/R^6$) without truncation.
  • They analyze Matrix Product States (MPS) and use perfect sampling algorithms to characterize the classical configurations within the disordered phase.

3. Key Contributions

1. Identification of the Disordered Phase as a Boundary Subsystem: The paper establishes that the disordered phase is not merely a "noise" region but a resource. Its intrinsic property of containing a superposition of local ordered structures makes it an "unbiased" boundary that can adapt to any adjacent bulk order.
2. Experimental Protocol: The proposal requires only local control of the on-site detuning $\delta_i$, a capability already present in modern Rydberg experiments (using local light shifts or spatial light modulators). It avoids the need for fine-tuning specific edge geometries.
3. Algebraic and Numerical Framework: The authors provide both numerical evidence and an algebraic analysis (in the Supplemental Material) demonstrating how flexible boundaries restore the continuous manifold of states in the floating phase.

4. Key Results

The efficacy of the protocol was demonstrated in both 1D and 2D systems:

• 2D Square Lattice (Star Phase Recovery):

  • Uniform Hamiltonian: On a $13 \times 13$ lattice, the ground state was dominated by a "square" phase due to boundary frustration, even in parameter regimes where the "star" phase is thermodynamically stable.
  • Non-Uniform Hamiltonian: By introducing a disordered boundary (4 rings of atoms with linearly varying $\delta_i$), the "star" phase order parameter ($O_{star}$) was restored across the entire relevant parameter space ($1.65 \le R_b \le 1.9$). The boundary atoms adopted a configuration commensurate with the bulk star order, eliminating frustration.
  • Striated Phase: The method also correctly recovered the subtle "striated" phase at $R_b = 1.6$, distinguishing it from the square phase by allowing weak density fluctuations on the $(1,1)$-sublattice, a feature lost in uniform finite systems.

• 1D Chains (Floating Phase Restoration):

  • Uniform Hamiltonian: The wavevector $k$ of density fluctuations was strictly quantized ($k/2\pi \sim z/L$), preventing access to the continuous floating phase.
  • Non-Uniform Hamiltonian: With a disordered boundary, the wavevector $k$ varied continuously with the interaction strength $R_b$.
  • Efficiency: A modest lattice of $L=121$ atoms with disordered boundaries reproduced the continuous $k(R_b)$ evolution of a thermodynamic-limit system, effectively mimicking a system an order of magnitude larger ($L \sim 1000$).

5. Significance

• Bridging Theory and Experiment: This work provides a practical solution to the "finite-size effect" problem, allowing current experimental platforms (which are limited to hundreds of atoms) to faithfully simulate bulk quantum phases that were previously thought to require much larger systems.
• General Applicability: The strategy is "agnostic" to the specific bulk ground state. It works for ordered crystalline phases, critical gapless phases (floating phase), and potentially topological orders, making it a versatile tool for quantum simulation.
• New Research Avenues: By stabilizing bulk orders in finite arrays, this technique opens the door to studying dynamical phenomena, topological edge states, and complex phase transitions in regimes previously inaccessible to experiment.

In summary, the paper demonstrates that by intentionally engineering the boundary into a correlated disordered state, one can effectively "decouple" the finite edges from the bulk, allowing small quantum simulators to exhibit the physics of infinite systems.