Universal Spin Squeezing Dynamical Phase Transitions across Lattice Geometries, Dimensions, and Microscopic Couplings

Abstract: Recent work has identified a dynamical squeezing phase transition in power-law interacting bilayer XXZ spin models, separating a fully collective phase with Heisenberg-limited squeezing from a partially-collective phase with universal critical scaling. Here we test and establish the universality of this transition along two qualitatively different microscopic axes: lattice geometry, by studying square, triangular, and honeycomb $2\mathrm{D}$ bilayers as well as $1\mathrm{D}$ ladders, and a symmetry-preserving rescaling $λ$ of the interlayer couplings relative to the intralayer ones. Combining a Bogoliubov instability analysis with discrete truncated Wigner simulations, we find that the transition persists across all four lattice geometries and over a wide range of $λ$ with critical exponents consistent within error, providing strong evidence for a genuine non-equilibrium universality class. The Bogoliubov theory recovers the previously identified scaling $a_Z^* \propto L$ in the long-range interacting regime $α< d+2$, and yields an analytical scaling $a_Z^* \propto L^{2/(α-d)}$ for the critical aspect ratio with system size for $α>d+2$, with $α$ the power-law exponent in dimension $d$. This uncovers a previously unrecognized sub-linear regime for short-range interactions. By tuning $λ$ we vary the interlayer coupling strength at fixed layer spacing, demonstrating that the dynamical transition can be driven purely through interaction engineering without modifying the underlying geometry. These findings provide a versatile route toward controlling entanglement generation in Rydberg-array, polar molecule, and trapped-ion platforms with applications in quantum sensing and simulation.

• The paper presents a universal dynamical phase transition between fully collective and partially collective spin squeezing regimes across varied lattice geometries and interaction strengths.
• It employs Bogoliubov instability analysis and discrete truncated Wigner approximation simulations to reveal critical exponents and scaling laws that govern the transition.
• The findings offer practical guidelines for controlling quantum entanglement and enhancing metrological precision in systems like Rydberg arrays, polar molecules, and trapped ions.

Universal Dynamical Phase Transitions in Spin Squeezing Across Lattice Geometries and Couplings

Introduction

This work presents a comprehensive study of dynamical phase transitions in spin-squeezing dynamics within bilayer spin-1/2 models featuring power-law interactions. The primary focus is on the universality of these transitions as system parameters—specifically lattice geometry, interaction range, and interlayer coupling—are varied. Notably, the work establishes that the previously identified transition between fully collective and partially collective dynamical regimes is not confined to any specific geometry or interaction structure but constitutes a genuine non-equilibrium universality class. Results from Bogoliubov instability analysis and discrete truncated Wigner approximation (dTWA) simulations are demonstrated to be consistent across a diverse set of lattices and microscopic couplings.

Model and Analytical Framework

The model consists of spin-1/2 systems on bilayer lattices—square, triangular, honeycomb (2D), and 1D ladders—with power-law decaying spin-spin interactions of the form $V_{ij} \propto |r_i-r_j|^{-\alpha}$, where the exponent $\alpha$ reflects the interaction range. The interactions are separated into intralayer Heisenberg-type and interlayer XX-type, with the relative interlayer strength controlled by a tunable parameter $\Lambda$. This allows the examination of dynamical transitions as a function of either geometric parameters (aspect ratio $a_z/L$) or interaction ratios, potentially independent of geometric manipulation—a scenario readily accessible in Floquet-engineered platforms.

To analytically probe the onset of dynamical instability and the phase boundary, the authors use Holstein-Primakoff bosonization at the collective and quadratic (Bogoliubov) levels. The instability of nonzero-momentum modes—diagnosed via the Bogoliubov spectrum—demarcates the boundary between the fully collective and partially collective phases. The dTWA simulations, validated against tensor-network and exact diagonalization benchmarks, complement the analytic predictions by capturing nonlinear dynamics and finite-size corrections.

Characterization and Universality of the Dynamical Transition

A key result is the robust emergence of a dynamical transition separating two distinct regimes of spin squeezing:

• Fully Collective Phase: Only the $k=0$ mode is dynamically unstable; two-mode squeezing dominates, resulting in a non-extensive, system-size independent squeezed variance and Heisenberg-limited sensitivity ($\propto N^{-2}$).
• Partially Collective Phase: Multiple $k$ modes become unstable, yielding a system-size dependent minimum variance and scalable (but sub-Heisenberg) enhancement.

The transition boundary can be controlled by the aspect ratio $a_z/L$ or by tuning $\Lambda$. Across square, triangular, and honeycomb bilayers in 2D, as well as 1D ladders, and for a wide range of $\alpha$, both analytic (Bogoliubov) and numerical (dTWA) treatments reveal the same critical exponents within error. Notably, Bogoliubov theory is found to provide a reliable, computationally tractable criterion for the phase boundary; however, it systematically underestimates the collective phase's stability due to higher-order nonlinear effects observed in dTWA. These findings confirm that the dynamical phase transition is a universal feature of this symmetry class, robust to geometrical and coupling-structure perturbations.

Scaling Theory and the Role of Interaction Range

A significant contribution of the study is the analytical resolution of the scaling of the critical aspect ratio with system size, previously established empirically:

• Long-range regime ($\alpha$0): The critical aspect ratio scales linearly with system size, $\alpha$1, confirming $\alpha$2 as a universal control parameter for the transition.
• Short-range regime ($\alpha$3): The scaling crosses over to $\alpha$4; $\alpha$5 is no longer the correct scaling variable. For example, in 1D with $\alpha$6, $\alpha$7.

This analytical insight, supported by numerical data, identifies a previously unrecognized scaling regime and has direct implications for experimental design, e.g., in dipolar ($\alpha$8) or van der Waals ($\alpha$9) coupled systems.

Critical Exponents and Experimental Implications

The scaling behavior of the minimum squeezed variance during evolution is captured by the ansatz

$\Lambda$0

where $\Lambda$1 are critical exponents. Detailed collapses show universality: the extracted exponents are insensitive to changes in lattice geometry (square, triangular, honeycomb) or to the interlayer coupling strength $\Lambda$2, provided the underlying (spin) symmetry is preserved. This universal scaling persists over orders of magnitude in system size and interaction strengths.

Experimentally, these results show that the dynamical phase transition—and thus the qualitative nature of entanglement generation and quantum enhancement—can be controlled either by geometric parameters or purely via Floquet-engineered coupling ratios. The latter is highly advantageous, as it circumvents demanding requirements on lattice manipulation, and is compatible with current quantum simulation platforms such as Rydberg arrays, polar molecules, and trapped ion bilayers.

Implications and Future Directions

This work firmly establishes a non-equilibrium universality class of dynamical spin squeezing transitions in a wide family of power-law interacting spin systems. The results are theoretically significant for the classification of non-thermal fixed points and dynamically generated entanglement in many-body quantum systems. The explicit identification of the symmetry class (intralayer SU(2), interlayer U(1)) invites generalization to cases where these symmetries are perturbed, potentially connecting to a broader theory of non-equilibrium criticality.

Furthermore, the analytic and computational methods demonstrated (Bogoliubov theory/dTWA) provide a practical toolkit for predicting metrologically relevant dynamical phase boundaries in large quantum systems. The predictions can be immediately tested on current platforms utilizing Floquet engineering and bilayer geometries.

Potential extensions include systematic exploration of symmetry-breaking perturbations, the relationship of the observed exponents to those at non-thermal fixed points in other spin models, and experimental verification of the predicted universal scaling behavior in engineered quantum simulators.

Conclusion

The universality of dynamical squeezing phase transitions in power-law interacting spin bilayers is demonstrated to transcend specific microscopic details such as lattice geometry and interaction structure. The transition is characterized by critical exponents and scaling laws that persist across these variations, underlining a robust non-equilibrium universality class. These insights advance the understanding of entanglement generation in synthetic quantum matter and provide clear guidelines for experimental realization and control in quantum metrology and simulation platforms (2605.13969).