Power-Law Interacting XXZ Models

• Power-law interacting XXZ models are quantum spin systems where interactions decay with distance following a power law, bridging short- and infinite-range behaviors.
• They exhibit nontrivial critical phenomena and anomalous scaling in excitation spectra, entanglement, and fluctuations across various coupling regimes.
• Experimental platforms such as trapped ions, Rydberg atom arrays, and NV centers leverage these models for quantum simulation, metrology, and many-body physics.

Power-law interacting XXZ models are quantum spin systems in which exchange interactions decay with distance according to a power law, providing a bridge between short-range and infinite-range models. These models are of central interest for quantum many-body physics, with direct experimental relevance in systems such as trapped ions, Rydberg atom arrays, polar molecules, and solid-state spin ensembles. Power-law XXZ models exhibit a rich phase structure, anomalous critical phenomena, and nontrivial entanglement scaling, and underpin practical schemes for generating collective quantum resources such as spin squeezing and nonlocal entanglement.

1. Model Definition and Coupling Regimes

The canonical power-law XXZ Hamiltonian for spin-½ systems on a $d$-dimensional lattice is

$$H = \sum_{i<j} \frac{J}{|r_i - r_j|^{\alpha}} \left( S_i^x S_j^x + S_i^y S_j^y + \Delta S_i^z S_j^z \right)$$

where $J$ sets the interaction scale, $\alpha$ governs the decay, and $\Delta$ is the exchange anisotropy. For $\alpha \to \infty$, the model reduces to nearest-neighbor XXZ; for $\alpha = 0$, it becomes all-to-all coupled (Lieb–Mattis limit).

The interplay between spatial dimension $d$, decay exponent $\alpha$, and symmetry (XY, Heisenberg, Ising) yields three distinct regimes (Frérot et al., 2017):

• Short-range-like (SR), $\alpha > d + \sigma_c$: Behavior matches that of finite-range models; conventional critical exponents, standard correlation decay, and standard area laws for entanglement.
• Medium-range (MR), $d < \alpha < d + \sigma_c$: Intrinsic to power-law models, with continuously varying dynamical exponent $z = (\alpha - d)/\sigma_c$ ($\sigma_c = 2$ for XY, $1$ for Ising). Correlations, entanglement, and fluctuation scaling are non-universal and depend explicitly on $\alpha$.
• Long-range-like (LR), $\alpha < d$: The model is effectively infinite-range; fluctuations are suppressed and the excitation spectrum becomes flat.

This trichotomy leads directly to nontrivial scaling of observables and determines the universality class of critical points (Frérot et al., 2017, Adelhardt et al., 2024).

2. Excitation Spectra and Anomalous Scaling

Elementary excitations in power-law XXZ models exhibit nonstandard dispersion governed by the dynamical exponent $z$:

• Gapped (Néel) phase: $$\omega(\mathbf{k}) \approx \Delta_g + c|\mathbf{k}|^z$$
• Gapless (XY) phase: $\omega(\mathbf{k}) \sim |\mathbf{k}|^z$

The variation of $z$ across regimes results in anomalous Goldstone modes; for instance, in 2D, $z = (\alpha-2)/2$ for $2 < \alpha < 4$ and $z=1$ for $\alpha \geq 4$ (Adelhardt et al., 2024, Frérot et al., 2017). In the mean-field regime (small $\alpha$), all symmetry classes exhibit mean-field exponents ($\nu=1/2$, $\beta=1/2$, $\eta=0$); as $\alpha$ increases, exponents interpolate to their short-range values via a "crossover" scenario controlled purely by $\alpha$.

This continuously varying criticality was confirmed via pCUT+Monte Carlo methods for quantum-critical properties and by linear spin-wave and tensor-network approaches for spectral and correlation functions (Adelhardt et al., 2024, Frérot et al., 2017, Muleady et al., 2023).

3. Entanglement, Correlation, and Fluctuation Structure

Power-law XXZ models support a range of nontrivial correlation and entanglement behaviors, with universal features directly tied to $\alpha$, $d$, and model symmetry (Frérot et al., 2017, Li et al., 12 Mar 2025):

• Two-party entanglement: In finite power-law chains, two-spin concurrence $C(r)$ decays algebraically, $C(r) \sim r^{-\eta(\alpha,\Delta)}$. The exponent $\eta$ increases with $\alpha$ and $\Delta$, and the overall entanglement distribution exhibits novel, piecewise scaling not seen in finite-range models (Li et al., 12 Mar 2025).
• Fluctuation scaling: In the XY phase, spin fluctuations scale as $\langle(\delta S^x)^2\rangle \sim L^{\max(d,2z)}$ and $\langle(\delta S^y)^2\rangle \sim L^{d+z}$; in the gapped phase, area-law scaling dominates with, for example, $\langle(\delta S^z_A)^2\rangle \sim L_A^{d-1}$ (Frérot et al., 2017).
• Entanglement entropy: For the gapless XY regime, the entropy contains a universal subleading logarithm, $$S_A = a L_A^{d-1} + \frac{(d-z)}{2}\log L_A + \cdots$$, with the log prefactor continuously tuned by $\alpha$. In the fully long-range ($\alpha < d$) limit, this reduces to a pure log law $S_A \sim \frac{d}{2} \log L_A$.

These findings demonstrate that power-law models interpolate continuously between short-range area laws and long-range "Dicke-state" entanglement, and allow tuning violation of area/volume laws through $\alpha$.

4. Spin Squeezing and Metrological Implications

Power-law XXZ models support scalable spin squeezing—collective quantum correlations useful for quantum-enhanced metrology—if interactions are sufficiently long-range and disorder is controlled. The optimal squeezing parameter $\xi^2_{\rm opt}$ typically scales as a power law with system size, $\xi^2_{\rm opt} \sim N^{-\nu}$, with $\nu$ approaching $2/3$ in the all-to-all ("one-axis twisting") limit and interpolating to $2/5$ for clean power-law models at large $N$ (Begg et al., 15 Jan 2026, Muleady et al., 2023).

Semi-classical phase-space simulations (discrete truncated Wigner approximation, dTWA) and tensor-network time evolution (TDVP + global subspace expansion) were shown to yield quantitatively consistent scaling of $\xi^2$ in large 2D lattices (Muleady et al., 2023). Experimentally relevant disorder, such as positional vacancies, induces a disorder-driven loss of scalable squeezing above a critical vacancy threshold $p_c(\Delta)$ set by both interaction range and anisotropy (Begg et al., 15 Jan 2026). For dipolar couplings ($\alpha=3$) and $\Delta=-1$, $p_c \simeq 0.72$.

Further, two-mode squeezing and entanglement at the Heisenberg limit ($\sim1/N$ sensitivity scaling) can be generated in power-law coupled bilayer systems, especially when spatio-temporal engineering (Floquet protocols) is used to maximize the effective pair-production channel between collective spin modes (Duha et al., 2024).

5. Disorder, Many-Body Localization, and Dynamical Properties

Disorder strongly influences the dynamical and entanglement properties of power-law XXZ models. In 1D, strong random fields promote a many-body localized (MBL) phase for sufficiently large $\alpha$, with critical values $\alpha_c \approx 2$ (Heisenberg, $\Delta=1$) and $\alpha_c\sim 1-1.5$ (XY, $\Delta=0$) (Safavi-Naini et al., 2018).

In the localized regime, entanglement entropy grows subballistically as $S(t) \sim t^{1/\alpha}$ (contrasting with the logarithmic growth in short-range MBL), while the quantum Fisher information grows only logarithmically, reflecting that long-distance entanglement builds quickly but does not accumulate as strongly in two-point fluctuations (Safavi-Naini et al., 2018). This asymmetry in diagnostics is a signature of the nonlocal character of power-law interaction-induced entanglement.

Emergent critical lines in the disorder-decay parameter space demarcate MBL, thermal, and long-range-ordered phases. Experimental systems such as NV centers in diamond with high vacancy fractions provide concrete examples where these mechanisms govern the absence or presence of scalable squeezing and robust order (Begg et al., 15 Jan 2026).

6. Quantum Simulation, Experimental Realizations, and Outlook

Power-law XXZ models are experimentally realized in numerous platforms:

• Trapped ions: Achieve tunable $\alpha$ through the detuning of phonon-mediated spin-spin interactions.
• Rydberg atom arrays, polar molecules: Natural dipolar ($\alpha=3$) or van der Waals couplings implement XXZ Hamiltonians in 2D and 3D.
• NV centers and other solid-state ensembles: Provide random realizations of power-law XXZ models, with disorder as an intrinsic parameter (Begg et al., 15 Jan 2026).

Recent work has emphasized the necessity of both long-range interactions ($\alpha$ below a platform-dependent threshold) and disorder control (e.g., patterned defect placement or alternative host materials) to access robust collective quantum phases and scalable metrological enhancements (Begg et al., 15 Jan 2026). Phase diagram mapping via semi-classical and tensor-network techniques enables the quantitative design of large entangled resources in realistic architectures (Muleady et al., 2023, Duha et al., 2024, Adelhardt et al., 2024).

Ongoing research explores the universal aspects of entanglement distribution, violations of area/volume laws, and new forms of nonlocal order enabled by both tunable interaction range and disorder, with continuing theoretical and practical significance for quantum simulation and quantum information science.