Nonlocally Interacting Spin Systems

• Nonlocally interacting spin systems are defined by spin interactions that extend over macroscopic distances, often emerging from long-range potentials or auxiliary couplings.
• They exhibit unique phenomena such as ensemble inequivalence, negative specific heat, and unconventional critical behavior not seen in short-range spin models.
• These systems underpin experimental platforms like cold-atom simulators and inform theories in quantum information, many-body localization, and statistical mechanics.

Nonlocally interacting spin systems are spin models in which the effective interactions between spin degrees of freedom extend over macroscopic distances, rather than decaying rapidly with separation. These models can arise natively from long-range potentials, through effective emergent mechanisms such as coupling to auxiliary degrees of freedom, or via constraints that lead to effective nonlocality. They exhibit a range of phenomena—ensemble inequivalence, nonadditivity, anomalous localization, and unconventional critical behavior—that contrast sharply with those of conventional short-range spin systems. Nonlocal spin interactions are central to the understanding of collective phenomena in quantum information science, statistical mechanics, many-body localization, and quantum simulation platforms.

1. Microscopic Origin of Nonlocal Interactions

Nonlocality in spin interactions emerges via several physical mechanisms:

• Explicit long-range coupling: Power-law interactions, $J_{ij}\propto|r_i - r_j|^{-\alpha}$ with $0 \leq \alpha < d$, provide direct nonlocality. The nonadditive limit is realized when the interaction range diverges with the system size, such as in systems with dipolar or van der Waals coupling subject to the Kac prescription to maintain extensivity (Mori, 2012).
• Emergent nonlocality through constraint or auxiliary fields: In models with auxiliary fields (e.g., lattice distortions or gauge fields), eliminating these via constraints or integrating out fast variables generates effective all-to-all or power-law spin spin-couplings. For example, the elastic spin model exhibits emergent long-range coupling purely from local spin-lattice interactions when the system is observed on timescales unable to break microscopic bonds (Mori, 2013). In lattice gauge theory simulators, imposing Gauss's law on local fermion-gauge field models induces effective infinite-range "tilted" interactions among remaining spin or charge degrees of freedom (Mallick et al., 2024).
• Ancilla-induced nonlocal effective Hamiltonians: Mediating spins through a common quantum ancilla—electronic, bosonic, or mechanical—generates effective all-to-all Ising- or Heisenberg-like couplings even if the direct spin-spin interactions are negligible. Stroboscopic (Floquet) driving can further enhance or stabilize these nonlocal effects (Geng et al., 2021).
• Boundary-induced nonlocality from coupling to critical bulk: At the edge of a critical 2D SPT (e.g., the AKLT state), coupling to gapless bulk modes induces effective power-law (nonlocal) interactions between otherwise local 1D edge spins, modifying their criticality and accessible phases (Jian et al., 2020).

2. Mathematical Formulation and Scaling

The generic Hamiltonian for a nonlocally interacting spin system takes the form:

$$H = -\frac{1}{2} \sum_{i,j} J_{ij} S^a_i S^a_j + \cdots$$

where $J_{ij}$ may scale with system size and distance:

• Kac scaling: To enforce extensivity, the strength of long-range terms is scaled as $J_{ij} = L^{-d}\phi\left((r_i - r_j)/L\right)$, ensuring $\sum_{i,j} J_{ij}=O(N)$ (Mori, 2013, Mori, 2012).
• Variable fall-off exponent: In spin-$s$ Ising models, $J_{ij} = J|i-j|^{-\alpha}$ interpolates between nonlocal (NL), quasi-local (QL), and local regimes as $\alpha$ increases (Ghosh et al., 2023).
• Shape-anisotropic and inhomogeneous couplings: Models arising from Schwinger-like gauge constraints feature $J_{ij}$ that can grow linearly with the site index or otherwise vary non-uniformly in space (Mallick et al., 2024).

Effective nonlocality can also be achieved via interaction mediated by ancillary systems, resulting in Hamiltonians such as $H_{\mathrm{eff}}\sim L_z^2$ (collective Ising interaction) or higher-body terms (Geng et al., 2021).

3. Thermodynamic and Ensemble Properties

Nonlocal interactions generically render the system nonadditive, giving rise to novel thermodynamic anomalies:

• Ensemble Inequivalence: Negative specific heat ($C<0$ in the microcanonical ensemble) or negative susceptibility ($\chi<0$ in restricted canonical ensembles) manifest in parameter regimes where additivity fails. The phase diagrams obtained in different ensembles may not coincide (Mori, 2013, Mori, 2012).
• Mean-field theory exactness and its breakdown: In unconstrained canonical ensembles, the free energy is rigorously given by variational minimization over uniform magnetization, and mean-field (Curie–Weiss) theory becomes exact for all $0 \leq \alpha < d$. With additional constraints (e.g., fixed magnetization components), inhomogeneous solutions and nontrivial spatial structures can arise, and mean-field theory can break down in ensemble-inequivalent regimes (Mori, 2012).
• Order parameters, correlations, and negative anomalies: The presence of nonlocality modifies critical exponents, allows for phase separation, and impacts scaling of correlations. Susceptibilities can diverge or become negative at instability boundaries determined by the Hessian of the free energy functional.
• Volume-law entanglement and multipartite signatures: Nonlocal models can sustain volume-law scaling for bipartite entanglement, high mutual information at large spatial separation, and extreme multipartite entanglement, unlike local models that obey area laws (Ghosh et al., 2023).

4. Dynamical and Quantum Information Features

Nonlocally interacting spin systems exhibit distinctive properties in out-of-equilibrium and quantum information contexts:

• Hilbert-space localization and nonmonotonicity: In systems with site-dependent or spatially inhomogeneous nonlocal interactions (e.g., "tilted" interactions from Gauss's law), the participation ratio and localization properties in Hilbert space vary nonmonotonically with the range of interaction, reflecting competition between clustering (localization in subspaces) and delocalization through tilt-induced potentials (Mallick et al., 2024).
• Floquet stabilization and synchronization via nonlocality: Ancilla-assisted discrete time crystals arise as nonlocal (all-to-all) Ising couplings stabilize Floquet subharmonic response against errors that would destabilize time-crystalline order in noninteracting spins. Ancilla-mediated nonlocality also allows "remote synchronization" of spin ensembles by coupling their mediating ancillae, a phenomenon verified both analytically and experimentally (Geng et al., 2021).
• Transitions between NL, QL, and L regimes: Varying the range exponent $\alpha$ in variable-range spin-$s$ chains produces sharp dynamical crossovers in entropy and entanglement measures, with universal critical exponents for the NL$\rightarrow$QL transition ($\alpha_d^* \sim \log_2 d$) and model-dependent QL$\rightarrow$L points (Ghosh et al., 2023).
• Boundary critical behavior and deconfined transitions with nonlocal interactions: Nonlocal couplings at critical boundaries can induce continuous transitions between distinct ordered phases (e.g., Néel-VBS in SO(3) systems) not accessible with purely local interactions, exhibiting modified critical exponents and scaling forms for correlation functions (Jian et al., 2020).

5. Experimental Realizations and Theoretical Implications

Nonlocal spin models are central to diverse experimental and conceptual developments:

• Quantum simulators and programmable matter: Cold-atom, Rydberg-atom, and trapped-ion arrays can realize nonlocal spin models with tunable interaction range, disorder strength, and control over constraints, enabling direct observation of both thermodynamic and dynamical regimes described above (Mallick et al., 2024, Geng et al., 2021).
• Lattice gauge theories and effective models: Implementation of constraints analogous to Gauss's law in digital or analog simulators naturally leads to nonlocal effective spin models, as demonstrated in truncated Schwinger models and spin-charge converted systems (Mallick et al., 2024).
• Elastic or structural solids: Solids with spin crossover or strong spin-lattice coupling can exhibit emergent nonlocal spin interactions through global lattice distortion even in the absence of explicit long-range forces (Mori, 2013).
• Boundary phenomena and SPT physics: Edge states of higher-dimensional SPT phases at criticality realize effective 1D spin chains with nonlocal boundary-induced interactions, allowing tests of boundary conformal field theory and RG predictions (Jian et al., 2020).

A plausible implication is that nonlocally interacting spin systems provide a robust platform for exploring phases and transitions in which additivity, ensemble equivalence, and local order break down; they offer routes to engineer new quantum matter and test fundamental principles of nonequilibrium and quantum statistical mechanics.

6. Diagnostic Techniques and Transitions Between Regimes

Transitions between nonlocal, quasi-local, and local phases can be precisely diagnosed and quantified using:

• Weighted-graph state entanglement spectra: Explicit computation of time-averaged mutual information, bipartite entanglement entropy, and generalized geometric measure (GGM) distinguishes NL, QL, and L regimes. Critical exponents for the onset of QL behavior ($\alpha_d^*$) scale logarithmically with local Hilbert space dimension. The emergence of a saturation plateau in GGM with increasing system size signals entry into the fully local regime (Ghosh et al., 2023).
• Participation ratio and spectral statistics: Exact diagonalization of nonlocally interacting Hamiltonians and analysis of the participation ratio and energy gap statistics expose localization/delocalization crossovers and anomalous nonmonotonic dependence on the interaction range (Mallick et al., 2024).
• RG flow analysis and operator scaling: Field-theoretic renormalization group methods give access to stable fixed points, scaling dimensions of correlation functions, and the precise conditions under which nonlocality fundamentally alters critical behavior (e.g., direct Néel–VBS transitions in 1D) (Jian et al., 2020).

7. Outlook and Broader Implications

Nonlocally interacting spin systems exemplify the key role of long-range quantum and classical correlations in many-body physics. Their study rigorously delineates the precise boundaries of mean-field theory validity, clarifies the physical origins of nonadditivity and ensemble inequivalence, and reveals new critical collective and dynamical phenomena unattainable with local interactions alone. These findings directly inform experimental schemes for quantum information processing, quantum simulation, and the engineering of exotic many-body phases, and motivate further investigation into the interplay between locality, quantum constraints, and collective order (Mori, 2013, Mori, 2012, Ghosh et al., 2023, Mallick et al., 2024, Geng et al., 2021, Jian et al., 2020).