Long-Range Interacting Lattice Models

• Long-range interacting lattice models are defined by power-law decaying interactions that establish strong nonlocal correlations and modify critical exponents and phase transitions.
• These systems exhibit anomalous nonequilibrium dynamics, including the breakdown of standard light-cone propagation and emergence of sub-ballistic and ballistic correlation behaviors.
• Advanced numerical methods, such as optimized Monte Carlo schemes and stochastic cutoffs, enable efficient simulation of rich static and dynamic phenomena in these models.

Long-range interacting lattice models are systems in which the pairwise interactions between degrees of freedom—such as spins, particles, or fermions—decay with distance as a power law or more slowly, so that distant sites may remain strongly correlated. Unlike short-range models, where only neighboring sites couple significantly, long-range lattice models exhibit fundamentally different static and dynamical properties, affecting critical behavior, correlation propagation, phase transitions, simulation complexity, and emergent collective phenomena.

1. Model Formulation and Classification

Long-range interacting lattice models are formulated by defining a Hamiltonian or action where the coupling $J_{ij}$ between sites $i$ and $j$ decays with their spatial separation $r_{ij}$:

$$H = -\frac{1}{2} \sum_{i,j} J_{ij} \sigma_i \sigma_j,$$

with $J_{ij} \propto r_{ij}^{-(d+\sigma)}$ for a $d$-dimensional lattice and decay parameter $\sigma$ (Aiudi et al., 2023). Variants may include quantum (commuting or non-commuting spin operators), particle (density-density), or fermionic (BCS-like) Hamiltonians.

Classification by Decay Exponent $\sigma$ and Dimension $d$:

• Mean-Field Regime ($\sigma < d/2$): Static and dynamic critical exponents are equal to mean-field values.
• Non-Trivial Long-Range Regime ($d/2 < \sigma < 2-\eta_{\mathrm{SR}}$): Critical exponents vary continuously with $\sigma$ and depend on the details of the algebraic decay.
• Short-Range Regime ($\sigma > 2-\eta_{\mathrm{SR}}$): Universal properties coincide with those of nearest-neighbor models (Aiudi et al., 2023).

Here $\eta_{\mathrm{SR}}$ is the anomalous dimension for the corresponding short-range system. In equilibrium, the crossover to short-range universality occurs at $\sigma = 2-\eta_{\mathrm{SR}}$.

Quantum models may include, for instance, transverse-field Ising or XXZ chains with interactions decaying as $1/|i-j|^{\alpha}$, where $\alpha$ plays the role of $\sigma$ and controls locality of dynamics and correlations (Maghrebi et al., 2015, Despres, 4 Oct 2024).

2. Static Critical Behavior and Phase Transitions

The presence of long-range interactions modifies thermodynamic properties and critical phenomena:

• Static Critical Exponents: As established by Sak and others, susceptibility and magnetization exponents interpolate between mean-field and non-trivial values as $\sigma$ is tuned. In the long-range regime $(d/2 < \sigma < 2-\eta_{\mathrm{SR}})$, for Ising models the magnetization susceptibility exponent $y$ satisfies $y = \sigma$ (Aiudi et al., 2023).
• Phase Transition Structure: Depending on parameters, models can exhibit first-order transitions and ensemble inequivalence (e.g., non-differentiable entropy), especially in microcanonical setups for lattice gases (Aristoff et al., 2016). A typical variational principle for microcanonical entropy takes the form

$$S(\xi, \rho) = \sup_{f \in \mathcal{Y}_{\xi, \rho}} [-H(f)],$$

with $H(f)$ a functional of occupancy profiles and $\mathcal{Y}_{\xi, \rho}$ imposing energy and density constraints.

• Topological and Exotic Phases: In certain quantum models with power-law interactions, the interplay between lattice discreteness and long-range couplings can give rise to exotic paired states, e.g., chiral $d+p$-wave superconductivity with nontrivial Chern number (Buchheit et al., 2022).
• Lattice Structure Selection: In variational problems modeling diblock copolymers or generalized Coulomb systems, the minimization of the regular part of a non-local kernel (e.g., Green's function of $(-\Delta)^{-3/2}$) selects specific lattice shapes (BCC in 3D) (Ren et al., 2022).

3. Nonequilibrium Dynamics and Correlation Spreading

The propagation of information and correlations in long-range models is dramatically altered compared to their short-range counterparts:

• Breakdown of Ballistic "Light Cones": For interactions decaying as $1/r^\alpha$ with $\alpha>D$ ($D$ the spatial dimension), a generalized Lieb-Robinson bound holds, but the causal region is no longer linear; for $\alpha>D$, it grows only logarithmically or as a power law (Eisert et al., 2013, Despres, 4 Oct 2024). For $\alpha<D$, even this bound fails and correlations can appear instantaneously across the system.
• Twofold Causality Structure: After global quenches, the spread of equal-time connected correlations displays a twofold universal algebraic structure:
  • A correlation edge (CE), propagating sub-ballistically ($t^* \propto R^{\beta_{CE}}$, $\beta_{CE}>1$), whose scaling depends on the low-momentum structure of the excitation spectrum (e.g., $\beta_{CE} = 3-\alpha$ for 1D transverse Ising chains in the gapped phase).
  • Internal extrema (local maxima), propagating ballistically ($t_{\max} \propto R$), independent of $\alpha$ in the gapped regime (Despres, 4 Oct 2024).
• Hydrodynamics with Multiple Conservation Laws: In systems with center-of-mass conservation, long-range pair-hopping processes can lead to subdiffusive, diffusive, or superdiffusive relaxation depending on the competition between the decay exponents $\alpha$ (pairing) and $\beta$ (hopping), with the dynamical exponent $z$ varying continuously: $z = \alpha+1$ or $z = \beta-1$ depending on which process is long-range (Morningstar et al., 2023).
• Non-Markovian Relaxation: For lattices with fixed sites and momentum degrees of freedom, the relaxation timescale $\tau_r$ scales polynomially with system size $N$: for power-law $1/r^\alpha$ interactions, distinct thresholds at $\alpha=d/2$ and $\alpha=d$ separate regimes with fundamentally different relaxation behavior; non-Markovian memory effects dominate for finite $N$ and slow decay (Filho et al., 2019).

4. Numerical Methods and Computational Schemes

Simulating long-range interacting lattice models is computationally demanding due to all-to-all couplings:

• Predecision Schemes (Metropolis Updates): Efficient algorithms reduce the computational complexity of traditional Monte Carlo updates:
  • For potentials $V(r) = r^{-d-\sigma}$, the "predecision" method reduces the per-sweep cost from $O(N^2)$ to $O(N^{2-\sigma/d})$ for $\sigma < d$ and to $O(N)$ for $\sigma > d$ by safely aborting the summation of small couplings once the move can be accepted or rejected exactly (Müller et al., 13 Aug 2025).
  • The algorithm preserves the exact Markov chain as a traditional implementation, applicable to O(n)-spin models (Ising, XY, spin glass) and adaptable for both equilibrium and nonequilibrium studies.
• Stochastic Cutoff and Parallelized Schemes: Stochastic elimination of weak pairs, combined with distributed vertex coloring (e.g., Kuhn–Wattenhofer algorithm), allows for dividing the lattice into noninteracting sublattices, enabling highly efficient parallel updates (e.g., >100$\times$ speedup for 2D dipolar systems using 288 processors) (Endo et al., 2015).
• Dynamical Lévy Lattices: Local dynamics is achieved by dynamically sampling the interaction network at each update step according to a Lévy (power-law) probability, thus mimicking long-range statistics while keeping each update local (Aiudi et al., 2023).

5. Emergent Entanglement and Topology

Long-range interactions affect both the nature and scaling of quantum entanglement:

• Area Law Stability: For quantum $D$-dimensional lattices with two-site interactions decaying as $1/r^\alpha$, entanglement entropy of any subsystem increases at most with the boundary area (not volume) for $\alpha > D+1$ during time evolution. The ground-state entanglement area law remains valid under adiabatic deformations provided $\alpha > 2D+2$ (Gong et al., 2017).
• Dynamical and Quantum Phase Transitions: Reducing $\alpha$ below these thresholds can induce transitions from area-law to volume-law entanglement scaling during dynamical processes, indicating the presence of new types of non-equilibrium dynamical phase transitions not detected by conventional spectral gap closures (Gong et al., 2017).
• Topological Phases: Exact continuum representations using, for instance, the Epstein zeta function decomposition make it possible to analyze gap equations and uncover transitions to topological phases such as $d+p$ or $d+s$ wave superconductors (Buchheit et al., 2022).

6. Localization, Fragmentation, and Nonergodic Phases

Long-range interactions, especially with "tilted" or position-dependent strength, induce nontrivial effects in Hilbert space structure:

• Nonmonotonic Localization Volume: In chains where the interaction strength increases linearly with position (as in certain staggered Schwinger models), the many-body participation entropy (a measure of localization volume in Hilbert space) exhibits a nonmonotonic dependence on the interaction range $R$. There exists an intermediate regime in $R$ where the localization volume is minimized due to competition between Hilbert space fragmentation and global connectivity (Mallick et al., 25 Jan 2024).
• Hilbert Space Fragmentation: The two-body terms can partition the Hilbert space into disjoint or weakly connected clusters, leading to nonergodic or scarred many-body eigenstates even in disorder-free systems.
• Implications for Quantum Simulation: These effects are highly relevant for implementations of quantum simulators of lattice gauge theories with effective long-range interactions, where screening or tuning of $R$ can be used to experimentally access regimes with enhanced localization or ergodicity breaking (Mallick et al., 25 Jan 2024).

7. Impact, Applications, and Outlook

Long-range interacting lattice models provide a unified framework for describing systems as diverse as quantum simulators of lattice gauge theories, ultracold atomic arrays, frustrated and glassy magnets, long-range Ising and XY models, diblock copolymers, and unconventional superconductors.

Their key features—the breakdown of strict causality, anomalous non-Markovian relaxation, rich dynamical phase diagrams, anomalous entanglement scaling, and nontrivial localization phenomena—have broad theoretical and experimental implications:

• Critical and Dynamical Scaling: Enable access to universality classes not accessible in short-range models (Aiudi et al., 2023, Maghrebi et al., 2015).
• Simulation Algorithms: Enable efficient paper of systems with millions of degrees of freedom, facilitating detailed exploration of equilibrium and nonequilibrium phenomena (Müller et al., 13 Aug 2025, Endo et al., 2015).
• Quantum Information and Transport: Predict and analyze regimes with anomalous spreading of quantum information, security for quantum communication, and new mechanisms for nonergodic protection (Eisert et al., 2013, Despres, 4 Oct 2024, Mallick et al., 25 Jan 2024).
• Material Design and Pattern Formation: Explain and engineer the emergence of crystal lattices (pine BCC/FCC) or novel superconducting phases via the interplay of local and long-range energetics (Ren et al., 2022, Buchheit et al., 2022).

Future directions include systematically classifying dynamical phase transitions as a function of decay exponent and dimension, extending coarse-graining and continuum methods to multicomponent and quantum systems, and leveraging parallel and randomized algorithms for exploration of complex long-range models in high-dimensional parameter spaces. Experimental advances, particularly in ultracold gases and Rydberg platforms, provide fertile ground for realizing and testing many of these predictions.