--- title: Long-Range Interacting Lattice Models url: https://www.emergentmind.com/topics/long-range-interacting-lattice-models type: topic --- # Long-Range Interacting Lattice 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 $\sigma$ [2303.18057]. 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 [2303.18057]. 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 [1508.00906, 2410.03125]. ## 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 $y = \sigma$ [2303.18057]. - **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 [1604.01337]. 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 [2201.11101]. - **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) [2208.00528]. ## 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 [1309.2308, 2410.03125]. For $\alpha1$), 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 [2410.03125]. - **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 [2304.12354]. - **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 [1906.03535]. ## 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 [2508.09775]. - 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) [1503.03295]. - **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 [2303.18057]. ## 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 $\alpha > 2D+2$ [1702.05368]. - **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 [1702.05368]. - **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 [2201.11101]. ## 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 [2401.14369]. - **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 [2401.14369]. ## 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 [2303.18057, 1508.00906]. - **Simulation Algorithms**: Enable efficient study of systems with millions of degrees of freedom, facilitating detailed exploration of equilibrium and nonequilibrium phenomena [2508.09775, 1503.03295]. - **Quantum Information and Transport**: Predict and analyze regimes with anomalous spreading of quantum information, security for quantum communication, and new mechanisms for nonergodic protection [1309.2308, 2410.03125, 2401.14369]. - **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 [2208.00528, 2201.11101]. 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. Source: https://www.emergentmind.com/topics/long-range-interacting-lattice-models