Long-Range Quantum Ising Models

• Long-Range Quantum Ising Models are quantum spin systems with algebraically decaying interactions that interpolate between nearest-neighbor and infinite-range regimes.
• They exhibit rich quantum criticality with continuously varying universality classes and unique dynamic scaling as the decay exponent α tunes non-local interactions.
• Advanced numerical methods and machine learning techniques are used to simulate phase transitions, entanglement, and non-equilibrium dynamics in these models.

Long-range quantum Ising models are quantum spin systems in which the Ising (spin-spin) interactions decay algebraically with distance as $J_{ij} \propto |i-j|^{-\alpha}$, interpolating between the nearest-neighbor transverse-field Ising model (TFIM) and the infinite-range Lipkin–Meshkov–Glick (LMG) model. The decay exponent $\alpha$ serves as a non-locality tuning parameter, fundamentally altering equilibrium phase diagrams, universality classes, dynamical scaling, entanglement structure, and susceptibility to competing orders. These models are realized in platforms such as trapped ions, Rydberg arrays, frustrated Josephson-junction networks, and are central to understanding and benchmarking quantum simulation, optimization, and non-equilibrium critical phenomena.

1. Hamiltonians and Interaction Structure

The canonical long-range quantum Ising Hamiltonian is

$$H = -\sum_{i<j} J_{ij} \sigma^z_i \sigma^z_j - h \sum_i \sigma^x_i,$$

with $J_{ij} = J/|i-j|^\alpha$ ($J<0$ ferromagnet, $J>0$ antiferromagnet), and $h$ the transverse field amplitude. The exponent $\alpha$ controls the interaction range:

• $\alpha \to \infty$: Short-range (nearest-neighbor limit).
• $\alpha < d$ (space dimension $d$): Strong long-range regime; interactions non-additive.
• $\alpha=0$: Infinite-range (LMG/Gaudin-Dicke limit).

This algebraic decay structure generalizes to higher-dimensional square and triangular lattices (Koziol et al., 2021, Humeniuk, 2016, Naik et al., 9 Dec 2025), and can incorporate anisotropy or staggered sign modulations to stabilize complex orders or unfrustrated antiferromagnetic chains (Herráiz-López et al., 3 Sep 2024).

2. Quantum Criticality and Universality Classes

The nature of the quantum phase transition (QPT) from the polarized paramagnet to the ordered phase is highly sensitive to $\alpha$, lattice dimension, and the sign/frustration of $J_{ij}$:

Ferromagnetic Regime

Three critical regimes arise for both 1D and 2D systems (Koziol et al., 2021, Gonzalez-Lazo et al., 2021, Shiratani et al., 2023):

• Short-range universality ($\alpha \gg d+2-\eta_{SR}$): Critical exponents coincide with nearest-neighbor (e.g., 1D: $\nu=1$, $\beta=1/8$; 2D: $\nu\approx 0.63$, $\beta\approx 0.326$).
• Continuously varying regime ($\alpha_{LG}<\alpha<\alpha_{SR}$): Exponents interpolate monotonically; no simple closed form. E.g., in 1D at $\alpha=1.5$, $\nu\approx 0.78$, $\beta\approx 0.21$.
• Long-range (mean-field/Gaussian) regime ($\alpha<\alpha_{LG}$; $d=1$: $\alpha<2/3$, $d=2$: $\alpha<4/3$): Exponents become $\nu=1/\sigma$, $\eta=2-\sigma$, $z=\sigma/2$, $\beta=1/2$, with $\sigma=\alpha-d$.

Binder cumulant crossings and advanced finite-size-scaling forms incorporating dangerous irrelevant variables are required in the mean-field regime to extract physical exponents correctly (Koziol et al., 2021, Shiratani et al., 2023).

Antiferromagnetic and Frustrated Regimes

For the 1D long-range antiferromagnetic Ising chain, all transitions remain in the standard Ising universality class ($\mu=2$, $\nu=1$) for any $\alpha>0$ (Sun, 2017). In higher dimensions and on non-bipartite lattices, however, frustration can stabilize "clock" (three-sublattice-ordered) phases in triangular geometries and lift classical degeneracy via "order by disorder" (Humeniuk, 2016, Saadatmand et al., 2018, Koziol et al., 2019). In such cases, phase boundaries and the nature of broken symmetry are highly geometry- and $\alpha$-dependent.

First-Order and Tricritical Transitions

Long-range unfrustrated antiferromagnetic chains with staggered (even-odd) structure stabilize a tricritical point and first-order transitions at strong long-range ($\alpha$ small), controlled by the competition of ferromagnetic intrasublattice and antiferromagnetic intersublattice couplings. As $\alpha$ increases, the first-order segment shrinks and vanishes for $\alpha \gtrsim 2.5-3$, yielding a purely Ising-type second-order transition elsewhere (Herráiz-López et al., 3 Sep 2024).

Regime	Order of Transition	Universality Class / Notes
Ferromagnetic, $\alpha\gg d$	2nd (Ising)	Standard $(d+1)$D Ising universality
Ferromagnetic, $\alpha<\alpha_{LG}$	2nd (MF)	Mean-field exponents, dangerous irrelevance
AF chain, any $\alpha>0$	2nd (Ising)	$(\mu,\nu)=(2,1)$ (Sun, 2017)
Staggered AF, $\alpha<\alpha^*$	1st/tricritical	Tricritical point at $g_{tp},h_{tp}$
Triangular AF, large $\alpha$	2nd (clock/XY)	3D-XY/clock universality; order by disorder
Triangular AF, small $\alpha$	Suppressed order	Clock/stripe phase boundary at $\alpha_c$

3. Finite-Size Scaling, Numerical Methods, and Machine Learning

Unbiased ground-state properties and critical exponents are extracted via:

• Stochastic Series Expansion (SSE) QMC: Efficient for both 1D and 2D, incorporating Ewald summation for minimizing artifacts in long-range interactions; scales to $\sim 10^3$ spins (Koziol et al., 2021, Humeniuk, 2016).
• Path-Integral Monte Carlo (PIMC): Allows direct access to finite-$T$ transitions and thermal scaling (Gonzalez-Lazo et al., 2021).
• DMRG and MPS/iMPS/iDMRG: Essential for validating criticality, entanglement, and correlation scaling in 1D chains and quasi-1D 2D cylinders (Saadatmand et al., 2018, Jaschke et al., 2016, Sun, 2017).
• Variational Machine Learning Wavefunctions: Vision Transformer (ViT) ansatz achieves high accuracy for mapping phase diagrams (order parameters, critical exponents, entanglement) and outperforms RBM-like wavefunctions, especially in regimes with strong long-range correlations (Roca-Jerat et al., 5 Jul 2024). In 2D, neural quantum states (e.g., convolutional resnet architectures) enable simulation of quench dynamics beyond the reach of conventional techniques (Naik et al., 9 Dec 2025).

Stochastic parameter optimization techniques have allowed for automated, high-precision determination of universality boundaries (e.g., $\sigma^*=7/4$ in 1D) and exponents using finite-size scaling at isotropic points in $(\xi_x/L, \xi_\tau/K)$ (Shiratani et al., 2023).

4. Dynamical Phenomena and Kibble–Zurek Scaling

Non-equilibrium critical dynamics under parameter ramps through the quantum critical point exhibit Kibble–Zurek scaling, with exponents dynamically renormalized by long-range interactions:

• Defect density after a linear quench scales as $n_{\rm def} \sim \tau_q^{-\nu/(1+z\nu)}$ (Puebla et al., 2019, Jaschke et al., 2016, Li et al., 2022).
• The dynamical critical exponent $z$ transitions from $z=1$ (short-range) to $z=\sigma/2$ (mean-field long-range), leading to continuous change in the scaling of freeze-out length scales and defect production as a function of $\alpha$ (or $\sigma$).
• Experimental work in trapped-ion chains with tunable $\alpha$ up to $N=61$ ions validates this scaling through KZM exponents, with finite-size scaling collapse confirming theory in the thermodynamic limit (Li et al., 2022).
• For antiferromagnetic chains, Kibble–Zurek scaling persists with Ising exponents, but frustration-induced gap suppression can hinder defect observation for large $N$ (Jaschke et al., 2016, Li et al., 2022).

5. Correlations, Entanglement, and Breakdown of Conformal Invariance

Long-range quantum Ising models present rich correlation behavior:

• Away from criticality, connected correlators decay as a product of exponential and power-law forms, with the power-law exponent $\gamma(\alpha)$ tracing the interaction tail; for $\alpha<1$, exponential decay disappears, yielding pure algebraic decay even in gapped phases (Vodola et al., 2015).
• Entanglement entropy can violate the area law (logarithmic scaling) for sufficiently slow decay ($\alpha\lesssim 1$), especially in the “quasi-critical” paramagnetic regime.
• Along critical lines, effective central charge evolves as $c_{\rm eff}(\alpha)$, interpolating from $1/2$ (Ising) to $1$ as $\alpha \to 0$; dispersion relations and dynamical exponents signal loss of conformal invariance for $\alpha<2$.

Edge phenomena are also enhanced: for $\alpha\lesssim 1$, massive Majorana edge modes and quasi-topological edge states appear in both spin and fermionic representations, with gap closures no longer required for phase transitions (Vodola et al., 2015).

6. Frustration, Geometry, and Extended Lattice Realizations

Frustrated lattices and constraints (e.g., triangular, Kagome, Josephson arrays with $0$–$\pi$ junctions) map onto generalized long-range Ising Hamiltonians:

• Frustration, sublattice structure, and topological constraints lead to highly anisotropic, sign-oscillating couplings and novel ordered phases (stripe, clock/clock-like, and nematic orders).
• On triangular lattices, extended and stripe-ordered, clock-ordered, and gapless Kosterlitz–Thouless-like phases can be stabilized, with the phase boundary and universality class tunable by both $\alpha$ and lattice circumference (Humeniuk, 2016, Saadatmand et al., 2018, Koziol et al., 2019).
• In frustrated Josephson junction networks, the emergent Ising Hamiltonian presents long-range, algebraically decaying, anisotropic couplings with rich thermodynamic and quantum entanglement behavior under topological constraints (Neyenhuys et al., 2023).

7. Outlook and Experimental Relevance

Long-range quantum Ising models serve as paradigmatic testbeds for fundamental aspects of quantum phase transitions, dynamics, and quantum simulation. They enable:

• Emulation and exploration of non-trivial universality classes, including continuously varying, mean-field/crossover, and unconventional first-order transitions.
• Quantitative benchmarking of experimental simulations in trapped-ion and Rydberg-atom systems, enabling measurement and verification of critical exponents, correlation decay, entanglement, and dynamical scaling with system size and tunable interaction range (Li et al., 2022, Pagano et al., 2019).
• Investigation of open-system and non-equilibrium phenomena, including aging, non-equilibrium scaling, and universality classes beyond equilibrium Model A (Halimeh et al., 2018).
• Theoretical frameworks accommodating the coexistence of multiple disorder phases and emergent tricriticality under strong correlations and higher-order interactions (Kumar et al., 2021, Herráiz-López et al., 3 Sep 2024).

Advances in machine learning-based wavefunction ansätze and scalable QMC/MPS algorithms now allow for systematic paper in large system sizes and higher dimensions, with neural quantum states providing access to dynamic and non-equilibrium regimes (Naik et al., 9 Dec 2025, Roca-Jerat et al., 5 Jul 2024). The field continues to develop rigorously quantitative methods to map out and understand the emergent physics arising from competing locality, interaction geometry, and quantum fluctuations in long-range Ising systems.

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References: (Koziol et al., 2021, Humeniuk, 2016, Gonzalez-Lazo et al., 2021, Puebla et al., 2019, Shiratani et al., 2023, Sun, 2017, Vodola et al., 2015, Herráiz-López et al., 3 Sep 2024, Roca-Jerat et al., 5 Jul 2024, Naik et al., 9 Dec 2025, Saadatmand et al., 2018, Koziol et al., 2019, Li et al., 2022, Jaschke et al., 2016, Kumar et al., 2021, Román-Roche et al., 2023, Neyenhuys et al., 2023, Pagano et al., 2019, Halimeh et al., 2018).