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Monte Carlo based techniques for quantum magnets with long-range interactions

P. Adelhardt, J. A. Koziol, A. Langheld, K. P. Schmidt

arXiv:2403.00421v1cond-mat.str-elquant-ph

TL;DR

Long-range interactions complicate the study of quantum-critical behavior, including scaling above the upper critical dimension. This review explains two Monte Carlo-based approaches and surveys their use in quantum magnets, emphasizing generalized hyperscaling and the scope of systems with extensive ground-state energy.

  • Problem

    Long-range interactions can alter quantum criticality, and above the upper critical dimension standard finite-size scaling and hyperscaling are not applicable, complicating extraction of critical properties.

    Source: §1 · p. 4

  • Method

    The review explains pCUT+MC for thermodynamic-limit series expansions and SSE QMC for finite systems whose results are analyzed with finite-size scaling.

    Source: §1 · p. 4

  • Results

    The review presents a generalized hyperscaling relation and describes long-range systems whose upper critical dimension can fall to the spatial dimension, yielding mean-field critical behavior for d ≥ duc.

    Source: §2 · p. 7, §2.1 · p. 9

  • Takeaways & Limitations

    The review uses long-range quantum magnets as test beds for studying above-upper-critical-dimension transitions and their scaling in low-dimensional systems.

    Source: §1 · p. 4, §2 · p. 7

    The review excludes σ ≤ 0 strong long-range interactions, focusing instead on weak interactions or competing antiferromagnetic strong interactions with an extensive ground-state energy.

    Source: §1 · p. 3

Abstract

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Long-range interactions are relevant for a large variety of quantum systems in quantum optics and condensed matter physics. In particular, the control of quantum-optical platforms promises to gain deep insights in quantum-critical properties induced by the long-range nature of interactions. From a theoretical perspective, long-range interactions are notoriously complicated to treat. Here, we give an overview of recent advancements to investigate quantum magnets with long-range interactions focusing on two techniques based on Monte Carlo integration. First, the method of perturbative continuous unitary transformations where classical Monte Carlo integration is applied within the embedding scheme of white graphs. This linked-cluster expansion allows to extract high-order series expansions of energies and observables in the thermodynamic limit. Second, stochastic series expansion quantum Monte Carlo which enables calculations on large finite systems. Finite-size scaling can then be used to determine physical properties of the infinite system. In recent years, both techniques have been applied successfully to one- and two-dimensional quantum magnets involving long-range Ising, XY, and Heisenberg interactions on various bipartite and non-bipartite lattices. Here, we summarise the obtained quantum-critical properties including critical exponents for all these systems in a coherent way. Further, we review how long-range interactions are used to study quantum phase transitions above the upper critical dimension and the scaling techniques to extract these quantum critical properties from the numerical calculations.

1. Introduction

The review examines how long-range interactions shape quantum phase transitions and how Monte Carlo-based methods can quantify their critical properties. It focuses on pCUT+MC and SSE QMC, while restricting its discussion to settings with an extensive ground-state energy.

1. Introduction

  • The discussion is restricted to weak long-range interactions or competing antiferromagnetic strong long-range interactions, where an extensive ground-state energy can be defined without rescaling the coupling.For strong long-range interactions with σ ≤ 0, common thermodynamic-limit definitions of internal energy and entropy do not apply; this regime is not reviewed.Source: §1 · p. 3
  • Long-range quantum magnets are relevant to quantum-optical platforms, including trapped ions, cavity-coupled neutral atoms, Rydberg simulators, and ultracold dipolar gases.Many of these platforms can map atomic degrees of freedom onto quantum spin models.Source: §1 · p. 3
  • Long-range interactions can change criticality with their decay strength, lower the upper critical dimension, and make standard finite-size scaling and hyperscaling inapplicable in some regimes.Above the upper critical dimension, dangerous irrelevant variables must be handled to extract critical exponents from finite systems.Source: §1 · pp. 3–4
  • The review explains pCUT+MC, which uses classical Monte Carlo integration in linked-cluster graph embedding to obtain energies and observables in the thermodynamic limit.It also covers SSE QMC for suitable finite systems and the use of finite-size scaling to infer infinite-system properties.Source: §1 · pp. 3–4
  • The review surveys quantum-critical properties of long-range Ising, XY, and Heisenberg magnets, including ferromagnetic and antiferromagnetic models on bipartite and non-bipartite lattices.Its coverage includes scaling above the upper critical dimension and quantum phase transitions in one- and two-dimensional systems.Source: §1 · p. 5

2. Quantum Phase Transitions

This section introduces quantum phase transitions and their critical behavior, then explains why long-range interactions make scaling above the upper critical dimension accessible in low-dimensional systems. It also describes how finite-size observables encode thermodynamic criticality below that dimension.

2. Quantum Phase Transitions

  • A quantum phase transition is a non-analyticity in the ground-state energy as a function of a tuning parameter, and the review focuses on continuous transitions involving diverging characteristic length and time scales.Near a second-order transition, the ground state changes from preserving a Hamiltonian symmetry to breaking it.Source: §2 · p. 6
  • For systems with non-competing long-range interactions, sufficiently small decay exponents lower the upper critical dimension; when d ≥ duc, critical exponents are mean-field, while standard finite-size scaling and hyperscaling do not apply.Long-range interactions can therefore make above-upper-critical-dimension transitions accessible in low-dimensional systems.Source: §2 · p. 7

2.1. Critical exponents in the thermodynamic limit

  • Critical exponents characterize power-law singularities, and universality classes group transitions with the same exponents despite differing microscopic Hamiltonians.For the quantum transitions considered, the section relates these ideas to RG fixed points and scaling forms.Source: §2 · p. 6, §2.1 · p. 8
  • The authors treat the usual hyperscaling relation as a special case of a generalized relation, with particular relevance to the long-range models that enter above the upper critical dimension.They note that the generalized relation reduces to the usual one below that dimension.Source: §2.1 · p. 9

2.2. Finite-size scaling below the upper critical dimension

  • Smaller chains deviate from the thermodynamic-limit susceptibility farther from criticality, and their peak positions shift farther from the critical point.Figure 1 compares linear system sizes from L = 64 to L = 724.Source: Figure 1 · p. 10
  • Finite systems round and shift thermodynamic singularities: susceptibility peaks broaden and move from the critical point, with the rounding tied to ν and peak magnitude to the relevant critical exponents.Finite-size scaling uses these size-dependent observables to relate finite systems to the criticality of the infinite system.Source: §2.2 · pp. 9–10

Appendix B.

The appendix relates renormalization-group scaling to finite-size scaling, then explains how dangerous irrelevant variables modify scaling above the upper critical dimension. In that regime, the characteristic length can grow faster than the system’s linear size, as illustrated by long-range Ising data.

Appendix B.

  • The generalized homogeneity laws derive observable scaling from the free-energy density, while setting the leading irrelevant coupling to zero relies on analyticity.Magnetization homogeneity follows by differentiating the free energy with respect to the symmetry-breaking field.Source: Appendix B · pp. 10–11
  • Scaling exponents are interdependent: the appendix derives relations among α, β, γ, and ν from the homogeneity laws.The stated relations are 2 − α = (d + z)ν, 2 − α = 2β + γ, and γ = β(δ − 1).Source: Appendix B · p. 11
  • Finite-size scaling fits observables across system sizes near criticality to estimate critical exponents and the critical point; at a known critical point, size dependence can yield exponent ratios.For relatively small systems, corrections to scaling may need to be included.Source: Appendix B · pp. 11–12

2.3. Finite-size scaling above the upper critical dimension

  • Above the upper critical dimension, the free-energy density is singular at u = 0, making u a dangerous irrelevant variable whose scaling modifies the finite-size scaling powers.The scaling of u is absorbed into the scaling of the other variables.Source: §2.3 · p. 12
  • Above the upper critical dimension, modified scaling powers are determined only up to a common nonzero factor, so their ratios—not absolute values—are fixed.The chosen absolute values do not affect finite-size scaling or the physics.Source: §2.3 · p. 13
  • A data collapse for the long-range transverse-field Ising chain with σ = 0.3 supports ξ_L ∼ L^ϟ, showing that the characteristic length can exceed L.The passage reports ϟ ≈ 2.2 and contrasts this result with the prior belief that ξ_L was bounded by L.Source: §2.3 · p. 14

3. Monte Carlo Integration

Monte Carlo integration estimates integrals or sums by sampling configurations and averaging reweighted function values. The review describes Markov-chain sampling, including the conditions for convergence and the Metropolis-Hastings construction used in many applications.

3. Monte Carlo Integration

  • Monte Carlo integration estimates an integral or discrete sum as an expectation over sampled configurations, using function values reweighted by the sampling probability.In this review, it is used for white-graph embedding in series expansions and for calculating thermodynamic observables with SSE.Source: §3 · p. 15
  • A Markov-chain random walk samples configurations according to relative weights, which can be used without knowing their full normalization.The transition probabilities determine each next state from the current configuration.Source: §3 · p. 15
  • A chain needs the desired stationary distribution, irreducibility, and aperiodicity; the latter two conditions ensure convergence to that distribution.Irreducibility means every configuration can be reached from any other in finitely many steps.Source: §3 · p. 15
  • Metropolis-Hastings constructs transitions satisfying detailed balance, and symmetric proposal probabilities reduce its acceptance rule to Metropolis acceptance.The review identifies Metropolis-Hastings as the basis of most applications of Monte Carlo integration discussed.Source: §3 · p. 16
  • For Boltzmann weights, proposed moves to lower energy are always accepted, while higher-energy moves are accepted with a temperature-dependent probability.This makes low-temperature sampling favor low-energy states and high-temperature sampling favor the maximum-entropy state.Source: §3 · p. 16

4. Series-expansion Monte Carlo embedding

This section develops linked-cluster expansions for long-range systems using pCUT, white graphs, and Monte Carlo integration of the embedding problem, including thermodynamic-limit energies and excitations. It explains the role of particle additivity in linked-cluster subtraction and when non-additive perturbation methods limit or permit the expansion.

4. Series-expansion Monte Carlo embedding

  • The pCUT+MC approach combines high-order linked-cluster expansions using white graphs with Monte Carlo integration to handle long-range graph embedding.The stated goal is to obtain energies and observables in the thermodynamic limit.Source: §4 · p. 16

4.1. Motivation and basic concepts

  • Linked-cluster expansions obtain thermodynamic-limit contributions by summing connected clusters when the quantity is cluster additive; ground-state energy is an example, but excited-state energies are not generally additive.The section describes restoring additivity for excited states by subtracting ground-state contributions.Source: §4.1 · p. 18

4.2. Perturbation method – perturbative continuous unitary transformations

  • pCUT transforms the Hamiltonian order by order into a quasiparticle-conserving effective Hamiltonian, and the effective Hamiltonian and observables retain cluster additivity.The effective Hamiltonian is block diagonal in quasiparticle number, reducing the problem to separate quasiparticle sectors.Source: §4.2 · pp. 18, 20, 21

4.3. Unraveling cluster additivity

  • The section extends cluster additivity to the quasiparticle basis through particle additivity, because an N-particle restriction on disconnected clusters includes contributions from N + 1 particle distributions and is not itself cluster additive.The formalism is developed for excited-state energies and observables; N-particle states can distribute particles between disconnected clusters in multiple ways.Source: §4.3 · pp. 22–24
  • Particle additivity lets irreducible particle-sector contributions be extracted by an inductive subtraction scheme, starting at N = 0 and using each sector to determine the next.Each irreducible N-particle contribution defines the corresponding operator and its other particle channels.Source: §4.3 · p. 25
  • Effective observables require an additional particle-number-change index d, with their additive contributions likewise determined inductively by subtraction.Positive d creates quasiparticles and negative d annihilates them.Source: §4.3 · pp. 25–26
  • Non-cluster-additive perturbation methods allow quasiparticle processes between disconnected clusters, leaving disconnected contributions and lower-than-expected graph orders in a linked-cluster expansion [162].The review notes exceptions for excited-state energies when the excitation cannot couple to a lower particle channel because it lies in a distinct symmetry sector.Source: §4.3 · p. 26
  • For the transverse-field Ising model high-field expansion, the first excited state avoids direct coupling to the ground state because the perturbation has T−2, T0, and T2 terms but no T±1 terms.The absence of T±1 follows from symmetry.Source: §4.3 · p. 26
  • A projective cluster additive transformation preserves cluster additivity and can calculate high-order perturbative series more efficiently, although this review uses pCUT.Source: §4.3 · p. 26

4.4. Calculating cluster contributions

This section explains how pCUT graph-based linked-cluster calculations isolate irreducible cluster contributions to derive energies, excitation gaps, and spectral weights, and how white-graph decomposition makes calculations with long-range interactions feasible by deferring bond-specific information until embedding.

4.4. Calculating cluster contributions

  • pCUT subtracts subcluster contributions to prevent overcounting, yielding reduced quantities that vanish on disconnected clusters for additive operators and support linked-cluster expansions.For Hamiltonian contributions, the subtraction depends on particle channel and process; ground-state energy is subtracted for diagonal processes, while N = 0 effective observables are already irreducible. Higher-order calculations use a computer program, with maximal orders usually ranging from 8 to 20.Source: §4.4 · pp. 27–29

4.5. Energy spectrum and observables

  • The pCUT framework calculates the ground-state energy and excitation gap, as well as observables such as spectral weights derived from the dynamic structure factor.The overview treats quantities on a suitably designed cluster rather than requiring a full graph decomposition.Source: §4.5 · p. 29

4.5.1. Ground-state energy and elementary excitation gap

  • High-order ground-state energy series yield energy per site and control-parameter susceptibility, while one-quasiparticle hopping amplitudes give the dispersion and excitation gap.The cluster is chosen to accommodate fluctuations through the maximal order; the energy series is valid in the thermodynamic limit up to that order, and the susceptibility and gap show power-law behavior near the critical point.Source: §4.5.1 · pp. 29–30

4.5.2. Spectral properties

  • The pCUT method obtains one-quasiparticle spectral weights from effective observables, which are already irreducible; near critical momentum, the weight diverges with exponent (2 − z − η)ν.The dynamic structure factor describes momentum- and energy-resolved spin correlations and is relevant to inelastic neutron scattering.Source: §4.5.2 · pp. 31–33

4.6. White graph decomposition

  • White-graph decomposition classifies graphs without treating edge colours as a topological attribute, a feature identified as essential for long-range interactions.The review uses white-graph decomposition for its graph decompositions.Source: §4.6 · p. 33

4.6.1. Graph theory

  • Graphs are finite vertex-and-edge structures; connected simple undirected graphs are topologically equivalent when an isomorphism preserves adjacency.An automorphism maps a graph onto itself while preserving its edge set; the number of automorphisms gives its symmetry number.Source: §4.6.1 · pp. 33–34
  • Monomorphisms count embeddings of subgraphs onto the infinite lattice, while vertex colours must be preserved under graph mappings.Colour attributes can distinguish hopping vertices and reduce the coloured graph’s symmetry number.Source: §4.6.1 · p. 34, Figure 8 · p. 35

4.6.2. Graph generation

  • Graph generation uses canonical representatives to avoid duplicate isomorphs, while long-range interactions require generating every connected graph up to the chosen edge order.Graphs and symmetry numbers are stored as bond lists; process-specific coloured graphs track symmetry-equivalent contributions for hopping and spectral-weight calculations.Source: §4.6.2 · pp. 35–36

4.6.3. White-graphs for long-range interactions

  • White graphs avoid infinitely many interaction-specific graph types by assigning abstract expansion parameters to edges and restoring distance-dependent interactions during lattice embedding.This reduces the number of graphs at a given order; multiple interaction types can be represented by multiple parameters per edge. An efficient representation also uses nested containers and bitwise edge tracking to avoid repeated state calculations and explicitly subtracting subcluster contributions.Source: §4.6.3 · pp. 37–39, Figure 9 · p. 37

4.7. Monte Carlo embedding of white graphs

The section describes Monte Carlo methods for embedding linked-cluster white-graph contributions to obtain thermodynamic-limit quantities for long-range systems. It covers configuration sums for energies and quasiparticle observables, including a Markov-chain approach using importance sampling and shift and rift moves.

4.7. Monte Carlo embedding of white graphs

  • White-graph embedding computes each graph contribution once, substitutes physical interaction parameters for each embedding, and weights and sums contributions to recover lattice quantities.The embedding factor counts graph embeddings; white graphs avoid edge-colour matching, but their abstract expansion parameters must be replaced with the interactions for each embedding.Source: §4.7 · pp. 39–40

4.7.1. Conventional nearest-neighbour embedding

  • For nearest-neighbour lattices, the embedding factor counts monomorphisms divided by graph automorphisms, while reduced subgraph contributions are subtracted to obtain energy and quasiparticle observables.The framework yields ground-state energy per site, one-quasiparticle hopping amplitudes and dispersion, and spectral weight as perturbative series.Source: §4.7.1 · pp. 40–42

4.7.2. Embedding for models with long-range interactions

  • For long-range ground-state energy, contributions are grouped by graph vertex count and evaluated over non-overlapping configurations, substituting each embedded edge with its algebraically decaying interaction in a Monte Carlo sum.Grouping by vertex count permits graph contributions to share an integrand because the long-range embedding contribution depends on relative vertex positions.Source: §4.7.2 · pp. 43–44
  • Because long-range interactions allow infinitely many hoppings at each order, the dispersion is evaluated as a perturbative series at selected momenta rather than as an analytic function of momentum.The hopping embedding can be treated like the energy embedding by summing configurations and accounting for hopping constraints with the coloured-graph symmetry number; spectral-weight evaluation is analogous.Source: §4.7.2 · pp. 45–47

4.7.3. Monte Carlo algorithm for the long-range embedding problem

  • The long-range Monte Carlo sum samples vertex positions on the fully connected lattice, excluding overlaps because vertices do not self-interact and overlap would imply infinite interaction strength.The configuration sum comprises one lattice-position sum per graph vertex, subject to the non-overlap restriction.Source: §4.7.3 · p. 47
  • Importance sampling rewrites the target sum as an expectation under a probability weight, then uses an analytically solvable reference sum to eliminate the unknown partition function.The reference integrand should resemble the target integrand and contribute in similar regions of configuration space to promote convergence.Source: §4.7.3 · pp. 48, 50
  • Each Monte Carlo step uses shift moves for confined fluctuations and rift moves that draw new vertex separations from ζ-function distributions to capture long-range asymptotic behavior.The two move types are illustrated for a linear graph on a two-dimensional lattice in Figure 11.Source: §4.7.3 · pp. 50–52, Figure 11 · p. 51
  • The pCUT+MC approach reaches order 10 for 1qp spectral weight, order 11 for the 1qp gap, and order 13 for ground-state energy; its run count scales quadratically with perturbative order.Typical runs use 10–20 seeds for 12–24 hours.Source: §4.7.3 · p. 53
  • The algorithm currently applies only to lattices with a trivial single-site unit cell; extension requires intra-cell moves and handling matrix-valued dispersion, which may also affect convergence.The reference sum may need adaptation for the changed Monte Carlo dynamics.Source: §4.7.3 · p. 54

4.8. Series extrapolation

DlogPadé extrapolation uses high-order perturbative series to estimate critical points and exponents, while the pCUT+MC workflow generates and embeds graph contributions to obtain thermodynamic-limit series. The resulting series are extrapolated to extract critical quantities and construct dispersion near criticality.

4.8. Series extrapolation

  • DlogPadé extrapolants use high-order series to estimate critical points and exponents, including beyond the series’ radius of convergence.For a second-order transition, the physical extrapolant pole estimates λc and its residuum estimates θ; biased extrapolants can use a known λc.Source: §4.8 · pp. 54–55
  • A single-quantity series expansion cannot capture level crossings, leaving first-order phase transitions inaccessible to this critical-property extraction.Source: §4.8 · p. 54
  • Reliable estimates combine the highest-order extrapolants from sufficiently populated, more-diagonal families, while their spread gives a subjective rather than rigorous error.Families are grouped by L − M, usually with |L − M| ≤ 2 or 3.Source: §4.8 · p. 55

4.9. Workflow of series expansion Monte Carlo embedding

  • The pCUT+MC workflow generates graphs once, calculates pCUT white-graph contributions, converts them into embeddable functions, and Monte Carlo samples them on the lattice.Graph and process symmetries are accounted for in the embedding; averaging runs across seeds produces the perturbative series.Source: §4.9 · pp. 55, 57, Figure 12 · p. 56
  • The averaged perturbative series yield critical points and exponents for ground-state energy, the 1qp gap, and 1qp spectral weight, and can construct dispersion near criticality.The associated exponents are α, zν, and (2 − z − η)ν, respectively.Source: §4.9 · p. 57

5. Stochastic series expansion quantum Monte Carlo

This section describes stochastic-series-expansion quantum Monte Carlo methods for transverse-field Ising and unfrustrated Heisenberg models, including configuration sampling, updates, and measurements of observables, responses, and correlations. It also addresses finite-size scaling and effectively zero-temperature simulations in the presence of long-range correlations, and summarizes path-integral QMC findings in extended Bose-Hubbard models.

5. Stochastic series expansion quantum Monte Carlo

  • SSE represents thermal expectation-value sampling through basis states and imaginary-time operator sequences whose non-negative weights permit Markov-chain Monte Carlo.A sufficiently large fixed sequence-length cutoff is used because longer sequences are exponentially suppressed.Source: §5 · pp. 57, 59

5.1. Algorithm for arbitrary transverse-field Ising models

  • For arbitrary Ising couplings, the chosen operator decomposition avoids the sign problem by adding positive constants to Ising operators and satisfies the no-branching condition.Nonzero-weight configurations place ferromagnetic and antiferromagnetic bonds on aligned and antialigned spins, respectively.Source: §5.1 · pp. 60–61
  • The transverse-field Ising SSE update alternates diagonal operator exchanges with nonlocal off-diagonal cluster updates to preserve configuration weight.Field operators must occur an even number of times per site to preserve imaginary-time periodicity; the cluster update addresses this and the risk of invalidating Ising-operator weights.Source: §5.1 · pp. 61, 63
  • The off-diagonal update partitions configurations into disjoint space–imaginary-time clusters, each flipped with probability 1/2 without changing the configuration weight.Clusters are built deterministically; spins with no operators acting on them can also be flipped with probability 1/2.Source: §5.1 · p. 65
  • For arbitrary transverse-field Ising models, combining diagonal and off-diagonal cluster updates samples the entire configuration space.Source: §5.1 · p. 67

5.2. Algorithm for unfrustrated Heisenberg Models

  • Unfrustrated Heisenberg models have no lattice loop with an odd number of antiferromagnetic bonds, enabling non-negative SSE weights and loop updates that preserve configuration weights.Flipping each constructed loop independently with probability 1/2 satisfies detailed balance; the method covers unfrustrated long-range antiferromagnetic ladders and bilayers.Source: §5.2 · pp. 67, 70

5.3. Observables

  • Diagonal observables can be estimated by averaging over imaginary time, while the mean energy has a simple estimator; at low temperatures, heat capacity subtracts large numbers.Source: §5.3 · p. 72

5.3.1. Linear response and correlation functions

  • For diagonal operators, SSE samples imaginary-time correlations for linear responses; measuring all site pairs costs O(βN^3), reducible to O(βN) when correlating one fixed site with all sites.Off-diagonal operators require case-specific estimators; imaginary-time separations correspond to a binomial distribution of SSE propagation-step separations.Source: §5.3.1 · pp. 73–75
  • Long-range correlations decay algebraically even away from criticality, making finite-system characteristic lengths difficult to define and crucial for testing Q-FSS scaling ξL ∼ L^ϟ rather than L [34].Some definitions can yield ξ∞ = ∞ away from criticality.Source: §5.3.1 · pp. 75–76

5.4. Sampling at effectively zero temperature

  • Beta-doubling checks observable convergence across successively doubled inverse temperatures, capturing L^zϟ scaling and corrections without requiring prior knowledge of z.The method can be tested where the finite-size gap is expected to be smallest, then the validated temperature used across the parameter range; Figure 18 illustrates the procedure.Source: §5.4 · p. 77, Figure 18 · p. 78

5.5. Overview: Path integral quantum Monte Carlo algorithms for long-range models

  • PI QMC studies find that long-range interactions stabilize more solid phases than nearest-neighbour models and produce supersolid ground states in extended Bose-Hubbard models [64].These quantitative simulations are relevant to dipolar-gas experiments, and the authors expect SSE implementation for these models to be straightforward using directed loops.Source: §5.5 · p. 79

6. Long-range transverse-field Ising models

This section reviews long-range transverse-field Ising models and their quantum criticality, covering ferromagnetic transitions across three universality regimes and antiferromagnets on bipartite and non-bipartite lattices. It describes how long-range interactions relate to low-field order and phase behavior, including debated critical behavior at small decay exponents and competition between stripe and clock order.

6.2. Antiferromagnetic long-range transverse-field Ising models on bipartite lattices

  • For antiferromagnetic interactions on bipartite lattices, the transition separates an antiferromagnetic low-field phase from a symmetric x-polarised high-field phase, with no indicated change in universality class as interaction range varies.The review also notes that on non-bipartite lattices, long-range interactions may substantially alter the nearest-neighbour phase diagram.Source: §6 · p. 80

6.1. Ferromagnetic long-range transverse-field Ising models

  • Ferromagnetic long-range interactions stabilize the ordered phase, shifting the transition to larger transverse fields, and produce short-range, non-trivial long-range, and long-range mean-field critical regimes.As σ → 0, the critical field diverges because the single-spin-flip energy cost becomes extensive.Source: §6.1 · pp. 80–81, §6 · p. 80

6.1.1. ϕ4-theory for quantum rotor models with long-range interactions

  • Long-range ϕ4 theory gives a lower critical dimension dlc = σ/2 and Gaussian η = 2 − σ; its Gaussian fixed point is stable for d ≥ duc = 3σ/2.Below duc, the theory flows to a non-trivial fixed point; for σ < 2, z ≠ 1 and η remains 2 − σ. The short-range boundary is σ = 2 − ηSR.Source: §6.1.1 · pp. 82–83

6.1.2. Critical exponents and critical points for one- and two-dimensional systems

  • For the ferromagnetic chain, pCUT+MC and SSE QMC cover all three regimes and reproduce their limiting critical behavior, but finite length scales and logarithmic corrections complicate resolving regime boundaries.The boundary may shift from σ = 2 to 2 − ηSR; the reviewed methods could not establish this shift reliably, especially in two dimensions.Source: §6.1.2 · pp. 83, 85

6.1.3. Scaling above the upper critical dimension

  • Above the upper critical dimension, standard finite-size scaling and hyperscaling fail; quantum Q-FSS instead modifies the correlation-length scaling, and SSE measurements of its exponent agree with Q-FSS predictions.For the long-range regime, duc = 3σ/2 and the model is above it when σ < 2d/3.Source: §6.1.3 · pp. 87–88

6.2. Antiferromagnetic long-range transverse-field Ising models on bipartite lattices

  • For (d + σ)>0, the two nearest-neighbor low-field ground states remain adiabatically connected despite competing long-range couplings; at (d + σ)=0, every zero-z-magnetization state is a ground state at h=0 [24] [25] [26] [29] [31] [32].At the all-to-all point, any finite transverse field places the ground state in the x-polarized phase [30] [40].Source: §6.2 · p. 88
  • Critical fields decrease with σ, reaching zero at (d + σ)=0; chains show (1+1)D Ising criticality for σ≥1.25, while square lattices show (2+1)D Ising criticality for σ>0.5 [24] [25] [26] [29] [31] [32].For chains, a later DMRG study suggests Ising universality for any σ>−1 [26], challenging earlier indications of changing exponents below σ=1.25 [24].Source: §6.2 · p. 89
  • For chains below σ=1.25, early studies suggest changing exponents [24], but later DMRG finds Ising behavior for σ>−1 [26]; Monte Carlo methods struggle in this regime [25] [31] [32].SSE simulations become impractical as interaction competition grows, while high-field series extrapolations become challenging as the critical field decreases.Source: §6.2 · p. 89

6.3. Antiferromagnetic long-range transverse-field Ising models on non-bipartite lattices

  • On the triangular lattice, an infinitesimal transverse field selects Z2 × Z3 clock order by order-by-disorder; its transition from the polarized phase occurs at h/J=1.65 ± 0.05 and is (2+1)D-XY.The nearest-neighbor low-field degeneracy is broken through an effective quantum dimer model.Source: §6.3 · pp. 92–93
  • Long-range interactions stabilize a gapped, sixfold-degenerate stripe phase, distinct from clock order; the clock phase lies between stripes and the high-field phase, and both neighboring phases expand as long-range interactions strengthen [27] [29] [30].The stripe states remain stable against finite transverse fields [30] [110].Source: §6.3 · p. 93
  • At small σ, iDMRG suggests clock order vanishes in favor of a direct transition to stripes [27], but the phase diagram remains inconclusive and no method reliably resolves this regime [27] [29] [30].A first-order transition from the polarized phase to stripes is considered possible, but high-field series cannot detect it from the excitation gap.Source: §6.3 · pp. 93–94
  • On Kagome and Ruby Rydberg lattices, full long-range interactions produce gapped zero-field orders distinct from fluctuation-selected orders [110], while a Ruby-lattice J1-J2-J3 model can select the same order through both mechanisms.A general theory of when long-range interactions and quantum fluctuations select the same or different orders remains unknown.Source: §6.3 · p. 95

7. Long-range transverse-field XY chain

The section contrasts particle-conserving isotropic XY chains, whose high-field dispersion is exactly solvable, with anisotropic chains requiring pCUT+MC. It reviews how ferromagnetic and antiferromagnetic interactions shape critical exponents and phase boundaries.

7.1. Isotropic long-range XY chain in a transverse-field

  • For β = 0, particle conservation makes the high-field ground state and first-order quasiparticle dispersion exact, allowing gap closing and critical dispersion to determine λc, z, and ν.The ferromagnetic gap closes at λc = (2ζ(1 + σ))^-1; the antiferromagnetic gap closes at λc = (2η(1 + σ))^-1.Source: §7 · p. 97, §7.1 · pp. 98, 100
  • For ferromagnetic β = 0, σ > 2 gives z = 2 and ν = 1/2, while for σ < 2, z decreases linearly and ν increases as 1/z.The field-theory predictions agree with exponents extracted from the high-field excitation gap.Source: §7.1 · pp. 98–99
  • For antiferromagnetic β = 0, z = 2 and ν = 1/2 hold for all σ, while λc diverges as σ approaches −1.The gap is defined for σ > −1 because the Dirichlet eta function converges there.Source: §7.1 · p. 100

7.2. Ferromagnetic anisotropic long-range XY chain in a transverse field

  • For ferromagnetic β > 0, pCUT+MC finds long-range mean-field behavior for σ < 2/3, nearest-neighbour 2D-Ising behavior at large σ, and continuously varying exponents between them.At β = 0, zν = 1 for all σ; for β > 0, increasing β stabilizes the low-field phase, and exact diagonalization agrees with pCUT+MC at β = 1/2.Source: §7.2 · p. 100
  • For ferromagnetic β > 0, the regime boundaries are not accurately resolved, and logarithmic corrections occur at σ = 2/3, the upper critical dimension.The reported biased extrapolations at σ = 2/3 show qualitatively good agreement for different β values.Source: §7.2 · pp. 100–101

7.3. Antiferromagnetic anisotropic long-range XY chain in a transverse-field

  • For antiferromagnetic β > 0, λc shifts upward as σ decreases and diverges near −1; gap exponents are reliably extracted only to about σ = 0.5.Within method limitations, results support zν = 1 and the nearest-neighbour 2D-Ising universality class across considered σ; a separate small-system study claims crossover behavior at σ = −0.4.Source: §7.3 · pp. 101–102

8. Long-range Heisenberg models

This section reviews long-range Heisenberg models and transitions toward Néel order in dimer systems and chains, focusing on their critical properties and how these depend on symmetry, dimension, and interaction decay exponent. Numerical results characterize distinct critical regimes, while discrepancies with other predictions leave some properties unsettled.

8. Long-range Heisenberg models

  • For continuous-symmetry Heisenberg order, the Hohenberg-Mermin-Wagner theorem excludes one-dimensional long-range order for σ > 2, while long-range interactions can produce anomalous or gapped Goldstone modes [129–131,294–296,320].The cited dynamic study reports a generalized Higgs regime for antiferromagnetic interactions at σ ≤ −2 and evidence that it occurs already for σ ≤ 0.2 while the model remains extensive.Source: §8 · pp. 104–105

8.1. Staggered antiferromagnetic long-range Heisenberg square lattice bilayer model

  • SSE QMC and pCUT+MC critical points agree excellently for the square-lattice bilayer, and both approaches confirm the three O(3) quantum-rotor critical regimes within their limitations.The QMC exponents better capture regime boundaries, while pCUT+MC overestimates exponents in the short-range regime; the methods extract different exponents, preventing direct comparison.Source: §8.1 · p. 107, Figure 32 · p. 108

8.2. Staggered antiferromagnetic long-range Heisenberg ladder models

  • The proposed deconfined ladder transition is disfavored by the trivial rung-singlet phase and reported exponents, while an observed dynamical gap may reflect finite-size effects from anomalous Goldstone dispersion.The section also notes that anomalous Goldstone modes and the ladder critical behavior need further clarification.Source: §8.2 · pp. 111–112
  • Long-range Heisenberg ladders show long-range mean-field and continuously varying critical regimes, but their lower critical decay exponent appears smaller and non-universal relative to the field-theory prediction.The exponents diverge near the lower critical boundary in FRG, whereas pCUT+MC gives ν ≈ 1; extraction is difficult, especially for α and near σ = 2/3, where logarithmic corrections occur.Source: §8.2 · pp. 111–112

8.3. Staggered antiferromagnetic long-range Heisenberg chain

  • The nearest-neighbour chain starts from a QLRO ground state with fractionalized spinons, while non-frustrating long-range interactions induce Néel order.The QLRO-to-ordered transition is expected only for σ < 2; its low-energy description uses the k = 1 Wess-Zumino-Witten nonlinear σ model with a long-range coupling [22].Source: §8.3 · p. 112
  • The critical line diverges near σ ≈ 1.8, terminates at the marginal point σ = 1, λ = 0, and is bounded by the HMW theorem.Unlike the ladder, the chain’s QLRO phase exists only for σ > 1; for σ < 1, λ is irrelevant and the system is always antiferromagnetically ordered [22].Source: §8.3 · p. 112
  • η = 2 − σ matches field-theoretical expectations for 1 ≤ σ ≤ 1.3, then SSE QMC results deviate as σ increases.The dynamical exponent is z = 1 at σ = 1 and quickly drops to about 0.75, remaining consistent with that value through σ = 1.7; this differs from the RG prediction z = 1. Finite-size-scaling estimates from ED and DMRG agree very well with QMC at λ = 1, and the chain does not show long-range mean-field behaviour.Source: Figure 35 · p. 113, §8.3 · p. 114
  • The numerical and field-theoretical descriptions remain discrepant, so the chain’s critical properties are not yet fully settled.In particular, the observed z differs from the RG prediction, and QMC estimates of ν were not accurate enough for comparison with the RG result [22].Source: §8.3 · p. 114

9. Summary and outlook

The review synthesizes quantum-critical properties of long-range quantum magnets and surveys complementary Monte Carlo methods, their constraints, and future directions. It also discusses quantum finite-size scaling above the upper critical dimension.

Complementary methods

  • pCUT+MC produces thermodynamic-limit series through white-graph embedding, while SSE QMC studies large finite systems and uses finite-size scaling to infer infinite-system properties.Both approaches include the full long-range interaction and can be used across spatial dimensions and geometries.Source: §9 · p. 114

Review scope

  • The review organizes phase diagrams and critical exponents for one- and two-dimensional long-range Ising, XY, and Heisenberg magnets, and surveys scaling above the upper critical dimension.It reports that the methods generally apply to unfrustrated systems, with exceptions for the one-dimensional isotropic XY and Heisenberg chains.Source: §9 · p. 114

Scope and limitations

  • For frustrated systems, both Monte Carlo techniques apply to Ising interactions, but pCUT+MC is generally the applicable method for other interactions; both approaches struggle at small σ.The stated exception to the review's general applicability is the one-dimensional isotropic XY and Heisenberg chains.Source: §9 · pp. 114–115

Research directions

  • Long-range interactions combined with geometric frustration are presented as a route to exotic phases, including a Z2 quantum spin liquid on the Ruby lattice.The review anticipates further development through analogue quantum simulation platforms such as programmable Rydberg atom simulators.Source: §9 · p. 115

Method outlook

  • Future method work includes extending pCUT+MC to arbitrary unit cells, larger spins, and multispin interactions, and developing SSE QMC for long-range interactions with frustration.The review also identifies SSE QMC applications to extended long-range Bose-Hubbard models as a natural development.Source: §9 · p. 115

Scaling outlook

  • The review identifies the small-temperature crossover between classical and quantum quantum-finite-size scaling as an open question.Quantum scaling above the upper critical dimension has only been treated successfully in recent years.Source: §9 · p. 115

Abbreviations

The abbreviations cover quantum Monte Carlo methods, phase-transition terminology, long-range model classes, and related concepts.

Numerical methods

  • DMRG means Density Matrix Renormalisation Group, FRG means Functional Renormalisation Group, and FSS means Finite-Size Scaling.Source: Abbreviations · p. 117

Methods

  • pCUT means perturbative Continuous Unitary Transformation, MCI means Monte Carlo Integration, and LCE means Linked-cluster Expansion.Source: Abbreviations · p. 117

Model terminology

  • LRI means Long-range Interactions, and LRTFIM means Long-Range Transverse-Field Ising Model.Source: Abbreviations · p. 117

Methods

  • QMC means Quantum Monte Carlo, SSE means Stochastic Series Expansion, and MC means Monte Carlo.Source: Abbreviations · p. 117

Phase-transition terminology

  • QPT means Quantum Phase Transition, QLRO means Quasi Long-Range Order, and qp means quasiparticle.Source: Abbreviations · p. 117

Related concepts

  • DIV means Dangerous Irrelevant Variable, VBS means Valence bond solid, and HMW theorem means Hohenberg-Mermin-Wagner theorem.Source: Abbreviations · p. 117

Appendix A. Short-range O(n) transition ϕ4-theory

The short-range O(n) transition is represented by a field theory whose order-parameter field and couplings can be analyzed through scaling and renormalisation-group flows. This framework relates fixed points and correlators to critical exponents, with different behavior above and below the upper critical dimension.

Appendix A. Short-range O(n) transition ϕ4-theory

  • The coarse-grained field ϕ represents the order parameter, and tuning r changes the action from a symmetric to a symmetry-broken minimum.The field theory is presented as a distilled description of short-range Ising criticality; its fields may have multiple components for O(n) symmetry breaking.Source: Appendix A · pp. 117–118
  • Gaussian power counting gives z = 1, [ϕ] = (d − 1)/2, and [r] = 2 for the short-range theory.The action’s Gaussian part retains its structure under scaling when the field and mass coupling are rescaled appropriately.Source: Appendix A · p. 119
  • The quartic coupling has scaling dimension [u] = 3 − d, making d = 3 the upper critical dimension.For d > 3, u flows toward zero; below the upper critical dimension it must be included perturbatively.Source: Appendix A · pp. 119–120
  • Below the upper critical dimension, the coupled flow has a nontrivial Wilson-Fisher fixed point distinct from the Gaussian fixed point.The fixed point and linearized flow can be treated perturbatively in ϵ = d_uc − d.Source: Appendix A · p. 121
  • Self-energy contributions can alter the propagator’s small-q behavior, producing an anomalous dimension that modifies the field’s naive scaling dimension.For short-range models with vanishing self-energy, such as the cited superfluidity-onset theory, this field-dimension modification is unnecessary.Source: Appendix A · p. 121
  • The correlation-length exponent ν is the inverse relevant flow eigenvalue at the transition fixed point, while η characterizes the propagator’s momentum dependence.The field-theory correlator and auxiliary-field derivatives connect to magnetisation, susceptibility, and the fluctuation-dissipation relation.Source: Appendix A · pp. 121–122

Appendix B. Generalised homogenious functions

This appendix collects definitions and properties of generalised homogeneous functions (GHFs), reproducing material from. It describes their scaling powers and behavior under several operations.

Appendix B. Generalised homogenious functions

  • A GHF scales as f(λ^a1x1, ..., λ^anxn) = λ^af f(x1, ..., xn), with nonzero scaling powers for its variables.The appendix attributes its definitions, theorems, and comments on GHFs to.Source: Appendix B · p. 122
  • A GHF with n variables has only n independent scaling powers, since its scaling relation can be normalized when af is nonzero.The appendix gives the corresponding normalized relation using p = 1/af.Source: Appendix B · p. 122
  • Products of GHFs add their scaling powers, while sums remain homogeneous only when the summands have equal scaling powers.The scaling power of a qualifying sum is the common scaling power of its terms.Source: Appendix B · p. 122
  • The appendix states that a GHF has power-law singularities along principal axes, with exponent af/ai for approach along xi.The stated axis behavior is proportional to |xi|^(af/ai).Source: Appendix B · p. 123
  • Fourier transformation in one variable preserves generalised homogeneity while changing the function and transformed-variable scaling powers.The transformed powers are given as ãf = af − dãj and ãj = −aj.Source: Appendix B · p. 123

Appendix C. Walker’s method of alias

Walker’s alias method preprocesses a discrete probability distribution into acceptance probabilities and aliases. Sampling then draws from the target distribution in constant time after a linear-time table setup.

Appendix C. Walker’s method of alias

  • Walker’s method samples a discrete distribution in O(1) time after an O(N) table setup, unlike rejection sampling’s O(N) expected steps.The tables are constructed once before simulation, so the method does not alter the overall algorithm’s complexity.Source: Appendix C · p. 124
  • The setup rescales probabilities to Pj = N · qj, then redistributes weights from entries with Pj ≥ 1 to fill those with Pj < 1.The transferred weight determines each alias and its associated shortfall probability.Source: Appendix C · p. 124
  • Each draw selects an index uniformly, accepts it when u < Pk, and otherwise returns its precomputed alias Ak.The acceptance probabilities and aliases reproduce the original distribution qj.Source: Appendix C · pp. 124–125

Appendix D. Critical exponents for long-range Heisenberg ladders

The figure presents critical exponents for the Néel-ordering transition in unfrustrated antiferromagnetic long-range Heisenberg ladders. It juxtaposes pCUT+MC data, rotor-model data, and long-range mean-field references across the indicated σ regime.

Appendix D. Critical exponents for long-range Heisenberg ladders

  • The panels display α, β, γ, δ, η, ν, and z for the Néel-ordering transition in unfrustrated long-range Heisenberg ladders.The caption labels β as first-row left, the same position it assigns to α.Source: Figure A2 · p. 126
  • pCUT+MC points compare parallel with parallel-plus-diagonal interactions, alongside one-dimensional O(3) rotor-model exponents.The caption attributes the pCUT+MC data to Ref. [35] and the rotor-model data to Ref. [243].Source: Figure A2 · p. 126
  • The blue shading marks the σ regime where long-range mean-field criticality is expected, and dashed lines show its exponents.Source: Figure A2 · p. 126
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