Graph lattice sums and graph zeta functions for long-range interacting quantum lattice models

1. Introduction

• Exact diagonalization is limited to about 25 sites, while the infinite-lattice limit remains a central target for quantum lattice simulations.Symmetries and supercomputing have reached about 50 spin-1/2 sites.Source: §1 · p. 2
• Linked-cluster expansions give formally exact series within their convergence radius, but their coefficients require high-dimensional graph lattice sums that Monte Carlo evaluates with cluster-scale runtimes and stochastic errors.For dispersion relations, the sums are oscillatory and must be recomputed at each wavevector.Source: §1 · pp. 2–3
• Earlier work computed circle-graph lattice sums with linear complexity in node number; this paper extends the reduction to all graphs appearing in long-range quantum-lattice simulations.The framework combines graph theory with generalized zeta functions and high-dimensional lattice-sum methods.Source: §1 · p. 3
• The framework covers block factorization, analytic evaluation of basic blocks, semi-analytic construction of series-parallel blocks, and tensor-network bucket elimination for highly connected graphs.The paper benchmarks the methods and applies them to reproduce published Monte Carlo results for the long-range transverse-field Ising model.Source: §1 · p. 4

2. Problem statement and graph lattice sums

• Graph lattice sums assign summable edge kernels to a connected two-terminal graph and sum over lattice positions with a momentum-dependent phase; power-law kernels with compactly supported corrections define graph zeta functions.For graph zeta functions, the regularized power laws must satisfy min_e,j Re(ν_ej) > d for the kernels to be summable.Source: §2 · pp. 5, 7, 8
• Graph lattice sums are absolutely convergent and uniformly bounded in momentum, independent of the pinned node and position, and simplify when terminals coincide or momentum is zero.Translational invariance explains the removal of the pinned vertex from the sum over relative positions.Source: §2 · p. 5
• Parallel edges can be replaced by one edge whose kernel is the orientation-adjusted Hadamard product of the original kernels, without changing the graph lattice sum.The merged graph remains simple and connected, and the merged kernel remains summable.Source: §2 · p. 7
• Exact diagonalization scales exponentially with site count, while graph decompositions represent Taylor coefficients of energy density and dispersion through iterated lattice sums whose dimension grows with expansion order.High-order corpus sums can extract physical quantities for gapped quantum lattice models [4].Source: §2 · pp. 8, 10
• Linked-cluster methods express perturbative coefficients as finite weighted sums of graph lattice sums, with graph corpora independent of the lattice and interaction kernel.The section identifies pCUT [34] [33] and its long-range white-graph extensions [27] [25], as well as pCAT, as examples of such methods.Source: §2 · p. 10
• The evaluation challenges are exponential per-graph complexity, arbitrarily slow convergence as the power-law exponent ν approaches d, recomputation at every momentum, and exponential corpus growth with order.The following sections solve the first three issues and severely reduce the impact of corpus growth.Source: §2 · p. 12
• Runtime falls from tens of thousands of core-hours for Monte Carlo sums at one momentum to minutes on a standard laptop for the full momentum grid at improved precision.The method addresses evaluation issues (1)–(3) and severely reduces the impact of exponential corpus growth; it begins by factoring high-dimensional graph lattice sums into blocks.Source: §2 · p. 12

3. Block factorization of graph lattice sums

• Naive lattice discretization scales as O(N^((|V|-1)d)), whereas block factorization reduces the exponential dependence on graph size to dependence on the largest block.For tree graphs, the scaling becomes linear in the number of nodes.Source: §3 · p. 12
• A cutvertex is shared by multiple blocks, and removing it disconnects the graph; the blocks and cutvertices form a block-cut tree.A block is a maximal connected subgraph without a cutvertex, or a single bridge.Source: §3 · pp. 12–13
• At a cutvertex, the graph lattice sum splits into a product of sums for the two subgraphs, with an attachment evaluated at zero momentum when both terminals lie in the other subgraph.The split follows by pinning the shared vertex using translational invariance and separating the edge products.Source: §3 · p. 13
• The terminal-to-terminal path through the block-cut tree forms the spine, and its blocks carry the routed momentum; the remaining attachments are momentum-independent.The full decomposition follows recursively from cutvertex splitting.Source: §3 · pp. 13–14
• Articulation vertices and biconnected blocks can be found in linear time and storage, and tree graph sums reduce to lattice Fourier transforms of the interaction kernel.For power-law interactions, the tree result reduces to the Epstein zeta function.Source: §3 · p. 14

Fourier transforms and Epstein zeta functions

• Epstein zeta functions represent lattice Fourier transforms of power-law kernels, with derivatives and rapidly decaying Fourier series extending the approach to general interaction kernels.The Epstein zeta function is the lattice Fourier transform of a homogeneous kernel; higher-order k derivatives allow general anisotropies.Source: §4 · pp. 15–16

Analytic blocks

• Bridge blocks are analytic Fourier transforms of the combined edge kernels, with power-law contributions evaluated through Epstein zeta functions.For equal terminals, the bridge result is obtained by setting k = 0.Source: §4 · pp. 16–17
• Cycle blocks without momentum have a full-precision representation at linear numerical cost, extending to general summable interaction kernels [9].Source: §4 · p. 17

Block census and caching

• 70.4% of 0qp blocks and 83.9% of 1qp blocks are analytic bridges or cycles; after caching, these types account for 4.7% and 7.7% of blocks requiring computation.The corpora contain 19,207 0qp block occurrences and 70,000 1qp occurrences before caching.Source: §4 · pp. 18–19
• Block caching reuses recurring block evaluations, reducing 19,207 0qp occurrences to 2,574 distinct blocks and 70,000 1qp occurrences to 2,868.The corresponding reuse factors are 7.5 and 24.4.Source: §4 · p. 18

Graph operations and composition

• Series composition corresponds to pointwise multiplication, while parallel composition corresponds to periodic convolution of graph lattice sums.A 1-sum attachment is represented by multiplying by a zero-momentum evaluation.Source: §5 · p. 20

Series-parallel construction

• The operations generate exactly the series-parallel two-terminal multigraphs; after block factorization, this covers graphs whose spine and attachment blocks meet the corresponding terminal conditions.Source: §5 · p. 22
• A series-parallel graph lattice sum can be built from edge transforms using |V| − 2 multiplications and |E| − |V| + 1 convolutions, or |E| − 1 operations total.Equal terminals require one additional zero-momentum evaluation.Source: §5 · p. 22

Treewidth characterization and scope

• A block is series-parallel precisely when its augmented graph has treewidth at most 2, but for distinct terminals the classification depends on their positions.A block with treewidth at most 2 alone need not be series-parallel for a given terminal pair.Source: §5 · pp. 23–24

Semi-analytical evaluation

• Algebraic tails make repeated convolutions difficult to evaluate at full precision; the section's semi-analytical algebra addresses this with linear cost in operations and log-linear cost in discretization parameters.Source: §5 · p. 24

• Power-law graph lattice sums are evaluated by separating leading non-analyticities into Epstein zeta functions and representing the regular part as a rapidly decaying Fourier series.This structure supports kernels with tails whose exponents are close to the lattice dimension and scales linearly in the number of operations and log-linearly in the number of discretization points.Source: §6 · p. 25
• Both convolution and pointwise multiplication preserve semi-analyticity, retaining the singular terms analytically while placing faster-decaying contributions in the Fourier series.Convolutions use the closure of Epstein zeta functions under convolution; multiplication analytically identifies leading product singularities and transfers the remainder to the Fourier series.Source: §6 · pp. 26–27
• O(|E| nd log(nd)) operations suffice for a series-parallel graph zeta function with uniform power-law kernels when Epstein terms are compressed after each operation.With cutoff νmax = d + σmax, only terms combining at most σmax/σ edges survive; the number of retained exponents is bounded independently of graph size after equal exponents are merged.Source: §6 · p. 30
• Terms with exponents in d + 2N are compressed before multiplication, while product exponents landing there generate no Epstein term because their singular product is polynomial and belongs in the Fourier series.Terms sufficiently close to these poles also require care because large components can cancel and amplify truncation error.Source: §6 · p. 31
• For uniform exponents ν = d + σ with σmax ≥ σ + 2 and no exponent in d + 2N, series-parallel block truncation error scales as n^-(d+σ+2), uniformly in k.Multiplication is exact on the sampled momentum grid, while truncating regular parts introduces tail errors in later operations; the stated block scaling is also observed numerically in Fig. 5.Source: §6 · p. 31
• The algorithm constructs graph lattice sums by applying convolution to eliminate parallel edges and multiplication to eliminate degree-2 nodes, compressing large-exponent Epstein terms after each operation.Compression moves terms above a threshold into the Fourier series while retaining leading-order singularities analytically and keeping the Epstein-term count bounded.Source: §6 · pp. 29, 31
• 93.70% of 0qp blocks and 97.96% of 1qp blocks are computable by the methods discussed so far.The remaining blocks have genuine treewidth at least 3 and require a different evaluator.Source: §6 · p. 32
• For the 0qp corpus through order r = 13, 70.35 % are analytic blocks and 23.34 % (4 483) are additional series-parallel blocks.Source: §6 · p. 32
• For the 1qp corpus through order r = 11, 83.87 % are analytic blocks and 14.09 % (9 863) are series-parallel blocks.Source: §6 · p. 32
• The method covers graph lattice sums for graphs whose blocks are bridges or series-parallel, corresponding to augmented blocks with tw(˜G) ≤2.Source: §6 · p. 32

Scope and practical performance

• For the LRTFIM corpora, treewidth never exceeds 4, and the method makes remaining blocks computable to high precision on a standard desktop, including in three dimensions with interaction exponents near d.Source: §7 · p. 32

Bucket elimination and scaling

• Tensor-network bucket elimination makes work exponential in elimination width rather than vertex count, and translation-invariant edge kernels enable FFT convolution that reduces scaling exponents.The method computes all momentum-grid values at the cost of a single momentum value, up to logarithmic scaling corrections.Source: §7 · p. 34

Numerical cost and benchmark graphs

• More than 99% of the benchmark graphs attain the optimal bound; for the hardest corpus graphs, with augmented treewidth 4, work scales as O(N^3 log N), and the full momentum grid costs one momentum evaluation.Source: §7 · p. 36

Reduction before contraction

• Reducing series-parallel parts before tensor contraction improves discretization accuracy because the remaining non-terminal nodes have degree at least three.The reduced blocks can be resolved on a small lattice grid to precision comparable to their series-parallel counterparts.Source: §7 · p. 36

Runtime and accuracy

• The full LRTFIM corpora can be evaluated on a standard laptop in minutes at better precision than leading Monte Carlo approaches, which require 24 hours on a cluster for one momentum value [4].Source: §7 · p. 36

8. Numerical Results

• The numerical evaluation combines block factorization with methods selected by block structure, including analytic zeta forms, semi-analytic algebra, and tensor-network bucket elimination.For series-parallel blocks, error scaling agrees with the predicted |n|^-(d+σ+2) bound.Source: §8 · p. 36, §8.1 · pp. 37, 40
• For dense cores with tw(G) ≥3, numerical work scales no faster than |Λn|^3 across the LRTFIM corpus.The reported tensor-network benchmarks compare torus truncation with box truncation and Richardson extrapolation.Source: §8 · p. 37, §8.2 · p. 41

8.1. Analytic and numerical benchmarks for series-parallel graphs.

• For series-parallel graphs, runtime grows linearly with node count or operation count, while direct summation grows exponentially.The zeta-algebra error for cycles is stable as σ approaches zero, unlike a simple Fourier approach, and singularities remain resolved after 12 operations.Source: §8.1 · pp. 37, 39, Figure 4 · p. 40

8.2. Benchmarks for tw ≥3 graphs evaluated through the tensor-network.

• For K4 and K5 on Z and Z2 at σ = 0.5, 1, and 2, tensor-network box truncation agrees with explicit nested summation to a maximum relative deviation of 5.3 × 10^-15.The comparison covered box half-widths up to L = 28, or 2.1 × 10^8 explicit terms.Source: §8.2 · p. 41

8.3. Comparison against published Monte Carlo data.

• Across 753 remaining LRTFIM coefficients, the deviation from Monte Carlo never exceeds 3.1 σMC, has median 0.12 σMC, and is within 1 σMC for 90%.The graph-zeta collection took 5 minutes on 8 cores; a cluster Monte Carlo run requires approximately 10^4–10^5 core-hours for one series.Source: Table 2 · p. 43

9. Physical application: Anomalous dispersion relation in a 3D quantum antiferromagnet with long-range interactions

• FFT yields the complete momentum grid at the same numerical cost as one momentum evaluation, unlike Monte Carlo runs that typically take a day per momentum.The application resolves a regime where the paper says Monte Carlo data are unavailable.Source: §9 · p. 45
• The order-10 result is a finely resolved dispersion for the cubic-lattice 3D LRTFIM at σ = 1 and antiferromagnetic λ = −0.03.The grid contains 4,096 momentum points, and the energies are expressed in units of the bare gap.Source: §9 · p. 45, Figure 8 · p. 46
• The dispersion reaches its global maximum at Γ, has a local minimum at M, and reaches its global minimum at R.M and R are equivalent to standard high-symmetry points by cubic symmetry.Source: §9 · pp. 45–46
• The full momentum grid required approximately 10 minutes on a single Apple M1 Max core.The calculation used 7,136 graphs and sums up to 30 dimensions.Source: §9 · p. 46, Figure 8 · p. 46
• At σ = 1, the dispersion has a |k| nonanalyticity at Γ, recovered analytically from simple Epstein zeta bridges.The paper identifies this anomalous dispersion as a direct fingerprint of a long-range interaction tail.Source: §9 · p. 46
• Figure 9 compares high-symmetry-path cuts at λ = −0.01, −0.02, and −0.03, and Γ-point dispersion across orders r = 4, 7, and 10.The λ = −0.03 cut corresponds to Figure 8.Source: Figure 9 · p. 47

10. Outlook

• The framework computes high-dimensional lattice sums for high-order series expansions of observables in gapped quantum lattice models.The conclusion describes applicability to general-dimensional lattices, interactions, and gapped models.Source: §10 · p. 47
• The method reduces numerical work substantially, with the largest share of objects evaluated at log-linear rather than exponential cost in graph-node number.It reproduces published Monte Carlo data for 1D, 2D, and 3D lattices within minutes on a laptop, at improved precision.Source: §10 · p. 47
• Future work will incorporate anisotropic generalized zeta functions and adapt the method to multi-atomic lattices, including frustrated systems.Other proposed directions include extensions to other models and perturbative approaches, finite temperature, and correlated disorder.Source: §10 · p. 48