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

Technical Summary: Graph Lattice Sums and Graph Zeta Functions for Long-Range Interacting Quantum Lattice Models

Problem Statement
The simulation of gapped quantum lattice models with long-range interactions (decaying as $|x|^{-\nu}$) is hindered by the exponential growth of the Hilbert space dimension with system size. While high-order linked-cluster expansions offer a rigorous route to solving the eigenvalue problem for infinite systems, they require the computation of high-dimensional, oscillatory lattice sums associated with graph structures. Currently, these sums are evaluated using Monte Carlo methods, which suffer from stochastic errors and cluster-scale runtimes (often requiring thousands of core-hours for a single momentum point). Furthermore, resolving the full dispersion relation requires recomputing these sums for every wavevector $k$, compounding the computational cost. The primary challenges are:

1. Exponential Complexity: Direct discretization scales exponentially with the number of graph nodes.
2. Slow Convergence: For power-law kernels with exponents $\nu$ close to the lattice dimension $d$, convergence is arbitrarily slow.
3. Momentum Grid Resolution: Standard methods cannot efficiently recover the full Brillouin zone.

Methodology
The authors propose a deterministic, high-performance framework that replaces stochastic sampling with a combination of graph theory, generalized zeta functions, and tensor networks. The core of the method involves factorizing graph lattice sums into smaller, computable components based on the graph's topology.

1. Block Factorization:
   The graph lattice sum $Z_{\Lambda, G}(k)$ is decomposed over the graph's blocks (maximal 2-connected subgraphs) and cutvertices. Using translational invariance, the sum factorizes into a product of sums over these blocks. Only the "spine" blocks (those on the path between terminals) depend on the momentum $k$; attachment blocks are evaluated at $k=0$. This reduces the complexity from exponential in the total number of nodes to exponential in the size of the largest block.

2. Analytic Evaluation of Basic Blocks:

   • Bridges: For bridges (two nodes connected by edges), the sum is the lattice Fourier transform of the interaction kernel. For power-law kernels, this is the Epstein zeta function, which is computed analytically or via high-performance libraries (EpsteinLib).
   • Cycles: For cycles without momentum dependence, the sum is represented as an integral over the Brillouin zone of the product of Fourier transforms, computable via Duffy transforms or Gauss quadrature.

3. Semi-Analytic Algebra for Series-Parallel Blocks:
   For blocks with treewidth $tw \le 2$ (series-parallel graphs), the authors construct the graph lattice sum using pointwise multiplication and convolution of base objects (bridges).

   • To handle power-law tails and singularities at $k=0$, they introduce a semi-analytic representation consisting of a rapidly decaying Fourier series (for the regular part) and a finite sum of Epstein zeta functions (for the singular parts).
   • Theorems 6.2 and 6.3 prove that this algebra is closed under multiplication and convolution. The singularities are tracked analytically, while the regular parts are handled via FFT.
   • A compression strategy discards Epstein terms with high exponents (which become smooth) into the Fourier series, preventing the explosion of terms and avoiding numerical cancellation errors.

4. Tensor-Network Bucket Elimination for Highly Connected Graphs:
   For blocks with $tw \ge 3$ (where series-parallel reduction stalls), the method employs tensor-network bucket elimination.

   • The lattice sum is represented as a tensor network where edge kernels act as convolution operators.
   • By exploiting translational invariance, the authors reduce the numerical scaling. The cost scales polynomially with the momentum grid size $N$ as $O(N^{tw})$ (or $O(N^{tw-1} \log N)$ with FFT), rather than exponentially with the number of nodes.
   • Crucially, this allows the evaluation of the entire momentum grid at the cost of a single momentum evaluation.

Key Contributions

• Resolution of Graph Lattice Sums: The paper provides the first method to compute all required graph lattice sums (graph zeta functions) for power-law kernels deterministically, eliminating the need for Monte Carlo sampling.
• Complexity Reduction: The method reduces the evaluation time for state-of-the-art series expansions from tens of thousands of core-hours to minutes on a single core.
• Full Momentum Resolution: Through the use of FFT and tensor networks, the full dispersion relation is recovered at the computational cost of a single $k$-point evaluation.
• Graph Zeta Library (GZL): The authors provide an open-source, high-performance implementation (GZL) that reproduces all results on standard hardware.

Results

• Benchmarking: The method was benchmarked against analytic solutions for simple graphs and against published Monte Carlo data for the long-range transverse-field Ising model (LRTFIM) on 1D, 2D, and 3D lattices.
• Agreement: The results show full agreement with Monte Carlo data within the statistical errors of the references. The maximum deviation across 998 series coefficients was $3.1\sigma_{MC}$, with a median of $0.12\sigma_{MC}$.
• Performance:
  • A 3D dispersion relation on a grid of 4096 momentum points was computed in approximately 10 minutes on a single core.
  • In contrast, reproducing the same data via Monte Carlo required roughly $10^4$–$10^5$ core-hours on a cluster.
• Precision: The method achieves controlled numerical precision even for interaction exponents $\nu$ close to the system dimension $d$, where direct summation fails.

Significance and Claims
The paper claims to resolve a fundamental bottleneck in the simulation of long-range interacting quantum systems. By rendering high-dimensional lattice sums computable with controlled precision and polynomial scaling, the method enables:

• Rigorous Access to Infinite Systems: It allows for the extraction of physical quantities (ground state energy, dispersion relations) for infinite lattices via high-order perturbation theory without stochastic errors.
• Experimental Relevance: The companion paper [22] utilizes this framework to reproduce experimental observations in quantum materials, specifically the dispersion of the triangular-lattice quantum Ising magnet $KTmSe_2$.
• Anomalous Dispersion: The method successfully resolves non-analytic features in the dispersion relation (scaling as $|k|^\sigma$) that are fingerprints of long-range interactions, which are inaccessible to standard Monte Carlo approaches due to the inability to resolve the full momentum grid efficiently.

The authors position this work as a mathematical foundation that, combined with linked-cluster perturbation theory, creates a comprehensive framework for studying gapped quantum lattice models, particularly in regimes (frustrated systems, long-range interactions) where other methods like DMRG or Quantum Monte Carlo are ineffective.