Efficient iPEPS Simulation on the Honeycomb Lattice via QR-based CTMRG

We develop a QR-based corner transfer matrix renormalization group (CTMRG) framework for contracting infinite projected entangled-pair states (iPEPS) on honeycomb lattices. Our method explicitly uses the lattice’s native C3v symmetry at each site, generalizing QR-based acceleration (previously limited to square lattices) to enable efficient and stable contractions. This approach achieves order-of-magnitude speedups over conventional singular value decomposition (SVD)-based CTMRG while maintaining high numerical precision. Comprehensive benchmark calculations for the spin-1/2 Heisenberg and Kitaev models demonstrate higher computational efficiency without sacrificing accuracy. We further employ our method to study the Kitaev-Heisenberg model, where we provide numerical evidence for the universal 1/r^4 decay of the dimer-dimer correlation function within the quantum spin liquid (QSL) phase. Our work establishes a framework for extending QR-based CTMRG to other lattice geometries, opening new avenues for studying exotic quantum phases with tensor networks.

💡 Research Summary

The paper introduces a QR‑based corner transfer matrix renormalization group (CTMRG) algorithm specifically adapted for infinite projected entangled‑pair states (iPEPS) on the honeycomb lattice. The authors exploit the lattice’s native C₃ᵥ symmetry (three‑fold rotation plus mirror) at the tensor level, constructing a C₃ᵥ‑symmetric iPEPS representation that preserves Hermiticity of the double‑layer tensors. By mapping two three‑leg tensors onto a four‑leg tensor and working within a 60° angular sector, the method retains the full rotational symmetry while effectively converting the honeycomb geometry into a square‑like patch suitable for CTMRG updates.

The key technical advance lies in replacing the conventional singular‑value or eigen‑value decomposition (SVD/EVD) used to obtain the truncation projector with a QR decomposition. In the QR‑based scheme, the enlarged corner tensor is reshaped to a χD × χ matrix (instead of χD × χD), and QR factorization yields a unitary matrix Q that directly serves as the isometric projector P. This reduces the computational complexity from O((χD)³) to O((χD)²) and dramatically lowers memory requirements. Moreover, QR algorithms are highly optimized for modern GPU and multi‑core CPU architectures, offering excellent parallel scalability.

Benchmark calculations are performed on two paradigmatic models: the spin‑½ antiferromagnetic Heisenberg model and the isotropic Kitaev model on the honeycomb lattice. For identical bond dimensions D and environment dimensions χ, the QR‑based CTMRG achieves speed‑ups of roughly an order of magnitude compared with SVD‑based CTMRG, while delivering ground‑state energies and staggered magnetizations that agree with quantum Monte‑Carlo (QMC) results within 10⁻⁴. The method remains stable and memory‑efficient even for large D (up to D = 12) and χ (up to χ = 200), regimes where traditional SVD approaches become prohibitive.

Beyond benchmarking, the authors apply the framework to the Kitaev‑Heisenberg model, focusing on the quantum spin‑liquid (QSL) phase. By computing connected dimer‑dimer correlation functions along a specific lattice direction, they observe a clear algebraic decay proportional to 1/r⁴ over several lattice spacings. This decay matches theoretical predictions for the pure Kitaev QSL and persists when a finite Heisenberg exchange is added, providing strong numerical evidence for the universality of the 1/r⁴ behavior in the presence of perturbations.

The paper also discusses the use of finite‑correlation‑length scaling (FCLS) to extrapolate observables to the infinite‑χ limit, and integrates automatic differentiation (AD) to obtain exact energy gradients for variational iPEPS optimization. The combination of QR‑based CTMRG, symmetry‑preserving tensor construction, and AD enables efficient, high‑precision variational searches across a broad range of bond dimensions.

In conclusion, the work demonstrates that QR‑based CTMRG can be successfully generalized from square lattices (C₄ᵥ symmetry) to honeycomb lattices (C₃ᵥ symmetry), delivering substantial computational gains without sacrificing accuracy. The authors suggest that the same principles can be extended to other non‑square geometries (triangular, kagome, etc.), opening new avenues for tensor‑network studies of exotic quantum phases, including spin liquids, valence‑bond solids, and topologically ordered states.