Variational optimization of projected entangled-pair states on the triangular lattice

TL;DR

The paper tackles the challenge of simulating frustrated quantum magnets on triangular-based lattices with high accuracy. It introduces a triangular-lattice CTMRG algorithm for infinite PEPS and uses automatic differentiation to perform direct variational energy minimization on the native lattice, avoiding square-lattice mappings. Key contributions include a detailed environment construction with six corners $C$ and twelve edges $T$ and a two-stage projector-based truncation scheme, yielding a dominant contraction cost of $O(\chi_E^2 \chi_B^9 p + \chi_E^3 \chi_B^6) \sim O(\chi_B^{13} p)$ when $\chi_E \sim \chi_B^2$. Benchmark results on the spin-$\frac{1}{2}$ Heisenberg models on triangular and kagome lattices show improved variational energies at fixed $\chi_B$ and robust correlation-length scaling, supporting a $120^{\circ}$ ordered state on the triangular lattice and a non-magnetic quantum spin liquid–like ground state on kagome. Overall, the method provides a scalable path to study spin liquids and frustrated magnetism on triangular-based lattices and opens avenues for extensions to other lattices and symmetry-enhanced implementations.

Abstract

We introduce a general corner transfer matrix renormalization group algorithm tailored to projected entangled-pair states on the triangular lattice. By integrating automatic differentiation, our approach enables direct variational energy minimization on this lattice geometry. In contrast to conventional approaches that map the triangular lattice onto a square lattice with diagonal next-nearest-neighbour interactions, our native formulation yields improved variational results at the same bond dimension. This improvement stems from a more faithful and physically informed representation of the entanglement structure in the tensor network and an increased number of variational parameters. We apply our method to the antiferromagnetic nearest-neighbour Heisenberg model on the triangular and kagome lattice, and benchmark our results against previous numerical studies.

Figures (14)

• Figure 1: Comparison of the number of variational parameters in a single real tensor for projected entangled-pair states defined on the square lattice (SL) and triangular lattice (TL), assuming a physical dimension of $p = 2$.
• Figure 2: Definition of all six corner tensors $C$ and twelve edge tensors $T$ in the triangular CTMRG procedure. The triangular lattice is spanned by lattice vectors $\mathbf a_1$ and $\mathbf a_2$. The edge tensors are labeled such that, in each of the six lattice directions, both tensors connect to the same PEPS tensor and $T_{i,a}$ appears before $T_{i,b}$ when traversing the network clockwise.
• Figure 3: Single-site norm of the PEPS state vector computed from the minimal set of CTMRG environment tensors, which only includes all six corner tensors.
• Figure 4: The kagome lattice has an underlying triangular Bravais lattice with a three-site basis, highlighted with green triangles. It can be simulated with a TL PEPS ansatz and a physical dimension of $p^3 = 8$ for a spin-$1/2$ system.
• Figure 5: Ground state energy $E_0$ and magnetic order parameter $m$ for the nearest-neighbour Heisenberg model on the triangular lattice. The data points are labeled by the PEPS bond dimension $\chi_B$ for a better comparison. Our results are compared against DMRG Huang2024, spiral iPEPS Hasik2024 and coarse-grained iPEPS Chi2022.