⚛️ quantum physics

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

The authors present a native corner transfer matrix renormalization group algorithm enhanced by automatic differentiation for optimizing projected entangled-pair states on the triangular lattice, which achieves superior variational results for the Heisenberg model by avoiding artificial lattice mappings and better capturing the system's entanglement structure.

Original authors: Jan Naumann, Jens Eisert, Philipp Schmoll

Published 2026-01-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jan Naumann, Jens Eisert, Philipp Schmoll

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a giant, three-dimensional puzzle where every piece is connected to its neighbors in a complex web. This puzzle represents the behavior of tiny particles (like electrons) in a material. Physicists call this a "quantum many-body problem." The goal is to find the most stable, lowest-energy arrangement of these pieces—the "ground state."

For decades, scientists have used a clever trick called Tensor Networks to solve these puzzles. Think of a Tensor Network as a digital scaffold that holds the puzzle together. The most popular version of this scaffold was designed for a square grid (like a chessboard).

However, nature isn't always a chessboard. Some materials, like those with a triangular or kagome (a pattern of interlocking triangles) structure, are naturally "frustrated." In these materials, the particles can't all be happy at the same time because their triangular neighbors pull them in conflicting directions.

The Old Way: Squaring the Circle

Previously, to study these triangular materials, scientists had to force the triangular puzzle onto a square grid.

The Analogy: Imagine trying to describe a perfect circle by drawing it on a piece of graph paper using only square blocks. You can get close, but you have to add "diagonal" connections between the blocks to make it look round.
The Problem: This mapping distorts the natural connections. The "diagonal" links are artificial, and the digital scaffold has to work extra hard to simulate the natural triangular geometry. This limits how accurately the simulation can capture the true physics, especially the "entanglement" (the deep, spooky connection between particles).

The New Way: Native Triangular Optimization

The authors of this paper, Jan Naumann, Jens Eisert, and Philipp Schmoll, have built a new scaffold specifically designed for triangles.

Instead of forcing a square grid onto a triangle, they built a system that natively understands the triangular shape.

The Analogy: Instead of drawing a circle on graph paper, they are now using a piece of paper that is already cut into a hexagon or triangle shape. The pieces fit together naturally without needing awkward diagonal bridges.
The "Automatic Differentiation" Boost: They also added a feature called "automatic differentiation." Think of this as a super-smart GPS for the optimization process. Instead of the computer guessing which way to move the puzzle pieces to find the lowest energy, this GPS calculates the exact slope of the hill and guides the pieces straight to the bottom. This makes the search for the best solution much faster and more precise.

Why This Matters: More Room to Breathe

The paper highlights two main advantages of this new triangular approach:

More Variational Parameters: In the old square-grid method, the "virtual" connections between pieces were limited. In the new triangular method, each piece has more connections (six neighbors instead of four).
- Analogy: Imagine the square grid is a small room with four doors. The triangular grid is a larger room with six doors. Even if the room size (bond dimension) is the same, the triangular room allows for more complex traffic flow and better representation of how the particles interact.
Better Entanglement Representation: The paper shows that the triangular scaffold can hold more "quantum information" (entanglement) than the square one of the same size.
- Analogy: If the square grid is a standard backpack, the triangular grid is a backpack with extra pockets and straps. It can carry more complex loads without getting heavier.

The Results: Testing the New Scaffold

The team tested their new method on two famous "frustrated" puzzles:

The Triangular Lattice: A material where spins (magnetic directions) arrange themselves in a 120-degree pattern.
The Kagome Lattice: A more complex triangular pattern that is suspected to host a "Quantum Spin Liquid"—a state where particles never settle down, even at absolute zero.

The Findings:

Triangular Lattice: The new method found a lower energy state (a more stable solution) than previous methods, matching the best known results but with a more natural representation of the physics.
Kagome Lattice: The new method provided a much clearer picture of the "Quantum Spin Liquid." It suggested the system is likely a "gapless" liquid (where particles can move freely) and did so with more confidence and less "noise" than the old square-grid mapping.

The Trade-off

There is a catch. Because the triangular scaffold is more complex (more connections, more math), it requires more computer power to run.

Analogy: The new triangular backpack is heavier and harder to pack than the old square one. However, the paper shows that the extra weight is worth it because the backpack holds the "treasure" (the correct physical answer) much better.

Conclusion

In short, this paper introduces a specialized tool for simulating triangular quantum materials. By abandoning the "square grid" workaround and building a native triangular system, the authors achieved more accurate results with the same amount of computational resources. They proved that when you respect the natural geometry of the problem, you get a clearer, more faithful picture of the quantum world.

Technical Summary: Variational Optimization of Projected Entangled-Pair States on the Triangular Lattice

Problem Statement
The simulation of frustrated quantum many-body systems, particularly on triangular and kagome lattices, remains a significant challenge in condensed matter physics. While Projected Entangled-Pair States (PEPS) have become a standard framework for simulating two-dimensional quantum systems, their variational optimization on triangular geometries has historically been constrained. Conventional approaches typically map the triangular lattice onto a square lattice, introducing diagonal next-nearest-neighbor interactions. This mapping forces correlations along the diagonal direction to be routed through the remaining lattice directions, which is an inefficient representation of the system's intrinsic entanglement structure. Furthermore, existing methods often rely on indirect mappings that fail to fully capture the rotational symmetry and specific connectivity of the triangular Bravais lattice, potentially limiting the accuracy of variational energy minimizations at fixed bond dimensions.

Methodology
The authors introduce a general Corner Transfer Matrix Renormalization Group (CTMRG) algorithm specifically tailored for infinite PEPS on the triangular lattice. Key methodological components include:

Native Triangular Formulation: Instead of mapping to a square lattice, the algorithm operates directly on the triangular geometry. The PEPS ansatz is defined using seven-index tensors (one physical index and six virtual indices connecting to all nearest neighbors), reflecting the coordination number of the triangular lattice.
Automatic Differentiation (AD): The approach integrates AD to compute energy gradients efficiently, enabling direct variational energy minimization without the need for manual derivation of complex update rules.
Radial CTMRG Procedure: Unlike the square lattice CTMRG, which utilizes orthogonal directions, the triangular variant employs a radial absorption scheme. The environment consists of six corner tensors ( $C$ ) and twelve edge tensors ( $T$ ). The algorithm performs iterative power methods to compute fixed-point environment tensors.
Absorption and Renormalization: The absorption of PEPS tensors into the environment is performed radially. Due to the non-orthogonal lattice vectors, updates to corner and edge tensors are interdependent. The method utilizes a two-step renormalization process involving two sets of projectors ( $P$ and $Q$ ) to truncate the enlarged bond dimensions back to a fixed environment bond dimension $\chi_E$ , ensuring numerical stability.
Symmetry Handling: The algorithm does not assume rotational symmetry of the input tensors, allowing it to accommodate arbitrary states, including symmetry-broken and spiral states. It can also be extended to incorporate physical symmetries to improve efficiency.

Key Contributions

Algorithmic Development: The paper presents the first computationally efficient and accurate method for contracting and variationally optimizing PEPS directly on the triangular lattice, bypassing the need for square-lattice mappings.
Enhanced Representability: By utilizing the native geometry, the triangular PEPS (TL-PEPS) ansatz captures quantum correlations along three independent lattice directions more faithfully than square-lattice (SL-PEPS) counterparts. Theoretical analysis shows that for a given bond dimension $\chi_B$ , the entanglement entropy upper bound for TL-PEPS is significantly higher (scaling as $(8L-2)\log\chi_B$ ) compared to SL-PEPS (scaling as $4L\log\chi_B$ ).
Increased Variational Parameters: At a fixed bond dimension, the TL-PEPS ansatz possesses a larger number of variational parameters ( $p\chi_B^6$ ) compared to the SL-PEPS ( $p\chi_B^4$ ), offering greater expressive power for the same computational resources in terms of bond dimension.
Benchmarking Framework: The authors provide a rigorous benchmarking protocol comparing native TL-PEPS against mapped SL-PEPS, ensuring fair comparisons by matching variational parameters or analyzing convergence with respect to environment bond dimensions.

Results
The method was applied to two paradigmatic antiferromagnetic spin-1/2 Heisenberg models:

Triangular Lattice Heisenberg Model:
- The TL-PEPS approach yielded lower variational energies than the SL-PEPS at the same bond dimensions.
- At $\chi_B = 5$ , the TL-PEPS achieved a ground state energy of $E = -0.55136$ , which is the lowest variational tensor network value reported to date for this model, improving upon previous extrapolated DMRG and iPEPS results.
- Extrapolation to the infinite bond dimension limit yielded $E = -0.55164$ , consistent with SL results but obtained with improved convergence properties.
- The magnetic order parameter $m$ showed a systematic reduction in TL-PEPS ( $m \approx 0.157$ ) compared to SL-PEPS ( $m \approx 0.182$ ), attributed to the ansatz's improved sensitivity to the entanglement structure.
Kagome Lattice Heisenberg Model:
- The TL-PEPS consistently provided lower variational energies across all accessible bond dimensions compared to the SL formulation.
- The extrapolated ground state energy was $E = -0.43929$ , slightly improving upon previous DMRG and exact diagonalization results.
- The TL-PEPS data exhibited a cleaner scaling behavior with respect to the inverse correlation length ( $1/\xi$ ), facilitating more reliable extrapolations even at lower bond dimensions where SL-PEPS data showed inaccuracies.
- The results support a non-magnetic ground state consistent with a quantum spin liquid (QSL), though the gapped vs. gapless nature remains an open question in the literature.

Significance and Claims
The paper claims that the native triangular lattice formulation offers a significant advantage in variational accuracy for frustrated quantum magnets. The improvements are attributed to:

Faithful Entanglement Representation: The ansatz naturally respects the lattice geometry and rotational symmetries, leading to a more accurate representation of the entanglement structure.
Parameter Efficiency: The increased number of variational parameters at a fixed bond dimension allows the ansatz to explore the Hilbert space more effectively.
Feasibility: Despite the higher computational cost per step (scaling as $O(\chi_B^{13})$ compared to the square lattice), the method remains feasible and competitive, particularly when matched against square lattice simulations with equivalent variational parameter counts.

The authors conclude that this method provides an unbiased perspective on open problems in frustrated quantum magnetism and paves the way for studying other triangular-based lattices (e.g., honeycomb, dice, maple-leaf) and models with next-to-nearest-neighbor interactions without complex routing. The implementation is available in the open-source variPEPS library.