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High-Performance Simulations of Higher Representations of Wilson Fermions

The paper introduces HiRep v2, an open-source, GPU-accelerated software suite that enables high-performance lattice field theory simulations of dynamical Wilson fermions in higher representations of $SU(N_g)$ gauge groups, offering optimized algorithms and excellent scalability for physics beyond the Standard Model.

Original authors: Vincent Drach, Sofie Martins, Claudio Pica, Antonio Rago

Published 2026-06-17

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Vincent Drach, Sofie Martins, Claudio Pica, Antonio Rago

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 the universe is built from a giant, invisible 3D grid, like a massive digital chessboard where every single square holds a piece of information about how particles interact. Physicists call this Lattice Field Theory. To understand the "glue" that holds atomic nuclei together (a force called Quantum Chromodynamics, or QCD), they need to run incredibly complex simulations on this grid.

However, most standard simulations only look at the most common type of particle interaction, like studying only the "King" and "Queen" pieces in chess. But to discover new physics beyond our current understanding (like the mysterious "Composite Higgs" theories), scientists need to study the "Rooks," "Bishops," and even pieces that don't exist in the standard game. They need to simulate particles in higher representations, which is like trying to play chess with a completely new set of rules and pieces that are much harder to calculate.

Enter HiRep v2. Think of this as a brand-new, super-powered video game engine designed specifically for these difficult simulations.

Here is what the paper says this new engine can do, explained simply:

1. It's Built for Speed (The GPU Accelerator)

The old version of this software was like a bicycle; it worked, but it was slow. The new version, HiRep v2, is a Formula 1 race car.

The Analogy: The authors have rewritten the code to run on GPUs (Graphics Processing Units). These are the same chips found in high-end video game cards, but here they are used to crunch numbers.
The Result: The software can now run on both NVIDIA and AMD graphics cards, making the simulations run incredibly fast. It's like switching from calculating a math problem with a pencil to using a supercomputer that does millions of calculations at once.

2. It Handles "Higher Representations" (The Specialized Toolbox)

Standard physics software is great at simulating the standard "SU(3)" group (the rules for normal matter). But HiRep v2 is built to handle SU(N) groups, where "N" can be any number.

The Analogy: Imagine a standard toolbox that only has a hammer and a screwdriver. HiRep v2 is a universal workshop that can build anything, from a simple birdhouse to a skyscraper, regardless of how complex the blueprint is. It allows scientists to simulate theories where the "colors" of particles (a property of the strong force) are not just 3, but 4, 5, or even 100.

3. It Doesn't Get Tired (Scalability)

One of the biggest problems in supercomputing is that when you add more computers to a job, they often spend more time talking to each other than doing the actual work.

The Analogy: Imagine a team of 1,000 people trying to move a mountain. If they all stop to argue about who moves which rock, the job never gets done. HiRep v2 is like a perfectly choreographed dance troupe. The paper shows that even when using up to 1,024 GPUs at once, the software stays efficient. It minimizes the "talking" (communication) and maximizes the "moving" (calculation), ensuring that adding more computers actually makes the job faster, not slower.

4. It's a Complete Package (The Swiss Army Knife)

This isn't just a calculator; it's a full laboratory.

The Features: It includes tools to:
- Clean up the data: Just like a photographer uses filters to remove noise from a photo, HiRep uses "smearing" techniques to smooth out the rough edges of the simulation grid.
- Measure everything: It can calculate the "mass" of particles, how they scatter, and even the "topological charge" (a measure of the grid's twistiness).
- Run different algorithms: It uses advanced math tricks (like "Hasenbusch acceleration") to make the simulations converge faster, saving time and energy.

5. It's Open and Honest (Open Source)

The paper emphasizes that this is open-source software.

The Analogy: Instead of a secret recipe locked in a vault, the "cookbook" for HiRep v2 is available to everyone on GitHub. Any scientist can look at the code, check the math, and improve it. This ensures that the results are trustworthy and reproducible.

Summary

In short, HiRep v2 is a high-performance, open-source software suite that allows physicists to simulate complex, non-standard versions of the universe's fundamental forces. By harnessing the power of modern graphics cards and optimizing how thousands of computers work together, it provides a robust tool for exploring "Beyond the Standard Model" physics—essentially helping scientists look for new rules of the universe that we haven't discovered yet.

The paper does not claim this software cures diseases or predicts the weather; its sole purpose is to provide the computational power needed to solve the hardest math problems in theoretical particle physics.

Technical Summary: High-Performance Simulations of Higher Representations of Wilson Fermions (HiRep v2)

Problem Statement
Lattice Field Theory is essential for the non-perturbative understanding of strongly coupled quantum field theories, particularly for precision tests of the Standard Model (QCD). While robust simulation libraries exist for $SU(3)$ gauge groups, direct searches for physics beyond the Standard Model (BSM) require software capable of simulating general gauge groups $SU(N_g)$ and fermions in higher representations (e.g., adjoint, symmetric, antisymmetric). Existing open-source libraries often lack the flexibility to support general $N_g$ with higher representations, or they lack the specific combination of high-performance features (such as GPU acceleration and advanced algorithms) required for large-scale simulations of composite Higgs and technicolor theories. There is a specific need for a library that implements Wilson fermions in higher representations with general numbers of colors, optimized for modern heterogeneous CPU-GPU architectures.

Methodology
The paper presents HiRep v2, an open-source software suite (written in C, CUDA C, and C++) designed to address these limitations. The core methodology involves a comprehensive re-engineering of the lattice field theory simulation pipeline with a focus on the Dirac operator, which is the primary computational bottleneck.

Dirac Operator and Linear Algebra: The implementation generalizes the Wilson-Dirac operator to arbitrary representations $R$ and gauge groups $SU(N_g)$ . To optimize performance, the code employs generic macro templates for linear algebra operations and utilizes "fused" kernels (e.g., linear combinations) to reduce memory movement.
GPU Acceleration and Memory Layout: A significant portion of the work involves porting the code to NVIDIA and AMD GPUs. The authors introduce a "computation-optimized geometry" layout where the local lattice is stored as a single contiguous block, separate from send/receive buffers. This contrasts with the previous "memory-optimized" layout. This change facilitates efficient memory coalescing and strided access patterns on GPUs, where threads access field components in blocks (e.g., 32 sites) rather than site-by-site.
Parallelization and Affinity: The code supports hybrid parallelization using MPI, OpenMP, and GPU threads. It implements topology-aware communicators to reshape MPI process associations with lattice sub-blocks, optimizing intra-node communication. It also addresses NUMA (Non-Uniform Memory Access) affinity by pinning processes to specific CPU/GPU domains and using "first touch" policies to ensure memory pages are allocated on the correct NUMA node.
Algorithms: HiRep v2 supports the Hybrid Monte Carlo (HMC) and Rational HMC (RHMC) algorithms. It includes advanced features such as Hasenbusch acceleration, even-odd preconditioning, and multi-shift conjugate gradient inverters. The suite supports various actions, including Symanzik improved gauge actions (Lüscher-Weisz), Clover and exponential Clover improvements, and Twisted Mass actions.
Observables: The library includes tools for spectroscopy (connected and disconnected contributions), scattering methods (Lüscher method), non-perturbative renormalization (RI'-MOM), and gauge smearing (APE, Stout, HYP, Wilson Flow).

Key Contributions

GPU Porting and Optimization: The primary contribution is the full support for GPU acceleration (NVIDIA and AMD) for simulations involving higher representations of Wilson fermions. The authors demonstrate that their implementation of the Dirac operator achieves excellent performance, reaching 80–90% of the peak theoretical bandwidth on modern GPUs.
Scalability: The software exhibits excellent strong and weak scaling properties across multiple GPUs and nodes. Benchmarks on the LUMI-G supercomputer show scaling up to 1024 GPUs with minimal efficiency loss, achieving a peak bandwidth of 908 TB/s.
Generalization to $N_g$ : The code supports general numbers of colors ( $N_g$ ) and various representations (Fundamental, Adjoint, Symmetric, Antisymmetric). The authors introduce a "Large- $N_g$ improved kernel" that parallelizes over the fermion dimension ( $D_R$ ) in addition to lattice sites. This approach mitigates register spilling and performance degradation that occurs in standard site-parallelized kernels as $N_g$ increases.
Algorithmic Robustness: The suite integrates advanced numerical techniques, including double-double arithmetic for high-precision reductions (to mitigate error accumulation in large volumes) and various integrators (Leapfrog, Omelyan) with hierarchical structures.

Results

Performance Benchmarks: On single GPUs (V100, A100, H100, MI250x), the linear algebra routines achieve up to 90% of peak bandwidth. The Wilson-Dirac operator shows that for large $N_g$ , the $D_R$ -site-parallelized kernel significantly outperforms the standard site-parallelized kernel, avoiding the bandwidth drop caused by register spilling.
Scaling: Weak scaling tests on LUMI-G and JUWELS demonstrate near-perfect scaling up to 1024 GPUs. Strong scaling shows expected efficiency losses for very small local lattices where communication overhead cannot be masked by computation.
Algorithmic Tests: The code passes standard algorithmic tests, including the Creutz equality ( $\langle e^{-\Delta H} \rangle = 1$ ) and the scaling of Hamiltonian violations with step size ( $\langle \Delta H \rangle \propto \delta\tau^4$ ). Reversibility tests confirm that the algorithm is reversible up to inverter precision.
Physics Validation: The authors validate the spectroscopy code by comparing CPU and GPU results for an $SU(2)$ gauge theory with adjoint fermions. The effective masses and correlation functions match, confirming the correctness of the real-number representation handling and the overall implementation.

Significance
The paper claims that HiRep v2 is currently the most flexible open-source high-performance lattice software suite supporting simulations of higher representations of Wilson fermions with general gauge groups. Its significance lies in enabling quantitative studies of BSM physics, such as composite Higgs models and the large- $N_g$ limit, which were previously hindered by a lack of optimized software. By providing a robust, scalable, and GPU-accelerated tool, HiRep v2 facilitates large-scale simulations of light fermionic masses on large lattices, essential for precision tests of theories beyond the Standard Model. The authors note that the software is suitable for deployment on state-of-the-art supercomputers and represents a significant step forward in the computational capabilities available for lattice field theory.