Signal-to-noise improvement through neural network contour deformations for 3D $SU(2)$ lattice gauge theory (2309.00600v1)

Abstract: Complex contour deformations of the path integral have been demonstrated to significantly improve the signal-to-noise ratio of observables in previous studies of two-dimensional gauge theories with open boundary conditions. In this work, new developments based on gauge fixing and a neural network definition of the deformation are introduced, which enable an effective application to theories in higher dimensions and with generic boundary conditions. Improvements of the signal-to-noise ratio by up to three orders of magnitude for Wilson loop measurements are shown in $SU(2)$ lattice gauge theory in three spacetime dimensions.

• The paper introduces neural network-driven contour deformations that enhance the signal-to-noise ratio by up to three orders of magnitude in 3D SU(2) lattice gauge theory.
• It employs a U-net architecture to optimize the path integral deformations and facilitate transfer learning across different lattice volumes and boundary conditions.
• The study applies maximal tree gauge fixing for effective variance reduction, paving the way for improved Wilson loop measurements and potential extensions to higher-dimensional gauge theories.

Signal-to-Noise Improvement through Neural Network Contour Deformations for 3D SU(2) Lattice Gauge Theory

In the presented study, the authors address the prevalent issue of statistical noise in lattice field theory, particularly focusing on signal-to-noise (StN) improvements in 3D SU(2) lattice gauge theory. This work builds on the foundation of complex contour deformations for reducing noise in path integrals, a technique that has shown promise in two-dimensional gauge theories.

Key Contributions

The paper introduces several advancements that allow the application of contour deformations to higher-dimensional theories with generic boundary conditions. Central to this approach is the use of neural networks to define the deformation of the path integral contour. This method is shown to enhance the StN ratio by up to three orders of magnitude in 3D SU(2) gauge theories, as evidenced by Wilson loop measurements.

The authors outline a novel methodology that includes the following steps:

1. Contour Deformations: By employing complex contour deformations for path integrals, the authors redefine observables that maintain the same expectation values but typically exhibit differing variance properties. These deformations act directly on link variables, thus accommodating theories with periodic boundaries.
2. Neural Network Optimization: The optimization of contour deformation is achieved through neural networks, employing a specific architecture called U-net. This framework not only aids in finding optimal deformations but also allows for transfer learning across different lattice volumes, substantially reducing computational efforts.
3. Gauge Fixing with Maximal Tree: By implementing a maximal tree gauge fixing approach, the authors ensure effective variance reduction. This technique constrains gauge configurations, allowing for more manageable deformations and improved results.

Numerical Results and Implications

The numerical results reported highlight the profound impact of their techniques, especially in terms of variance reduction for larger Wilson loops across various lattice sizes. The improvement is predominantly determined by the loop geometry rather than the total lattice volume, which suggests the method's scalability and robustness.

These findings offer substantial implications for lattice gauge theory computations. Practically, the enhanced StN ratio enables more precise measurements of Wilson loops, facilitating the study of confinement and related phenomena in lattice gauge theories. Theoretically, this approach underscores the potential of machine learning techniques in tackling complex problems characteristic of quantum field theory simulations.

Future Directions

The study sets the stage for several promising directions. Extending these methods to 4D gauge theories, particularly SU(3), appears a natural progression. The adaptability of U-net architectures to higher dimensions and their application to different gauge groups might offer significant advancements in simulating realistic quantum chromodynamics (QCD) environments.

Furthermore, exploring alternative gauge fixing schemes or deploying contour deformations without fixing the gauge could yield further improvements. Incorporating more dynamic, real-time contour deforming architectures might be another venture towards enhancing simulation fidelity and efficiency.

In conclusion, the paper significantly progresses in employing machine learning for statistical noise reduction in lattice gauge theories. It offers a well-defined pathway towards enhanced computational models that would be crucial for further understanding of fundamental physical theories.