Flow-based sampling for fermionic lattice field theories

Open Access

Flow-based sampling for fermionic lattice field theories

Michael S. Albergo, Gurtej Kanwar, Sébastien Racanière, Danilo J. Rezende, Julian M. Urban, Denis Boyda, Kyle Cranmer, Daniel C. Hackett, and Phiala E. Shanahan

Phys. Rev. D 104, 114507 – Published 15 December 2021

Abstract

Algorithms based on normalizing flows are emerging as promising machine learning approaches to sampling complicated probability distributions in a way that can be made asymptotically exact. In the context of lattice field theory, proof-of-principle studies have demonstrated the effectiveness of this approach for scalar theories, gauge theories, and statistical systems. This work develops approaches that enable flow-based sampling of theories with dynamical fermions, which is necessary for the technique to be applied to lattice field theory studies of the Standard Model of particle physics and many condensed matter systems. As a practical demonstration, these methods are applied to the sampling of field configurations for a two-dimensional theory of massless staggered fermions coupled to a scalar field via a Yukawa interaction.

Received 22 June 2021
Accepted 12 November 2021

DOI:https://doi.org/10.1103/PhysRevD.104.114507

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Lattice field theory Machine learning

Statistical Physics & ThermodynamicsParticles & Fields

Authors & Affiliations

Michael S. Albergo ^1,*, Gurtej Kanwar ^2,3,†, Sébastien Racanière^4,‡, Danilo J. Rezende ^4,§, Julian M. Urban ^5,∥, Denis Boyda^6,2,3, Kyle Cranmer¹, Daniel C. Hackett ^2,3, and Phiala E. Shanahan^2,3

¹Center for Cosmology and Particle Physics, New York University, New York, New York 10003, USA
²Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
³The NSF AI Institute for Artificial Intelligence and Fundamental Interactions, Cambridge, Massachusetts 02139-4307, USA
⁴DeepMind, London N1C 4DJ, United Kingdom
⁵Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
⁶Argonne Leadership Computing Facility, Argonne National Laboratory, Lemont, Illinois 60439, USA

^*albergo@nyu.edu
^†gurtej@mit.edu
^‡sracaniere@google.com
^§danilor@google.com
^∥urban@thphys.uni-heidelberg.de

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Issue

Vol. 104, Iss. 11 — 1 December 2021

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Images

Figure 1
Diagrams illustrating the four types of sampling schemes described in Sec. 3. Blue circles/ellipses depict the current state of the Markov chain. Yellow boxes depict exactly sampleable densities either produced from generative models or by Equation (16). Green boxes correspond to Metropolis accept/reject steps using the acceptance probabilities defined in the text. Dotted lines indicate the Markov chain, whereas solid lines correspond to the internal operations of each Markov chain step.
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Figure 2
Schematic overview of a coupling-layer-based flow model architecture. Masks $m_{i}$ and their complements ${\bar{m}}_{i} = 1 - m_{i}$ split the degrees of freedom into a subset to be updated and a subset that is frozen and used as input to the context function (red box), which provide the parameters for the invertible transformation $f_{i}$ applied within each coupling layer $g_{i}$ . Here, neural networks (NN) are used to implement the context functions. The generic variables $z$ , $x_{i}$ , and $x$ may generally be high-dimensional field configurations in a lattice field theory setting. Particular implementations of translationally equivariant convolutional networks are defined in Secs. 4b1 and 4b2. Constructions of coupling layers with affine transformations based on these convolutional networks are defined in Sec. 4b3.
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Figure 3
Architectures for the flow-based models defined in Sec. 4c for each sampling approach. Note that each coupling layer $g_{k}^{p}$ or $g_{k}^{ap}$ employs masking of the updated field as shown in Fig. 2, such that the frozen components of the field are included as input to context functions. Superscripts on coupling layers indicate the translational equivariance structure of coupling layer inputs and outputs (either consistently transforming as P fields or AP fields). Many other choices of architectures are possible to model each distribution; the figure reflects the choices utilized in the numerical study undertaken in Sec. 5a. (a) $ϕ -$ Marginal architecture based on convex potential flows (Sec. 4c1), (b) Fully joint architecture for $q (ϕ, φ)$ based on coupling layers (Sec. 4c4), (c) $ϕ -$ Conditional model $q (ϕ, φ)$ dened via a restricted joint architecture (Sec. 4c2), and (d) Autoregressive model $q (ϕ) q (φ | ϕ)$ defined via coupling layers and linear flows (Sec. 4c3).
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Figure 4
Connected two-point correlation functions of the scalar field (projected to zero spatial momentum in each time slice $t$ ) for each model and choice of action parameters, computed from 100 Markov chains with 9k configurations each. Error estimates are obtained using data blocking with a bin size of 100 and applying statistical jackknife. Left: $g = 0.1$ , right: $g = 0.3$ . Bottom panels show the ratio of each data point to the HMC baseline, where the shaded regions correspond to the $1 σ$ and $2 σ$ uncertainty bands of the HMC results. For both the scalar correlators here and the fermionic ones in Fig. 5, Hotelling’s t-squared statistic [107] comparing each flow model result to the HMC baseline finds results to be consistent with correlated statistical fluctuations.
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Figure 5
Average fermionic two-point correlation in the time direction for each model and choice of action parameters using the same configurations and data blocking as for Fig. 4. Left: $g = 0.1$ , right: $g = 0.3$ . The choice of odd $t$ selects staggered spinor components at the sinks that give a nonzero average correlation with the source at $x = 0$ . Shaded regions in the bottom panel again depict the $1 σ$ and $2 σ$ uncertainties of the HMC baseline results.
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