FeynmanDiagram

FeynmanDiagram.jl is a Julia package designed to efficiently encode Feynman diagrams --- essential elements of Quantum Field Theory (QFT) —-- into compact computational graphs for fast evaluation. It employs Taylor-mode Automatic Differentiation (AD) specifically to implement field-theoretic renormalization schemes, a pivotal technique in QFT that significantly improves the convergence of Feynman diagrammatic series. This approach underscores the synergy between QFT and AI technologies, effectively addressing the sophisticated computational challenges in QFT.

Key Features

• Computational Graphs for Feynman Diagrams: Utilizes computational graphs analogous to those in Machine Learning (ML) architectures, where nodes represent mathematical operations and edges denote the tensor flow.

• Compiler Architecture: Implements a three-stage compiler process, analogous to modern programming language compilers, to transform Feynman diagrams into executable code, significantly enhancing the adaptability and computational efficiency of QFT calculations.

• Interdisciplinary Approach: Bridges QFT and AI tech stack by adapting ML algorithms, such as Taylor-mode AD, for QFT calculations, enhancing computational efficiency with tools like JAX, TensorFlow, PyTorch, and Mindspore.

Compiler Architecture Overview

In general, Feynman diagrams represent high-order integrals. The integrands are propagators and interactions composed with basic arithmetic operations (multiplication, addition, power, etc). The sequence of calculating the integrand by combining propagators and interactions with arithmetic operations can be represented as an algebraic computational graph. In this sense, the computational graph serves as an intermediate representation that standardizes the treatment of various diagram types, ensuring a consistent approach across different QFT calculations.

Drawing from these insights, the architecture of our compiler is intentionally crafted to process Feynman diagrams via a strategic, three-stage process that reflects the advanced design principles of the LLVM compiler architecture. This approach enhances the compiler's flexibility for a wide array of QFT computations and significantly improves its computational efficiency. The workflow is organized into three critical stages:

• Front End: In this initial phase, Feynman diagrams are generated and transformed into a standardized intermediate representation, taking the form of static computational graphs. This stage features two key algorithms for generating generic Feynman diagrams for weak-coupling expansions:

  • The Parquet module systematically organizes higher-order Feynman diagrams into a concise hierarchical structure of sub-diagrams, enabling efficient evaluation of repeated sub-diagrams. This module introduces algorithms for constructing streamlined computational graphs for two-, three-, and four-point vertex functions by exploiting the perturbative representations of the Dyson-Schwinger and parquet equations. This approach facilitates the analysis of a wide array of observables in quantum many-body problems.
  • The GV module, specifically aimed at many-electron problems with Coulomb interactions, incorporates the algorithm proposed in Nat Commun 10, 3725 (2019).
  • The front end also allows for the integration of new diagram types by users, enhancing its adaptability.

• Intermediate Representation: At this stage, the compiler applies optimizations and Automatic Differentiation (AD) to the static computational graph. This process is geared towards refining the graph for thorough analysis and optimization. The optimizations are aimed at streamlining the graph by removing redundant elements, flattening nested chains, and eliminating zero-valued nodes, among other strategies. The incorporation of Taylor-mode AD is critical for efficiently calculating high-order derivatives, essential for deriving diagrams for specific heat, RG flow equations, or renormalized Feynman diagrams.

• Back End: This final phase is responsible for translating the optimized graph into executable code that is compatible with a broad spectrum of computing environments. It supports various programming languages and facilitates seamless integration with different software and hardware ecosystems, significantly extending the compiler's utility across multiple platforms.

Usage

Installation

To install the package, you can simply use the following command in the Julia package manager:

using Pkg
Pkg.add("FeynmanDiagram")

Example: Self-energy in FrontEnds

The algorithms in FrontEnds generate Feynman diagrams for weak coupling expansions, supporting various diagram types, such as self-energy, polarization, 3-point vertex function, and 4-point vertex function diagrams. The internal degrees of freedom can be either the loop variables (e.g., momentum or frequency) or the site variables (e.g., imaginary-time or lattice site).

The example below demonstrates how to generate all two-loop self-energy diagrams by Parquet and optimize their computational graphs.

Diagram generation with the Parquet algorithm

using FeynmanDiagram
import FeynmanDiagram.FrontEnds: NoHartree

# Define a parameter for two-loop self-energy diagrams in the momentum and the imaginary-time representation. Exclude any diagrams containing Hartree subdiagrams. 
para = Parquet.DiagPara(type = Parquet.SigmaDiag, innerLoopNum = 2, hasTau = true, filter=[NoHartree,]);

# Construct Feynman diagrams within a DataFrame utilizing the parquet algorithm. The resulting sigmadf DataFrame comprises two components: the instantaneous part and the dynamic part of the self-energy.
sigmadf = Parquet.build(para) 
2×4 DataFrame
 Row │ diagram                            extT    hash   type
     │ Graph…                             Tuple…  Int64  Analytic…
─────┼─────────────────────────────────────────────────────────────
   1 │ 18721-ΣIns, k[1.0, 0.0, 0.0], t(…  (1, 1)  18721  Instant
   2 │ 18722-ΣDyn, k[1.0, 0.0, 0.0], t(…  (1, 2)  18722  Dynamic

# Optimize the Graph for the given Feynman diagrams.
optimize!(sigmadf.diagram); 

Construct renormalized Feynman diagrams using Taylor-mode AD

The example code below demonstrates how to build renormalized Feynman diagrams for the self-energy including Green's function and interaction counterterms using Taylor-mode AD.

# Set the renormalization orders. The first element is the maximum order of the Green's function counterterms, and the second element is the maximum order of the interaction counterterms.
renormalization_orders = [2, 1];

# Define functions that determine how the differentiation variables depend on the properties of the leaves in your graphs, identifying `BareGreenId` and `BareInteractionId` properties as the Green's function and interaction counterterms, respectively.
leaf_dep_funcs = [pr -> pr isa FrontEnds.BareGreenId, pr -> pr isa FrontEnds.BareInteractionId];

# Generate the Dict of Graph for the renormalized self-energy diagrams with the Green's function counterterms and the interaction counterterms.
dict_sigma = taylorAD(sigmadf.diagram, renormalization_orders, leaf_dep_funcs)
Dict{Vector{Int64}, Vector{Graph}} with 6 entries:
  [0, 0] => [18822Ins, k[1.0, 0.0, 0.0], t(1, 1)=0.0=Ⓧ , 19019Dyn, k[1.0, 0.0, 0.0], t(1, 2)=0.0=⨁ ]
  [2, 1] => [18823Ins, k[1.0, 0.0, 0.0], t(1, 1)[2, 1]=0.0=Ⓧ , 19020Dyn, k[1.0, 0.0, 0.0], t(1, 2)[2, 1]=0.0=⨁ ]
  [2, 0] => [18824Ins, k[1.0, 0.0, 0.0], t(1, 1)[2]=0.0=Ⓧ , 19021Dyn, k[1.0, 0.0, 0.0], t(1, 2)[2]=0.0=⨁ ]
  [1, 1] => [18825Ins, k[1.0, 0.0, 0.0], t(1, 1)[1, 1]=0.0=Ⓧ , 19022Dyn, k[1.0, 0.0, 0.0], t(1, 2)[1, 1]=0.0=⨁ ]
  [1, 0] => [18826Ins, k[1.0, 0.0, 0.0], t(1, 1)[1]=0.0=Ⓧ , 19023Dyn, k[1.0, 0.0, 0.0], t(1, 2)[1]=0.0=⨁ ]
  [0, 1] => [18827Ins, k[1.0, 0.0, 0.0], t(1, 1)[0, 1]=0.0=Ⓧ , 19024Dyn, k[1.0, 0.0, 0.0], t(1, 2)[0, 1]=0.0=⨁ ]

Example: Compile Feynman diagrams to different programming languages

The Back End architecture enables the compiler to output source code in a range of other programming languages and machine learning frameworks. The example code below demonstrates how to use the Compilers to generate the source code for the self-energy diagrams in Julia, C, and Python.

# Access the self-energy data for the configuration with 2nd-order Green's function counterterms and 1st-order interaction counterterms.
g_o21 = dict_sigma[[2,1]]; 

# Compile the selected self-energy into a Julia RuntimeGeneratedFunction `func` and a `leafmap`.
# The `leafmap` associates vector indices of leaf values with their corresponding leaves (propagators and interactions). 
func, leafmap = Compilers.compile(g_o21);

# Export the self-energy's source code to a Julia file.
Compilers.compile_Julia(g_o21, "func_o21.jl");

# Export the self-energy's source code to a C file.
Compilers.compile_C(g_o21, "func_o21.c");

# Export the self-energy's source code to a Python file.
Compilers.compile_Python(g_o21, "func_o21.py");

Computational Graph visualization

Tree-type visualization using ETE3

To visualize tree-type structures of the self-energy in the Parquet example, install the ETE3 Python package, a powerful toolkit for tree visualizations.

Execute the following command in Julia for tree-type visualization of the self-energy generated in the above Parquet example:

plot_tree(sigmadf.diagram, depth = 3)

For installation instructions on using ETE3 with PyCall.jl, please refer to the PyCall.jl documentation on how to configure and use external Python packages within Julia.

Graph visualization using DOT

The Back-End's Compilers module includes functionality to translate computational graphs into .dot files, which can be visualized using the dot command from the Graphviz software package. Below is an example code snippet that illustrates how to visualize the computational graph of the self-energy from the previously mentioned Parquet example.

# Convert the self-energy graph from the `Parquet` example into a dot file.
Compilers.compile_dot(sigmadf.diagram, "sigma_o2.dot");

# Use Graphviz's dot command to create an SVG visualization of the computational graph.
run(`dot -Tsvg sigma_o2.dot -o sigma_o2.svg`);

The resulting computational graphs are depicted as directed acyclic graphs by the dot compiler. In these visualizations, the leaves are indicated by green oval nodes, while the intermediate nodes take on a rectangular shape. On the penultimate level of the graph, the left blue node with the Sum operator signifies the dynamic part of the self-energy, and the right blue node denotes the instantaneous part of the self-energy.

License

FeynmanDiagram.jl is open-source, available under the MIT License. For more details, see the license.md file in the repository.