Drudge is a symbolic algebra system built on top of SymPy, focusing primarily on tensorial and noncommutative algebras. It is motivated by tedious symbolic derivations in quantum chemistry and many-body theory but is also useful for any symbolic manipulation and simplification involving indexed quantities, symbolic summations, and noncommutative algebras.

Leveraging the generic algorithms implemented in libcanon for canonicalization of combinatorial objects, such as strings and graphs, Drudge can find canonical forms for mathematical expressions involving tensors with symmetries and symbolic summations. For example, considering a fourth-order tensor $u$ satisfying

$$ u_{abcd} = -u_{bacd} = -u_{abdc} = u_{badc}, $$

an expression like

$$ \sum_{cd} u_{acbd} \rho_{dc} - \sum_{cd} u_{cabd} \rho_{dc} + \sum_{cd} u_{cdbc} \rho_{cd} $$

can be automatically simplified to a single term,

$$ 3 \sum_{cd} u_{acbd} \rho_{dc}, $$

irrespective of the names and ordering of the summation indices.

In addition to fully considering the permutational symmetries of tensors and summations, Drudge provides a general framework for handling noncommutative algebraic systems. Currently, Drudge natively supports the canonical commutation relations (CCR) and canonical anticommutation relations (CAR) algebras for fermions and bosons, respectively, in many-body theory, general Clifford algebras, and the su(2) algebra in its Cartan-Killing basis. Other noncommutative algebraic systems can be added with ease.

Using symbolic expressions from Drudge, the companion package gristmill can automatically optimize and generate numerical code. Optimal computational complexity can be achieved for various methods in quantum chemistry, which heavily rely on tensor contractions and their sums.

Drudge can be installed directly from the GitHub repository using uv (recommended)

uv pip install git+https://github.com/DrudgeCAS/drudge.git

or pip

pip install git+https://github.com/DrudgeCAS/drudge.git

Note: Native Windows builds are currently not working. Please use WSL (Windows Subsystem for Linux) to install and run Drudge on Windows for now.

Please refer to the documentation at https://drudgecas.github.io/drudge/. Additional examples can be found in the ./docs/examples directory.

If you use Drudge in your work, please cite this repository and Jinmo Zhao's Ph.D. thesis:

1. The Drudge GitHub repository

@misc{DrudgeCAS,
  author       = {Jinmo Zhao and Guo P. Chen and Gaurav Harsha and Matthew Wholey and Thomas M. Henderson and Gustavo E. Scuseria},
  title        = {Drudge: A symbolic algebra system for tensorial and noncommutative algebras},
  publisher    = {GitHub},
  year         = {2016--2025},
  url          = {https://github.com/DrudgeCAS/drudge},
  note         = {GitHub repository}
}

2. Jinmo Zhao’s Ph.D. thesis

@phdthesis{Zhao2018Drudge,
  author       = {Jinmo Zhao},
  title        = {Symbolic Solution for Computational Quantum Many-Body Theory Development},
  school       = {Rice University},
  year         = {2018},
  month        = {April},
  address      = {Houston, Texas, USA},
  type         = {PhD thesis},
  url          = {https://www.proquest.com/openview/61a9a86c07dbb6e5270bdeb1c84384db/1?pq-origsite=gscholar&cbl=18750&diss=y}
}

Link: Symbolic Solution for Computational Quantum Many-Body Theory Development — Jinmo Zhao (2018)

You may also use the CITATION.cff file provided in this repository, which is compatible with citation managers such as Zotero and Mendeley.

Drudge was originally developed by Jinmo Zhao during his Ph.D. at Rice University, under the supervision of Prof. Gustavo E. Scuseria. The project was supported as part of the Center for the Computational Design of Functional Layered Materials, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award DE-SC0012575. The package is currently maintained by Guo P. Chen, Gaurav Harsha, Matthew Wholey, and other members of the Scueria group.