SNEG - Mathematica package for calculations with noncommuting operators of the second-quantization algebra

Copyright (C) 2002-2026 Rok Zitko

SNEG is a Wolfram Language package for symbolic calculations with operators of second quantization, with particular emphasis on anticommuting fermionic operators. It defines operator algebras through transformation rules and provides utilities for constructing operators, states, bases, and matrix representations.

At its core is nc, a noncommutative multiplication operation that automatically rewrites operators into a canonical form according to selected commutation and anticommutation rules. Conventional normal ordering places creation operators before annihilation operators; particle-hole and no-reordering conventions are also available. Canonical ordering simplifies expressions and supports efficient matrix-element evaluation in a chosen basis.

SNEG automates sign-sensitive manipulations that are otherwise tedious and error-prone.

• Operator declarations and canonical algebras for fermionic, spinless fermionic, bosonic, Majorana, and spin operators, with EMPTY, SEA, and NONE ordering conventions.

• High-level constructors for particle number, spin, isospin, Nambu operators, Hubbard interactions, ordinary and anomalous hopping, projection operators, and spin-spin and charge-charge couplings.

• Canonical conjugation, commutators and anticommutators, spin inversion, normal ordering, vacuum expectation values, and Wick expansion.

• Occupation-number representations of states, efficient application of operator strings to states, and conversion between state and product-of-operators representations.

• Generation of symmetry-adapted bases with well-defined Q and S_z, Q and S, or I and S, with optional reflection parity.

• Basis manipulation, transformation, merging, decomposition, and orthogonalization utilities.

• Dense and sparse matrix representations of operators in a chosen basis.

• Symbolic sums over free and dummy indexes, including automatic index renaming and simplification of Kronecker deltas.

• Dirac bra-ket notation that can be combined with second-quantization operator expressions.

• Particle-hole-aware ordering and Wick contractions for filled Fermi seas.

• Anticommuting Grassmann variables and fermionic coherent states.

• Finite single- and multi-mode phonon bases and operators, including fermion-phonon tensor-product bases.

• Selected Baker-Hausdorff simplifications and noncommutative operator power-series expansions.

SNEG provides the symbolic foundation of NRG Ljubljana, a framework for numerical renormalization group calculations for quantum impurity problems such as Kondo and Anderson impurity models. It has also been used for exact diagonalization of Hubbard clusters, high-order perturbation theory, and commutators of complex operator expressions.

The package is also suitable for teaching second-quantization methods: it exposes the algebra while automating lengthy sign-sensitive steps. The examples directory contains notebooks that can be adapted to more involved calculations.

SNEG is a Wolfram Language package and requires no compilation. With Mathematica 12.1 or newer, install or update the latest release directly from Mathematica:

PacletInstall[
  "https://github.com/rokzitko/sneg/releases/latest/download/SNEG.paclet"
]

Load the installed package and check its version with:

Needs["sneg`"]
$SnegVersion

The canonical capitalization Needs["Sneg"]` is also supported.

Remove the installed paclet with PacletUninstall["SNEG"].

For Mathematica versions 7 through 12.0, download the main branch archive or clone the repository. Place it in the per-user application directory returned by:

FileNameJoin[{$UserBaseDirectory, "Applications", "sneg"}]

The directory must be named sneg; rename an extracted sneg-main directory if necessary. Load a legacy installation with `<< sneg``. A checkout at any location can also be loaded directly:

Get["/absolute/path/to/sneg/sneg.m"]

Function-reference notebooks are available in docs/Documentation/English, including a legacy quickstart.nb. They were created for Mathematica's former help browser and should be opened directly; the old "Rebuild Help Index" integration is not supported by modern Mathematica versions.

Additional material includes:

• Example notebooks and scripts in examples.
• The early minimalist SNEG manual.
• The bundled preprint of the Computer Physics Communications article.

When publishing work that uses SNEG, please cite Rok Zitko, "SNEG - Mathematica package for symbolic calculations with second-quantization-operator expressions," Computer Physics Communications 182 (2011), 2259-2264, doi:10.1016/j.cpc.2011.05.013.

SNEG is free software distributed under the GNU General Public License, version 2 or, at your option, any later version. See LICENSE for the license text.

Bug reports, fixes, new regression tests, and documentation improvements are welcome through GitHub issues and pull requests.

Run the core regression suite from the repository root with:

wolframscript -file test-sneg.m

The exact-diagonalization example has a separate regression suite:

wolframscript -file examples/test-diagonalize.m

Build the release paclet with:

wolframscript -file build-paclet.m

The current SNEG code requires Mathematica 7 or newer. It has been tested with Mathematica versions through 15.

GitHub repository: https://github.com/rokzitko/sneg

• Rok Zitko
• "Jozef Stefan" Institute, F1 - Theoretical physics
• Jamova 39, SI-1000 Ljubljana, Slovenia
• Email: rok.zitko@ijs.si