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

Copyright (C) 2006-2023 Rok Zitko

The SNEG library is a package for Mathematica computer algebra system. It provides a framework for performing calculations using the operators of the second quantisation with an emphasis on the anti-commuting fermionic operators. It consists of a collection of transformation rules that define the algebra of operators and a number of utility functions.

The foundation is a definition of non-commutative multiplication with automatic reordering of operators in a standard form (usually the conventional normal ordering with creation operators preceding the annihilation operators), which takes into account selected (anti)commutation rules. Standard form reordering allows simplifications of expressions and the choice of normal ordering permits efficient evaluation of matrix elements in a given basis.

The library makes otherwise tedious calculations a routine operation. Especially, it prevents inauspicious sign errors when commuting fermionic operators.

Features

  • Collection of utility functions that generate various operator expressions, such as electron number, electron spin and isospin, 1-electron and 2-electron hopping, projection operators, spin-spin and charge-charge inter-site coupling, etc. These functions can be applied to construct the Hamiltonian and operators for observables.

  • Manipulation of operator expressions: canonical conjugation, spin inversion.

  • Calculation of vacuum expectation values of operator expressions.

  • Occupation-number representation of states and evaluation of operator-vector expressions. Occupation-number representations allows great speed-up in applying a string of operators on a basis state.

  • Transformations from product-of-operators to occupation-number representations of states and vice-versa.

  • Generation of basis states with well-defined particle number Q and spin projection Sz, well-defined number Q and spin S, or well-defined isospin I and spin S. For models with reflection symmetry, a parity quantum number can also be introduced.

  • Utility functions for manipulating sets of basis states: conversions between various representations, mapping a function to each state, transformations of basis, merging several sets of basis states, orthogonalization, etc.

  • Generation of matrix representations of operators in a given basis

  • Support for free (dummy) indexes and summed-over indexes: it is easy to write multiple sums over wave-numbers k_i and spins sigma_i. Automatic simplifications can be performed in such sums, which take into account that multiple summed-over indexes can be interchanged, etc.

  • Support for Dirac's bra-ket notation. Bra-ket notation can be intermixed with the second-quantization operators notation.

  • Distinction between particle and hole operators. This distinction is used in the standard normal ordering (creation operators are those that create a particle or a hole) and in the applications of the Wick theorem (see next entry).

  • Simplifications using Wick's theorem, in particular calculation of the ground state (vacuum) expectation values.

  • Support for commuting bosonic operators.

  • Support for anti-commuting Grassman variables and fermionic coherent states.

  • Support for real (Majorana) fermions.

  • Support for spin operators.

  • Automatic simplification of expressions with exponential functions of operators using the Baker-Campbell-Hausdorff formula.

  • Built-in support for pretty printing of operator expressions, obviating the need to use the Notation package. Colors are used to further improve readability.

  • Code for rewritting an operator expression in terms of higher-level functions, such as number, hopping, electron-electron repulsion, spin, etc. operators.

  • Support for converting compact ASCII operator-string expressions to the SNEG internal representation and vice-versa. This functionality is currently in the testing stage. See the examples in the file snegtoascii_asciitosneg.nb.

Applications

SNEG forms the basis of "NRG Ljubljana", a framework for performing numerical renormalization group calculations for quantum impurity problems, such as Kondo and Anderson impurity models (http://nrgljubljana.ijs.si/). In the past, it has also been applied to perform exact diagonalizations on Hubbard clusters, perturbation theory to higher orders and calculation of commutators of complex operator expressions. It should also be suitable for educational purposes, since it simplifies tedious calculations with second-quantization operators, much like Mathematica simplified learning calculus. A number of examples is included in the SNEG library distribution; they can easily be extended to non-trivial calculations.

Installation

Download the latest version from the main branch on github or here. The package can be installed by extracting to your '$InstallationDirectory\AddOns\Applications' directory. The installation directoy can be found by running $InstallationDirectory in a Wolfram language kernel. Alternatively, the package can be installed through the Mathematica provided GUI by going to file > instal... > type: Application > Source: From directory and selecting the downloaded Sneg folder.

The package is correctly installed if one can run

<< Sneg`

Documentation

Documentation for SNEG library is located in Documentation/English directory. This directory should be copied to your local Applications directory, then merged in your Mathematica help system using Rebuild Help Index in Help menu. The relevant pages are then located in "Add-ons & Links" tab, in the section "sneg documentation".

In the directory docs/ there is an older version of a SNEG manual and a long version of an article describing the SNEG package.

A number of example Mathematica notebooks can be found in the directory examples/.

License

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA

The full text of the GPL General Public License can be found in file LICENSE.

Contributing to SNEG

If you make improvements to SNEG, you are encouraged to share them with other users. Bug reports (and fixes) are very welcome as well. The contact information is in the next section.

Interesting directions for possible further extensions are improved support for bosonic operators (basis construction, simplifications in the case of bosons); performance improvements; code simplifications; and improved documentation.

Compatibility

SNEG was mostly developed and tested using Mathematica 5.2. It was also tested to work under Mathematica 5.0 and 5.1, as well as under new versions 6, 7, and 8. The author tries his best to make SNEG compatible across different versions of Mathematica.

Contact information:

SNEG library home-page: http://nrgljubljana.ijs.si/sneg

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

Rok Zitko "Jozef Stefan" Institute F1 - Theoretical physics Jamova 39 SI-1000 Ljubljana Slovenia

[email protected] (preferred contact address)