SNEG library

SNEG library is a Mathematica package that provides a framework for performing calculations using the operators of the second quantization with an emphasis on the anti-commuting fermionic operators in the context of solid-state and atomic physics. It consists of a collection of transformation rules that define the algebra of operators and a comprehensive library of utility functions. The cornerstone of SNEG 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.

Examples

Note: you need to have Mathematica or Mathematica Player installed in order to view the following examples.

Additional examples can be found in the Further examples sections of the Help notebooks for SNEG functions. See also Quick start manual in the form of Mathematica notebook.

News

19.4.2022: updated version (2.0.0). The sneg code is now hosted on github. This version also fixes a bug in snegrealconstants (thanks for spotting this one, Luka!), and adds some new functions. There is also a citable Zenodo snapshot.
24.1.2019: updated version (1.250). Maintenance release, tested to work with Mathematica 11.3.
29.7.2011: SNEG - Mathematica package for symbolic calculations with second-quantization-operator expressions, R. Zitko, Comp. Phys. Comm. 182 2259, (2011) has been published. (PDF). Please cite this work if you make extensive use of SNRG in your research.
18.3.2011: updated version (1.228). Maintenance release, tested to be compatible with Mathematica 7 and 8.
9.4.2008: updated version (1.191). It improves compatibility with Mathematica 6 and adds support for spin operators. Otherwise version 1.191 is only a minor update.
6.6.2007: updated version (1.168). In addition to much improved documentation (now fully integrated in Mathematica Help Browser) and additional examples, a considerable number of new features has been implemented.
14.7.2006: first public release of the library.

Features

Collection of utility functions that generate various operator expressions, such as electron number, electron spin and isospin, one-electron and two-electron hopping, projection operators, spin-spin and charge-charge inter-site coupling, etc. This is heavily used in the "NRG Ljubljana" for building expressions for the Hamiltonian and other operators.
Manipulation of operator expressions: canonic 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 S_z, 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. These sets of basis states are used in the "NRG Ljubljana" code to perform exact diagonalizations in various invariant subspaces.
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 σ_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. It is used to represent degrees of freedom in addition to the fermionic ones in the "NRG Ljubljana, in particular phonons and spin. 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 the Wick 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.
Automatic canonical ordering of fermionic operators in the case of Fermi sea vacuum.
Support for symbolic sums: automatic renaming of dummy indexes when name conflicts appear, better simplification of expressions involving sums.
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 spin operators.
Functions for basis generation for some additional cases: conserved total charge, conserved total spin, spinless fermions, no conserved quantum numbers.

Applications

The SNEG library is useful beyond the NRG calculations. It has 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 eased learning calculus. A number of examples is included in the SNEG library distribution; they can easily be extended to non-trivial calculations.