Efficient numerical computation of the Pfaffian for dense and banded skew-symmetric matrices
This work provides practical, optimized algorithms for physicists needing to compute Pfaffians, a common but computationally challenging task in condensed matter physics.
The authors develop efficient numerical methods for computing the Pfaffian of skew-symmetric matrices by reducing them to tridiagonal form, with implementations in Fortran, Python, Matlab, and Mathematica. They demonstrate the methods on a class D nanowire, showing numerical equivalence of topological charge definitions based on Hamiltonian and scattering matrix.
Computing the Pfaffian of a skew-symmetric matrix is a problem that arises in various fields of physics. Both computing the Pfaffian and a related problem, computing the canonical form of a skew-symmetric matrix under unitary congruence, can be solved easily once the skew-symmetric matrix has been reduced to skew-symmetric tridiagonal form. We develop efficient numerical methods for computing this tridiagonal form based on Gauss transformations, using a skew-symmetric, blocked form of the Parlett-Reid algorithm, or based on unitary transformations, using block Householder transformations and Givens rotations, that are applicable to dense and banded matrices, respectively. We also give a complete and fully optimized implementation of these algorithms in Fortran, and also provide Python, Matlab and Mathematica implementations for convenience. Finally, we apply these methods to compute the topological charge of a class D nanowire, and show numerically the equivalence of definitions based on the Hamiltonian and the scattering matrix.