symveig: Verified eigenvalue enclosures for symmetry-decomposed Hermitian matrices

Abstract

Exact diagonalization of quantum lattice Hamiltonians returns floating-point eigenvalues whose accuracy is not certified: rounding error, eigensolver behaviour, and ill-conditioning can corrupt a result without warning. We present \texttt{symveig}, a pure NumPy/SciPy package that computes rigorous, machine-checkable enclosures of all eigenvalues of a Hermitian matrix, with an optional symmetry-sector decomposition. For a matrix that commutes with an abelian conserved quantity diagonal in the working basis (for example the total magnetization of a spin model), the package verifies each symmetry sector independently. Because each sector block is much smaller than the full matrix, this yields enclosures that are both tighter (by a factor of $3$-$9$ across system sizes $L = 4$-$12$) and dramatically faster (a wall-clock speedup of up to $130\times$ at $L = 12$) than verifying the full matrix, while never forming or diagonalizing it. Every enclosure half-width is a guaranteed upper bound on the distance from a computed eigenvalue to the nearest true eigenvalue under IEEE~754 round-to-nearest arithmetic, obtained by explicit floating-point error analysis with no heuristic slack. The implementation requires neither INTLAB nor MATLAB, bringing rigorously verified eigenvalue enclosure into the standard scientific-Python stack used in computational physics. We validate the package on $1$D Heisenberg (open and periodic), $J_1$-$J_2$ Heisenberg, and $2$D Heisenberg lattices, confirming that every computed eigenvalue is contained in its enclosure across all tested configurations.