Excitation spectra and rank tomography of linear matrix product tangent spaces

We formulate a tangent-space method for algebraic varieties of matrix product states (MPS) to study excitation spectra of non-uniform quantum many-body systems with open boundary conditions. We further introduce a rank tomography of the MPS tangent space, which characterizes its expressivity in terms of particle-sector rank profiles of the underlying MPS variety. Using the Bose–Hubbard model as a benchmark, we illustrate that the method reproduces low-lying excitations and captures finite-size precursors of the Mott-insulator to superfluid transition.