Researchers have developed a new tangent-space method to study the excitation spectra of non-uniform quantum many-body systems with open boundary conditions. The technique focuses on algebraic varieties of matrix product states (MPS), an efficient representation of quantum states that allows for the simulation of complex systems. This advance is crucial for understanding the dynamics of heterogeneous quantum systems, where properties vary spatially, a common scenario in quantum devices and advanced materials.

1The method introduces a "rank tomography" of the MPS tangent space, which quantifies the expressive power of these states in terms of the particle-sector rank profiles of the underlying MPS variety. This characterization allows for the evaluation of the fidelity with which MPS can describe system excitations, providing a tool to optimize simulations and understand their limitations. The ability to analyze expressivity is fundamental for the design and interpretation of experiments with many-body quantum systems.

To validate the methodology, it was applied to the Bose-Hubbard model, a benchmark system in condensed matter physics that describes interacting bosons on a lattice. The results demonstrate that the method accurately reproduces low-lying excitations and, significantly, captures finite-size precursors of the Mott-insulator to superfluid transition. This ability to identify phase transitions in finite systems is an important step towards understanding emergent quantum phenomena in real materials and engineering new quantum devices.