Research directions

I am currently following these research directions.

Degree of tensor train varieties via integral geometry

Tensor networks are widely used in many-body quantum physics. However, only a few algebraic properties of their corresponding parametrization are known. In this project we derive a formula, or rather algorithmic computation, for the degree of tensor train varieties. Joint work with Andrea Rosana.

Geometry of excited states via linearised tangent method of TT

Using an algebro-geometric tangent-space formulation of matrix product states (MPS) one can probe excitation spectra in strongly correlated lattice systems. Starting from the Gutzwiller mean-field description, we derive a correlated generalization that incorporates quantum fluctuations within the MPS manifold. Excitations appear as tangent vectors to the variational manifold, providing a unified geometric picture of collective modes and their dispersion. Our finite-size, open-boundary implementation for the Bose–Hubbard model extends previous approaches restricted to infinite, translation-invariant systems. The geometric structure of the MPS space is further analyzed through its interpretation as a projective variety, linking tensor-network methods to concepts from algebraic geometry and clarifying the role of gauge redundancies in variational wave-function representations. Joint work with Iacopo Carusotto.

Homotopy hypersurface region for continuation of ground states on varieties

Homotopy continuation between two Hamiltonians allows to follow solution of an initial Hamiltonina H0 to a final Hamiltonian H1. However, such a continuation is not always possible. To determine when a continuation of the ground state solution is possible, two discriminants can be determined symbolically or numerically using pseudo-witness sets. Joint work with Oskar Henriksson.