Post Matrix Product State Methods: from low-energy dynamics to thermalization

In this talk, I will review the concept of so-called "Post Matrix Product State (MPS) methods", i.e. algorithms based on the geometrical concept of the MPS manifold and its tangent space. The MPS tange space appears naturally when applying the Dirac-Frenkel time-dependent variational principle (TDVP) to the set of MPS, with provides an alternative algorithm to study time evolution and is becoming increasingly more popular, because it can easily deal with long-range interactions and does not suffer from some of the drawbacks of methods like the Time-Evolving Block Decimation. However, time evolution still leads to unbounded growth of entanglement, and when interested in the low-energy dynamics, the MPS tangent space and its extensions provides alternative methods to target e.g. the elementary quasi-particle excitations on top of the strongly correlated MPS ground state, and to compute their dispersion relation, their scattering matrix, and their contribution to spectral functions. More recently, the symplectic properties of the TDVP evolution have also made it a potential candidate to transform the unitary dynamics of closed quantum systems into an effective chaotic semi-classical dynamics that can be used to study thermalization.