Projector formalism for kept and discarded spaces of matrix product states

Any matrix product state $\mathrm{\mid }\mathrm{\Psi }⟩$ has a set of associated kept and discarded spaces, needed for the description of $\mathrm{\mid }\mathrm{\Psi }⟩$, and changes thereof, respectively. These induce a partition of the full Hilbert space of the system into mutually orthogonal spaces of irreducible $n$-site variations of $\mathrm{\mid }\mathrm{\Psi }⟩$. Here, we introduce a convenient projector formalism and diagrammatic notation to characterize these $n$-site spaces explicitly. This greatly facilitates the formulation of MPS algorithms that explicitly or implicitly employ discarded spaces. As an illustration, we derive an explicit expression for the $n$-site energy variance and evaluate it numerically for a model with long-range hopping. We also describe an efficient algorithm for computing low-lying $n$-site excitations above a finite MPS ground state.