This chapter discusses the central models of interest, dubbed Yang–Mills matrix models. We explain how quantum spaces are obtained as nontrivial backgrounds or vacua of these models. Their quantization is discussed, both from a perturbative as well as a nonperturbative point of view.

The tangent space of a tensor network state is a manifold that parameterizes the variational trajectory of local tensors. It leads to a framework to accommodate the time-dependent variational principle (TDVP), which unifies stationary and time-dependent methods for dealing with tensor networks. It also offers an ideal platform for investigating elementary excitations, including nontrivial topological excitations, in a quantum many-body system under the single-mode approximation. This chapter introduces the tangent-space approaches in the variational determination of MPS and PEPS, starting with a general discussion on the properties of the tangent vectors of uniform MPS. It then exemplifies TDVP by applying it to optimize the ground state MPS. Finally, the methods for calculating the excitation spectra in both one and two dimensions are explored and applied to the antiferromagnetic Heisenberg model on the square lattice.