  Based on the MPS formalism, we introduce an ansatz for capturing excited
states in finite systems with open boundary conditions, providing a very
efficient method for computing, e.g., the spectral gap of quantum spin chains.
This method can be straightforwardly implemented on top of an existing DMRG or
MPS ground-state code. Although this approach is built on open-boundary MPS, we
also apply it to systems with periodic boundary conditions. Despite the
explicit breaking of translation symmetry by the MPS representation, we show
that momentum emerges as a good quantum number, and can be exploited for
labeling excitations on top of MPS ground states. We apply our method to the
critical Ising chain on a ring and the classical Potts model on a cylinder.
Finally, we apply the same idea to compute excitation spectra for 2-D quantum
systems on infinite cylinders. Again, despite the explicit breaking of
translation symmetry in the periodic direction, we recover momentum as a good
quantum number for labeling excitations. We apply this method to the 2-D
transverse-field Ising model and the half-filled Hubbard model; for the latter,
we obtain accurate results for, e.g., the hole dispersion for cylinder
circumferences up to eight sites.
