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Based on the MPS formalism, we introduce an ansatz for capturing excitedstates in finite systems with open boundary conditions, providing a veryefficient method for computing, e.g., the spectral gap of quantum spin chains.This method can be straightforwardly implemented on top of an existing DMRG orMPS ground-state code. Although this approach is built on open-boundary MPS, wealso apply it to systems with periodic boundary conditions. Despite theexplicit breaking of translation symmetry by the MPS representation, we showthat momentum emerges as a good quantum number, and can be exploited forlabeling excitations on top of MPS ground states. We apply our method to thecritical 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 quantumsystems on infinite cylinders. Again, despite the explicit breaking oftranslation symmetry in the periodic direction, we recover momentum as a goodquantum number for labeling excitations. We apply this method to the 2-Dtransverse-field Ising model and the half-filled Hubbard model; for the latter,we obtain accurate results for, e.g., the hole dispersion for cylindercircumferences up to eight sites.

Efficient MPS methods for extracting spectral information on rings and cylinders