  We develop tangent space methods for projected entangled-pair states (PEPS)
that provide direct access to the low-energy sector of strongly-correlated
two-dimensional quantum systems. More specifically, we construct a variational
ansatz for elementary excitations on top of PEPS ground states that allows for
computing gaps, dispersion relations, and spectral weights directly in the
thermodynamic limit. Solving the corresponding variational problem requires the
evaluation of momentum transformed two-point and three-point correlation
functions on a PEPS background, which we can compute efficiently by using a
contraction scheme. As an application we study the spectral properties of the
magnons of the Affleck-Kennedy-Lieb-Tasaki model on the square lattice and the
anyonic excitations in a perturbed version of Kitaev's toric code.
