We present several algorithms for the simulation of translationally invariant (TI) quantum spin chains with periodic
boundary conditions (PBC). By using TI matrix product states (MPS) we can in general reduce the computational
cost by a factor N, with N the number of spins in the chain.
First we present an algorithm for the approximation of ground states (GS) that is based on the computation of the
gradient of the energy [1]. We achieve a scaling of the computational cost of O(D3n2) + O(D3mn), where D is the
virtual bond dimension of the MPS and m and n are some parameters that will be explained in more detail in the
talk. There is a tradeoff between the parameters n and m and we show how to find the optimal balance. The analysis
of the numerical results confirms previous observations regarding the induced correlation length of MPS with finite D
[2, 3]. Furthermore we observe a crossover between the finite-N scaling and finite-D scaling in the context of critical
quantum spin chains similar to the one observed by Nishino [4] in the context of classical two dimensional systems.
Next we present an algorithm for the approximation of dispersion relations that uses as an ansatz MPS-based states
with well defined momentum [5]. Here, we achieve a scaling of the computational cost of O(D6N2). Due to the large
D scaling we are restricted to comparatively small D. Nonetheless we obtain very good approximations of one-particle
excitations. The numerical results yield some insight into the interpretation of the quasiparticles that occur in the
exact solution of the Quantum Ising Model with PBC.