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We present a matrix product state (MPS) algorithm to approximate groundstates of translationally invariant systems with periodic boundary conditions.For a fixed value of the bond dimension D of the MPS, we discuss how tominimize the computational cost to obtain a seemingly optimal MPS approximationto the ground state. In a chain of N sites and correlation length \xi, thecomputational cost formally scales as g(D,\xi /N)D^3, where g(D,\xi /N) is anontrivial function. For \xi << N, this scaling reduces to D^3, independent ofthe system size N, making our algorithm N times faster than previous proposals.We apply the method to obtain MPS approximations for the ground states of thecritical quantum Ising and Heisenberg spin-1/2 models as well as for thenoncritical Heisenberg spin-1 model. In the critical case, for any chain lengthN, we find a model-dependent bond dimension D(N) above which the polynomialdecay of correlations is faithfully reproduced throughout the entire system.

Exploiting translational invariance in Matrix Product State simulations of spin chains with periodic boundary conditions