  We present some exact results for the optimal Matrix Product State (MPS)
approximation to the ground state of the infinite isotropic Heisenberg spin-1/2
chain. Our approach is based on the systematic use of Schmidt decompositions to
reduce the problem of approximating for the ground state of a spin chain to an
analytical minimization. This allows to show that results of standard
simulations, e.g. density matrix renormalization group and infinite time
evolving block decimation, do correspond to the result obtained by this
minimization strategy and, thus, both methods deliver optimal MPS with the same
energy but, otherwise, different properties. We also find that translational
and rotational symmetries cannot be maintained simultaneously by the MPS ansatz
of minimum energy and present explicit constructions for each case.
Furthermore, we analyze symmetry restoration and quantify it to uncover new
scaling relations. The method we propose can be extended to any translational
invariant Hamiltonian.
