  We explore matrix product state approximations to wavefunctions which have
spontaneously broken symmetries or are critical. We are motivated by the fact
that symmetries, and their associated conservation laws, lead to block-sparse
matrix product states. Numerical calculations which take advantage of these
symmetries run faster and require less memory. However, in symmetry-broken and
critical phases the block sparse ansatz yields less accurate energies. We
characterize the role of conservation laws in matrix product states and
determine when it is beneficial to make use of them.

Quantum Gases;Strongly Correlated Electrons