Violation of the area law and long range correlations in infinite matrix product states

We propose to construct a family of states of one-dimensional spin chains by replacing the finite dimensional matrices in matrix product states by infinite dimensional operators constructed from bosonic annihilation and creation operators. The resulting states are demonstrated to violate the area law and to exhibit long range correlations. In addition, we propose an efficient way to prepare the states experimentally, in which the spins interact sequentially with an ancilla system.