Violation of the area law and long-range correlations in infinite-dimensional-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.