Path selection rule in matrix product systems

A matrix product system chooses a special path in the phase diagram to undergo a quantum phase transition (QPT) and shows different behaviors compared with a traditional QPT, such as the symmetry behavior of some physical observables described in this paper. An equation is established, which (i) helps one to understand the special behaviors of a matrix product state (MPS)-QPTs, and (ii) can be used to detect the QPT point of a MPS, much simpler than usual procedures of calculating the transfer matrix or density matrix of the system. The equation acts as a selection rule for the path of the MPS-QPT and is believed to be the essence of distinguishing a MPS-QPT from a traditional QPT. Furthermore, the discontinuity of the derivative of an observable is found to be connected directly to the turning point in the path of the MPS, but not the phase boundary point in the phase diagram, though the two are in accordance with each other in many cases.