Properties of Parity Non-conserved and Parity Conserved States in Matrix Product Systems

In terms of reflection transformation of a matrix product state (MPS), the parity of the MPS is defined. Based on the reflective parity non-conserved MPS pair we construct the even-parity state vertical bar Psi(e)> and the odd-parity state vertical bar Psi(o)>. It is interesting to find that the parity non-conserved reflective MPS pair have no long-range correlations; instead the even-parity state vertical bar Psi(e)> and the odd-parity state vertical bar Psi(o)> constructed from them have the same long-range correlations for the parity non-conserved block operators. Moreover, the entanglement between a block of n contiguous spins and the rest of the spin chain for the states vertical bar Psi(e)> and vertical bar Psi(o)> is larger than that for the reflective MPS pair except for n = 1, and the difference of them approaches 1 monotonically and asymptotically from 0 as n increases from 1. These characteristics indicate that MPS parity as a conserved physical quantity represents a kind of coherent collective quantum mode, and that the parity conserved MPSs contain more correlation, coherence, and entanglement than the parity non-conserved ones.