Dynamics of local symmetry correlators for interacting many-particle systems

Recently [P. A. Kalozoumis et al. Phys. Rev. Lett. 113, 050403 (2014)] the concept of local symmetries in one-dimensional stationary wave propagation has been shown to lead to a class of invariant two-point currents that allow to generalize the parity and Bloch theorem. In the present work, we establish the theoretical framework of local symmetries for higher-dimensional interacting many-body systems. Based on the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, we derive the equations of motion of local symmetry correlators which are off-diagonal elements of the reduced one-body density matrix at symmetry related positions. The natural orbital representation yields equations of motion for the convex sum of the local symmetry correlators of the natural orbitals as well as for the local symmetry correlators of the individual orbitals themselves. An alternative integral representation with a unique interpretation is provided. We discuss special cases, such as the bosonic and fermionic mean field theory, and show in particular that the invariance of two-point currents is recovered in the case of the non-interacting one-dimensional stationary wave propagation. Finally we derive the equations of motion for anomalous local symmetry correlators which indicate the breaking of a global into a local symmetry in the stationary non-interacting case.