Dynamics of Local Symmetry Correlators for Interacting Many-Particle Systems

TL;DR

This paper extends the concept of local symmetries and invariant currents from one-dimensional stationary waves to higher-dimensional interacting many-particle systems, providing a theoretical framework based on the BBGKY hierarchy.

Contribution

It develops a comprehensive theoretical framework for local symmetry correlators in higher-dimensional interacting systems, including their equations of motion and special cases.

Findings

Invariant two-point currents are recovered in non-interacting 1D cases.

Derived equations of motion for local symmetry correlators in many-body systems.

Provided an alternative integral representation with a clear interpretation.

Abstract

Recently (PRL 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 (BBGKY) 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 invariant two-point currents are 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.