Current algebra for a generalized two-sites Bose-Hubbard model

Abstract

I present a current algebra for a generalized two-sites Bose-Hubbard model and use it to get the quantum dynamics of the currents. Different choices of the Hamiltonian parameters yield different dynamics. The current algebra is isomorphic to the SO(3)-algebra of the angular momentum. Using the wave functions I discuss the symmetries of the system. The Hamiltonian has one conserved quantity, the total number of atoms N, that is related to its global U(1) gauge symmetry. The $\mathbb {Z}_2$ symmetry is associated with the parity of the wave function and is related to the parity of N. I generalize the Heisenberg equation of motion to write the second time derivative of any operator.