  We study the ground-state properties of bosons loaded into the $p$-band of a
one dimensional optical lattice. We show that the phase diagram of the system
is substantially affected by the anharmonicity of the lattice potential. In
particular, for a certain range of tunneling strength, the full many-body
ground state of the system becomes degenerate. In this region, an additional
symmetry of the system, namely the parity of the occupation number of the
chosen orbital, is spontaneously broken. The state with nonvanishing staggered
angular momentum, which breaks the time-reversal symmetry, becomes the true
ground state of the system.
