Z(2) x Z(2) symmetry and Z(4) Berry phase of bosonic ladders

Bose gas on a two-leg ladder exhibits an interesting topological phase. We show the presence of a bosonic symmetry-protected-topological (SPT) phase protected by Z(2) x Z(2) symmetry. This symmetry leads to Z(4) fractional quantization of the Z(4) Berry phase, which is a topological order parameter to identify the bulk. Using the Z(4) Berry phase, we show that the interacting bosonic system possesses rich topological phases depending on the particle density and strength of interaction. Based on the bulk-edge correspondence, each edge state of the SPT phases is discussed in relation to the Z(4) Berry phases. In particular, we have found an intermediate phase that is not adiabatically connected to a simple adiabatic limit that possesses unconventional edge states, which we numerically demonstrate by employing the density-matrix renormalization-group algorithm.