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Bose gas on a two-leg ladder exhibits an interesting topological phase. Weshow the presence of a bosonic symmetry-protected-topological (SPT) phaseprotected by $Z_2\times Z_2$ symmetry. This symmetry leads to $Z_4$ fractionalquantization of $Z_4$ Berry phase, that is a topological order parameter toidentify the bulk. Using the $Z_4$ Berry phase, we have shown that theinteracting bosonic system possesses rich topological phases depending onparticle density and strength of interaction. Based on the bulk-edgecorrespondence, each edge state of the SPT phases is discussed in relation tothe $Z_4$ Berry phases. Especially we have found an intermediate phase that isnot adiabatically connected to a simple adiabatic limit, that possessesunconventional edge states, which we have numerically demonstrate by employingthe density-matrix-renormalization group algorithm.

$Z_2\times Z_2$ symmetry and $Z_4$ Berry phase of bosonic ladder