Identification of topologically different valence bond states in spin ladders

We discuss relation between three different methods to identify topologically distinct short-range valence-bond ground states in spin-1/2 two-leg ladders: the string order parameters, the level-crossing method, and the ground-state expectation value of the twist operator. For this purpose we reexamine a phase transition in the frustrated spin ladder. We propose a proper bosonization of the string order parameters. Then, these three methods are shown to be equivalent reflecting the Z2xZ2 symmetry breaking of the order and the disorder fields of the two-dimensional Ising model