  We study quantum phase transitions between competing orders in
one-dimensional spin systems. We focus on systems that can be mapped to a
dual-field double sine-Gordon model as a bosonized effective field theory. This
model contains two pinning potential terms of dual fields that stabilize
competing orders and allows different types of quantum phase transition to
happen between two ordered phases. At the transition point, elementary
excitations change from the topological soliton of one of the dual fields to
that of the other, thus it can be characterized as a topological transition. We
compute the dynamical susceptibilities and the entanglement entropy, which
gives us access to the central charge, of the system using a numerical
technique of infinite time-evolving block decimation and characterize the
universality class of the transition as well as the nature of the order in each
phase. The possible realizations of such transitions in experimental systems
both for condensed matter and cold atomic gases are also discussed.
