Fig. 2: Topological half-charge pumping.
a The two-period time evolution in the two-level system. b The sketch of a half-charge pumping chain with \(v(t)={v}_{0}{\sin }^{2}\frac{\omega t}{2}\) and \(w(t)={w}_{0}\sin \frac{2\pi t}{T}\) and the cyclic evolution in v-w space. The black and red circles mean for the topologically nontrivial case \(v(t)=-\frac{{v}_{0}}{2}+{v}_{0}{\sin }^{2}\frac{\omega t}{2}\) and trivial case \(v(t)=m+{v}_{0}{\sin }^{2}\frac{\omega t}{2}\) with m > 0. c The energy dispersion for the gapless one-dimensional chain in Eq. (3) in two time period T. d The energy dispersion for the gapped one-dimensional chain in Eq. (3) with a nonzero mass in \(v(t)=m+{v}_{0}{\sin }^{2}\frac{\omega t}{2}\) in two time period T. e Electric polarization as a function of t in two time period T for the gapless one dimensional chain. f Electric polarization as a function of t in one time period T for the gapped cases of \(v(t)=m+{v}_{0}{\sin }^{2}\frac{\omega t}{2}\) and \(v(t)=-\frac{{v}_{0}}{2}+{v}_{0}{\sin }^{2}\frac{\omega t}{2}\).
