Topology in the space-time scaling limit of quantum dynamics

We investigate the role of topology in the space-time scaling limit of quantum quench dynamics, where both time and system size tend to infinity at a constant ratio. There, while the standard topological characterization relying on local unitary transformations becomes ill defined, we show how a different dynamical notion of topology naturally arises through a dynamical winding number encoding the linear response of the Berry phase to a magnetic flux. Specifically, we find that the presence of a locally invisible constant magnetic flux is revealed by a dynamical staircase behavior of the Berry phase, whose topologically quantized plateaus characterize the space-time scaling limit of a quenched Rice-Mele model. These jumps in the Berry phase are also shown to be related to the interband elements of the DC current operator. We outline possible experimental platforms for observing the predicted phenomena in finite systems.

Figure 1

(a) Illustration of a Rice-Mele lattice model on a ring [see Eq. (6)], subject to a quench by a sudden variation of the intercell hopping amplitude $\gamma r^i\to \gamma r^f$. (b) Schematic representation of the closed loops formed in the complex plane by the Bloch state overlap ${\xi }_{-}$ [see Eq. (1)] as a function of quasimomentum $k$. Nontrivial loops (right panel) may form in the STSL regime at critical values ${\eta }_m^{*}$ [see Eq. (3)] of the ratio $\eta =2\pi t/L$. Solid lines represent the zeroth-order contribution ${\xi }_{-}^{\left(0\right)}(k,\eta )$ [see Eq. (2)]; shaded areas visualize the subleading contribution $2\pi {\xi }_{-}^{\left(1\right)}(k,\eta )/L$. When $\eta <{\eta }_1^{*}$ (left panel), the winding number $\nu$ vanishes, while for ${\eta }_1^{*}<\eta <{\eta }_2^{*}$ (right panel) $\nu =2$. The dashed line depicts the unit circle as a guide to the eye.

Figure 2

The linear response $\mathrm{\Delta }{\phi }_B/\mathrm{\Phi }$ of the Berry phase to an applied magnetic flux is plotted, in units of $e/\hslash =2\pi /{\mathrm{\Phi }}_0$, as a function of $\eta =2\pi t/L$ in the STSL regime after quantum quenches in finite Rice-Mele lattices with PBCs [see Eq. (6)]. In all quenches, the energy scale $\gamma$ is fixed to a constant value throughout the entire protocol, together with the ratio of the staggered potential $u=0.1$. The ratio $r$ of the staggered hopping amplitudes is instead quenched from $r^i=0.5$ to $r^f=2$, while the magnetic flux, when present, is constant and equal to $\mathrm{\Phi }/{\mathrm{\Phi }}_0=1/10$. System sizes are $L=40$ (blue), $L=80$ (red), and $L=400$ (green). The plateaus at $\nu =2$, 4, 6, and 8 are clearly visible already for $L=40$; they do not depend on system size and abruptly change at critical ratios ${\eta }_m^{*}$. The fluctuations are, instead, system size dependent and are suppressed with increasing $L$.