Stroboscopic symmetry-protected topological phases

Symmetry-protected topological (SPT) phases of matter have been the focus of many recent theoretical investigations, but controlled mechanisms for engineering them have so far been elusive. In this work, we demonstrate that by driving interacting spin systems periodically in time and tuning the available parameters, one can realize lattice models for bosonic SPT phases in the limit where the driving frequency is large. We provide concrete examples of this construction in one and two dimensions, and discuss signatures of these phases in stroboscopic measurements of local observables.

Figure 1

Couplings in the leading term of the Magnus expansion (4) as functions of the scaled driving amplitude $\lambda$. White and gray regions correspond, respectively, to trivial and stroboscopic SPT phases.

Figure 2

Competing dimerization patterns in the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ SPT chain. Dotted and solid lines account, respectively, for the dominant dimerization patterns in the trivial and SPT phases. When only the 3-spin term is present in the model, a dangling spin is localized on the edges.

Figure 3

Stroboscopic evolution of $\langle {\sigma }_i^z\left(t\right)\rangle$ for $i=1,2,$ and 5 for an eight-site chain, where the initial state is chosen to be the product state $|{\mathrm{\Psi }}_0\rangle =|\uparrow \uparrow \cdots \uparrow \rangle$. Panels (a)–(c) depict stroboscopic evolution at $\omega =\infty$ [i.e., as defined in Eq. (9)], while panels (d)–(f) depict the exact stroboscopic evolution [i.e., as defined in Eq. (10a)] for $\omega =100h$. In (a), $2\lambda$ equals the second zero of the Bessel function $J_0\left(x\right)$ in Fig. 1, and the longitudinal symmetry-breaking field $h_z=0$. In (b), $\lambda =2.6$ and $h_z=0$, while in (c), $\lambda =2.6$ and $h_z=0.01$ in units of the bare transverse field $h$. The plots in panels (d), (e), and (f) use the same parameters as the ones in panels (a), (b), and (c), respectively.

Figure 4

Finite-frequency corrections to stroboscopic evolution of $\langle {\sigma }_i^z\left(t\right)\rangle$. The system size and initial state are the same as in Fig. 3. Panel (a) shows the evolution depicted in Fig. 3 over a longer time, so that the envelope time scale discussed in the main text is more apparent. Panel (b) uses the same parameters as Figs. 3 and 3, but at frequency $\omega =10h$.

Figure 5

Couplings in the first-order correction to the Magnus expansion, cf. Eq. (A8), as functions of the scaled driving amplitude $\lambda$. White and gray regions are used to distinguish the trivial and stroboscopic SPT phases, as in Fig. 1, and the Bessel function $J_0\left(2\lambda \right)$ is plotted for reference.

Figure 6

Triangular lattice where the model Eq. (B2) and the driven Hamiltonian Eq. (B6a) are defined. The sites nearest neighbors to site $j$ are labeled 1 through 6.