Experimental observation of dynamical bulk-surface correspondence in momentum space for topological phases

Characterization of topological invariants in experiments is at the heart of understanding topological quantum phases. We experimentally demonstrate a dynamical classification approach for investigation of topological quantum phases using a solid-state spin system through nitrogen vacancy (NV) center in diamond. Similar to the bulk-boundary correspondence in real space at equilibrium, we observe a dynamical bulk-surface correspondence in the momentum space from a dynamical quench process. An emergent dynamical topological invariant is precisely measured in experiment by imaging the dynamical spin textures on the recently defined band-inversion surfaces, with high topological numbers being implemented. Importantly, the dynamical classification approach is shown to be independent of quench ways and robust to the dephasing effects, offering a powerful strategy for dynamical topology characterization, especially for high-dimensional gapped topological phases.

Figure 1

Schematics of experimental dynamical classification protocol. (a) The band structure of a 2D quantum anomalous model in the topological regime. (b) Spin dynamics under quenching along the $z$ axis. The blue arrow denotes the initial ground state of the initial Hamiltonian ${\mathcal{H}}^i\left(\mathbf{k}\right)$. The pink arrow denotes the final Hamiltonian ${\mathcal{H}}^f\left(\mathbf{k}\right)$. The dynamics under other quenching directions can be visualized through a simple coordinate transformation. (c) The location of band inversion surface (BIS) detected through observing the time averaged spin polarization along the quenching direction ($z$ axis here), if ${\mathcal{H}}^f$ is topological. ${\mathbf{k}}_{\perp }$ denotes the norm vector of BIS. (d) Spin polarizations of different momentum points along ${\mathbf{k}}_{\perp }$. (e) The experimental system: an NV center in the diamond. (f) The energy diagrams of the NV center used for simulation. A microwave pulse is applied to simulate the 2D quantum anomalous Hall model at each momentum via controlling its detuning frequency ($\delta$), amplitude ($B_{\mathrm{mw}}$), and phase ($\varphi$). (g) A visualization of the driven quantum system in the Bloch sphere.

Figure 2

Topological invariant emerging on BISs. (a) The BIS identified by measuring time-averaged spin polarization $\overline{\langle {\sigma }_z(\mathbf{k},t)\rangle }$ in quenching along the $z$ axis. (b) Linear dependence of spin polarizations $\overline{\langle {\sigma }_{x,y}\rangle }$ on momentum points along the ${\vec{k}}_{\perp }$ as shown in panel (a). The dashed lines are linear fit of the experimental results (dots). From the slope we obtain the corresponding $g_x$ and $g_y$. (c) The measured emergent dynamical field ($g_x,g_y,0$) on the BIS. (d) Redraw of panel (c) to count how many rounds the dynamical spin-texture field rotates around the central point. (e) The BIS of spin polarization $\overline{\langle {\sigma }_x\rangle }$ observed through quenching along the $x$ axis. [(f), (g)] The measured spin-texture field ($g_0,g_y,g_z$) on the BIS. [(h), (i)] The corresponding winding picture of panels (f) and (g).

Figure 3

Characterizing high topological invariant. (a) The BIS of spin polarizations $\overline{\langle {\sigma }_z\rangle }$ observed through quenching along the $z$ axis. (b) The measured spin-texture field ($g_x,g_y,0$) and (c) the corresponding winding number of panel (b).

Figure 4

Classifying topological invariant in the presence of decoherence. (a) The results of topological characterization in the presence of decoherence through quenching along the $z$ axis. The postquench parameters are $m_z=t_0=2t_{\mathrm{so}}$ as in Figs. 2–3 but with a smaller value of $t_{\mathrm{so}}=0.8$ MHz to enhance the effect of noise. Each pixel data is an average of the Rabi oscillation data more than 10 full periods as shown in panel (c). (b) Linear dependence of extracted decay rate on $n_z$ as observed by experiments and predicted by theory [Eq. (3)]. [(c), (d), (e)] Rabi oscillations of $\langle {\sigma }_{x,y,z}\rangle$ which agree well with theoretical prediction, from which the decay rate is extracted.