Dynamical detection of topological charges

We propose a scheme to characterize topological phases with integer invariants through detecting topological charges via quench dynamics. The essential advantage of the current scheme is that only a single-(pseudo)spin component needs to be measured to extract the complete information of topological charges which determine the bulk topology of the phase. A topological charge depicts the chirality of a monopole at Dirac or Weyl point captured by the node of the (pseudo)spin-orbit coupling, and the topological phases are determined by total charges enclosed by the so-called band-inversion surfaces (BISs). We show a duality that measuring the $i\mathrm{th}$ spin component in quenching $j\mathrm{th}$ spin axis precisely equals to measuring the $j\mathrm{th}$ spin component in quenching the $i\mathrm{th}$ spin axis. With this we find that characterizing the complete information of topological charges is precisely mapped to measuring the quench (pseudo)spin dynamics of a single spin component caused by quenches along all spin axes. Therefore, the detection of topological physics involves measuring only a single spin component and has great experimental advantages. We numerically examine two concrete models, and propose a cold-atom experimental setup with high feasibility for measurement. This work paves the way to dynamically classify topological phases.

Figure 1

Scheme of dynamical detection. (a) Topological phases can be characterized by the total charges in the region enclosed by BISs. (b) Quenching $h_i$ corresponds to initializing a polarized state along this axis (blue arrow) for spin precession about the post-quench vector field $\mathbf{h}$ (red arrow). Time-averaged spin polarization $\overline{\langle \mathbf{\gamma }\rangle }$ reflects the $h_i$ component: The surface with $h_i\left(\mathbf{k}\right)=0$ would exhibit no polarization, while for $h_i\left(\mathbf{k}\right)<0$ (or $h_i\left(\mathbf{k}\right)>0$), the time-averaged spin orientation points to the (or opposite) direction of $\mathbf{h}$. (c) BISs can be identified as the surfaces with vanishing spin polarization in the time-averaged spin texture ${\overline{\langle {\gamma }_0\left(\mathbf{k}\right)\rangle }}_0$ obtained by quenching $h_0$, which also emerges in ${\overline{\langle {\gamma }_0\left(\mathbf{k}\right)\rangle }}_i$ by quenching $h_{i>0}$ (dashed lines). In spin textures ${\overline{\langle {\gamma }_0\left(\mathbf{k}\right)\rangle }}_{i>0}$, other surfaces besides the BIS exhibiting no polarization are the surfaces $h_i\left(\mathbf{k}\right)=0$. The topological monopoles are located at the intersections of all the $h_{i>0}\left(\mathbf{k}\right)=0$ surfaces, and the charges can be characterized via the spin textures ${\overline{\langle {\gamma }_0\left(\mathbf{k}\right)\rangle }}_i$.

Figure 2

Numerical results of 2D QAH model. (a)–(c) Time-averaged spin textures $\overline{\langle {\sigma }_z\left(\mathbf{k}\right)\rangle }$ via quenching $h_z$ (a), $h_y$ (b), and $h_x$ (c). In (a), the magnetization $m_z$ is quenched from $25t_0$ to $t_0$, with $m_x=m_y=0$; for (b) [or (c)], the quench is varying $m_y$ (or $m_x$) from $25t_0$ to 0 and $m_z$ from 0 to $t_0$, while keeping $m_x=0$ (or $m_y=0$). Here we take $t_{\mathrm{so}}=t_0$. In addition to the ring-shape structure, which characterizes the BIS, two lines with vanishing polarization emerge in the spin textures ${\overline{\langle {\sigma }_z\left(\mathbf{k}\right)\rangle }}_y$ (b) and ${\overline{\langle {\sigma }_z\left(\mathbf{k}\right)\rangle }}_x$ (c), giving the interfaces with $h_y\left(\mathbf{k}\right)=0$ and $h_x\left(\mathbf{k}\right)=0$, respectively. (d) The dynamical field $\mathbf{\Theta }\left(\mathbf{k}\right)$, constructed from the spin textures, characterizes the topological charges of the four monopoles: (blue) $\mathcal{C}=-1$ at $\mathrm{\Gamma }$ (0,0) and $M (-\pi ,-\pi )$; (red) $\mathcal{C}=+1$ at $X_1 (0,-\pi )$ and $X_2 (-\pi ,0)$. The dashed green line denotes the first BZ.

Figure 3

Dynamical detection of 3D topological phases. (a) Time-averaged spin textures determine the BIS with ${\overline{\langle {\gamma }_0\left(\mathbf{k}\right)\rangle }}_0=0$ (green surface) and identify the monopoles as the intersection points of the surfaces ${\overline{\langle {\gamma }_0\left(\mathbf{k}\right)\rangle }}_{i>0}=0$. (b) Topological charges can be characterized by the constructed dynamical field, with $\mathcal{C}=1$ (red) at ${\mathcal{O}}_{1,3,5,7}$ and $\mathcal{C}=-1$ (blue) at ${\mathcal{O}}_{2,4,6,8}$. The details can be found in Appendix pp2.

Figure 4

The realization of dynamical detection with optical Raman lattice. (a) The process of quenching $h_{x,y}$ can be realized by changing the relative symmetry between the Raman and the lattice potentials. Before the quench, on-site spin flipping is permitted, generating a large Zeeman term $m_x{\sigma }_x$ or $m_y{\sigma }_y$; after the quench, spin-flipped hopping is produced, realizing the QAH model. (b) Experimental setup. A pair of laser beams ${\mathbf{E}}_x$ and ${\mathbf{E}}_y$, reflected by two mirrors $R_{1,2}$, produce square lattice and two independent Raman potentials for atoms. The two EOMs are used to manipulate the relative symmetry of the Raman potentials and realize the quench process.

Figure 5

Geometric interpretation of topological charges. The vector $\mathbf{h}\left(\mathbf{k}\right)$ traces a closed parameter surface $\mathcal{D}$, passing through the negative ${\gamma }_0$ axis at several interaction points ${\mathbf{\varrho }}_n$. These interactions are what we call “monopoles,” with the charges being the the orientations “$\pm$”.

Figure 6

Numerical results of the 3D model. (a) The observed time-averaged spin textures ${\overline{\langle {\gamma }_0\left(\mathbf{k}\right)\rangle }}_i=0$ determine the BIS and the locations of topological monopoles. (b) and (c) Topological charges can be seen from the dynamical field on the ${\overline{\langle {\gamma }_0\rangle }}_i=0$ ($i=1,2,3$) surfaces. These surfaces coincide with the planes $k_{x,y,z}=0$ (b) and $-\pi$ (c), denoted by $S_0^{x,y,z}$ and $S_f^{x,y,z}$, respectively. The dynamical field takes the same pattern on $S_0^{x,y,z}$ or on $S_f^{x,y,z}$, and can be constructed by the spin textures, with an example shown in (d) and (e). (d) and (e) $S_0^z$ and $S_f^z$ are the two planes with ${\overline{\langle {\gamma }_0\rangle }}_3=0$. While the spin texture ${\overline{\langle {\gamma }_0\left(\mathbf{k}\right)\rangle }}_0$ (d1, e1) can determine the BIS, the other two ${\overline{\langle {\gamma }_0\left(\mathbf{k}\right)\rangle }}_1$ (d2, e2) and ${\overline{\langle {\gamma }_0\left(\mathbf{k}\right)\rangle }}_2$ (d3, e3) are used to construct the dynamical field $\mathbf{\Theta }\left(\mathbf{k}\right)$ with the components ${\mathrm{\Theta }}_{1,2}\propto {\overline{\langle {\gamma }_0\rangle }}_{1,2}$ (d4, e4). Here $t_{\mathrm{so}}=t_0$.

Figure 7

Raman and lattice couplings for ${}^{40}\mathrm{K}$ atoms. Here the light beams are of wavelength between the $D_1$ and $D_2$ lines such that ${\mathrm{\Delta }}_{D_1}>0$ and ${\mathrm{\Delta }}_{D_2}<0$.