Linking invariant for the quench dynamics of a two-dimensional two-band Chern insulator

We discuss the topological invariant in the ($2+1$)-dimensional quench dynamics of a two-dimensional two-band Chern insulator starting from a topological initial state (i.e., with a nonzero Chern number $c_i$), evolved by a postquench Hamiltonian (with Chern number $c_f$). This process is classified by the torus homotopy group ${\tau }_3\left(S^2\right)$. In contrast to the process with $c_i=0$ studied in previous works, this process cannot be characterized by the Hopf invariant that is described by the sphere homotopy group ${\pi }_3\left(S^2\right)=\mathbb{Z}$. It is possible, however, to calculate a variant of the Chern-Simons integral with a complementary part to cancel the Chern number of the initial spin configuration, which at the same time does not affect the ($2+1$)-dimensional topology. We show that the modified Chern-Simons integral gives rise to a topological invariant of this quench process, i.e., the linking invariant in the ${\mathbb{Z}}_{2c_i}$ class: $\nu =(c_f-c_i)mod\left(2c_i\right)$. We give concrete examples to illustrate this result and also show the detailed deduction to get this linking invariant.

Figure 1

The distribution of the Bloch vectors ${\mathbf{n}}_i\left(\mathbf{k}\right)$ for the initial state with Chern number $c_i=1$ (a), and the vectors of the postquench Hamiltonians ${\mathbf{h}}^f\left(\mathbf{k}\right)$ with different Chern numbers $c_f$ (b)–(d). The squares in the figures represent the first Brillouin zone $\left\{(k_x,k_y)\right|k_x\in [-\pi ,\pi ),k_y\in [-\pi ,\pi )\}$. The arrows are colored by the $z$ components $n_z$ (or $h_z$) of the corresponding vectors $\mathbf{n}$ (or $\mathbf{h}$).

Figure 2

Extracting the linking numbers for different quench processes starting with a topological initial state with $c_i=1$, evolved by different postquench Hamiltonians with $c_f=0$ (a),(b), $c_f=1$ (c), and $c_f=2$ (d). The figures show the preimage loops of two points on the Bloch sphere with $(\theta ,\varphi )=(\pi /4,\pi /5)$ and $(3\pi /4,6\pi /5)$ (the antipodal point of the former). The straight lines going along the time direction are the auxiliary lines used to define the linking of two loops. Some other auxiliary lines in lighter colors are added to make the separated lines connected (to form closed loops). Such auxiliary lines can cancel in pairs because of the periodicity of the torus (note that their directions are always opposite). The postquench Hamiltonians are the same for (a) and (b), and the only difference between them is the different choice the positions of the auxiliary straight lines along the time direction. To facilitate the identification of the linking numbers, we have also made schematic illustrations on the top right of each figures, which show the circles that are homeomorphic to the corresponding ones in the main figures. The sign of the linking number is determined by the right-hand rule.

Figure 3

Illustration of the operation of clinging a $D^2$ to the Brillouin zone (which is a two-torus denoted as $T^2$) at a quasimomentum point $\mathbf{s}$. The arrows indicate the spin directions ($\mathbf{n}$ on $T^2$ and $\tilde{\mathbf{n}}$ on $D^2$). We assume the magnetic field $\tilde{\mathbf{h}}$ on the sphere is consistent with the magnetic field $\mathbf{h}$ on the Brillouin zone. We further assume the magnetic field for the sphere is always pointing out along the normal vector direction of the sphere, which always gives Chern number ${\tilde{c}}_f=-1$. Then (a) shows the case that $\mathbf{n}(\mathbf{s},t=0)$ is parallel to $\mathbf{h}\left(\mathbf{s}\right)$, and (b) shows the case that $\mathbf{n}(\mathbf{s},t=0)$ is antiparallel to $\mathbf{h}\left(\mathbf{s}\right)$. In both cases, the total Chern numbers for the initial spin configurations on the modified Brillouin zone ${\tilde{T}}^2=(T^2\setminus \mathbf{s})\cup D^2$ (which is still a two-torus) are zero.

Figure 4

Continuous deformation of the spin configuration ${\mathbf{n}}_i$ (a) and magnetic field ${\mathbf{h}}^f$ (b) on the disk. The $r$ axes in both (a) and (b) are in unit of arbitrary unit, and the $h_z$ axis in (b) is in unit of the ${\mathbf{h}}^f$ outside the disk. Before the deformation, both ${\mathbf{n}}_i$ (c1) and ${\mathbf{h}}^f$ (d1) are uniform on the disk, pointing to the south pole direction. After the deformation, ${\mathbf{n}}_i(r,\mathrm{\Phi })$ is rotated to the direction ${\mathbf{n}}_i(r,\mathrm{\Phi })=(sin\theta \left(r\right),0,cos\theta \left(r\right))$ [with $\theta \left(r\right)$ given by (a)] (c2), and the magnitude of ${\mathbf{h}}^f$ is increased as $|{\mathbf{h}}^f(r,\varphi )|=h_z\left(r\right)$ [with $h_z\left(r\right)$ given by (b)] (d2). The spin and magnetic field configurations for the attached $D^2$ in (c2) and (d2) are also deformed to make them continuous at the clinging point: Compared with (c1) and (d1), the corresponding vector components are deformed as $x\to x,y\to -y,z\to -z$ on $D^2$. We see that the continuous deformations of ${\mathbf{n}}_i$ and ${\mathbf{h}}^f$ give rise to a change of linking numbers (for any two preimages of any two points, except for the north or south poles, on the Bloch sphere $S^2$) from 0 (e1) to $2n{c}_i$ (which equals 4 for the current case with $n=2$ and $c_i=1$) (e2).