Chern number in one-dimensional system As the title, could we define Chern number for condensed matter systems with one spatial dimension? E.g. the 1D Su-Schrieffer–Heeger (SSH) model.
 A: You can not define Chern number for 1D system, but you can define the winding number (another topological index) for 1D system, which can be used to characterize the topological phases in the SSH model. One still start with the Berry connection $A=\langle u_k|\mathrm{i}\partial_k|u_k\rangle\mathrm{d}k$ (in terms of the differential 1-form). In 1D, the Berry connection has only one component. Then define winding number
$$P_1=\frac{1}{2\pi}\int_\text{BZ}A.$$
Unlike the Chern number which is quantized, and any different integer value of the Chern number labels a different topological phase. The winding number is not quantized in general, and two winding numbers differed by $1$ are equivalent (i.e. $P_1\sim P_1+1$). This is where symmetry becomes important. Under time-reversal symmetry (of the symmetry class DIII), the winding number is reversed $P_1\to-P_1$. So if the symmetry is preserved, the winding number can only take two values $P_1=0,1/2$ (note that $1/2=-1/2$ because of the mod $1$ periodicity). In this case, the winding number labels different phases in the SSH model. $P_1=0$ corresponds to the trivial phase and $P_1=1/2$ corresponds to the topological phase.
