Biot-Savart law in quantum matter

We study the topological nature of a class of lattice systems whose Bloch vector can be expressed as the difference of two independent periodic vector functions (knots) in an auxiliary space. We show exactly that each loop as a degeneracy line generates a $\mathbf{P}$ field, which is defined as the gradient of the Zak phase in parameter space, obeying the Biot-Savart law: The degeneracy line acts as a current-carrying wire, while the $\mathbf{P}$ field corresponds to the generated magnetic field. Applying the Amp\`ere's circuital law on a nontrivial topological system, we find that two Bloch knots entangle with each other, forming a link with the linking number being the value of Chern number of the energy band. In addition, two lattice models, an extended Qi-Wu-Zhang (QWZ) model and a quasi-one-dimensional model with magnetic flux, are proposed to exemplify the application of our approach. In the aid of the Biot-Savart law, the pumping charge as a dynamic measure of Chern number is obtained numerically from quasi-adiabatic processes.