Biot-Savart law in quantum matter

TL;DR

This paper reveals a topological analogy between degeneracy lines in lattice systems and magnetic fields generated by currents, linking the Chern number to the linking number of knots, with applications demonstrated in two lattice models.

Contribution

It introduces a novel topological framework connecting degeneracy lines to the Biot-Savart law, providing new insights into the Chern number via knot linking in lattice systems.

Findings

Degeneracy lines generate polarization fields following the Biot-Savart law.

The linking number of knots equals the Chern number of the energy band.

Numerical results confirm the topological interpretation in specific lattice models.

Abstract

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 polarization field, obeying the Biot-Savart law: The degeneracy line acts as a current-carrying wire, while the polarization field corresponds to the generated magnetic field. Applying the Ampere'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 QWZ model and a quasi-1D 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.