Hunting Down Magnetic Monopoles in 2D Topological Insulators

Contrary to the electric charge that generates the electric field, magnetic charge (namely magnetic monopoles) does not exist in the elementary electromagnetism. Consequently, magnetic flux lines only form loops and cannot have a source or a sink in nature. It is thus extraordinary to find that magnetic monopoles can be pictured conceptually in topological materials. Specifically in the 2D topological insulators, the topological invariant corresponds to the total flux of an effective magnetic field (the Berry curvature) over the reciprocal space.It is thus tempting to wrap the 2D reciprocal space into a compact manifold--a torus, and imagine the total flux to originate from magnetic monopoles inside the torus with a quantized total charge. However, such a physically appealing picture has not been realized quantitatively: other than their existence in a toy (actually misleading) picture, the properties of the magnetic monopoles remain unknown. Here, we will address this long-standing problem by hunting down the magnetic monopoles in the reciprocal $k$-space. We will show that a simple and physically useful picture will arrive upon analytically continuing the system to a third imaginary momentum space. We then illustrate the evolution of the magnetic monopoles across the topological phase transition and use it to provide natural explanations on: 1) discontinuous jump of integer topological invariants, 2) the semi-metallic nature on the phase boundary, and 3) how a change of global topology can be induced via a local change in reciprocal space.