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We know that there is an effective "magnetic field" in terms of Berry curvature, which leads to Luttinger's anomalous velocity, etc; and similarly there is also a "magnetic flux", which appears in Dirac point of graphene. I am wondering whether there is any analog of magnetic monopole in the same context? If there is, what is its application in condensed matter physics?

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Yes, magnetic monopoles of Berry curvature do exist in condensed matter physics. They are Weyl points (Weyl nodes) in Weyl semimetals. Weyl semimetals have been experimentally realized (such as TaAs), and the Weyl points are also identified through the observation of surface Fermi arcs (a phenomenon closely related to the monopoles of Berry curvature).

Weyl semimetals are 3D materials, with 3D Brillouin zones, which can host monopoles of Berry curvature. The monopoles are located at specific momentum points in the Brillouin zone, which are called Weyl points. Weyl points must appear in positive-negative pairs, since the Brillouin zone is a compact manifold, and the total Berry flux must vanish. The electronic band structure around a Weyl point is described by the Weyl fermion Hamiltonian:

$$H=\sum_{\mathbf{k}}c_\mathbf{k}^\dagger v_F \mathbf{k}\cdot\mathbf{\sigma} c_\mathbf{k},$$

where $\mathbf{k}$ is the momentum deviation from the Weyl point. Given this band structure, it is easy to show that the Berry curvature around the Weyl point is given by:

$$\mathbf{\Omega}=\frac{\mathbf{k}}{2k^3},$$

which describes a monopole at the Weyl point $\mathbf{k}=0$, which is giving out $2\pi$ Berry flux in total.

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If I correctly understand your question, then I will say that I am currently aware of two condensed matter systems which contain magnetic monopoles.

  • Spin Ice, in which frustrated exchange interactions of localized spins lead to a highly fluctuating ground state. The excitations of this system are naturally dipolar, but due to the highly fluctuating spin environment (of the groundstate), the two "monopoles" which make up the dipolar exciation can become deconfined and fractionalize, thus propagating through the material independently.
  • Weyl Semimetals, where breaking time reversal or crystal inversion symmetry splits the degenerate Dirac Points by their chirality. These points, called Weyl nodes, then act like monopole sources of Berry curvature in momentum space. These features first were experimentally observed Summer 2015, and so are currently of high interest.

As far as technical details regarding either, I'm afraid that I am still those learning myself.

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  • $\begingroup$ Nice answer. However I must say that Weyl nodes are not "low-energy quasi-particles excitations". $\endgroup$ – Everett You Dec 3 '15 at 4:48
  • $\begingroup$ I didn't mean to imply them as such, so I appreciate you pointing that out. $\endgroup$ – Todd R Dec 3 '15 at 4:54
  • $\begingroup$ Thank you Todd! It seems that Weyl semimetal is what I was looking for. $\endgroup$ – David Sun Dec 3 '15 at 4:58

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