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Magnetic Skyrmions are topological objects (protected by a topological number and an energy gap and robust under disturbances) that have been originally proposed in high energy theory, yet recently observed in real condensed matter systems. They have a viable application as a next generation storage devices and have been actively studied theoretically and experimentally. Due to extended nature of the Skyrmions, most of the studies rely on phenomenological models.

There is so-called Ward identities that only rely on symmetries present in the system and thus conservation equations. We have generalized this to derive a more powerful set of Ward identities that captures the effects of topological charge on 2+1 dimensions in journal in the arXiv site (Original article in the journal, Physics Review Letter). From mathematical point of view, the contribution of the topological charges comes from a central extension in momentum momentum correlator, which can not be captured by conservation equations.

We provide several experimental signatures. For example, the thermal Hall conductivity is directly related to the topological charge density of Skyrmions when the system is insulating. This relation is further extended in the presence of a magnetic field and a conserved current. Interestingly, the topological charge density produces a distinct signature in the electric Hall conductivity, which is identified in existing experimental data.

We emphasize that the Hall Viscosity can play an important role in the magnetic Skyrmion experiments. We propose a simple and clear way to measure the Hall viscosity once the thermal Hall conductivity is measure as a function of momentum. For insulating materials with translation invariance, the Hall viscosity is nothing but the Skyrmion density multiplied by the ratio between the slope (numerator) and intercept (denominator) at zero momentum! We also provide a similar relation to the conducting materials in the presence of charge carriers as well.

The left graph qualitatively explains the non-zero Hall viscosity when the measured Hall conductivity has non-zero slope at zero mometum. Quantitatively, Hall viscosity is Skyrmion topological charge density multiplied by the ratio of the slope and the intercept of the Hall conductivity as a function of momentum squared.

This work has been presented in several different institutes including the Berkeley Center for Theoretical Physics, Berkeley; Korea Institute for Advanced Study, Seoul; Great Lakes Strings Conference, Ann Arbor; SPOCK regional string meeting, Cincinnati. Here is the link for full length YouTube video.

Recently, I was invited to give a talk at the 62nd Annual Conference on Magnetism and Magnetic Materials (2017 MMM conference), Pittsburgh, PA. The conference review paper written for wide range of audiences can be found in the arXiv site and also in the AIP journal site.

- B. S. Kim, "Skyrmions and Hall viscosity," AIP Advances 8, 055601 (2018). [arXiv:1712.05032 [cond-mat.str-el]]
- B. S. Kim and A. Shapere, “Skyrmions and Hall Transport,” Phys. Rev. Lett. 117, 116805 (2016). [arXiv:1506.08199 [cond-mat.str-el]]
- C. Hoyos, B. S. Kim and Y. Oz, “Ward Identities for Transport in 2+1 Dimensions,” JHEP 1503, 164 (2015). [arXiv:1501.05756 [hep-th]]
- C. Hoyos, B. S. Kim and Y. Oz, “Ward Identities for Hall Transport,” JHEP 1410, 054 (2014). [arXiv:1407.2616 [hep-th]]

$40,000 Scholarships are available in physics with CPaMS Scholars