We derive a generalized set of Ward identities that captures the effects of topological charge on Hall transport. The Ward identities follow from the (2+1)-dimensional momentum algebra, which includes a central extension proportional to the topological charge density. In the presence of topological objects like Skyrmions, we observe that the central term leads to a direct relation between the thermal Hall conductivity and the topological charge density. We extend this relation to incorporate the effects of a magnetic field and an electric current. The topological charge density produces a distinct signature in the electric Hall conductivity, which is identified in existing experimental data and yields further novel predictions. For insulating materials with translation invariance, the Hall viscosity can be directly determined from the Skyrmion density and the thermal Hall conductivity to be measured as a function of momentum.
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This work is partially supported by NSF Grant No. PHY-1214341.
See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.117.116805 for the derivation of the full Ward identity in the presence of a magnetic field and current.
Kim, Bom Soo and Shapere, Alfred D., "Skyrmions and Hall Transport" (2016). Physics and Astronomy Faculty Publications. 437.