We discuss a geometric perspective on chiral ferromagnetism. Much like gravity becomes the effect of spacetime curvature in theory of relativity, the Dzyaloshinski-Moriya interaction arises in a Heisenberg model with nontrivial spin parallel transport. The Dzyaloshinskii-Moriya vectors serve as a background SO(3) gauge field. In 2 spatial dimensions, the model is partly solvable when an applied magnetic field matches the gauge curvature. At this special point, solutions to the Bogomolny equation are exact excited states of the model. We construct a variational ground state in the form of a skyrmion crystal and confirm its viability by Monte Carlo simulations. The geometric perspective offers insights into important problems in magnetism, e.g., conservation of spin current in the presence of chiral interactions.
Bibliographical noteFunding Information:
The authors thank Sayak Dasgupta, Se Kwon Kim, Volodymyr Kravchuk, Predrag Nikolic, Zo-har Nussinov, Yuan Wan, and Shu Zhang for discussions. D.H. and O.T. have been supported by the US DOE Basic Energy Sciences, Materials Sciences and Engineering Award DE-SC0019331. This work has been done in part at the Aspen Center for Physics supported by the US NSF Grant PHY-1607611 and at the Kavli Institute for Theoretical Physics, supported by the US NSF Grant PHY-1748958. V.S. thanks the Max Planck Institute for Mathematics in the Sciences, Leipzig, for hospitality and acknowledges support by the Leverhulme grant RPG-2018-438.
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