The index theorem implies that there are fermionic states localized on a soliton. Presence of these modes may significantly alter the pattern of interaction between the solitons. As a particular example we investigate the chiral magnetic Skyrmions coupled to spin-isospin fermions. It is shown that there are sequences of fermionic modes localized on the Skyrmions. We investigate the pattern of interaction between the soltions with localized modes and proved the existence of stable system of magnetic Skyrmions bounded by the strong attractive dipole interaction mediated by the chargeless fermionic modes.
$\mathbb{C}\mathrm{P}^2$ Skyrmion with Fermion Backreaction
Research and Education Center for Natural Sciences, Keio University（自然科学研究与教育中心，庆应义塾大学）；Department of Physics, Keio University（物理学系，庆应义塾大学）；Department of Physics and Astronomy, Faculty of Science and Technology, Tokyo University of Science（理学院物理与天文系，东京科学大学）
Topological Charge Asymmetry in a $\mathbb{C}\mathrm{P}^N$ Skyrmion-Fermion Coupled System
Research and Education Center for Natural Sciences, Keio University（自然科学与教育中心，庆应义塾大学）；Department of Physics, Keio University（物理系，庆应义塾大学）；Department of Physics and Astronomy, Faculty of Science and Technology, Tokyo University of Science（理学院物理与天文系，东京科学大学）
Fermionic domain-wall Skyrmions of QCD in a magnetic field
Department of Physics, Yamagata University（山形大学物理系）；Research and Education Center for Natural Sciences, Keio University（庆应义塾大学自然科学研究与教育中心）；Department of Physics, Keio University（庆应义塾大学物理系）
Fermi gas of domain-wall Skyrmions in QCD in a strong magnetic field
International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM 2 ), Hiroshima University（可持续发展与扭结手性超物质国际研究所）；Hiroshima University（广岛大学）；Department of Physics, Yamagata University（山形大学物理系）；Yamagata University（山形大学）；Research and Education Center for Natural Sciences, Keio University（庆应义塾大学自然科学与教育中心）；Keio University（庆应义塾大学）；Department of Physics, Keio University（庆应义塾大学物理系）
