14 Topology in Magnetism
401
105. D. Kleckner, W.T.M. Irvine, Creation and dynamics of knotted vortices. Nat. Phys. 9(4),
253–258
106. P.J. Ackerman, I.I. Smalyukh, Diversity of knot solitons in liquid crystals manifested by
linking of preimages in torons and hopfions. Phys. Rev. X 7, 011006 (2017)
107. P.J. Ackerman, I.I. Smalyukh, Static three-dimensional topological solitons in fluid chiral
ferromagnets and colloids. Nat. Mater. 16(4), 426–432 (2017)
108. I. Georgescu, Out of the shadows. Nat. Phys. 13(3), 208 (2017)
109. I. Dzyaloshinsky, A thermodynamic theory of weak ferromagnetism of antiferromagnetics.
J. Phys. Chem. Solids 4(4), 241–255 (1958)
110. T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120,
91–98 (1960)
111. J.-S.B. Tai, I.I. Smalyukh, Static Hopf solitons and knotted emergent fields in solid-state
noncentrosymmetric magnetic nanostructures. Phys. Rev. Lett. 121, 187201 (2018)
112. Y. Liu, R.K. Lake, J. Zang, Binding a hopfion in a chiral magnet nanodisk. Phys. Rev. B 98,
174437 (2018)
113. P. Sutcliffe, Hopfions in chiral magnets. J. Phys. A: Math. Theor. 51(37), 375401 (2018)
114. F.N. Rybakov, N.S. Kiselev, A.B. Borisov, L. Döring, C. Melcher, S. Blügel, Magnetic hopfions in solids. Preprint, page arxiv:1904.00250 (2019)
115. M.R. Hestenes, E. Stiefel, Methods of conjugate gradients for solving linear systems. J. Res.
Nat. Bureau Stand. 49, 409–436 (1952)
116. J.H.C. Whitehead, An expression of Hopf’s invariant as an integral. Proc. Natl. Acad. Sci.
33(5), 117–123 (1947)
117. S. Zhang, S.S.-L. Zhang, Generalization of the Landau-Lifshitz-Gilbert equation for conducting ferromagnets. Phys. Rev. Lett. 102, 086601 (2009)
118. J. Gladikowski, M. Hellmund, Static solitons with nonzero Hopf number. Phys. Rev. D 56,
5194–5199 (1997)
119. R. Bott, L.W. Tu, Differential Forms in Algebraic Topology (Graduate Texts in Mathematics.
Springer, New York, 1995)
120. C.P. McNally, Divergence-free interpolation of vector fields from point values – exact ∇ · B =
0 in numerical simulations. Month. Not. Roy. Astronomic. Soc.: Lett. 413(1), L76–L80 (2011)
121. X. Zhang, J. Xia, Y. Zhou, D. Wang, X. Liu, W. Zhao, M. Ezawa, Control and manipulation
of a magnetic skyrmionium in nanostructures. Phys. Rev. B 94, 094420 (2016)
122. F. Zheng, H. Li, S. Wang, D. Song, C. Jin, W. Wei, A. Kovács, J. Zang, M. Tian, Y. Zhang,
D. Haifeng, R.E. Dunin-Borkowski, Direct imaging of a zero-field target skyrmion and its
polarity switch in a chiral magnetic nanodisk. Phys. Rev. Lett. 119, 197205 (2017)
123. C.L. Kane, E.J. Mele, Z 2 topological order and the quantum spin Hall effect. Phys. Rev. Lett.
95, 146802 (2005)
124. B. Andrei Bernevig, T.L. Hughes, S-C. Zhang, Quantum spin Hall effect and topological
phase transition in HgTe quantum wells. Science 314(5806), 1757–1761 (2006)
125. M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L.W. Molenkamp, X.-L. Qi, S.-C.
Zhang, Quantum spin hall insulator state in HgTe quantum wells. Science 318(5851), 766–770
(2007)
126. M.Z. Hasan, C.L. Kane, Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067
(2010)
127. X.-L. Qi, S.-C. Zhang, Topological insulators and superconductors. Rev. Mod. Phys. 83,
1057–1110 (2011)
128. C.L. Kane, T.C. Lubensky, Topological boundary modes in isostatic lattices. Nat. Phys. 10(1),
39–45 (2014)
129. F.D.M. Haldane, S. Raghu, Possible realization of directional optical waveguides in photonic
crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008)
130. S. Fujimoto, Hall effect of spin waves in frustrated magnets. Phys. Rev. Lett. 103, 047203
(2009)
131. H. Katsura, N. Nagaosa, P.A. Lee, Theory of the thermal Hall effect in quantum magnets.
Phys. Rev. Lett. 104, 066403 (2010)
401
105. D. Kleckner, W.T.M. Irvine, Creation and dynamics of knotted vortices. Nat. Phys. 9(4),
253–258
106. P.J. Ackerman, I.I. Smalyukh, Diversity of knot solitons in liquid crystals manifested by
linking of preimages in torons and hopfions. Phys. Rev. X 7, 011006 (2017)
107. P.J. Ackerman, I.I. Smalyukh, Static three-dimensional topological solitons in fluid chiral
ferromagnets and colloids. Nat. Mater. 16(4), 426–432 (2017)
108. I. Georgescu, Out of the shadows. Nat. Phys. 13(3), 208 (2017)
109. I. Dzyaloshinsky, A thermodynamic theory of weak ferromagnetism of antiferromagnetics.
J. Phys. Chem. Solids 4(4), 241–255 (1958)
110. T. Moriya, Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120,
91–98 (1960)
111. J.-S.B. Tai, I.I. Smalyukh, Static Hopf solitons and knotted emergent fields in solid-state
noncentrosymmetric magnetic nanostructures. Phys. Rev. Lett. 121, 187201 (2018)
112. Y. Liu, R.K. Lake, J. Zang, Binding a hopfion in a chiral magnet nanodisk. Phys. Rev. B 98,
174437 (2018)
113. P. Sutcliffe, Hopfions in chiral magnets. J. Phys. A: Math. Theor. 51(37), 375401 (2018)
114. F.N. Rybakov, N.S. Kiselev, A.B. Borisov, L. Döring, C. Melcher, S. Blügel, Magnetic hopfions in solids. Preprint, page arxiv:1904.00250 (2019)
115. M.R. Hestenes, E. Stiefel, Methods of conjugate gradients for solving linear systems. J. Res.
Nat. Bureau Stand. 49, 409–436 (1952)
116. J.H.C. Whitehead, An expression of Hopf’s invariant as an integral. Proc. Natl. Acad. Sci.
33(5), 117–123 (1947)
117. S. Zhang, S.S.-L. Zhang, Generalization of the Landau-Lifshitz-Gilbert equation for conducting ferromagnets. Phys. Rev. Lett. 102, 086601 (2009)
118. J. Gladikowski, M. Hellmund, Static solitons with nonzero Hopf number. Phys. Rev. D 56,
5194–5199 (1997)
119. R. Bott, L.W. Tu, Differential Forms in Algebraic Topology (Graduate Texts in Mathematics.
Springer, New York, 1995)
120. C.P. McNally, Divergence-free interpolation of vector fields from point values – exact ∇ · B =
0 in numerical simulations. Month. Not. Roy. Astronomic. Soc.: Lett. 413(1), L76–L80 (2011)
121. X. Zhang, J. Xia, Y. Zhou, D. Wang, X. Liu, W. Zhao, M. Ezawa, Control and manipulation
of a magnetic skyrmionium in nanostructures. Phys. Rev. B 94, 094420 (2016)
122. F. Zheng, H. Li, S. Wang, D. Song, C. Jin, W. Wei, A. Kovács, J. Zang, M. Tian, Y. Zhang,
D. Haifeng, R.E. Dunin-Borkowski, Direct imaging of a zero-field target skyrmion and its
polarity switch in a chiral magnetic nanodisk. Phys. Rev. Lett. 119, 197205 (2017)
123. C.L. Kane, E.J. Mele, Z 2 topological order and the quantum spin Hall effect. Phys. Rev. Lett.
95, 146802 (2005)
124. B. Andrei Bernevig, T.L. Hughes, S-C. Zhang, Quantum spin Hall effect and topological
phase transition in HgTe quantum wells. Science 314(5806), 1757–1761 (2006)
125. M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L.W. Molenkamp, X.-L. Qi, S.-C.
Zhang, Quantum spin hall insulator state in HgTe quantum wells. Science 318(5851), 766–770
(2007)
126. M.Z. Hasan, C.L. Kane, Colloquium: Topological insulators. Rev. Mod. Phys. 82, 3045–3067
(2010)
127. X.-L. Qi, S.-C. Zhang, Topological insulators and superconductors. Rev. Mod. Phys. 83,
1057–1110 (2011)
128. C.L. Kane, T.C. Lubensky, Topological boundary modes in isostatic lattices. Nat. Phys. 10(1),
39–45 (2014)
129. F.D.M. Haldane, S. Raghu, Possible realization of directional optical waveguides in photonic
crystals with broken time-reversal symmetry. Phys. Rev. Lett. 100, 013904 (2008)
130. S. Fujimoto, Hall effect of spin waves in frustrated magnets. Phys. Rev. Lett. 103, 047203
(2009)
131. H. Katsura, N. Nagaosa, P.A. Lee, Theory of the thermal Hall effect in quantum magnets.
Phys. Rev. Lett. 104, 066403 (2010)