396
X. S. Wang and X. R. Wang
176], and the A-B-A bond is very promising for inducing a strong pseudodipolar
exchange interaction. Our model may also be realized in magnonic crystals in which
nanomagnets are arranged into a honeycomb lattice [177].
14.4 Conclusion
In conclusion [178], magnetism is an old and an important subject in physics. Our
long lasting interest on the subject in the past hundred years is mainly because the
studies of the subject keep providing us with new ideas, new concepts and new
principles, as well as because of its applications in modern information technologies. Topology, an old mathematical concept, is becoming a popular physics concept
since the recent discoveries of topologically protected surface states in both quantum
and classical systems. The concept of topology has been used in two ways in physics.
One is to classify structures of order parameters such as the magnetic domain walls,
vortices, skyrmions, etc., in magnetism. Namely, spin structures cannot be mapped to
a single magnetic domain under a continuous deformation of the structures. It should
be emphasized that nature does not respect topology, but the energy. Thus, a topologically non-trivial structure does not imply stability or a potential barrier between the
structure and other distinct ones. The second usage of topology refers to a class of
materials that support chiral surface states. These states can only be destroyed when
some topological charges existing in the bulk can be removed by external forces. This
second usage of topology leads to a new emerging field called topological materials that include topological insulators and topological semimetals. The concept of
topological protection provides a paradigm for stabilizing structures and excitations
because the kinetic paths to topologically distinct structures are greatly reduced even
in the case of only a finite potential barrier between different structures. Last but not
least, concepts and principles reviewed here can also be applied to other fields of the
physics.
References
1. S. Li, Origine de la boussole II. aimant et boussole. Isis 45(2), 175–196 (1954)
2. J.D. Jackson, Classical Electrodynamics, 3rd ed. edn. (Wiley, New York, NY, 1999)
3. J.J. Sakurai, Modern Quantum mechanics, rev edn. (Addison-Wesley, Reading, MA, 1994)
4. W. Ley, The galactic giants. Galaxy Science Fiction (1965), pp. 130–142
5. M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A.
Friederich, J. Chazelas, Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices.
Phys. Rev. Lett. 61, 2472–2475 (1988)
6. G. Binasch, P. Grünberg, F. Saurenbach, W. Zinn, Enhanced magnetoresistance in layered
magnetic structures with antiferromagnetic interlayer exchange. Phys. Rev. B 39, 4828–4830
(1989)
7. N. Kurtl, Selected Works of Louis Néel, 1st edition (CRC Press, 1988)
X. S. Wang and X. R. Wang
176], and the A-B-A bond is very promising for inducing a strong pseudodipolar
exchange interaction. Our model may also be realized in magnonic crystals in which
nanomagnets are arranged into a honeycomb lattice [177].
14.4 Conclusion
In conclusion [178], magnetism is an old and an important subject in physics. Our
long lasting interest on the subject in the past hundred years is mainly because the
studies of the subject keep providing us with new ideas, new concepts and new
principles, as well as because of its applications in modern information technologies. Topology, an old mathematical concept, is becoming a popular physics concept
since the recent discoveries of topologically protected surface states in both quantum
and classical systems. The concept of topology has been used in two ways in physics.
One is to classify structures of order parameters such as the magnetic domain walls,
vortices, skyrmions, etc., in magnetism. Namely, spin structures cannot be mapped to
a single magnetic domain under a continuous deformation of the structures. It should
be emphasized that nature does not respect topology, but the energy. Thus, a topologically non-trivial structure does not imply stability or a potential barrier between the
structure and other distinct ones. The second usage of topology refers to a class of
materials that support chiral surface states. These states can only be destroyed when
some topological charges existing in the bulk can be removed by external forces. This
second usage of topology leads to a new emerging field called topological materials that include topological insulators and topological semimetals. The concept of
topological protection provides a paradigm for stabilizing structures and excitations
because the kinetic paths to topologically distinct structures are greatly reduced even
in the case of only a finite potential barrier between different structures. Last but not
least, concepts and principles reviewed here can also be applied to other fields of the
physics.
References
1. S. Li, Origine de la boussole II. aimant et boussole. Isis 45(2), 175–196 (1954)
2. J.D. Jackson, Classical Electrodynamics, 3rd ed. edn. (Wiley, New York, NY, 1999)
3. J.J. Sakurai, Modern Quantum mechanics, rev edn. (Addison-Wesley, Reading, MA, 1994)
4. W. Ley, The galactic giants. Galaxy Science Fiction (1965), pp. 130–142
5. M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A.
Friederich, J. Chazelas, Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices.
Phys. Rev. Lett. 61, 2472–2475 (1988)
6. G. Binasch, P. Grünberg, F. Saurenbach, W. Zinn, Enhanced magnetoresistance in layered
magnetic structures with antiferromagnetic interlayer exchange. Phys. Rev. B 39, 4828–4830
(1989)
7. N. Kurtl, Selected Works of Louis Néel, 1st edition (CRC Press, 1988)