14 Topology in Magnetism
359
In the 18th century, Leonhard Euler studied Seven Bridges of Königsberg problem
and the properties of polyhedrons, which is considered as the birth of the topology
[14]. Topology became a well-accepted branch of mathematics in the early part of
the 20th century. The basic motivation of topology is to study preserved quantities and their possible relationships under continuous deformations, such as stretching/compressing and twisting, but not tearing or gluing. In 1940s, Shiing-Shen Chern
formulated the concept of Chern class and Chern numbers. The vector bundles on a
smooth manifold are classified into Chern classes with specific Chern numbers, and
within the same Chern class, the vector bundles are topologically equivalent. Almost
at the same time, physicists started to study gauge transformation and the gauge-field
origin of the fundamental forces in nature. For example, it was realized that nature
must have electromagnetic fields whose dynamics follows the Maxwell’s equation
if nature respects U (1) gauge symmetry. We know now that the gauge charges are
the Chern numbers of these fields [15]. In the 1970s and the 1980s, the interest on
topology proliferates from particle physics to condensed matter physics. David J.
Thouless, J. Michael Kosterlitz, F. Duncan M. Haldane and many others discovered topological phase transitions and possible topological phases of matter. David,
Michael, and Duncan were awarded the Nobel Prize in Physics in 2016 for these
pioneering work. Simply speaking, topology provides a way to certain invariants in
nature. Within each class, some common properties exist and these properties are
robust against continuous deformations. This robustness is called topological protection [16]. In many circumstances, an extra cost (energy, time, force, defects, etc.)
has to be paid to break the topological protection. Thus, the topologically protected
excitations are extraordinarily useful in practice.
In summary, magnetism and topology are academically interesting and practically
useful. The hybridization of the two subjects is the central theme of this chapter. Here
we review recent developments of two usages of topology in magnetism based on
our own results. We focus on ferromagnets although there are increasing interest in
anti-ferromagnetic systems. In Sect. 14.2, we introduce spin/magnetic textures with
nontrivial topology in real space, including domain walls in one-dimensional (1D)
wires to 3D bulk magnets, vortices and skyrmions in two-dimensional (2D) films and
hopfions in 3D magnets. It should be emphasized that topological structures with various non-zero topological number does not necessarily imply stability although they
cannot be mapped to a single domain with zero topological number under a continuous deformation. Nature respects energy, but not the topological deformations.
In Sect. 14.3, we introduce topological spin waves (or magnons, in this chapter we
use this two terms interchangeably). Spin waves are the propagation of small spin
fluctuation, and we concentrate on unidirectional surfaces waves topologically protected by the band gap in the reciprocal space. These topological spin waves very
often imply robustness because they can only be destroyed when the bulk topological
charges are removed or annihilated through the external forces that consume a finite
amount of energy.
359
In the 18th century, Leonhard Euler studied Seven Bridges of Königsberg problem
and the properties of polyhedrons, which is considered as the birth of the topology
[14]. Topology became a well-accepted branch of mathematics in the early part of
the 20th century. The basic motivation of topology is to study preserved quantities and their possible relationships under continuous deformations, such as stretching/compressing and twisting, but not tearing or gluing. In 1940s, Shiing-Shen Chern
formulated the concept of Chern class and Chern numbers. The vector bundles on a
smooth manifold are classified into Chern classes with specific Chern numbers, and
within the same Chern class, the vector bundles are topologically equivalent. Almost
at the same time, physicists started to study gauge transformation and the gauge-field
origin of the fundamental forces in nature. For example, it was realized that nature
must have electromagnetic fields whose dynamics follows the Maxwell’s equation
if nature respects U (1) gauge symmetry. We know now that the gauge charges are
the Chern numbers of these fields [15]. In the 1970s and the 1980s, the interest on
topology proliferates from particle physics to condensed matter physics. David J.
Thouless, J. Michael Kosterlitz, F. Duncan M. Haldane and many others discovered topological phase transitions and possible topological phases of matter. David,
Michael, and Duncan were awarded the Nobel Prize in Physics in 2016 for these
pioneering work. Simply speaking, topology provides a way to certain invariants in
nature. Within each class, some common properties exist and these properties are
robust against continuous deformations. This robustness is called topological protection [16]. In many circumstances, an extra cost (energy, time, force, defects, etc.)
has to be paid to break the topological protection. Thus, the topologically protected
excitations are extraordinarily useful in practice.
In summary, magnetism and topology are academically interesting and practically
useful. The hybridization of the two subjects is the central theme of this chapter. Here
we review recent developments of two usages of topology in magnetism based on
our own results. We focus on ferromagnets although there are increasing interest in
anti-ferromagnetic systems. In Sect. 14.2, we introduce spin/magnetic textures with
nontrivial topology in real space, including domain walls in one-dimensional (1D)
wires to 3D bulk magnets, vortices and skyrmions in two-dimensional (2D) films and
hopfions in 3D magnets. It should be emphasized that topological structures with various non-zero topological number does not necessarily imply stability although they
cannot be mapped to a single domain with zero topological number under a continuous deformation. Nature respects energy, but not the topological deformations.
In Sect. 14.3, we introduce topological spin waves (or magnons, in this chapter we
use this two terms interchangeably). Spin waves are the propagation of small spin
fluctuation, and we concentrate on unidirectional surfaces waves topologically protected by the band gap in the reciprocal space. These topological spin waves very
often imply robustness because they can only be destroyed when the bulk topological
charges are removed or annihilated through the external forces that consume a finite
amount of energy.