360
X. S. Wang and X. R. Wang
14.2 Topological Spin Textures
The order parameter of ferromagnets is magnetization, M. The magnetization at
different locations are strongly correlated. Generally speaking, a system is an object
occupied a real space of R with real-space dimension d. The order parameter of the
system has a dimension d
. d and d
are not related. d
of M is 2 when a ferromagnet
is far below the Curie temperature, but it can be 3 near the Curie temperature or
smaller than 2 when M is restricted. For ferrimagnets or anti-ferromagnets, the order
parameters can be more than one [17]. In the ambient conditions, thermodynamic
quantity M can be treated as a classical vector. When M at each point can take only
two discrete values of “up” and “down”, the system is usually called Ising-like with
dimension d
= 0. When M has a fixed magnitude and can only move in a plane or
on a surface, the magnet is XY-like with d
= 1 because M is on a 1D circle. For
most ferromagnets, M of fixed magnitude can point to any direction in real space
so their order parameters are unit vector direction m of M. d
= 2 and the system
can be modelled by a Heisenberg model or its generalizations consisting of various
anisotropies and all kinds of exchange interactions: symmetric or asymmetric, longor short-range interactions. The simplest spin texture is the one with all spins pointing
to the same direction. Since the spins in an Ising model are discrete, there is no
continuous deformation and magnetization m acts as an index to distinguish the two
states. For different d and d
≥ 1, one can classify various structures according to
their topology, as shown in Fig. 14.1. Different topological indices are defined for
different dimensionality.
In this section, we discuss various spin structures for domain walls (Sect. 14.2.1),
vortices, skyrmions (Sect. 14.2.2), and hopfions (Sect. 14.2.3) in d = 1 to 3. We
consider the continuous limit where the length scale of excitations is much larger
than atom-atom distance. The order parameter, m(r, t), is governed by the LandauLifshitz-Gilbert (LLG) equation [18],
∂m
∂t
= −γm × B eff + αm ×
∂m
∂t
,
(14.5)
where γ is the gyromagnetic ratio, B eff is the effective field due to all kinds of
anisotropic energies and exchange interactions, and α is the Gilbert damping. The
LLG equation describes generally the magnetization dynamics for an arbitrary d and
d
= 2.
14.2.1 Domain Walls
Domain walls (DWs) are transition regions between different domains. The minimal
model for a DW is on R
1
→ M
0 , as shown in Fig. 14.1. In this case, the Ising DW is
a topological defect [19] of trivial structure without a well-accepted dynamics. We
X. S. Wang and X. R. Wang
14.2 Topological Spin Textures
The order parameter of ferromagnets is magnetization, M. The magnetization at
different locations are strongly correlated. Generally speaking, a system is an object
occupied a real space of R with real-space dimension d. The order parameter of the
system has a dimension d
. d and d
are not related. d
of M is 2 when a ferromagnet
is far below the Curie temperature, but it can be 3 near the Curie temperature or
smaller than 2 when M is restricted. For ferrimagnets or anti-ferromagnets, the order
parameters can be more than one [17]. In the ambient conditions, thermodynamic
quantity M can be treated as a classical vector. When M at each point can take only
two discrete values of “up” and “down”, the system is usually called Ising-like with
dimension d
= 0. When M has a fixed magnitude and can only move in a plane or
on a surface, the magnet is XY-like with d
= 1 because M is on a 1D circle. For
most ferromagnets, M of fixed magnitude can point to any direction in real space
so their order parameters are unit vector direction m of M. d
= 2 and the system
can be modelled by a Heisenberg model or its generalizations consisting of various
anisotropies and all kinds of exchange interactions: symmetric or asymmetric, longor short-range interactions. The simplest spin texture is the one with all spins pointing
to the same direction. Since the spins in an Ising model are discrete, there is no
continuous deformation and magnetization m acts as an index to distinguish the two
states. For different d and d
≥ 1, one can classify various structures according to
their topology, as shown in Fig. 14.1. Different topological indices are defined for
different dimensionality.
In this section, we discuss various spin structures for domain walls (Sect. 14.2.1),
vortices, skyrmions (Sect. 14.2.2), and hopfions (Sect. 14.2.3) in d = 1 to 3. We
consider the continuous limit where the length scale of excitations is much larger
than atom-atom distance. The order parameter, m(r, t), is governed by the LandauLifshitz-Gilbert (LLG) equation [18],
∂m
∂t
= −γm × B eff + αm ×
∂m
∂t
,
(14.5)
where γ is the gyromagnetic ratio, B eff is the effective field due to all kinds of
anisotropic energies and exchange interactions, and α is the Gilbert damping. The
LLG equation describes generally the magnetization dynamics for an arbitrary d and
d
= 2.
14.2.1 Domain Walls
Domain walls (DWs) are transition regions between different domains. The minimal
model for a DW is on R
1
→ M
0 , as shown in Fig. 14.1. In this case, the Ising DW is
a topological defect [19] of trivial structure without a well-accepted dynamics. We