Topological stability vs. energetic stability A non-trivial topology does not in itself imply energetic stability. There is in fact no necessary relation between topology and energetic stability. Hence, one must be careful not to confuse 'topological stability,' which is a mathematical concept, with energy stability in real physical systems. Topological stability refers to the idea that in order for a system described by a continuous field to transition from one topological state to another, a rupture must occur in the continuous field, i.e. a discontinuity must be produced. For example, if one wishes to transform a flexible balloon doughnut (torus) into an ordinary spherical balloon, it is necessary to introduce a rupture on some part of the balloon doughnut's surface. Mathematically, the balloon doughnut would be described as 'topologically stable.' However, in physics, the free energy required to introduce a rupture enabling the transition of a system from one 'topological' state to another is always
finite. For example, it is possible to turn a rubber ballon into flat piece of rubber by poking it with a needle (and popping it!). Thus, while a physical system can be
approximately described using the mathematical concept of topology, attributes such as
energetic stability are dependent on the system's parameters—the strength of the rubber in the example above—not the topology per se. In order to draw a meaningful parallel between the concept of topological stability and the energy stability of a system, the analogy must necessarily be accompanied by the introduction of a non-zero phenomenological 'field rigidity' to account for the finite energy needed to rupture the field's topology. Modeling and then integrating this field rigidity can be likened to calculating a breakdown energy-density of the field. These considerations suggest that what is often referred to as 'topological protection,' or a 'topological barrier,' should more accurately be referred to as a 'topology-related energy barrier,' though this terminology is somewhat cumbersome. A quantitative evaluation of such a topological barrier can be obtained by extracting the critical magnetic configuration when the topological number changes during the dynamical process of a skyrmion creation event. Applying the topological charge defined in a lattice, the barrier height is theoretically shown to be proportional to the exchange stiffness.
Further observations It is important to be cognizant of the fact that magnetic n =1 structures are in fact not stabilized by virtue of their 'topology,' but rather by the field rigidity parameters that characterize a given system. However, this does not suggest that topology plays an insignificant role with respect to energetic stability. On the contrary, topology may create the
possibility for certain stable magnetic states to exist, which otherwise could not. However, topology in itself does not
guarantee the stability of a state. In order for a state to have stability associated with its topology, it must be further accompanied by a non-zero field rigidity. Thus, topology can be considered a necessary but insufficient condition for the existence of certain classes of stable objects. While this distinction may at first seem pedantic, its physical motivation becomes apparent when considering two magnetic spin configurations of identical topology n =1, but subject to the influences of only one differing magnetic interaction. For example, we may consider one spin configuration with, and one configuration without the presence of
magnetocrystalline anisotropy, oriented perpendicular to the plane of an ultra-thin magnetic film. In this case, the n =1 configuration that is influenced by the magnetocrystalline anisotropy will be more energetically stable than the n =1 configuration without it, in spite of identical topologies. This is because the magnetocrystalline anisotropy contributes to the field rigidity, and it is the field rigidity, not the topology, that confers the notable energy barrier protecting the topological state. Finally, it is interesting to observe that in some cases, it is not the topology which helps n =1 configurations to be stable, but rather the converse, as it is
the stability of the field (which depends on the relevant interactions) which favors the n =1 topology. This is to say that the most stable energy configuration of the field constituents, (in this case magnetic atoms), may in fact be to arrange into a topology which can be described as an n =1 topology. Such is the case for magnetic skyrmions stabilized by the
Dzyaloshinskii–Moriya interaction, which causes adjacent magnetic spins to 'prefer' having a fixed angle between each other (energetically speaking). Note that from a point of view of practical applications this does not alter the usefulness of developing systems with Dzyaloshinskii–Moriya interaction, as such applications depend strictly on the topology [of the skyrmions, or lack thereof], which encodes the information, and not the underlying mechanisms which stabilize the necessary topology. These examples illustrate why use of the terms 'topological protection' or 'topological stability' interchangeably with the concept of energy stability is misleading, and is liable to lead to fundamental confusion.
Limitations of applying the concept of topology One must exercise caution when making inferences based on topology-related energy barriers, as it can be misleading to apply the notion of topology—a description which only rigorously applies to
continuous fields— to infer the energetic stability of structures existing in
discontinuous systems. Giving way to this temptation is sometimes problematic in physics, where fields which are approximated as continuous become discontinuous below certain size-scales. Such is the case for example when the concept of topology is associated with the
micromagnetic model—which approximates the magnetic texture of a system as a continuous field—and then applied indiscriminately without consideration of the model's physical limitations (i.e. that it ceases to be valid at atomic dimensions). In practice, treating the spin textures of magnetic materials as vectors of a continuous field model becomes inaccurate at size-scales on the order of < 2 nm, due to the discretization of the atomic lattice. Thus, it is not meaningful to speak of magnetic skyrmions below these size-scales. ==Practical applications==