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A short guide to topological terms in the effective theories of condensed matter

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ABSTRACT

This article is meant as a gentle introduction to the topological terms that often play a decisive role in effective theories describing topological quantum effects in condensed matter systems. We first take up several prominent examples, mainly from the area of quantum magnetism and superfluids/superconductors. We then briefly discuss how these ideas are now finding incarnations in the studies of symmetry-protected topological phases, which are in a sense a generalization of the concept of topological insulators to a wider range of materials, including magnets and cold atoms.

No MeSH data available.


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Schematic illustration of a skyrmion configuration in a ferromagnet. Skyrmions are spin configurations for which the winding number , which counts the number of times the unit vector  (indicated by the arrows) wraps around the unit sphere, is nonzero. In the configuration depicted, .
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Figure 6: Schematic illustration of a skyrmion configuration in a ferromagnet. Skyrmions are spin configurations for which the winding number , which counts the number of times the unit vector (indicated by the arrows) wraps around the unit sphere, is nonzero. In the configuration depicted, .

Mentions: A very similar force also arises in the dynamics of skyrmions in 2D ferromagnets [17], which are topological objects that are recently receiving considerable attention due to their potential value toward applications. Here the topological term takes the form (in real time) , where ρ is the density of spins (whose spin quantum number is S and its direction is represented by the unit vector ) and is the Berry phase term associated with a spin residing at position . The skyrmion is a configuration for which the winding number associated with the snapshot configuration (i.e. instantaneous configuration)21is a nonzero integer. (Notice that mathematically this is the same winding number (though associated with a different base manifold) which appeared in the previous section when we discussed the Haldane gap of antiferromagnetic spin chains.) A typical example of a skyrmion with is depicted in figure 6. Since this integer-valued number cannot change continuously, a skyrmion is stable unless processes involving singular configurations (which usually are energetically too costly to be relevant) are allowed. As in the vortex dynamics problem, we assume that the time dependence of the field takes the form , where now stands for the center of the skyrmion (or more generally a collective coordinate of the skyrmion). Using the formula for the variation of the surface angle (see figure 5) , we readily find that22Thus a skyrmion in motion behaves like a charged particle in the presence of a magnetic field .


A short guide to topological terms in the effective theories of condensed matter
Schematic illustration of a skyrmion configuration in a ferromagnet. Skyrmions are spin configurations for which the winding number , which counts the number of times the unit vector  (indicated by the arrows) wraps around the unit sphere, is nonzero. In the configuration depicted, .
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5036487&req=5

Figure 6: Schematic illustration of a skyrmion configuration in a ferromagnet. Skyrmions are spin configurations for which the winding number , which counts the number of times the unit vector (indicated by the arrows) wraps around the unit sphere, is nonzero. In the configuration depicted, .
Mentions: A very similar force also arises in the dynamics of skyrmions in 2D ferromagnets [17], which are topological objects that are recently receiving considerable attention due to their potential value toward applications. Here the topological term takes the form (in real time) , where ρ is the density of spins (whose spin quantum number is S and its direction is represented by the unit vector ) and is the Berry phase term associated with a spin residing at position . The skyrmion is a configuration for which the winding number associated with the snapshot configuration (i.e. instantaneous configuration)21is a nonzero integer. (Notice that mathematically this is the same winding number (though associated with a different base manifold) which appeared in the previous section when we discussed the Haldane gap of antiferromagnetic spin chains.) A typical example of a skyrmion with is depicted in figure 6. Since this integer-valued number cannot change continuously, a skyrmion is stable unless processes involving singular configurations (which usually are energetically too costly to be relevant) are allowed. As in the vortex dynamics problem, we assume that the time dependence of the field takes the form , where now stands for the center of the skyrmion (or more generally a collective coordinate of the skyrmion). Using the formula for the variation of the surface angle (see figure 5) , we readily find that22Thus a skyrmion in motion behaves like a charged particle in the presence of a magnetic field .

View Article: PubMed Central - PubMed

ABSTRACT

This article is meant as a gentle introduction to the topological terms that often play a decisive role in effective theories describing topological quantum effects in condensed matter systems. We first take up several prominent examples, mainly from the area of quantum magnetism and superfluids/superconductors. We then briefly discuss how these ideas are now finding incarnations in the studies of symmetry-protected topological phases, which are in a sense a generalization of the concept of topological insulators to a wider range of materials, including magnets and cold atoms.

No MeSH data available.


Related in: MedlinePlus