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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.


The two low-energy paths relevant to the evaluation of the transition amplitude from the up-spin state to the down-spin state. The difference of the Berry phase for the two paths is well defined and equals S multiplied by , i.e. the surface area corresponding to half of the unit sphere.
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Figure 1: The two low-energy paths relevant to the evaluation of the transition amplitude from the up-spin state to the down-spin state. The difference of the Berry phase for the two paths is well defined and equals S multiplied by , i.e. the surface area corresponding to half of the unit sphere.

Mentions: Consider a spin moment with spin quantum number S. Suppose that there is a dominant easy-plane and a sub-dominant easy-plane anisotropy; we take for concreteness the Hamiltonian , with . If we denote the direction of the spin vector by the unit vector (the direction is only meaningful in a semi-classical sense since the spin components are non-commuting entities), the action is given by4The topological term is related to the Berry phase which is induced by the evolution of the spin orientation. If the motion of is such that its orientation coincides at the beginning and the end of the time interval under consideration, this term can be expressed as in the second line, where is the solid angle, i.e. the area which the trajectory of traces out on the surface of the unit sphere. In terms of the spherical coordinate , this is explicitly written as5Let us now consider the probability amplitude associated with the transition of from the north to the south pole, i.e. between the two lowest energy states [6, 7]. The path integral for this transition is dominated (see figure 1) by the two low energy paths connecting the poles, which are located along the longitudes (path A) and (path B). Though each path alone does not trace out a closed curve, their difference does. Noting that the two paths correspond to the same kinetic energy, we expect the transition amplitude to take the form6where, in the third line, we have used the fact that the surface area corresponding to half of the sphere is . (A detailed evaluation can be found in the original references [6, 7].) Thus if S is half of an odd integer (), the probability amplitude vanishes (destructive interference), while if S is integer-valued (), the amplitude is enhanced (constructive interference). This can have experimental consequences, for example, for switching effects in single-molecule nanomagnets such as Mn-acetate and Fe. The two interference patterns discussed above can manifest themselves as unavoided or avoided level crossings between the up-spin and down-spin levels (the former case corresponds to the suppression of hybridization) as a function of an external magnetic field applied along the z-axis. (It has been pointed out however that realistic situations tend to obscure this effect [8].) It is also interesting that essentially the same mechanism has been proposed to control the macroscopic tunneling of magnetic flux in a fabricated superconducting island (it is possible to map the low-energy sector onto an effective spin system), which may find applications to quantum information processes [9].


A short guide to topological terms in the effective theories of condensed matter
The two low-energy paths relevant to the evaluation of the transition amplitude from the up-spin state to the down-spin state. The difference of the Berry phase for the two paths is well defined and equals S multiplied by , i.e. the surface area corresponding to half of the unit sphere.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 1: The two low-energy paths relevant to the evaluation of the transition amplitude from the up-spin state to the down-spin state. The difference of the Berry phase for the two paths is well defined and equals S multiplied by , i.e. the surface area corresponding to half of the unit sphere.
Mentions: Consider a spin moment with spin quantum number S. Suppose that there is a dominant easy-plane and a sub-dominant easy-plane anisotropy; we take for concreteness the Hamiltonian , with . If we denote the direction of the spin vector by the unit vector (the direction is only meaningful in a semi-classical sense since the spin components are non-commuting entities), the action is given by4The topological term is related to the Berry phase which is induced by the evolution of the spin orientation. If the motion of is such that its orientation coincides at the beginning and the end of the time interval under consideration, this term can be expressed as in the second line, where is the solid angle, i.e. the area which the trajectory of traces out on the surface of the unit sphere. In terms of the spherical coordinate , this is explicitly written as5Let us now consider the probability amplitude associated with the transition of from the north to the south pole, i.e. between the two lowest energy states [6, 7]. The path integral for this transition is dominated (see figure 1) by the two low energy paths connecting the poles, which are located along the longitudes (path A) and (path B). Though each path alone does not trace out a closed curve, their difference does. Noting that the two paths correspond to the same kinetic energy, we expect the transition amplitude to take the form6where, in the third line, we have used the fact that the surface area corresponding to half of the sphere is . (A detailed evaluation can be found in the original references [6, 7].) Thus if S is half of an odd integer (), the probability amplitude vanishes (destructive interference), while if S is integer-valued (), the amplitude is enhanced (constructive interference). This can have experimental consequences, for example, for switching effects in single-molecule nanomagnets such as Mn-acetate and Fe. The two interference patterns discussed above can manifest themselves as unavoided or avoided level crossings between the up-spin and down-spin levels (the former case corresponds to the suppression of hybridization) as a function of an external magnetic field applied along the z-axis. (It has been pointed out however that realistic situations tend to obscure this effect [8].) It is also interesting that essentially the same mechanism has been proposed to control the macroscopic tunneling of magnetic flux in a fabricated superconducting island (it is possible to map the low-energy sector onto an effective spin system), which may find applications to quantum information processes [9].

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.