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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.


Trajectories corresponding to two slightly varying (imaginary) time evolutions of the unit vector . The difference in the solid angle that is traced out amounts to .
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Figure 5: Trajectories corresponding to two slightly varying (imaginary) time evolutions of the unit vector . The difference in the solid angle that is traced out amounts to .

Mentions: The kinetic term of the effective action should describe the tendency for nearby to align, and is found to take the form12in the continuum approximation, where g is a coupling constant which depends on the details of the starting lattice Hamiltonian (we have again set an additional coefficient with the dimension of velocity to unity for simplicity). In order to derive the topological term of the effective action, we begin by observing that the spin Berry phase term for a single spin moment is odd with respect to spin inversion, i.e. . The first equality follows by noting that the solid angle is an oriented surface area, whereas the second equality points to the fact that since this term always appears in the form of a phase factor, the portion is irrelevant (due to ). Thus the contributions from the Berry phase term for each spin moment in the antiferromagnet add up into13On taking the continuum limit in the second line, we have converted differences into derivatives, and further used the fact that the derivatives are contributed by every other link on the chain (resulting in the factor of 1/2). To proceed, we seek the help of figure 5, and obtain the final form of the topological term (which is often referred to in the literature of a θ term),14where , and15is an integer-valued winding number which counts the number of times wraps around the sphere as one probes through the entire (Euclidean) spacetime. The partition function therefore reads16In the second expression we have sorted the configurations entering the path integral according to the value of its winding number . Equation (16) suggests that the system behaves in a qualitatively different way for integer S (for which ) and half odd integer S (where ). For the latter case, the sign-alternating factors tend to lead to a destructive interference between configurations with nonzero (such spacetime configurations are often called instantons). Since instantons will apparently cause a strong disruption to the antiferromagnet order, we thus expect that antiferromagnetic spin chains with half-odd integer spin, for which case the instanton events are suppressed, should exhibit a stronger degree of spin ordering than those with integer spins, where instantons are not suppressed. It is by now well established that the former are indeed critically ordered (i.e. exhibit a power-law decaying spin-spin correlation), while the latter are strongly disordered (with exponentially decaying correlations).


A short guide to topological terms in the effective theories of condensed matter
Trajectories corresponding to two slightly varying (imaginary) time evolutions of the unit vector . The difference in the solid angle that is traced out amounts to .
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 5: Trajectories corresponding to two slightly varying (imaginary) time evolutions of the unit vector . The difference in the solid angle that is traced out amounts to .
Mentions: The kinetic term of the effective action should describe the tendency for nearby to align, and is found to take the form12in the continuum approximation, where g is a coupling constant which depends on the details of the starting lattice Hamiltonian (we have again set an additional coefficient with the dimension of velocity to unity for simplicity). In order to derive the topological term of the effective action, we begin by observing that the spin Berry phase term for a single spin moment is odd with respect to spin inversion, i.e. . The first equality follows by noting that the solid angle is an oriented surface area, whereas the second equality points to the fact that since this term always appears in the form of a phase factor, the portion is irrelevant (due to ). Thus the contributions from the Berry phase term for each spin moment in the antiferromagnet add up into13On taking the continuum limit in the second line, we have converted differences into derivatives, and further used the fact that the derivatives are contributed by every other link on the chain (resulting in the factor of 1/2). To proceed, we seek the help of figure 5, and obtain the final form of the topological term (which is often referred to in the literature of a θ term),14where , and15is an integer-valued winding number which counts the number of times wraps around the sphere as one probes through the entire (Euclidean) spacetime. The partition function therefore reads16In the second expression we have sorted the configurations entering the path integral according to the value of its winding number . Equation (16) suggests that the system behaves in a qualitatively different way for integer S (for which ) and half odd integer S (where ). For the latter case, the sign-alternating factors tend to lead to a destructive interference between configurations with nonzero (such spacetime configurations are often called instantons). Since instantons will apparently cause a strong disruption to the antiferromagnet order, we thus expect that antiferromagnetic spin chains with half-odd integer spin, for which case the instanton events are suppressed, should exhibit a stronger degree of spin ordering than those with integer spins, where instantons are not suppressed. It is by now well established that the former are indeed critically ordered (i.e. exhibit a power-law decaying spin-spin correlation), while the latter are strongly disordered (with exponentially decaying correlations).

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.