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

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.


The trajectory of a vortex executing a round excursion through a superfluid (of superconductor) thin film. The Berry phase associated with this process, which is recorded by the topological term, is proportional to the number of particles within the area bounded by the closed curve. In an analogy with the Aharonov–Bohm effect, this indicates that the vortex behaves like a charged particle in an external magnetic field.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC5036487&req=5

Figure 2: The trajectory of a vortex executing a round excursion through a superfluid (of superconductor) thin film. The Berry phase associated with this process, which is recorded by the topological term, is proportional to the number of particles within the area bounded by the closed curve. In an analogy with the Aharonov–Bohm effect, this indicates that the vortex behaves like a charged particle in an external magnetic field.

Mentions: The second term on the right-hand side can be derived from a standard Ginzburg–Landau type action using a phase-only approximation. (The coefficient K, which describes the rigidity against phase fluctuations, is proportional to the square of the amplitude of the order-parameter. We also note that we have set for simplicity a coefficient with the dimension of a velocity to unity.) The first term is the topological or Berry phase term, and resembles the -term which appears in the (imaginary-time) Lagrangian of a single particle whose dynamics is described in terms of a pair of canonically conjugate variables q (position) and p (momentum). Recalling that the canonical conjugate of the phase ϕ is the particle density of the superfluid condensate, we interpret the coefficient ρ as the superfluid density (or more precisely, the offset value that this term imposes on this physical quantity). To see how this term influences the physics of the superfluid, we will study the motion of a vortex moving about in the system; for simplicity we consider a 2D system, i.e. a superfluid thin film, with a constant value of ρ. Furthermore, we assume that the vortex returns after an excursion to its initial position (see figure 2), as is required from the periodicity in the imaginary time direction. It is easy to check that the action contributes the quantity each time the vortex goes around a (bosonic) particle in a superfluid condensate (or in superconductor language, a Cooper pair), where is the vorticity. Hence, if the total number of bosons which has been encircled by the vortex is , where A is the area bounded by the trajectory (note that in the 2D case ρ is the number of condensate particles per area, and generally is not an integer), the net outcome from the topological term becomes . In other words, it contributes to the Boltzmann weight entering the path integral a phase factor of8This may be viewed as an Aharonov–Bohm-like effect; vortices see the surrounding condensate particles as a sort of magnetic field (with an intensity proportional to ρ), as is clear from the phase accumulation that occurs when the vortex performs a round trip.


A short guide to topological terms in the effective theories of condensed matter
The trajectory of a vortex executing a round excursion through a superfluid (of superconductor) thin film. The Berry phase associated with this process, which is recorded by the topological term, is proportional to the number of particles within the area bounded by the closed curve. In an analogy with the Aharonov–Bohm effect, this indicates that the vortex behaves like a charged particle in an external magnetic field.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

Figure 2: The trajectory of a vortex executing a round excursion through a superfluid (of superconductor) thin film. The Berry phase associated with this process, which is recorded by the topological term, is proportional to the number of particles within the area bounded by the closed curve. In an analogy with the Aharonov–Bohm effect, this indicates that the vortex behaves like a charged particle in an external magnetic field.
Mentions: The second term on the right-hand side can be derived from a standard Ginzburg–Landau type action using a phase-only approximation. (The coefficient K, which describes the rigidity against phase fluctuations, is proportional to the square of the amplitude of the order-parameter. We also note that we have set for simplicity a coefficient with the dimension of a velocity to unity.) The first term is the topological or Berry phase term, and resembles the -term which appears in the (imaginary-time) Lagrangian of a single particle whose dynamics is described in terms of a pair of canonically conjugate variables q (position) and p (momentum). Recalling that the canonical conjugate of the phase ϕ is the particle density of the superfluid condensate, we interpret the coefficient ρ as the superfluid density (or more precisely, the offset value that this term imposes on this physical quantity). To see how this term influences the physics of the superfluid, we will study the motion of a vortex moving about in the system; for simplicity we consider a 2D system, i.e. a superfluid thin film, with a constant value of ρ. Furthermore, we assume that the vortex returns after an excursion to its initial position (see figure 2), as is required from the periodicity in the imaginary time direction. It is easy to check that the action contributes the quantity each time the vortex goes around a (bosonic) particle in a superfluid condensate (or in superconductor language, a Cooper pair), where is the vorticity. Hence, if the total number of bosons which has been encircled by the vortex is , where A is the area bounded by the trajectory (note that in the 2D case ρ is the number of condensate particles per area, and generally is not an integer), the net outcome from the topological term becomes . In other words, it contributes to the Boltzmann weight entering the path integral a phase factor of8This may be viewed as an Aharonov–Bohm-like effect; vortices see the surrounding condensate particles as a sort of magnetic field (with an intensity proportional to ρ), as is clear from the phase accumulation that occurs when the vortex performs a round trip.

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.