Topological Phase: Definition & Examples

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Discussion Overview

The discussion revolves around the concept of topological phases, particularly in the context of topological field theories. Participants explore definitions, examples, and the mathematical formalism associated with topological phases, including references to specific literature and derivations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions whether any system described by a topological field theory can be considered to reside in a topological phase, noting a lack of clear definitions.
  • Another participant cites Levin's definition of a topological phase, which includes being gapped, having ground state degeneracy dependent on the topology of the manifold, and exhibiting fractional statistics.
  • Similar definitions are referenced from Nayak et al. and Hansson et al., while Moore's notes provide a different definition, distinguishing between 'Thouless phases' and 'Wen-type phases.'
  • There is a request for references that explain the derivation of the Chern-Simons (CS) Lagrangian, indicating interest in the mathematical formalism.
  • One participant expresses a desire for a full-scale explanation of the CS Lagrangian, indicating their background as an amateur with a good mathematical foundation.
  • Another participant inquires about the existence of derivations of similar expressions for even-dimensional spaces, specifically asking about N = 4.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definition of topological phases, with multiple competing definitions and interpretations presented. The discussion remains unresolved regarding the broader applicability of topological phases beyond specific examples.

Contextual Notes

There are limitations in the discussion regarding the clarity of definitions and the scope of topological phases, particularly concerning the dimensionality of spaces involved in the derivations of the CS Lagrangian.

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Hi there!

Can anybody tell me, if generically any system, which is solely described by a topological field theory, resides in a topological phase? I can't find any clear notion of topological phase. Only topological phase of matter, but I mean any kind of system.

Thanks for your help.
 
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Levin gives a definition in this talk. He defines a topological phase as being gapped, having a ground state degeneracy that depends only on the topology of the manifold that the system is placed on, and has fractional statistics.

Similar definitions are given by Nayak et al (section III.A) and Hansson et al (section B of the introduction).

Moore's notes contain a different definition of topological phase ('Thouless phases'), which he distinguishes from the definition of topological phase ('Wen-type phases') used by Levin, Nayak and Hansson.
 
Last edited:
atyy said:
Levin gives a definition in this talk. He defines a topological phase as being gapped, having a ground state degeneracy that depends only on the topology of the manifold that the system is placed on, and has fractional statistics.

Similar definitions are given by Nayak et al (section III.A) and Hansson et al (section B of the introduction).

Moore's notes contain a different definition of topological phase ('Thouless phases'), which he distinguishes from the definition of topological phase ('Wen-type phases') used by Levin, Nayak and Hansson.


Very interesting. Do you know some references explaining the derivation of the CS Lagrangian (e.g. wenphases.pdf, page 1, (5))?
 
At what level would you like the explanation?
 
full scale please :)
 
Physics Monkey said:
At what level would you like the explanation?

Well, what can I say? The level of an amateur with a relatively good mathematical background. I was wondering about the formalism of that Lagrangian and interested by the fact that it is interpreted as a "topological one". I know that CS theories are developped in spaces with an odd number of dimensions (N = 3, 5...). Is there also in the published literature a derivation of a similar expression for spaces with an even number of dimensions (e.g. N = 4)?
 

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