Can anybody tell me, if generically any system, which is solely described by a topological field theory, resides in a topological phase? I cant find any clear notion of topological phase. Only topological phase of matter, but I mean any kind of system.

Thanks for your help.

  • $\begingroup$ Your first sentence/question makes absolutely zero sense. $\endgroup$ Aug 10 '12 at 1:15
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    $\begingroup$ @ChrisGerig, that's not entirely accurate; sometimes "topological phase" refers to stable states of matter with topological order, like quantum hall states... Which I think is maybe what the questioner is getting at, but the question is unclear to me also. :\ $\endgroup$
    – wsc
    Aug 10 '12 at 4:26
  • $\begingroup$ @Chris Gerig: No I do not mean picking up a "phase factor". $\endgroup$
    – Hamurabi
    Aug 10 '12 at 6:31
  • $\begingroup$ @wsc: Yes, I mean configurations like those you wrote down. However, they are as I have understood, associated to matter degrees of freedom, when correlation functions become diffinvariant. I am just wondering, if generically, systems described by topological field theories describe topological phases, when disregarding other parts of the action. $\endgroup$
    – Hamurabi
    Aug 10 '12 at 6:35

Topological order is a new kind of order in zero-temperature phase of quantum spins, bonsons, and/or electrons. The new order corresponds to pattern of long-range quantum entanglement. Topological order is beyond the Landau symmetry-breaking description. It cannot be described by local order parameters and long range correlations. However, topological orders can be described/defined by a new set of quantum numbers, such as ground state degeneracy, non-Abelian geometric phases of degenerate ground states, quasiparticle fractional statistics, edge states, topological entanglement entropy, etc. Fractional quantum Hall states and quantum string liquids are examples of topologically ordered phases.

The low energy effective theory of topological phases happen to be topological quantum field theory. In nature, topological quantum field theory always appears as the low energy effective theory of topological phase of quantum spins, bonsons, and/or electrons, etc. By definition topological phase is always a quantum phase of quantum spins, bonsons, electrons, etc. ie topological phase is always a quantum state of matter.

  • $\begingroup$ Thank you for the clarification! I would like to ask some more things attached to your answer: 1) How does the TQFT arise? Starting from a higher temperature and cooling down, how does the TFT Lagrangian emerge? Is this topological order destroyed when lifting to higher temperatures? 2) How does fractional spin arise? Is there some sort of spin-statistics theorem to make this statement rigid? How is this linked to the gauge group of the TFT, like SU(2) CS-theory at level k? Thank you :) $\endgroup$
    – Hamurabi
    Aug 15 '12 at 19:32
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    $\begingroup$ 1) How does the TQFT arise? FQH states have topological degeneracy and TQFT reproduce the topological degeneracy. 2) How does fractional spin arise? See the paper, Xiao-Gang Wen and A. Zee. Shift and spin vectors: New quantum numbers of the FQH states. Phys. Rev. Lett., 69:953, 1992. (E) Phys. Rev. Lett., 69:3000, 1992. $\endgroup$ Sep 29 '12 at 1:28

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