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Topological phase transition

  1. Jun 9, 2015 #1
    I have just been reading a classical paper on the formation of majorana edge states (MES) in quantum wires. The hamiltonian is Kitaev type with a superconducting and spin-orbit interacting and one finds that the energies have a gap that closes and reopens as we vary the magnetic field. According to the authors this indicates a topological phase transition and they then proceed to show that a Majorana edge state forms centered around the point, where the closes.
    But I must I am a bit confused with the language and terms in this article. What does a topological phase transition mean and what indicates that we are dealing with a topological phase transition?
    The article is this one: http://arxiv.org/abs/1003.1145
  2. jcsd
  3. Jun 9, 2015 #2


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    Usually we think of phase transitions in terms of symmetry breaking (Landau Ginzburg). You think about phase transitions in terms of an order parameter. However for a topological phase transition you don't have this. You can go from one phase to another without breaking symmetry. A topological state is characterized by a topological invariant which is some sort of winding number or other invariant. When you go through a topological phase transition, this winding number changes. For example, in the quantum Hall effect, if you keep raising the magnetic field, the system will reach a point at which is goes from an insulator to a metal and then back to an insulator. During this process the hall conductance goes from n to n+1.

    For the kitaev majorana chain you have two majoranas on a site and an onsite coupling term as well as a coupling term between sites. When the onsite term dominates, the system is trivial, all sites are paired. However for a critical value of the coupling when the hopping term between sites starts to dominate, the two majoranas at the end of the chain are unpaired. The vector space is fractionalized and you have fractional excitation at the ends. This is topologically nontrivial. You can then take the onsite term to zero without changing the topological invariant and see that in the thermodynamic limit you have gapless modes at the ends of the chain.
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