What is a topological phase transition and how is it characterized?

In summary, the article discusses the formation of Majorana edge states (MES) in quantum wires with a Kitaev-type hamiltonian. The authors find that the energy gap closes and reopens as the magnetic field is varied, indicating a topological phase transition. This type of phase transition is different from traditional ones as it does not involve symmetry breaking, but rather a change in a topological invariant. The article also compares this to the quantum Hall effect, where the hall conductance changes as the magnetic field is increased. The authors then explain how in the Kitaev majorana chain, there is a critical value of the coupling where the system becomes topologically nontrivial, resulting in gapless modes at the ends of the chain
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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
 
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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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1. What is a topological phase transition?

A topological phase transition is a type of phase transition that occurs in a system when there is a change in its topological properties. This means that the system undergoes a change in its global structure, but without any change in its local properties. Topological phase transitions are often accompanied by the emergence of new types of quantum states, known as topological phases.

2. How does a topological phase transition differ from other types of phase transitions?

Unlike conventional phase transitions, which are characterized by a change in the symmetry of a system, topological phase transitions are characterized by a change in the topology of the system. This means that the global structure of the system changes without any change in its local properties. Additionally, topological phase transitions can occur at zero temperature and are not associated with the breaking of any symmetries.

3. What are some examples of topological phase transitions?

One example of a topological phase transition is the quantum Hall effect, where a two-dimensional electron gas undergoes a topological phase transition to become a new type of insulator with quantized Hall conductance. Other examples include topological superconductors, topological insulators, and topological magnets.

4. What is the significance of studying topological phase transitions?

Studying topological phase transitions is important for understanding the underlying physics of complex systems and for developing new materials with unique properties. These transitions also have potential applications in quantum computing and topological quantum information processing.

5. Can topological phase transitions be observed in real-world systems?

Yes, topological phase transitions have been observed in various systems, including electronic systems, optical systems, and even cold atoms. These transitions can be experimentally detected through various techniques, such as measuring changes in conductivity or optical properties.

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