Topological Phase: Definition & Examples

In summary, Levin defines a topological phase as a phase that is gapped, has 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.
  • #1
complement
11
0
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.
 
Physics news on Phys.org
  • #2
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:
  • #3
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))?
 
  • #4
At what level would you like the explanation?
 
  • #5
full scale please :)
 
  • #6
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)?
 

1. What is a topological phase?

A topological phase is a type of phase of matter that is characterized by its unique topological properties, such as non-localized excitations and robust edge states. This phase cannot be described by traditional symmetry breaking methods and is instead defined by its mathematical topology.

2. What are some examples of topological phases?

Some examples of topological phases include topological insulators, topological superconductors, and fractional quantum Hall states. These phases have been observed in various materials, such as graphene, topological insulator materials, and certain superconducting materials.

3. How is a topological phase different from a conventional phase?

A topological phase is different from a conventional phase in that it is not described by traditional symmetry-breaking methods. Instead, it is defined by its non-trivial topology, which leads to unique properties that cannot be explained by conventional theories. It also has robust boundary states that are protected from local perturbations.

4. What are the potential applications of topological phases?

Topological phases have potential applications in quantum computing, as they have unique properties that could be harnessed for more efficient and robust quantum information processing. They also have potential applications in creating new types of electronic devices, such as topological transistors.

5. How are topological phases studied and characterized?

Topological phases are studied and characterized using various experimental techniques, such as angle-resolved photoemission spectroscopy, scanning tunneling microscopy, and transport measurements. Theoretical methods, such as topological band theory and topological field theory, are also used to understand and predict the properties of topological phases.

Similar threads

A How to read modern condensed matter field research
    • Atomic and Condensed Matter
Replies
1
Views
1K
A Black hole as topological insulator
    • Atomic and Condensed Matter
Replies
4
Views
2K
Solid State Texts on Topological Effects/Phases in Materials
    • Science and Math Textbooks
Replies
2
Views
1K
A Quantum Field theory vs. many-body Quantum Mechanics
    • Quantum Physics
2
Replies
36
Views
3K
A What Topological Vector Spaces have an uncountable Schauder basis?
    • Topology and Analysis
Replies
6
Views
1K
Computational Literature suggestions please (Topological Quantum Computers)
    • Science and Math Textbooks
Replies
1
Views
662
Induction motor windings, number of turns per phase and per pole
    • Electrical Engineering
Replies
2
Views
593
Kitaev's Periodic Table (of Topological Insulators & SCs)
    • Atomic and Condensed Matter
Replies
20
Views
9K
A Introduction to topological field theory?
    • Quantum Physics
Replies
1
Views
535
I Are SM B & L conservation violations through sphalerons possible?
    • High Energy, Nuclear, Particle Physics
Replies
1
Views
921

Hot Threads

Recent Insights

Back
Top