Topological order

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Topological sort
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In

phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy[2] and quantized non-Abelian geometric phases of degenerate ground states.[1] Microscopically, topological orders correspond to patterns of long-range quantum entanglement.[3]
States with different topological orders (or different patterns of long range entanglements) cannot change into each other without a phase transition.

Various topologically ordered states have interesting properties, such as (1)

fault-tolerant quantum computation.[11]

Topological insulators[12] and topological superconductors (beyond 1D) do not have topological order as defined above, their entanglements being only short-ranged, but are examples of symmetry-protected topological order
.

Background

Matter composed of

states of matter or phases. According to condensed matter physics and the principle of emergence, the different properties of materials generally arise from the different ways in which the atoms are organized in the materials. Those different organizations of the atoms (or other particles) are formally called the orders in the materials.[13]

Atoms can organize in many ways which lead to many different orders and many different types of materials. Landau symmetry-breaking theory provides a general understanding of these different orders. It points out that different orders really correspond to different symmetries in the organizations of the constituent atoms. As a material changes from one order to another order (i.e., as the material undergoes a phase transition), what happens is that the symmetry of the organization of the atoms changes.

For example, atoms have a random distribution in a liquid, so a liquid remains the same as we displace atoms by an arbitrary distance. We say that a liquid has a continuous translation symmetry. After a phase transition, a liquid can turn into a crystal. In a crystal, atoms organize into a regular array (a lattice). A lattice remains unchanged only when we displace it by a particular distance (integer times a lattice constant), so a crystal has only discrete translation symmetry. The phase transition between a liquid and a crystal is a transition that reduces the continuous translation symmetry of the liquid to the discrete symmetry of the crystal. Such a change in symmetry is called symmetry breaking. The essence of the difference between liquids and crystals is therefore that the organizations of atoms have different symmetries in the two phases.

Landau symmetry-breaking theory has been a very successful theory. For a long time, physicists believed that Landau Theory described all possible orders in materials, and all possible (continuous) phase transitions.

Discovery and characterization

However, since the late 1980s, it has become gradually apparent that Landau symmetry-breaking theory may not describe all possible orders. In an attempt to explain

high temperature superconductivity[14] the chiral spin state was introduced.[5][6] At first, physicists still wanted to use Landau symmetry-breaking theory to describe the chiral spin state. They identified the chiral spin state as a state that breaks the time reversal and parity symmetries, but not the spin rotation symmetry. This should be the end of the story according to Landau's symmetry breaking description of orders. However, it was quickly realized that there are many different chiral spin states that have exactly the same symmetry, so symmetry alone was not enough to characterize different chiral spin states. This means that the chiral spin states contain a new kind of order that is beyond the usual symmetry description.[15] The proposed, new kind of order was named "topological order".[1] The name "topological order" is motivated by the low energy effective theory of the chiral spin states which is a topological quantum field theory (TQFT).[16][17][18] New quantum numbers, such as ground state degeneracy[15] (which can be defined on a closed space or an open space with gapped boundaries, including both Abelian topological orders [19]
[20] and non-Abelian topological orders
topological entropy.[23][24]

But experiments[

which?] soon indicated[how?] that chiral spin states do not describe high-temperature superconductors, and the theory of topological order became a theory with no experimental realization. However, the similarity between chiral spin states and quantum Hall states allows one to use the theory of topological order to describe different quantum Hall states.[2]
Just like chiral spin states, different quantum Hall states all have the same symmetry and are outside the Landau symmetry-breaking description. One finds that the different orders in different quantum Hall states can indeed be described by topological orders, so the topological order does have experimental realizations.

The

superconductor, discovered in 1911, is the first experimentally discovered topologically ordered state; it has Z2 topological order.[notes 1]

Although topologically ordered states usually appear in strongly interacting boson/fermion systems, a simple kind of topological order can also appear in free fermion systems. This kind of topological order corresponds to integral quantum Hall state, which can be characterized by the

Chern number of the filled energy band if we consider integer quantum Hall state on a lattice. Theoretical calculations have proposed that such Chern numbers can be measured for a free fermion system experimentally.[28][29]
It is also well known that such a Chern number can be measured (maybe indirectly) by edge states.

The most important characterization of topological orders would be the underlying fractionalized excitations (such as

fermions
). Current research works show that the loop and string like excitations exist for topological orders in the 3+1 dimensional spacetime, and their multi-loop/string-braiding statistics are the crucial signatures for identifying 3+1 dimensional topological orders. [30] [31] [32] The multi-loop/string-braiding statistics of 3+1 dimensional topological orders can be captured by the link invariants of particular topological quantum field theory in 4 spacetime dimensions.[32]

Mechanism

A large class of 2+1D topological orders is realized through a mechanism called

string-net condensation.[33] This class of topological orders can have a gapped edge and are classified by unitary fusion category (or monoidal category
) theory. One finds that string-net condensation can generate infinitely many different types of topological orders, which may indicate that there are many different new types of materials remaining to be discovered.

The collective motions of condensed strings give rise to excitations above the string-net condensed states. Those excitations turn out to be

fractional statistics.[34]

The condensations of other extended objects such as "

fractals also lead to topologically ordered phases[37] and "quantum glassiness".[38][39]

Mathematical formulation

We know that group theory is the mathematical foundation of symmetry-breaking orders. What is the mathematical foundation of topological order? It was found that a subclass of 2+1D topological orders—Abelian topological orders—can be classified by a K-matrix approach.[40][41][42][43] The string-net condensation suggests that tensor category (such as fusion category or monoidal category) is part of the mathematical foundation of topological order in 2+1D. The more recent researches suggest that (up to invertible topological orders that have no fractionalized excitations):

  • 2+1D bosonic topological orders are classified by unitary modular tensor categories.
  • 2+1D bosonic topological orders with symmetry G are classified by G-crossed tensor categories.
  • 2+1D bosonic/fermionic topological orders with symmetry G are classified by unitary braided fusion categories over symmetric fusion category, that has modular extensions. The symmetric fusion category Rep(G) for bosonic systems and sRep(G) for fermionic systems.

Topological order in higher dimensions may be related to n-Category theory. Quantum operator algebra is a very important mathematical tool in studying topological orders.

Some also suggest that topological order is mathematically described by extended quantum symmetry.[44]

Applications

The materials described by Landau symmetry-breaking theory have had a substantial impact on technology. For example,

molecules find wide application in display technology. Crystals that break translation symmetry lead to well defined electronic bands which in turn allow us to make semiconducting devices such as transistors
. Different types of topological orders are even richer than different types of symmetry-breaking orders. This suggests their potential for exciting, novel applications.

One theorized application would be to use topologically ordered states as media for

fault tolerant.[11]

Topologically ordered states in general have a special property that they contain non-trivial boundary states. In many cases, those boundary states become perfect conducting channel that can conduct electricity without generating heat.[47] This can be another potential application of topological order in electronic devices.

Similarly to topological order, topological insulators[48][49] also have gapless boundary states. The boundary states of topological insulators play a key role in the detection and the application of topological insulators. This observation naturally leads to a question: are topological insulators examples of topologically ordered states? In fact topological insulators are different from topologically ordered states defined in this article. Topological insulators only have short-ranged entanglements and have no topological order, while the topological order defined in this article is a pattern of long-range entanglement. Topological order is robust against any perturbations. It has emergent gauge theory, emergent fractional charge and fractional statistics. In contrast, topological insulators are robust only against perturbations that respect time-reversal and U(1) symmetries. Their quasi-particle excitations have no fractional charge and fractional statistics. Strictly speaking, topological insulator is an example of

Haldane phase of spin-1 chain.[51][52][53][54]
But the Haldane phase of spin-2 chain has no SPT order.

Potential impact

Landau symmetry-breaking theory is a cornerstone of condensed matter physics. It is used to define the territory of condensed matter research. The existence of topological order appears to indicate that nature is much richer than Landau symmetry-breaking theory has so far indicated. So topological order opens up a new direction in condensed matter physics—a new direction of highly entangled quantum matter. We realize that quantum phases of matter (i.e. the zero-temperature phases of matter) can be divided into two classes: long range entangled states and short range entangled states.[3] Topological order is the notion that describes the long range entangled states: topological order = pattern of long range entanglements. Short range entangled states are trivial in the sense that they all belong to one phase. However, in the presence of symmetry, even short range entangled states are nontrivial and can belong to different phases. Those phases are said to contain SPT order.[50] SPT order generalizes the notion of topological insulator to interacting systems.

Some suggest that topological order (or more precisely,

elementary particles in our universe.[4]

See also

Notes

  1. ^ Note that superconductivity can be described by the Ginzburg–Landau theory with dynamical U(1) EM gauge field, which is a Z2 gauge theory, that is, an effective theory of Z2 topological order. The prediction of the vortex state in superconductors was one of the main successes of Ginzburg–Landau theory with dynamical U(1) gauge field. The vortex in the gauged Ginzburg–Landau theory is nothing but the Z2 flux line in the Z2 gauge theory. The Ginzburg–Landau theory without the dynamical U(1) gauge field fails to describe the real superconductors with dynamical electromagnetic interaction.[8][25][26][27] However, in condensed matter physics, superconductor usually refers to a state with non-dynamical EM gauge field. Such a state is a symmetry breaking state with no topological order.

References

  1. ^ a b c d Wen 1990
  2. ^ a b Wen & Niu 1990
  3. ^
    S2CID 14593420
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  4. ^ a b Levin & Wen 2005a See also Levin & Wen 2006a
  5. ^ a b Kalmeyer & Laughlin 1987
  6. ^ a b Wen, Wilczek & Zee 1989, pp. 11413–23
  7. PMID 10043303
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  8. ^
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  9. ^ a b Tsui, Stormer & Gossard 1982
  10. ^ a b Laughlin 1983
  11. ^ a b Kitaev 2003
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  13. ^ Xiao-Gang Wen, An Introduction of Topological Orders (PDF), archived from the original (PDF) on 29 Aug 2017
  14. S2CID 118314311
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  15. ^ a b Xiao-Gang Wen, Phys. Rev. B, 40, 7387 (1989), "Vacuum Degeneracy of Chiral Spin State in Compactified Spaces"
  16. ISSN 1618-1913, http://www.numdam.org/item?id=PMIHES_1988__68__175_0
  17. ISSN 0010-3616, http://projecteuclid.org/euclid.cmp/1104161738
  18. ^ Yetter 1993
  19. S2CID 17803056
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  20. S2CID 33537923
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  21. S2CID 10125789
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  22. S2CID 14662084
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  23. ^ Kitaev & Preskill 2006
  24. ^ Levin & Wen 2006
  25. doi:10.1103/PhysRevB.95.014508
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  26. doi:10.1016/j.aop.2004.05.006
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  27. S2CID 119204930
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  28. doi:10.1103/Physics.4.99
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  29. S2CID 14947121
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  30. S2CID 23104804
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  32. ^
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  33. ^ Levin & Wen 2005
  34. ^ Levin & Wen 2003
  35. ^ Hamma, Zanardi & Wen 2005
  36. ^ Bombin & Martin-Delgado 2007
  37. Higgs phases
    ; also introduced a dimension index (DI) to characterize the robustness of the ground state degeneracy of a topologically ordered state. If DI is less or equal to 1, then topological orders cannot exist at finite temperature.
  38. S2CID 118911031
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  39. ^ Chamon 2005
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  44. doi:10.3842/sigma.2009.051
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  45. ^ Dennis et al. 2002
  46. ^ Freedman et al. 2003
  47. ^ Wen 1991a
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References by categories

Fractional quantum Hall states

Chiral spin states

Early characterization of FQH states

  • Off-diagonal long-range order, oblique confinement, and the fractional quantum Hall effect, S. M. Girvin and A. H. MacDonald, Phys. Rev. Lett., 58, 1252 (1987)
  • Effective-Field-Theory Model for the Fractional Quantum Hall Effect, S. C. Zhang and T. H. Hansson and S. Kivelson, Phys. Rev. Lett., 62, 82 (1989)

Topological order

Characterization of topological order

Effective theory of topological order

Mechanism of topological order

Quantum computing

Emergence of elementary particles

Quantum operator algebra

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