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# Topological order

In physics, topological order is a new kind of order (a new kind of organization of particles) in a quantum state that 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 by a new set of quantum numbers, such as ground state degeneracy, quasiparticle fractional statistics, edge states, topological entropy, etc. Roughly speaking, topological order is a pattern of long-range quantum entanglement in quantum states. States with different topological orders can change into each other only through a phase transition.

## Background

Although all matter is formed by atoms, matter can have very different properties and appear in very different forms, such as solid, liquid, superfluid, magnet, etc. According to condensed matter physics and the principle of emergence, the different properties of materials originate 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.

Atoms can organize in many ways which lead to many different orders and many different types of materials. With so many different orders, we need a general understanding of the orders. Landau symmetry-breaking theory provides such a general understanding. 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 it 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, 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 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 is a very successful theory. For a long time, physicists believed that Landau symmetry-breaking theory describes all possible orders in materials, and all possible (continuous) phase transitions.

## The discovery and characterization of topological order

However, in last twenty years, it has become more and more apparent that Landau symmetry-breaking theory may not describe all possible orders. In 1987, physicists introduced chiral spin state in an attempt to explain high temperature superconductivity [Kalmeyer and Laughlin, 1987; Wen etal, 1989]. At first people 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. 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 symmetry description.[Wen, 1989] This new kind of order was named topological order.[Wen, 1990] (The name "topological order" is motivated by the low energy effective theory of the chiral spin states, which is a topological quantum field theory.[Witten, 1989]) New quantum numbers, such as ground state degeneracy [Wen, 1989] and the non-Abelian Berry's phase of degenerate ground states [Wen, 1990], were introduced to characterize the different topological orders in chiral spin states. Recently, it was shown that topological orders can also be characterized by topological entropy.[Kitaev and Preskill, 2006; Levin and Wen, 2006]

But experiments soon indicated 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.[Wen and Niu, 1990] Just like chiral spin states, different quantum Hall states all have the same symmetry and are beyond 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.

## Mechanism of topological order

A large class of topological orders is realized through a mechanism called string-net condensation.[Levin and Wen, 2003] This class of topological orders is described and classified by tensor 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 gauge bosons. The ends of strings are defects which correspond to another type of excitations. Those excitations are the gauge charges and can carry Fermi or fractional statistics.

The condensations of other extended objects such as membranes,[Hamma etal, 2005] brane-nets,[Bombin, M.A. Martin-Delgado, 2006] and fractals [Chamon, 2005] also lead to topologically ordered phases.

## Applications

The materials described by Landau symmetry-breaking theory have had a substantial impact on technology. For example, Ferromagnetic materials that break spin rotation symmetry can be used as the media of digital information storage. A hard drive made of ferromagnetic materials can store gigabytes of information. Liquid crystals that break the rotational symmetry of molecules find wide application in display technology; nowadays one can hardly find a household without a liquid crystal display somewhere in it. Crystals that break translation symmetry lead to well defined electronic bands which in turn allow us to make semiconducting devices such as transistors. Topologically ordered states are a new class of materials that are even richer than symmetry breaking states. This may suggest an exciting potential for applications.

One theorized application would be to use topologically ordered states as media for quantum computing. A topologically ordered state is a state with complicated non-local quantum entanglement. The non-locality means that the quantum entanglement in a topologically ordered state is distributed among many different particles. As a result, the pattern of quantum entanglements cannot be destroyed by local perturbations. This significantly reduces the effect of decoherence. This suggests that if we use different quantum entanglements in a topologically ordered state to encode quantum information, the information may last much longer.[Dennis etal, 2002] The quantum information encoded by the topological quantum entanglements can also be manipulated by dragging the topological defects around each other. This process may provide a physical apparatus for performing quantum computations.[Freedman etal, 2003] Therefore, topologically ordered states may provide natural media for both quantum memory and quantum computation. Such realizations of quantum memory and quantum computation may potentially be made fault tolerant.[Kitaev, 2003]

## Potential impact

Why is topological order important? 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. The exciting time of condensed matter physics is still ahead of us. Some suggest that topological order (or more precisely, string-net condensation) has a potential to provide a unified origin for photons, electrons and other elementary particles in our universe.

## References

Fractional quantum Hall states:

• Two-Dimensional Magnetotransport in the Extreme Quantum Limit, D. C. Tsui and H. L. Stormer and A. C. Gossard, Phys. Rev. Lett., 48, 1559 (1982)
• Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, R. B. Laughlin, Phys. Rev. Lett., 50, 1395 (1983)

Chiral spin states:

• Equivalence of the resonating-valence-bond and fractional quantum Hall states, V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett., 59, 2095 (1987)
• Chiral Spin States and Superconductivity, Xiao-Gang Wen, F. Wilczek and A. Zee, Phys. Rev., B39, 11413 (1989)

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:

• Quantum field theory and the Jones polynomial, E. Witten, Comm. Math. Phys., 121, 351 (1989)
• Vacuum Degeneracy of Chiral Spin State in Compactified Spaces, Xiao-Gang Wen, Phys. Rev. B, 40, 7387 (1989)
• Topological Orders in Rigid States, Xiao-Gang Wen, Int. J. Mod. Phys., B4, 239 (1990)
• Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces, Xiao-Gang Wen and Qian Niu, Phys. Rev. B41, 9377 (1990)

Characterization of topological order:

• Fractional Statistics and the Quantum Hall Effect, D. Arovas and J. R. Schrieffer and F. Wilczek, Phys. Rev. Lett., 53, 722 (1984)
• Gapless Boundary Excitations in the FQH States and in the Chiral Spin States, Xiao-Gang Wen, Phys. Rev. B, 43, 11025 (1991)
• Topological Entanglement Entropy, Alexei Kitaev and John Preskill, Phys. Rev. Lett. 96, 110404 (2006)
• Detecting Topological Order in a Ground State Wave Function, Michael Levin and Xiao-Gang Wen, Phys. Rev. Lett. 96, 110405 (2006)

Mechanism of topological order

• Photons and electrons as emergent phenomena, Michael A. Levin, Xiao-Gang Wen, Rev. Mod. Phys., 77, 871 (2005)
• String-net condensation: A physical mechanism for topological phases, Michael Levin, Xiao-Gang Wen, Phys. Rev. B, 71, 045110 (2005)
• Quantum Glassiness, Claudio Chamon, Phys. Rev. Lett. 94, 040402 (2005)
• String and Membrane condensation on 3D lattices, Alioscia Hamma, Paolo Zanardi, Xiao Gang Wen, Phys.Rev. B72 035307 (2005)
• Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates, H. Bombin, M.A. Martin-Delgado, cond-mat/0607736

Quantum computing:

• Fault-tolerant quantum computation by anyons, A. Yu. Kitaev Ann. Phys. (N.Y.), 303, 1 (2003)
• Topological quantum computation, Michael H. Freedman, Alexei Kitaev, Michael J. Larsen, and Zhenghan Wang, Bull. Amer. Math. Soc., 40, 31 (2003)
• Topological quantum memory, Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill, J. Math. Phys., 43, 4452 (2002)
• Proposed Experiments to probe the Non-Abelian nu=5/2 Quantum Hall State, Ady Stern and Bertrand I. Halperin, Phys. Rev. Lett., 96, 016802 (2006)

Emergence of elementary particles

• Quantum order from string-net condensations and origin of light and massless fermions, Xiao-Gang Wen, Phys. Rev. D68, 024501 (2003)
• A lattice bosonic model as a quantum theory of gravity, Zheng-Cheng Gu and Xiao-Gang Wen, gr-qc/0606100