Topological order

This article is about quantum physics. For the graph-theoretical concept, see topological sort.

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, non-Abelian Berry phases of degenerate ground states, quasiparticle fractional statistics, edge states, topological entanglement entropy, etc. Strictly speaking, topological order is a pattern of long-range quantum entanglement in quantum states, defined via local unitary transformations. States with different topological orders cannot change into each other without 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. 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 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 (integer times of 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 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, since 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 people introduced chiral spin state. 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 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. The proposed, new kind of order was named "topological order". (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)). New quantum numbers, such as ground state degeneracy and the non-Abelian Berry's phase of degenerate ground states, 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.

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. 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.

Fractional quantum Hall (FQH) states were discovered in 1982 before the introduction of the concept of topological order. But FQH states are not the first experimentally discovered topologically ordered states. Superconductors discovered in 1911 are the first, which have a Z2 topological order (note that superconductivity can be described by the Ginzburg-Landau theory WITH dynamical U(1) gauge field, which is a Z2 gauge theory -- 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 fail to describe the real superconductors with dynamical electromagnetic interaction. ).

Mechanism of topological order

A large class of topological orders is realized through a mechanism called string-net condensation. This class of topological orders can be classified by utilizing tensor 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 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", "brane-nets", and fractals also lead to topologically ordered phases and "quantum glassiness".

Mathematical foundation of topological order

We know that group theory is the mathematical foundation of symmetry breaking orders. What is the mathematical foundation of topological order? The string-net condensation suggests that tensor category (or monoidal category) theory may be the mathematical foundation of topological order. 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.

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.

Different types of Topologically 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 quantum computing in a technique known as topological 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. 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. 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.

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. This can be another potential application of topological order in electronic devices. For example, the topological insulator is a simple example of topological order. The boundary states of topological insulator play a key role in the detection and the application of topological insulators.

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. Some suggest that topological order (or more precisely, string-net condensation) and the local bosonic (spin) models have the potential to provide a unified origin for photons, electrons and other elementary particles in our universe.

See also

• AKLT Model
• Fractionalization
• Herbertsmithite
• Implicate Order
• Quantum phase transitions
• Quantum critical point
• Quantum topology
• String-net liquid
• String-net
• Topological defect
• Topological entropy in physics
• Topological quantum field theory
• Topological quantum number
• Topological string theory

References by categories

Fractional quantum Hall states

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

Chiral spin states

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

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

• Xiao-Gang Wen, Phys. Rev. B, 40, 7387 (1989), "Vacuum Degeneracy of Chiral Spin State in Compactified Spaces"
• Xiao-Gang Wen, Int. J. Mod. Phys., B4, 239 (1990), "Topological Orders in Rigid States"
• Xiao-Gang Wen, Quantum Field Theory of Many Body Systems—From the Origin of Sound to an Origin of Light and Electrons, Oxford Univ. Press, Oxford, 2004.

Characterization of topological order

• D. Arovas and J. R. Schrieffer and F. Wilczek, Phys. Rev. Lett., 53, 722 (1984), "Fractional Statistics and the Quantum Hall Effect"
• Xiao-Gang Wen and Qian Niu, Phys. Rev. B41, 9377 (1990), "Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces"
• Xiao-Gang Wen, Phys. Rev. B, 43, 11025 (1991), "Gapless Boundary Excitations in the FQH States and in the Chiral Spin States"
• Alexei Kitaev and John Preskill, Phys. Rev. Lett. 96, 110404 (2006), "Topological Entanglement Entropy"
• Michael Levin and Xiao-Gang Wen, Phys. Rev. Lett. 96, 110405 (2006), "Detecting Topological Order in a Ground State Wave Function"

Effective theory of topological order

• Quantum field theory and the Jones polynomial, E. Witten, Comm. Math. Phys., 121, 351 (1989)

Mechanism of topological order

• Michael Levin, Xiao-Gang Wen, Phys. Rev. B, 71, 045110 (2005), String-net condensation: A physical mechanism for topological phases,
• Chamon, C., Phys. Rev. Lett. 94, 040402 (2005), Quantum Glassiness in Strongly Correlated Clean Systems: An Example of Topological Overprotection
• Alioscia Hamma, Paolo Zanardi, Xiao-Gang Wen, Phys.Rev. B72 035307 (2005), String and Membrane condensation on 3D lattices
• H. Bombin, M.A. Martin-Delgado, cond-mat/0607736, Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates

Quantum computing

• Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, Sankar Das Sarma, http://www.arxiv.org/abs/0707.1889, 2007, "Non-Abelian Anyons and Topological Quantum Computation", Rev. Mod. Phys. 80, 1083 (2008).
• A. Yu. Kitaev, Ann. Phys. (N.Y.), 303, 1 (2003), Fault-tolerant quantum computation by anyons
• Michael H. Freedman, Alexei Kitaev, Michael J. Larsen, and Zhenghan Wang, Bull. Amer. Math. Soc., 40, 31 (2003), "Topological quantum computation"
• Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill, J. Math. Phys., 43, 4452 (2002), Topological quantum memory
• Ady Stern and Bertrand I. Halperin, Phys. Rev. Lett., 96, 016802 (2006), Proposed Experiments to probe the Non-Abelian nu=5/2 Quantum Hall State

Emergence of elementary particles

• Xiao-Gang Wen, Phys. Rev. D68, 024501 (2003), Quantum order from string-net condensations and origin of light and massless fermions
• M. Levin and Xiao-Gang Wen, Fermions, strings, and gauge fields in lattice spin models., Phys. Rev. B 67, 245316, (2003).
• M. Levin and Xiao-Gang Wen, Colloquium: Photons and electrons as emergent phenomena, Rev. Mod. Phys. 77, Nu 12:19, 9 April 2009 (UTC)871 (2005), 4 pages; also, Quantum ether: Photons and electrons from a rotor model., arXiv:hep-th/0507118,2007.
• Zheng-Cheng Gu and Xiao-Gang Wen, gr-qc/0606100, A lattice bosonic model as a quantum theory of gravity,

Quantum operator algebra

• Yetter D.N., TQFTs from homotopy 2-types, J. Knot Theory 2 (1993), 113—123.
• Landsman N. P. and Ramazan B., Quantization of Poisson algebras associated to Lie algebroids, in Proc. Conf. on Groupoids in Physics, Analysis and Geometry(Boulder CO, 1999)', Editors J. Kaminker et al.,159{192 Contemp. Math. 282, Amer. Math. Soc., Providence RI, 2001,(also math{ph/001005.)
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• Levin A. and Olshanetsky M., Hamiltonian Algebroids and deformations of complex structures on Riemann curves, hep-th/0301078v1.
• Xiao-Gang Wen, Yong-Shi Wu and Y. Hatsugai., Chiral operator product algebra and edge excitations of a FQH droplet (pdf),Nucl. Phys. B422, 476 (1994): Used chiral operator product algebra to construct the bulk wave function, characterize the topological orders and calculate the edge states for some non-Abelian FQH states.
• Xiao-Gang Wen and Yong-Shi Wu., Chiral operator product algebra hidden in certain FQH states (pdf),Nucl. Phys. B419, 455 (1994): Demonstrated that non-Abelian topological orders are closely related to chiral operator product algebra (instead of conformal field theory).
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