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This entry has some overlap with topological state of matter, however unlike that entry, this entry does not include symmetry protected trivial order, since symmetry protected trivial order contains no topological order by definition.
Topological order is an order in quantum phase of matter which is beyond Landau symmetry breaking order. At long distance and low energy (ie at macroscopic level), topological order is defined by topological degeneracy? (see Wikipedia) of the ground states and the non-Abelian geometric phases (see Wilczek & Zee, 1984) obtained by deforming the degenerate ground states. The low energy effective theory of a topologically ordered state is a topological quantum field theory. It has many universal properties that are (by definition) invariant under any small smooth deformations of space-time (or any small deformation of Hamiltonian). The excitations in a topologically ordered state typically have fractional or non-Abelian statistics (for most topological orders in 2+1D). At microscopic level, topological order corresponds to patterns of long-range entanglement in the ground state defined by the local unitary transformation?s (see Chen et al, 2010).
Examples: quantum Hall effect, non-Abelian quantum Hall state? (see Wikipedia), chiral spin liquid? (see Wikipedia), Z2 spin liquid? (see Wikipedia)
The mathematical frame work of topological order involves tensor category, or more precisely n-category, for topological orders in n+1 dimensions.
Related entries: topological state of matter, TQFT, quantum computing, quantum Hall effect, modular tensor category, entanglement
Wikipedia: topological order
Xiao-Gang Wen, Topological Orders and Edge Excitations in FQH States,
Advances in Physics 44, 405 (1995). cond-mat/9506066.
Chetan Nayak, Steven H. Simon, Ady Stern, M. Freedman, Sankar Das Sarma, Non-Abelian anyons and topological quantum computation, Rev Mod Phys 80:3 (Aug 2008) 1083–1159 MR2009g:81041 doi
Xiao-Gang Wen, An introduction of topological order 2009 (pdf slides, article)
Michel Fruchart, David Carpentier, An Introduction to Topological Insulators, Comptes Rendus Physique 14 (2013) 779-815 (arXiv:1310.0255)
Xiao-Gang Wen, Vacuum Degeneracy of Chiral Spin State in Compactified Spaces, Phys. Rev. B, 40, 7387 (1989).
Xiao-Gang Wen, Topological Orders in Rigid States, Int. J. Mod. Phys. B4, 239 (1990)
Xiao-Gang Wen and Qian Niu, Ground state degeneracy of the FQH states in presence of random potential and on high genus Riemann surfaces, Phys. Rev. B41, 9377 (1990)
E. Keski-Vakkuri and Xiao-Gang Wen, Ground state structure of hierarchical QH states on torus and modular transformation Int. J. Mod. Phys. B7, 4227 (1993).
Davide Gaiotto, Anton Kapustin, Spin TQFTs and fermionic phases of matter, arxiv/1505.05856
N. Read and Subir Sachdev, Large-N expansion for frustrated quantum antiferromagnets, Phys. Rev. Lett. 66 1773 (1991) (on $Z_2$ topological order)
Xiao-Gang Wen, Mean Field Theory of Spin Liquid States with Finite Energy Gap and Topological orders, Phys. Rev. B 44 2664 (1991). (on $Z_2$ topological order)
Xiao-Gang Wen, Non-Abelian Statistics in the FQH states
Phys. Rev. Lett. 66, 802 (1991).
Moore, Gregory; Read, Nicholas. Nonabelions in the fractional quantum hall effect Nuclear Physics B 360 (2–3): 362 (1991).
Xiao-Gang Wen and Yong-Shi Wu, Chiral operator product algebra hidden in certain FQH states
Nucl. Phys. B419, 455 (1994).
Alexei Yu. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303:1, January 2003; Anyons in an exactly solved model and beyond, Annals of Physics 321:1, January 2006
Michael Levin?, Xiao-Gang Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B, 71, 045110 (2005).
A. Kitaev, C. Laumann, Topological phases and quantum computation, arXiv/0904.2771
Alexei Kitaev, John Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, 110404 (2006)
Levin M. and Wen X-G., Detecting topological order in a ground state wave function, Phys. Rev. Letts.,96(11), 110405, (2006)
Xie Chen, Zheng-Cheng Gu, Xiao-Gang Wen, Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order Phys. Rev. B 82, 155138 (2010), arXiv
Jan Carl Budich, Björn Trauzettel, From the adiabatic theorem of quantum mechanics to topological states of matter, physica status solidi (RRL) 7, 109 (2013) arXiv:1210.6672
Kumar S. Gupta, Amilcar Queiroz, Anomalies and renormalization of impure states in quantum theories, arxiv/1306.5570
Frank Wilczek & A. Zee (1984); Appearance of gauge structure in simple dynamical systems; Physical Review Letters 52 (24), 2111–2114; pdf.
Discussion in terms of extended TQFT, the cobordism theorem and stable homotopy theory is in
Daniel Freed, Gregory Moore, Twisted equivariant matter, arxiv/1208.5055 (uses equivariant K-theory to classify free fermion gapped phases with symmetry)
Daniel Freed, Short-range entanglement and invertible field theories (arXiv:1406.7278)
Univ. of Maryland, joint quantum institute
Center for topological matter (Korea)
Microsoft Research Station Q (KITP, Santa Barbara)
seminar in Koeln Topological states of matter
Topological Phases of Matter: Simons Center, June 10-14, 2013, videos available
CECAM 2013, Topological Phases in Condensed Matter and Cold Atom Systems: towards quantum computations description
Last revised on November 17, 2018 at 03:20:45. See the history of this page for a list of all contributions to it.