# nLab topological order

Topological order

### Context

#### Topological physics

Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.

General theory:

# Topological order

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.

## Idea

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)

## Mathematical foundation

The mathematical frame work of topological order involves tensor category, or more precisely n-category, for topological orders in n+1 dimensions.

## Literature

### Early discovery articles

• 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)

• 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).

• 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.

• Amit Jamadagni, Hendrik Weimer, An Operational Definition of Topological Order (arXiv:2005.06501)

Discussion in terms of extended TQFT, the cobordism theorem and stable homotopy theory is in

### Conference and seminar cycles

• 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 February 15, 2021 at 01:14:06. See the history of this page for a list of all contributions to it.