nLab
topological state of matter

Idea

We include here a phase of matter which is, at long distance and low energy, described by a topological quantum field theory (invariant under small smooth deformations of space-time) as well as the phases related to symmetry protected trivial orders (SPT orders). For the first see topological order. The ground state is typically degenerate (for reasonably nontrivial orders).

Examples of topological states of matter: quantum Hall effect, topological insulator, quantum spin Hall effect

• applications in graphene, topological quantum computing, study of entanglement etc.

Involves study of symmetry breaking, tensor categories, K-theory classification of topological phases of matter

Relation between SPT order and topological order

This entry refers both topological order and SPT order, and here is an comparison between them.

• Both topological order and SPT order are beyond Landau symmetry breaking theory.

• SPT states are short-range entangled while topologically ordered states are long-range entangled.

• Topologically ordered states have emergent fractional charge, emergent fractional statistics, and emergent gauge theory. In contrast, SPT states have no emergent fractional charge/fractional statistics for finite-energy excitations, nor emergent gauge theory (due to their short-ranged entanglement).

• SPT orders are described by group cohomology theory while topological orders are described by n-category theory. The SPT orders for free fermions are described by K-theory.

Related entries

• phase of matter

• gapped Hamiltonian

• topological field theory

• tensor network state

• quantum Hall effect

• topological order

• topological insulator

• topological photonics

• topological quantum computing

• quantum spin Hall effect

• entanglement

References

Reviews

• Wikipedia, topological insulator, topological order

• Xiao-Gang Wen, Topological Orders and Edge Excitations in FQH States,

  Advances in Physics 44, 405 (1995). cond-mat/9506066. (for topological order)

• 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 (for topological order)

• M. Z. Hasan, C. L. Kane, Topological insulators, Reviews of Modern Physics 82 (4): 3045 (2010) arXiv:1002.3895 doi (for topological insulator)

• C. L. Kane, An insulator with a twist, Nature physics 4, May 2008, pdf (for topological insulator)

• Class for Physics of the Royal Swedish Academy of Sciences, Topological phase transitions and topological phases of matter, Scientific Background on the Nobel Prize in Physics 2016, (2+)26 pages, pdf

In the context of topological quantum computation:

• Eric Rowell, Zhenghan Wang, Section 4 of: Mathematics of Topological Quantum Computing, Bull. Amer. Math. Soc. 55 (2018), 183-238 (arXiv:1705.06206, doi:10.1090/bull/1605)

Early discovery articles

• Xiao-Gang Wen, 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) (for topological order described by TQFT)
• E. Keski-Vakkuri, Xiao-Gang Wen, Ground state structure of hierarchical QH states on torus and modular transformation Int. J. Mod. Phys. B7, 4227 (1993). (for topological order described by TQFT)
• Liang Fu, C. L. Kane, Topological insulators with inversion symmetry, Physical Review B 76 (4): 045302. arXiv:cond-mat/0611341 doi; Superconducting proximity effect and Majorana fermions at the surface of a topological insulator, Phys. Rev. Lett. 100: 096407, arXiv:0707.1692 doi (for topological insulator, a special case of SPT states)
• B. Andrei Bernevig, Taylor L. Hughes, Shou-Cheng Zhang, Quantum spin Hall effect and topological phase transition in HgTe quantum wells, Science 15 December 2006: 314, n. 5806, pp. 1757-1761 doi (for topological insulator, a special case of SPT states)

Classification and symmetries

• Michael Levin?, Xiao-Gang Wen, String-net condensation: A physical mechanism for topological phases, Phys. Rev. B, 71, 045110 (2005). (uses unitary fusion category to classify 2+1D topological order with gapped boundary)

• Alexei Kitaev, Periodic table for topological insulators and superconductors, Proc. L.D.Landau Memorial Conf. “Advances in Theor. Physics”, June 22-26, 2008, Chernogolovka, Russia, arxiv/0901.2686 (uses K-homology, Bott periodicity etc. to classify free fermion gapped phases with symmetry)

• Xie Chen, Zheng-Cheng Gu, Zheng-Xin Liu, Xiao-Gang Wen, Symmetry protected topological orders and the group cohomology of their symmetry group, arXiv:1106.4772; A short version in Science 338, 1604-1606 (2012) pdf (uses group cohomology of the symmetry group to classify gapped interacting bosonic states with symmetry and trivial topological order)

• Daniel Freed, Gregory Moore, Twisted equivariant matter, arxiv/1208.5055 (uses equivariant K-theory to classify free fermion gapped phases with symmetry)

• Juven Wang, Zheng-Cheng Gu, Xiao-Gang Wen, Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond, arxiv:1405.7689, Phys. Rev. Lett. 114, 031601 (2015)

• Daniel Freed, Short-range entanglement and invertible field theories (arXiv:1406.7278)

• Guo Chuan Thiang, On the K-theoretic classification of topological phases of matter, arXiv:1406.7366

• Edward Witten, Fermion path integrals and topological phases, arxiv/1508.04715

• Ping Ye, (3+1)d anomalous twisted gauge theories with global symmetry, arxiv/1610.08645

Classification of topological phases of matter via tensor network states:

• C. Wille, O. Buerschaper, Jens Eisert, Fermionic topological quantum states as tensor networks, Phys. Rev. B 95, 245127 (2017) (arXiv:1609.02574)

• Andreas Bauer, Jens Eisert, Carolin Wille, Towards a mathematical formalism for classifying phases of matter (arXiv:1903.05413)

Other articles

• N. Read, Subir Sachdev, Large-N expansion for frustrated quantum antiferromagnets, Phys. Rev. Lett. 66 1773 (1991)
• Xiao-Gang Wen, Mean Field Theory of Spin Liquid States with Finite Energy Gap and Topological orders, Phys. Rev. B 44 2664 (1991).
• 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
• 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)
• M. Levin, X-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Letts.,96(11), 110405, (2006)
• 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, Modern Physics Letters A29:13 (2014) arxiv/1306.5570 doi
• Yosuke Kubota, Controlled topological phases and bulk-edge correspondence, arxiv/1511.05314

Research groups

• Univ. of Maryland, joint quantum institute
• Topology and correlation in condensed matter (Germany)
• Center for topological matter (Korea)
• Microsoft Research Station Q (KITP, Santa Barbara)

Conference and seminar cycles

• seminar in Koeln Topological states of matter

• Topological Phases of Matter: Simons Center, June 10-14, 2013, videos available

• A. Kitaev, On the classification of short-range entangled states, video

• CECAM 2013, Topological Phases in Condensed Matter and Cold Atom Systems: towards quantum computations description