Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.
General theory:
In electromagnetism:
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
Involves study of symmetry breaking, tensor categories, K-theory classification of topological phases of matter
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.
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:
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)
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
Last revised on February 15, 2021 at 01:12:25. See the history of this page for a list of all contributions to it.