Topological Physics – Phenomena in physics controlled by the topology (often: the homotopy theory) of the physical system.
General theory:
In electromagnetism:
Symmetry Protected Trivial order (SPT order) (also known as Symmetry Protected Topological order) is a new kind of order in zero-temperature states of matter that have a symmetry and a finite energy gap. The SPT order has the following defining properties:
Using the notion of quantum entanglement, we can say that SPT states are short-range entangled states with a symmetry.
Using the notion of topological order, we can say that SPT states are symmetric states with trivial topological order.
The first example of SPT order is the Haldane phase of spin-1 chain. It is a SPT phase protected by the $SO(3)$ spin rotation group symmetry. Another example of SPT order is the topological insulator of non-interacting fermions, a SPT phase protected by U(1) and time reversal symmetry.
Recently, it was shown that the bosonic SPT orders are described by group cohomology theory: d+1D SPT states with on-site symmetry G are labeled by the elements in group cohomology class $H^{d+1} [G, U(1)]$. It was also shown that the fermionic SPT orders are described by group super-cohomology theory.
So the group (super-)cohomology theory may allow us to classify all SPT orders even for interacting systems, which include interacting topological insulator/superconductor.
Free fermion system can also have non-trivial SPT phases, such as topological insulators and topological superconductors. Those free fermion SPT phases are classified by K-theory.
Related entries: TQFT, topological order, group cohomology, entanglement
Discussion via higher dimensional WZW models is in
Daniel S. Freed, Gregory W. Moore, Twisted equivariant matter, Annales Henri Poincaré December 2013, Volume 14, Issue 8, pp 1927–2023 arxiv/1208.5055 (uses equivariant K-theory)
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.)
Last revised on February 15, 2021 at 01:14:20. See the history of this page for a list of all contributions to it.