The table was developed between 2008–2010[1] by the collaboration of Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki and Andreas W. W. Ludwig;[2][3] and independently by Alexei Kitaev.
The table is no longer valid when interactions are included.
[1] The topological insulators and superconductors are classified here in ten symmetry classes (A,AII,AI,BDI,D,DIII,AII,CII,C,CI) named after Altland–Zirnbauer classification, defined here by the properties of the system with respect to three operators: the time-reversal operator
The symmetry classes are ordered according to the Bott clock (see below) so that the same values repeat in the diagonals.
[5] The dimension indicates the dimensionality of the systes: 1D (chain), 2D (plane) and 3D lattices.
It can be extended up to any number of positive integer dimension.
Below, there can be four possible group values that are tabulated for a given class and dimension:[5] The non-chiral Su–Schrieffer–Heeger model (
The aforementioned discrete symmetries label 10 distinct discrete symmetry classes, which coincide with the Altland–Zirnbauer classes of random matrices.
A bulk Hamiltonian in a particular symmetry group is restricted to be a Hermitian matrix with no zero-energy eigenvalues (i.e. so that the spectrum is "gapped" and the system is a bulk insulator) satisfying the symmetry constraints of the group.
in the Brillouin zone; then the symmetry constraints must hold for all
while maintaining the symmetry constraint and gap (that is, there exists continuous function
the Hamiltonian has no zero eigenvalue and symmetry condition is maintained, and
in this case, physically if two materials with bulk Hamiltonians
Each equivalence class can be labeled by a topological invariant; two Hamiltonians whose topological invariant are different cannot be deformed into each other and belong to different equivalence classes.
negative eigenvalues deformed to -1; the resulting such matrices are described by the union of real Grassmannians
The strong topological invariants of a many-band system in
These groups are displayed in this table, called the periodic table of topological insulators: There may also exist weak topological invariants (associated to the fact that the suspension of the Brillouin zone is in fact equivalent to a
Furthermore, the table assumes the limit of an infinite number of bands, i.e. involves
The table also is periodic in the sense that the group of invariants in
In the case of no ant-iunitary symmetries, the invariant groups are periodic in dimension by 2.
The periodic table also displays a peculiar property: the invariant groups in
One can also imagine arranging each of the eight real symmetry classes on the Cartesian plane such that the
dimensions for a certain real symmetry class is the same as the invariant group in
dimensions for the symmetry class directly one space clockwise.
[1][10] The Bott clock can be understood by considering the problem of Clifford algebra extensions.
[1] Near an interface between two inequivalent bulk materials, the Hamiltonian approaches a gap closing.
is an added "mass term" that and anticommutes with the rest of the Hamiltonian and vanishes at the interface (thus giving the interface a gapless edge mode at
term for the Hamiltonian on one side of the interface cannot be continuously deformed into the
be an arbitrary positive scalar) the problem of classifying topological invariants reduces to the problem of classifying all possible inequivalent choices of
to extend the Clifford algebra to one higher dimension, while maintaining the symmetry constraints.