Periodic table of topological invariants
The periodic table of topological invariants is an application of topology to physics. It indicates the group of topological invariant for topological insulators and superconductors in each dimension and in each discrete symmetry class.
Discrete symmetry classes
There are ten discrete symmetry classes of topological insulators and superconductors, corresponding to the ten Altland–Zirnbauer classes of random matrices. They are defined by three symmetries of the Hamiltonian, : time reversal symmetry, particle hole symmetry, and chiral symmetry.Chiral symmetry is a unitary operator, that acts on, as a unitary rotation and satisfies,. A Hamiltonian possesses chiral symmetry when, for some choice of .
Time reversal is an antiunitary operator, that acts on, as,. It can be written as where is the complex conjugation operator and is a unitary matrix. Either or. A Hamiltonian with time reversal symmetry satisfies, or on the level of first-quantised matrices,, for some choice of.
Charge conjugation is also an antiunitary operator which acts on as, and can be written as where is unitary. Again either or depending on what is. A Hamiltonian with particle hole symmetry satisfies, or on the level of first-quantised Hamiltonian matrices,, for some choice of.
In the Bloch Hamiltonian formalism for periodic crystals, where the Hamiltonian acts on modes of crystal momentum, the chiral symmetry, TRS, and PHS conditions become, and.
It is evident that if two of these three symmetries are present, then the third is also present, due to the relation.
The aforementioned discrete symmetries label 10 distinct discrete symmetry classes, which coincide with the Altland–Zirnbauer classes of random matrices.
| Symmetry Class | Time reversal symmetry | Particle hole symmetry | Chiral symmetry |
| A | No | No | No |
| AIII | No | No | Yes |
| AI | Yes, | No | No |
| BDI | Yes, | Yes, | Yes |
| D | No | Yes, | No |
| DIII | Yes, | Yes, | Yes |
| AII | Yes, | No | No |
| CII | Yes, | Yes, | Yes |
| C | No | Yes, | No |
| CI | Yes, | Yes, | Yes |
Equivalence classes of Hamiltonians
A bulk Hamiltonian in a particular symmetry group is restricted to be a Hermitian matrix with no zero-energy eigenvalues satisfying the symmetry constraints of the group. In the case of dimensions, this Hamiltonian is a continuous function of the parameters in the Bloch momentum vector in the Brillouin zone; then the symmetry constraints must hold for all.Given two Hamiltonians and, it may be possible to continuously deform into while maintaining the symmetry constraint and gap. Then we say that and are equivalent.
However, it may also turn out that there is no such continuous deformation. in this case, physically if two materials with bulk Hamiltonians and respectively neighbour each other with an edge between them, when one continuously moves across the edge one must encounter a zero eigenvalue. This may manifest as a gapless zero energy edge mode or an electric current that only flows along the edge.
An interesting question is to ask, given a symmetry class and a dimension of the Brillouin zone, what are all the equivalence classes of 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.
Classifying spaces of Hamiltonians
For each of the symmetry classes, the question can be simplified by deforming the Hamiltonian into a "projective" Hamiltonian, and considering the symmetric space in which such Hamiltonians live. These classifying spaces are shown for each symmetry class:| Symmetry Class | Classifying Space | of Classifying Space |
| A | ||
| AIII | ||
| AI | ||
| BDI | ||
| D | ||
| DIII | ||
| AII | ||
| CII | ||
| C | ||
| CI |
For example, a Hamiltonian in symmetry class AI can have its positive eigenvalues deformed to +1 and its negative eigenvalues deformed to -1; the resulting such matrices are described by the union of real Grassmannians
Classification of invariants
The strong topological invariants of a many-band system in dimensions can be labeled by the elements of the -th homotopy group of the symmetric space. These groups are displayed in this table, called the periodic table of topological insulators:| Symmetry Class | |||||||||
| A | |||||||||
| AIII | |||||||||
| AI | |||||||||
| BDI | |||||||||
| D | |||||||||
| DIII | |||||||||
| AII | |||||||||
| CII | |||||||||
| C | |||||||||
| CI |
There may also exist weak topological invariants, which are not included in this table. Furthermore, the table assumes the limit of an infinite number of bands, i.e. involves Hamiltonians for.
The table also is periodic in the sense that the group of invariants in dimensions is the same as the group of invariants in dimensions. In the case of no antiunitary symmetries, the invariant groups are periodic in dimension by 2.
For nontrivial symmetry classes, the actual invariant can be defined by one of the following integrals over all or part of the Brillouin zone: the Chern number, the Wess Zumino winding number, the Chern–Simons invariant, the Fu–Kane invariant.
Dimensional reduction and Bott Clock
The periodic table also displays a peculiar property: the invariant groups in dimensions are identical to those in dimensions but in a different symmetry class. Among the complex symmetry classes, the invariant group for A in dimensions is the same as that for AIII in dimensions, and vice versa. One can also imagine arranging each of the eight real symmetry classes on the Cartesian plane such that the coordinate is if time reversal symmetry is present and if it is absent, and the coordinate is if particle hole symmetry is present and if it is absent. Then the invariant group in dimensions for a certain real symmetry class is the same as the invariant group in dimensions for the symmetry class directly one space clockwise. This phenomenon was termed the "Bott Clock" by Alexei Kitaev, in reference to the Bott periodicity theorem.The Bott Clock can be understood by considering the problem of Clifford algebra extensions. Near an interface between two inequivalent bulk materials, the Hamiltonian approaches a gap closing. To lowest order expansion in momentum slightly away from the gap closing, the Hamiltonian takes the form of a Dirac Hamiltonian. Here, are a representation of the Clifford Algebra, while is an added "mass term" that and anticommutes with the rest of the Hamiltonian and vanishes at the interface. The term for the Hamiltonian on one side of the interface cannot be continuously deformed into the term for the Hamiltonian on the other side of the interface. Thus 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.