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 formula_1, (where formula_2, and formula_3, are the annihilation and creation operators of mode formula_4, in some arbitrary spatial basis) : time reversal symmetry, particle hole (or charge conjugation) symmetry, and chiral (or sublattice) symmetry.
Chiral symmetry is a unitary operator formula_5, that acts on formula_2, as a unitary rotation (formula_7,) and satisfies formula_8. A Hamiltonian formula_9 possesses chiral symmetry when formula_10, for some choice of formula_5 (on the level of first-quantised Hamiltonians, this means formula_12 and formula_9 are anticommuting matrices).
Time reversal is an antiunitary operator formula_14, that acts on formula_15, (where formula_16, is an arbitrary complex coefficient, and formula_17, denotes complex conjugation) as formula_18. It can be written as formula_19 where formula_20 is the complex conjugation operator and formula_21 is a unitary matrix. Either formula_22 or formula_23. A Hamiltonian with time reversal symmetry satisfies formula_24, or on the level of first-quantised matrices, formula_25, for some choice of formula_21.
Charge conjugation formula_27 is also an antiunitary operator which acts on formula_15 as formula_29, and can be written as formula_30 where formula_31 is unitary. Again either formula_32 or formula_33 depending on what formula_31 is. A Hamiltonian with particle hole symmetry satisfies formula_35, or on the level of first-quantised Hamiltonian matrices, formula_36, for some choice of formula_31.
In the Bloch Hamiltonian formalism for periodic crystals, where the Hamiltonian formula_38 acts on modes of crystal momentum formula_39, the chiral symmetry, TRS, and PHS conditions become formula_40, formula_41 and formula_42.
It is evident that if two of these three symmetries are present, then the third is also present, due to the relation formula_43.
The aforementioned discrete symmetries label 10 distinct discrete symmetry classes, which coincide with the Altland–Zirnbauer classes of random matrices.
Equivalence classes of Hamiltonians.
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 case of formula_44 dimensions, this Hamiltonian is a continuous function formula_38 of the formula_46 parameters in the Bloch momentum vector formula_47 in the Brillouin zone; then the symmetry constraints must hold for all formula_48.
Given two Hamiltonians formula_49 and formula_50, it may be possible to continuously deform formula_49 into formula_50 while maintaining the symmetry constraint and gap (that is, there exists continuous function formula_53 such that for all formula_54 the Hamiltonian has no zero eigenvalue and symmetry condition is maintained, and formula_55 and formula_56). Then we say that formula_49 and formula_50 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 formula_49 and formula_50 respectively neighbour each other with an edge between them, when one continuously moves across the edge one must encounter a zero eigenvalue (as there is no continuous transformation that avoids this). 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:
For example, a (real symmetric) Hamiltonian in symmetry class AI can have its formula_61 positive eigenvalues deformed to +1 and its formula_62 negative eigenvalues deformed to -1; the resulting such matrices are described by the union of real Grassmannians formula_63
Classification of invariants.
The strong topological invariants of a many-band system in formula_46 dimensions can be labeled by the elements of the formula_46-th homotopy group of the symmetric space. 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 formula_66 sphere wedged with lower-dimensional spheres), which are not included in this table. Furthermore, the table assumes the limit of an infinite number of bands, i.e. involves formula_67 Hamiltonians for formula_68.
The table also is periodic in the sense that the group of invariants in formula_46 dimensions is the same as the group of invariants in formula_70 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 formula_46 dimensions are identical to those in formula_72 dimensions but in a different symmetry class. Among the complex symmetry classes, the invariant group for A in formula_46 dimensions is the same as that for AIII in formula_72 dimensions, and vice versa. One can also imagine arranging each of the eight real symmetry classes on the Cartesian plane such that the formula_75 coordinate is formula_76 if time reversal symmetry is present and formula_77 if it is absent, and the formula_78 coordinate is formula_79 if particle hole symmetry is present and formula_77 if it is absent. Then the invariant group in formula_46 dimensions for a certain real symmetry class is the same as the invariant group in formula_72 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 formula_83. Here, formula_84 are a representation of the Clifford Algebra formula_85, while formula_86 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 formula_87). The formula_86 term for the Hamiltonian on one side of the interface cannot be continuously deformed into the formula_86 term for the Hamiltonian on the other side of the interface. Thus (letting formula_90 be an arbitrary positive scalar) the problem of classifying topological invariants reduces to the problem of classifying all possible inequivalent choices of formula_91 to extend the Clifford algebra to one higher dimension, while maintaining the symmetry constraints.