Symmetry Classes and the Free-Fermion Periodic Table

The Altland–Zirnbauer table classifies gapped quadratic fermion Hamiltonians with internal time-reversal, particle–hole, and chiral constraints. Its answer depends on spatial dimension $d$, the squares of the antiunitary operations, and stable equivalence under addition of trivial bands. It is not a classification of crystalline phases, interacting phases, intrinsic topological order, or gapless nodes.

Required background. Time-reversal topological insulators supplies a concrete Z2 example; antiunitary symmetries supplies the distinction between linear and antiunitary actions.

The three spectral constraints

After shifting the Fermi energy to zero, the single-particle or Bogoliubov–de Gennes Hamiltonian can obey

$$T H(\mathbf k)T^{-1}=H(-\mathbf k),\quad T^2=\pm1,$$ $$C H(\mathbf k)C^{-1}=-H(-\mathbf k),\quad C^2=\pm1,$$ $$S H(\mathbf k)S^{-1}=-H(\mathbf k),\quad S=TC.$$

Here $T$ and $C$ are antiunitary, while $S$ is unitary. In a BdG problem, $C$ is the Nambu doubling constraint rather than an independently acting physical symmetry. Confusing it with charge conjugation of a number-conserving system changes the class.

Spectral flattening replaces a gapped $H$ by $Q=\operatorname{sgn}H$ without closing the gap, so $Q^2=1$. Classification becomes a homotopy problem for the space of symmetry-compatible flattened matrices. Adding decoupled positive- and negative-energy orbitals implements stable equivalence; Bott periodicity then gives period two for complex classes and period eight for real classes Kitaev 2009, pp. 22–30.

Dimensions zero through three

The following reduced stable groups use spatial dimension $d$ and the conventional K-theory generator. In spinful superconducting conventions an integer generator may be reported as $2\mathbb Z$ because the minimal physical block carries two units.

Class	$T^2$	$C^2$	$S$	$d=0$	$d=1$	$d=2$	$d=3$
A	–	–	–	$\mathbb Z$	$0$	$\mathbb Z$	$0$
AIII	–	–	$1$	$0$	$\mathbb Z$	$0$	$\mathbb Z$
AI	$+1$	–	–	$\mathbb Z$	$0$	$0$	$0$
BDI	$+1$	$+1$	$1$	$\mathbb Z_2$	$\mathbb Z$	$0$	$0$
D	–	$+1$	–	$\mathbb Z_2$	$\mathbb Z_2$	$\mathbb Z$	$0$
DIII	$-1$	$+1$	$1$	$0$	$\mathbb Z_2$	$\mathbb Z_2$	$\mathbb Z$
AII	$-1$	–	–	$\mathbb Z$	$0$	$\mathbb Z_2$	$\mathbb Z_2$
CII	$-1$	$-1$	$1$	$0$	$\mathbb Z$	$0$	$\mathbb Z_2$
C	–	$-1$	–	$0$	$0$	$\mathbb Z$	$0$
CI	$+1$	$-1$	$1$	$0$	$0$	$0$	$\mathbb Z$

For example, class A in $d=2$ gives the Chern integer; class AII in $d=2,3$ gives the strong Z2 index; class D in $d=1$ gives the Majorana-chain Z2 invariant. The same class in a different dimension can have a different answer.

What the table does not assert

The table assumes a free-fermion or mean-field BdG description and a spectral or mobility gap. Disorder-compatible formulations exist without translation symmetry, but crystalline indices require separate spatial-symmetry data. Interactions may reduce an integer classification, identify phases distinct in the free limit, or introduce phases with no band representative. In particular, one-dimensional BDI reduces from $\mathbb Z$ to $\mathbb Z_8$ for symmetry-preserving interacting Majorana chains Fidkowski and Kitaev 2011.

Exercise

A two-dimensional BdG Hamiltonian has particle–hole symmetry with $C^2=+1$ and neither $T$ nor $S$. Identify its class and stable group.

Solution

The symmetry data define class D. At spatial dimension $d=2$ the table gives $\mathbb Z$, represented by the BdG Chern number and the net chiral Majorana edge content.

References

• Lukasz Fidkowski and Alexei Kitaev, “Topological Phases of Fermions in One Dimension,” Physical Review B 83 (2011) 075103, doi:10.1103/PhysRevB.83.075103.
• Alexei Kitaev, “Periodic Table for Topological Insulators and Superconductors,” AIP Conference Proceedings 1134 (2009) 22–30, doi:10.1063/1.3149495.