Symmetry Classification Preview

Modern symmetry classification asks a structural question:

Given the symmetries and anti-symmetries of a Hamiltonian, what kinds of spectra, degeneracies, and topological phases are possible?

This page is a preview. It explains the vocabulary that connects ordinary quantum-mechanical discrete symmetries to the tenfold Altland–Zirnbauer classification used for noninteracting fermion systems and Bogoliubov–de Gennes Hamiltonians. Detailed classification by spatial dimension, topology, interactions, and crystalline symmetries belongs to quantum matter and field-theory treatments.

After resolving ordinary unitary symmetries and tracking how they interact with antiunitary operations, three kinds of constraints organize the problem:

• time reversal;
• particle–hole structure;
• chiral, or sublattice, symmetry.

Their presence or absence, and the signs of their squares, determine the ten Altland–Zirnbauer symmetry classes.

Classification Versus a Symmetry List

A list of symmetries says which transformations preserve a particular Hamiltonian. A classification scheme goes further: it groups Hamiltonians by the algebraic constraints those transformations impose.

For a Hamiltonian $H$, an ordinary unitary symmetry satisfies

$$UHU^{-1}=H.$$

If $U$ commutes with $H$, one can often decompose the Hilbert space into $U$ eigenspaces and study each block separately. One must also check whether an antiunitary operation preserves a block or exchanges it with another. For example, physical time reversal exchanges spin-up and spin-down blocks when $S_z$ is conserved; it need not be an internal symmetry of either block separately. The class assigned to an irreducible block uses the transformations actually acting within it.

This is why the classification is not just a longer list of parity, rotations, translations, and spin symmetries. It organizes the remaining internal antiunitary and spectral constraints. These operations are fixed by the physical model or allowed family of Hamiltonians. Diagonalizing one matrix and inventing complex conjugation in its eigenbasis would make every Hermitian matrix look “real,” but would not establish a physical time-reversal symmetry of the family.

Time Reversal

Time reversal is represented by an antiunitary operator, here denoted $\mathcal T$. For a Bloch or momentum-space Hamiltonian, the symmetry condition is

$$\mathcal T H(\mathbf k)\mathcal T^{-1} = H(-\mathbf k).$$

For a finite Hamiltonian with no momentum label, the same idea is simply

$$\mathcal T H\mathcal T^{-1}=H.$$

The square of time reversal matters:

$$\mathcal T^2=+I \qquad \text{or} \qquad \mathcal T^2=-I.$$

Spinless time reversal often has $\mathcal T^2=+I$. A single spin-$1/2$ degree of freedom has $\mathcal T^2=-I$, which leads to Kramers degeneracy when the Hamiltonian is time-reversal invariant.

In band theory, $\mathcal T$ maps a state at $\mathbf k$ to a state at $-\mathbf k$. At momenta equivalent to their negatives modulo a reciprocal lattice vector, the $\mathcal T^2=-I$ case forces Kramers pairs in the same momentum sector.

Particle–Hole Structure

Particle–hole symmetry in the Altland–Zirnbauer sense is represented by an antiunitary operator $\mathcal C$ satisfying

$$\mathcal C H(\mathbf k)\mathcal C^{-1} = -H(-\mathbf k).$$

For a finite Hamiltonian, this reads

$$\mathcal C H\mathcal C^{-1} = -H.$$

This is not the same as ordinary charge conservation. In many superconducting applications, $\mathcal C$ is a redundancy created by writing a Bogoliubov–de Gennes Hamiltonian in a Nambu basis containing both particle and hole variables. It guarantees a spectrum symmetric about the chosen zero of energy:

$$E \quad\longleftrightarrow\quad -E.$$

Like time reversal, particle–hole structure has two possible square signs in the tenfold classification:

$$\mathcal C^2=+I \qquad \text{or} \qquad \mathcal C^2=-I.$$

The sign depends on the representation and on the physical degrees of freedom included in the block.

Chiral Symmetry

Chiral symmetry, also called sublattice symmetry in many tight-binding models, is a unitary transformation $\mathcal S$ that anticommutes with the Hamiltonian:

$$\mathcal S H(\mathbf k)\mathcal S^{-1} = -H(\mathbf k).$$

We choose the standard normalization $\mathcal S^2=I$, so $\mathcal S$ is also Hermitian. Equivalently,

$$\{\mathcal S,H\}=0.$$

This also gives a spectrum symmetric around zero energy. If

$$H\lvert\psi\rangle=E\lvert\psi\rangle,$$

then

$$H\mathcal S\lvert\psi\rangle = -E\,\mathcal S\lvert\psi\rangle.$$

A simple source of chiral symmetry is a bipartite hopping Hamiltonian with no same-sublattice hopping. In a basis split into two sublattices, it has the block form

$$H = \begin{pmatrix} 0&A\\ A^\dagger&0 \end{pmatrix}, \qquad \mathcal S = \begin{pmatrix} I&0\\ 0&-I \end{pmatrix}.$$

Then $\mathcal S H\mathcal S^{-1}=-H$. The SSH chain is a standard reference model for this structure.

When both $\mathcal T$ and $\mathcal C$ are present, their product gives a chiral symmetry:

$$\mathcal S = \mathcal T\mathcal C,$$

up to an overall phase chosen to normalize $\mathcal S^2=I$, provided both operations act within the same block. Conversely, chiral symmetry together with either $\mathcal T$ or $\mathcal C$ supplies the other. This relation is why their presences are not three independent yes/no choices.

The Ten Classes

The Altland–Zirnbauer classes are labeled by whether $\mathcal T$, $\mathcal C$, and $\mathcal S$ are absent or present. For antiunitary $\mathcal T$ and $\mathcal C$, the table records the square as $+1$ or $-1$. A $0$ means the symmetry is absent in that class.

Class	$\mathcal T^2$	$\mathcal C^2$	$\mathcal S$	Typical description
A	$0$	$0$	$0$	no time reversal, no particle–hole, no chiral symmetry
AIII	$0$	$0$	$1$	chiral unitary class
AI	$+1$	$0$	$0$	spinless time reversal
BDI	$+1$	$+1$	$1$	time reversal and particle–hole with positive squares
D	$0$	$+1$	$0$	particle–hole structure, common for idealized superconductors
DIII	$-1$	$+1$	$1$	spinful time reversal plus particle–hole
AII	$-1$	$0$	$0$	spinful time reversal, Kramers structure
CII	$-1$	$-1$	$1$	time reversal and particle–hole with negative squares
C	$0$	$-1$	$0$	particle–hole with negative square
CI	$+1$	$-1$	$1$	time reversal positive, particle–hole negative

This table is not yet a table of topological phases. It is a table of symmetry constraints. To get topological classifications one must also specify spatial dimension, locality assumptions, whether interactions are included, and whether crystalline symmetries are being used.

Topological Phase Preview

In gapped noninteracting systems, a Hamiltonian is often classified up to continuous deformations that preserve:

• the energy gap;
• locality or smoothness in momentum space;
• the symmetry class.

If two Hamiltonians cannot be connected without closing the gap or breaking the symmetry, they belong to different topological phases.

Examples include:

• class A in two dimensions, where a Chern number can classify integer quantum Hall bands;
• class AII topological insulators in two dimensions, where time reversal with $\mathcal T^2=-I$ leads to the quantum spin Hall pattern;
• class D in one dimension, where the Kitaev chain is the standard superconducting model;
• chiral classes where winding numbers can appear in one-dimensional models such as the SSH chain.

The invariant is not determined by the class name alone. Dimension matters. The same symmetry class can have different possible invariants in different spatial dimensions.

Worked Two-Band Chain

With lattice spacing set to one and real hoppings $t_1,t_2$, the SSH Bloch Hamiltonian is

$$H(k)=(t_1+t_2\cos k)\sigma_x+t_2\sin k\,\sigma_y.$$

Although its matrix entries are generally complex, $H(k)^*=H(-k)$. It has $\mathcal T=K$, $\mathcal S=\sigma_z$, and $\mathcal C=\sigma_zK$, with $\mathcal T^2=\mathcal C^2=+I$. Thus this real-hopping model is BDI, not AIII. A model with only the chiral constraint and no antiunitary symmetry would be AIII. The class describes which constraints are retained when parameters are varied, including allowed perturbations.

The two energies are $E_\pm(k)=\pm|t_1+t_2e^{ik}|$. This same BDI family contains different winding numbers: the circle traced by $t_1+t_2e^{ik}$ encloses the origin when $|t_2|>|t_1|$ and does not when $|t_1|>|t_2|$. The gap closes at equality. The class name alone therefore does not decide the phase.

The usual periodic table uses a stable notion of equivalence, allowing addition of trivial bands as well as gap-preserving deformations. Fixed small band numbers, additional spatial symmetries, and interactions can require refinements. Continue with Topological Insulators or Topological Superconductors for gapped applications, Random Matrix Theory for spectral ensembles, and Symmetry-Protected Topological Phases for interacting systems.

Common Mistakes

• Treating particle–hole symmetry in a Bogoliubov–de Gennes Hamiltonian as ordinary charge conservation.
• Forgetting that $\mathcal T$ and $\mathcal C$ are antiunitary, so they conjugate complex coefficients.
• Calling a model class AII just because it has spin; the time-reversal operator must satisfy $\mathcal T^2=-I$ and preserve the Hamiltonian.
• Ignoring ordinary unitary symmetries that should be block-diagonalized before applying the tenfold classification.
• Assuming the tenfold way includes crystalline symmetries. It does not; crystalline symmetries refine or extend it.
• Thinking an Altland–Zirnbauer class by itself determines a topological invariant without specifying dimension and gap assumptions.

Exercises

1. Classify a spinless real Hamiltonian.

Suppose $H$ is a finite-dimensional real symmetric Hamiltonian with no particle–hole or chiral symmetry. Taking $\mathcal T=K$, which Altland–Zirnbauer class is suggested?

Solution

Complex conjugation satisfies

$$KHK^{-1}=H$$

for a real Hamiltonian, and

$$K^2=+I.$$

There is no particle–hole or chiral symmetry by assumption. The class is therefore AI.

2. Identify the time-reversal class for a spinful band.

A spin-$1/2$ Bloch Hamiltonian satisfies

$$\mathcal T H(\mathbf k)\mathcal T^{-1} = H(-\mathbf k), \qquad \mathcal T^2=-I,$$

and has no particle–hole or chiral symmetry. Which class is it?

Solution

The only Altland–Zirnbauer symmetry is time reversal with negative square:

$$\mathcal T^2=-I.$$

With $\mathcal C=0$ and $\mathcal S=0$, the class is AII.

3. Check chiral symmetry in a bipartite model.

For

$$H = \begin{pmatrix} 0&A\\ A^\dagger&0 \end{pmatrix}, \qquad \mathcal S = \begin{pmatrix} I&0\\ 0&-I \end{pmatrix},$$

show that $\mathcal S H\mathcal S^{-1}=-H$.

Solution

Since $\mathcal S^{-1}=\mathcal S$, multiply the matrices:

$$\mathcal S H\mathcal S = \begin{pmatrix} I&0\\ 0&-I \end{pmatrix} \begin{pmatrix} 0&A\\ A^\dagger&0 \end{pmatrix} \begin{pmatrix} I&0\\ 0&-I \end{pmatrix}.$$

The left multiplication changes the sign of the lower block row, and the right multiplication changes the sign of the right block column. The result is

$$\begin{pmatrix} 0&-A\\ -A^\dagger&0 \end{pmatrix} = -H.$$

4. Explain why particle–hole symmetry pairs energies.

If $\mathcal C H\mathcal C^{-1}=-H$ and $H\lvert\psi\rangle=E\lvert\psi\rangle$, show that $\mathcal C\lvert\psi\rangle$ has energy $-E$.

Solution

From $\mathcal C H\mathcal C^{-1}=-H$, one has

$$H\mathcal C = -\mathcal C H.$$

Apply this to $\lvert\psi\rangle$:

$$H\mathcal C\lvert\psi\rangle = -\mathcal C H\lvert\psi\rangle = -\mathcal C(E\lvert\psi\rangle).$$

Because $\mathcal C$ is antiunitary, the scalar is conjugated:

$$\mathcal C(E\lvert\psi\rangle) = E^*\mathcal C\lvert\psi\rangle.$$

For a self-adjoint Hamiltonian $E$ is real, so

$$H\mathcal C\lvert\psi\rangle = -E\,\mathcal C\lvert\psi\rangle.$$

Thus the spectrum is paired about zero energy.

References

• A. Altland and M. R. Zirnbauer, “Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures,” Physical Review B 55, 1142–1161, 1997. DOI.
• C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, “Classification of topological quantum matter with symmetries,” Reviews of Modern Physics 88, 035005, 2016.
• M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Reviews of Modern Physics 82, 3045–3067, 2010.
• A. Kitaev, “Periodic table for topological insulators and superconductors,” AIP Conference Proceedings 1134, 22–30, 2009.
• X.-L. Qi and S.-C. Zhang, “Topological insulators and superconductors,” Reviews of Modern Physics 83, 1057–1110, 2011.
• S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, “Topological insulators and superconductors: tenfold way and dimensional hierarchy,” New Journal of Physics 12, 065010, 2010. Author manuscript.
• A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, “Classification of topological insulators and superconductors in three spatial dimensions,” Physical Review B 78, 195125, 2008.