# Why do we use the anticommutation relation for particle-hole and chiral symmetries?

In physics we say that a quantity is conserved, if its operator commutes with Hamiltonian.

For example, in condensed matter systems, when the momentum $k$ commutes with the Hamiltonian $H$ as $[H,k]=0$, we say that it is a conserved quantity.

Now we take the time reversal symmetry operator $T$. When it commutes with our Hamiltonian as $[H,T]=0$, we say that time reversal symmetry is conserved for our system.

However, if we take the Particle hole symmetry $\mathcal{P}$ and Chiral symmetry operator $\mathcal{C}$, where they anti-commute with the Hamiltonian $\{H,\mathcal{P}\}=0,\{H,\mathcal{C}\}=0$, we say that particle-hole and chiral symmetries are conserved.

What I really don't understand is why we use the anti-commutation relation and not commutation relation to find whether particle-hole and chiral symmetries are conserved or not.

## 2 Answers

First, we do not say that the "time reversal symmetry is conserved". Momentum is conserved because it is the generator of a continuous transformation, the translations. Discrete symmetries like time reversal, which do not generate a continuous transformation, do not induce "conservation laws" in the usual sense of the word. For one, there is no Noether current for them, cf. this question and its answers.

A chiral "symmetry", i.e. one that anti-commutes with the Hamiltonian, is not a symmetry in the strict sense of, well, commuting with the Hamiltonian. Such chiral "symmetries" are not associated with conserved quantities (because, again, they do not commute with the Hamiltonian, which is what being conserved means).

Nevertheless, a chiral "symmetry" is useful because it implies things about the spectrum of the Hamiltonian. For instance, from $HC+CH=0$, you can directly show that non-zero eigenvalues of the Hamiltonian come in pairs: If $\psi_n$ is an eigenstate for eigenvalue $n$, then $C\psi_n$ is an eigenstate for $-n$.

Both the particle-hole operator and chiral operator do commute with the fully second quantized Hamiltonian. It is only for the single-particle Hamiltonian that they turn out to be anti-commuting.

To confirm this claim please look up.

• Topological insulators and superconductors: ten-fold way and dimensional hierarchy - Shinsei Ryu, Andreas Schnyder, Akira Furusaki, Andreas Ludwig
• Classification of topological quantum matter with symmetries - Ching-Kai Chiu, Jeffrey C.Y. Teo, Andreas P. Schnyder, Shinsei Ryu

These are the best two references which actually touch upon this point. It seems to be looked over or ignored (maybe on purpose) in most other places.