# Sublattice symmetry vs Particle hole symmetry

Sublattice symmetry and particle hole symmetry generally constrain a system's energy spectrum to be symmetric with respect to fermi level. My understanding is that they are both represented by an operator with the property $OHO^{-1}=-H$. For sublattice symmetry, $O$ is unitary and linear while for particle hole symmetry, $O$ is antinunitary and antililear. Here I have two questions:

1. Sometimes in literature, if a system's energy spectrum is symmetric with respect to fermi level, it is said to have particle hole symmetry. However, according to my understanding, this can also be attributed to sublattice symmetry. Does the term "particle hole symmetry" have some generalizations here?

2. I have never seen particle hole symmetry(in my definition) in a condensed matter system other than superconductors. Can anyone offer me an example?

Thanks!

• Regarding 2, particle-hole symmetry in general is approximate in real systems. In graphene near the Dirac point within the nearest neighbour tight binding regime we have particle hole symmetry. But adding further hoppings breaks this symmetry. The symmetry used often depends on the scale at which the phenomenon occurs and to what approximation one is okay with Commented Feb 7 at 18:52

Particle hole symmetry is charge conjugation symmetry. Charge conjugation implies $$C H^* C = -H$$ .Sublattice symmetry is Chiral symmetry. Chiral symmetry implies What you are thinking, namely $$O H O^\dagger = -H$$ with the additional fact that $$O^2 = 1$$.