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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!

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  • $\begingroup$ 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 $\endgroup$ Commented Feb 7 at 18:52

2 Answers 2

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@1, Sublattice symmetry also goes by a different name: Chiral symmetry, but I've not heard of it referred to as a particle-hole symmetry before, I think wherever you came across this, might have been mistaken, I'd need more context to answer fully.

The best(simple) introduction to the three fundamental symmetries (Chiral, Time and True Particle-hole symmetry) in terms of the symmetry of some energy spectrum can be found here: https://topocondmat.org/w1_topointro/0d.html

@2, There are different kinds of particle hole-symmetry. The one you refer to in superconductors arises from doubling the degrees of freedom of your Hamiltonian in the Bogoliubov approach, where the mean field description of the quasi-holes and quasi-electrons are bound together by the superconducting coupling. In that case the particle-hole symmetry exists between the "cooper-pair" or if you prefer the composite boson. Where as in a single particle description of a normal conductor you may have particle-hole symmetry between the quasi-electrons and quasi-holes, themselves, a particle-hole symmetry acting on the fermionic operators.

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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 $O^2 = 1$.

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