People often talk about particle-hole symmetry in solid state physics. What are the exact definition and physics picture of particle-hole symmetry? How to define the density of particles and holes?
You find particle hole symmetry (PHS) for example in superconductors, where you can take for the Hamiltonian the Bogoliubov-de Gennes (BdG) Hamiltonian as a mean-field approximation. This is the only experimental example I know. There are maybe other systems with particle hole symmetry that I don't know about, but I will use superconductors as an example in my explanation.
If you have a superconductor with N electrons in them (some electrons form Cooper pairs and some electrons are unpaired), the BdG Hamiltonian is of size $2N\times2N$ and the state of the system is written as a vector that consists of $N$ annihilation operators for electrons an $N$ creation operators for electrons above each other with the annihilation operators first. You can view the bottom creation operators for the electrons as annihilation operators for holes. You then have a complete vector of annihilation operators of length $2N$. Holes do the exact opposite of electrons and therefore you expect that the energy levels are complete symmetric with respect to the Fermi level. While you have $N$ electrons in the system, there are $2N$ energy levels which form pairs.
More technical now:
For superconductors the particle hole symmetry operator has the form $P = \tau_x K$, where $\tau_x$ is a Pauli matrix and $K$ means complex conjugation. The Hamiltonian then satisfies the following equation: $$P^\dagger H P = H \quad\Rightarrow\quad \tau_x H^* \tau_x = H$$ and $P^2$ is the identity matrix. Other Hamiltonians can also have particle hole symmetry and the PHS operator may then different, but is always squares to $1$ or $-1$.