Chiral transformation of lattice fermionic operators Consider the Hamiltonian of interacting particles on a lattice, e.g. the Hubbard Hamiltonian
$$
H = -t \sum_{i\sigma} ( c_{i,\sigma}^{\dagger} c_{i+1,\sigma} + \mathrm{h.c.} ) + U\sum_i n_{i\uparrow} n_{i\downarrow},
$$
where $c_{i\sigma}^{\dagger}$ creates a particle with spin $\sigma$ on the lattice site $i$.
I have heard about chiral transformation, which is a combination of particle-hole and time-reversal transformations, and I wonder how this Hamiltonian transforms under the action of the chiral transformation.
I suppose that it all boils down to understand how the operator $c_{i\sigma}^{\dagger}$ transforms under the action of this chiral transformation, which is actually not very clear to me after some research.
So my question is: how do fermionic operators on the real space of a lattice transform under the action of the chiral transformation?
To be clear, I know that for example the particle-hole transformation reads
$$
c_{i\sigma} \to c_{i\sigma}^{\dagger}; c_{i\sigma}^{\dagger} \to c_{i\sigma}
$$
and I would like to see a similar mapping for the action of a chiral transformation.
Thanks in advance for any help!
 A: It would be a combination of particle-hole and time-reversal, as you've pointed out, which switches creation and annihilation operators and is anti-unitary, sending $i \rightarrow -i$.  An example would be the Hubbard model on a bipartite lattice at half-filling:
$$
H = \sum_{R,\alpha,R',\beta,\sigma} -t_{\alpha,\beta} c^\dagger_{R,\alpha,\sigma} c_{R',\beta,\sigma} + U \sum_{R,\alpha} (n_{R,\alpha,\uparrow}-\frac{1}{2}) (n_{R,\alpha,\downarrow}-\frac{1}{2}).
$$ Here $R, R'$ denote unit cells, $\alpha, \beta$ the orbitals within a unit cell, and $n$ the density operators.  The bipartite lattice means that there is a way to partition the orbitals $\alpha$ into two sets, call them $L, {\tilde L}$, such that there is hopping only between the sublattices and no hopping within each sublattice (for example, the Lieb lattice).  Then the bipartite chiral symmetry reads
$$
c_{R,\alpha,\sigma} \rightarrow \epsilon(\alpha) c^\dagger_{R,\alpha,\sigma},
$$ where $\epsilon(\alpha)$ is $\pm 1$ depending on whether $\alpha$ is in $L$ or $\tilde L$.
