Why hopping term and overlap intergral always real in tight-binding model? I'm learning tight-binding model of polyacetylene (-CH- chain) and get confused by the off-diagonal matrix elements of Hamiltonian and overlap matrix.

For example, in all materials I can find, they all set $H_{AB}= \langle \psi_A|H|\psi_B\rangle = t$ where $t$ is a real number. But why? In general, I think t should be some complex number, I can't see why it has to be real. This also happend to the overlap matrix, again they set $S_{AB} = \langle \psi_A|\psi_B\rangle = s$ where s is real. Again, I think it can be some complex number.
This is Chapter 2 from Satio's <Physical Properties of Carbon Nanotubes> for example, and any other books are similar. Could anyone tell me why? Thank you very much.


Other materials for example:

*

*https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Introduction_to_Nanoelectronics_(Baldo)/06%3A_The_Electronic_Structure_of_Materials/6.05%3A_The_Tight_Binding_Approximation

*https://thiscondensedlife.files.wordpress.com/2016/01/polyacetylene.pdf
 A: The typical reasons to be able to take the off diagonal matrix elements
real are 1) The basis states can be taken to be real and the interaction
terms are real; 2) You have a tridiagonal Hermitian matrix. It looks
like you have both reasons.
In tight binding unless there are things like angular momentum dependent
pseudo potentials, the atomic Hamiltonians are real, so you can choose
their eigenstates to be real.  That is, if an eigenstate is complex,
it's real and imaginary parts separately, if they are different, are
eigenstates with the same eigenvalue.  Therefore you can choose the
basis real and if the interaction Hamiltonian is coulomb, it is also
real. Similarly the overlaps will be real.  This makes the matrix real
symmetric and its eigenvectors can be chosen to be real.
In general, if you have a Hermitian tridiagonal matrix, you can choose
the phases of the basis states so that it is real symmetric. This corresponds
to a diagonal unitary transformation.
It's easy to imagine cases with hopping around loops and with applied
magnetic fields, where you would need to use complex matrix elements,
but it appears that your cases are not those.
