It isn't true that in a non-Hermitian system the expectation value of any operator changes with time.

That the Hamiltonian isn't Hermitian doesn't imply that the evolution operator isn't unitary-and that's what ensures that the system is consistently closed. And it is that property that ensures that transformations that are symmetries at one time instant, remain symmetries.

The number operator of certain excitations can commute with the Hamiltonian-which expresses the fact that this quantity is conserved by the evolution; and the number operator of other quantities can fail to commute with the Hamiltonian-which expresses the fact that this quantity isn't conserved. The former describe symmetries of the system, the latter don't. They describe creation and annihilation of excitations, which leads to the questions of the rate with which these are created and annihilated.

So the physical reason is the symmetries of the Hamiltonian, that is the operators, functions of the dynamical variables, that commute with it. And it's the unitarity of the evolution operator that implies that if an operator commutes with it at one time, it will commute at all times.

Otherwise there isn't any physics to be described, since the mathematical description is incomplete.

What's important to realize is that it is the symmetries that are the starting point of any such discussion. The reason is that they're the foundation for ensuring that the description is complete, therefore sensible, or incomplete and, therefore, must be completed before any useful statement can be made.

(Incidentally the reason it is useful to consider anti-Hermitian operators is that they describe extended objects.)

Stam Nicolis

I don't get your answer. Take an example.

The following Hamiltonian is the Hatano-Nelson model without disorder term. Here, tL=te-g and tR=teg, and t and g both are real. This antisymmetric hopping makes the Hamiltonian non-Hermitian. Now, if we take the time evolution and we calculate the expectation value of the number operator for an eigenstate for different times, we will get different values. That is expected from quantum mechanics. Change in the expectation value of the number operator means the number of particles is changing with time. My question is, what is the physical reason for the change in the value of the number of particles in such kind of a system?

That the system isn't closed. (To define the Hamiltonian it's, of course, necessary to state what the boundary conditions are, incidentally, so that what's an eigenstate can be defined at all.)

For the moment, though, the fundamental issue is that the problem isn't fully defined at all-that's the real answer and the so-called time dependence doesn't mean anything by itself.

First of all, antisymmetric hopping means that tL=-tR, not that tL=1/tR.

Next, the expectation value assumes a definition of a scalar product, under which the eigenvectors have well-defined properties. If boundary conditions aren't specified, the calculation doesn't make sense.

These are part of the definition of the problem; it's possible to get any answer if the definition of the scalar product and of the boundary conditions is changed.

There's no point in writing some ``Hamiltonian'' and then asking questions; the symmetries of the Hamiltonian come first. And what matters, once more, isn't so much, whether the Hamiltonian is Hermitian, but whether the evolution operator, U=exp(-iHt), is unitary. IF H is Hermitian, THEN U is unitary. But U can be unitary, even if H isn't Hermitian. If U isn't unitary, the description of the system is incomplete and nothing useful can be deduced about its properties. One reason is that the expectation values don't make sense, since the probability measure they're supposed to be the expectation values of isn't well-defined.

Indeed the antisymmetry is due to not writing down the correct, Hermitian, Hamiltonian, that includes a Z2 valued gauge field in the hopping term. The fact that the occupancy number changes is due to the interaction of the particles with this external field.

Stam Nicolis

Thank you. Now, I get it.

In the paper, it is written that the system is in an imaginary gauge field. Have you any idea about the experimental realization of an imaginary gauge field? I have not much idea.