I am studying some properties of the one dimensional Kitaev chain, which has the following form:
$ H = -\mu \sum_n c_n^\dagger c_n - t \sum_n (c_{n+1}^\dagger c_n + h.c.) + \Delta \sum_n (c_n c_{n+1} +h.c.)$
Note that $ \mu, t, \Delta$ are taken as real number. Without further specification, this model could belong to two different classes. If no further symmetry other than particle hole symmetry is present ($\mathcal{P} = \tau_x \mathcal{K}$), the symmetry class would be D class with Z2 topological invariant. If time reversal symmetry is present ($\mathcal{T} = \mathcal{K}$), this model belongs to BDI class with Z topological invariant.
Since the Z2 topological invariant in D class would be robust to time reversal symmetry(TRS) breaking terms, I am trying to extend the original model . There are certainly many ways to do this, e.g. introduce several one dimensional Kitaev Chains with additional coupling terms of majorana fermions between the different chains. However I want to stay within the two band systems. Two apparent ways to break TRS would be to allow pairing and hopping terms to take complex numbers. Now the complex phase associated with hopping would be accounted for by a magnetic flux. I am wondering what physical meaning would a complex pairing term has.
Also, are there other ways to break TRS in the Kitaec chain other than the methods that I have mentioned?