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?


1 Answer 1


The main question is about the meaning of a complex-valued superconducting gap $\Delta = |\Delta| e^{i\theta}$. However, note that one can simply redefine [1] the fermions $c_n \to e^{-i \theta/2} c_n$, then $$\sum_n \left( \Delta c_n c_{n+1} + \Delta^* c_{n+1}^\dagger c_{n}^\dagger \right) \to |\Delta | \sum_n \left( c_n c_{n+1} + h.c. \right).$$ There is hence no real significance to taking a complex-valued $\Delta$ (at least if it is position-independent!).

With regard to other ways of breaking TRS: in the spinless case, TRS coincides with complex-conjugation, and since the fermions are real, the only (and hence most general) way of breaking it is by introducing complex-valued coupling constants.

As an aside: you mention that stacking chains would spoil the two-band picture: this is not necessarily true, since one can rewrite a stack of chains as a translation-invariant system by interlacing the chains into a single wire. We used this (very basic) trick in a work of ours: https://arxiv.org/abs/1709.03508 (see e.g. the picture for $H_2$ in Fig.1).

EDIT: things get more subtle when one has different ranges: $\sum_n \left( \Delta_1 c_n c_{n+1} + \Delta_2 c_n c_{n+2} + h.c. \right)$; one can use the above trick to make $\Delta_1$ real without loss of generality, but then one is stuck with the complex value of $\Delta_2$. I have not yet thought about its physical significance.

  • $\begingroup$ I see. Things would also get subtle if you have two hamiltonians with $\Delta_1$ and $\Delta_2$ having different phases and you quench $H_1$ with $H_2$, right? Because you can only set one phase to be zero. $\endgroup$
    – Kai Su
    Commented Sep 14, 2019 at 18:34

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