After learning the Kitaev model, I tried to reformulate it and encounter some conceptual loopholes of my own.
Here the setting:
Given the 1-D chain Hamiltonian (differed from original form proposed by Kitaev, here we slightly modify coefficient and the pairing terms with anticommutator $\{c^\dagger_{j+1},c^\dagger_j\}=0$ in order to get a more simple BdG Hamiltonian):
$$
H=\sum_{j=1}^{N}\bigg[\frac{-t}{2}\big(c^\dagger_{j+1}c_j+c^\dagger_{j}c_{j+1}\big)-\mu c^\dagger_{j} c_j+\frac{\Delta_0}{4}\big(c^\dagger_{j+1} c^\dagger_j-c^\dagger_j c^\dagger_{j+1} +\text{h.c.}\big)\bigg],
$$
where $t$ is a hopping coupling, $\mu$ is the chemical potential, and $\Delta_0$ an induced pairing gap function that we by convention chose as real and positive. One can perform a lattice Fourier transform with
$$
c^\dagger_{j}=\frac{1}{\sqrt{V}}\sum_ke^{ikR_j}c^\dagger_{k},
$$
where $R_j=aj$ and $a$ is the lattice constant set to one for simplicity. Thus,
$$
H=\sum_k\bigg[-t\cos k c^\dagger_{k}c_k-\mu c^\dagger_k c_k+\frac{\Delta_0}{4}\big(e^{ik}c^\dagger_{k}c^\dagger_{-k}-e^{-ik} c^\dagger_{k} c^\dagger_{-k}+\text{h.c.}\big) \bigg]\\
=\frac{1}{2}\sum_k \begin{pmatrix}
c^\dagger_{k} &c_{-k}
\end{pmatrix}\begin{pmatrix}
-t\cos k-\mu&i\Delta_0\sin k\\
-i\Delta_0\sin k &t\cos k+\mu
\end{pmatrix}\begin{pmatrix}
c_k\\c^\dagger_{-k}
\end{pmatrix}+\frac{1}{2}\sum_t \cos k +\mu;
$$
the above gives the energy spectrum
$$
E_\pm(k)=\pm\sqrt{\big(t\cos k +\mu\big)^2+\Delta_0^2\sin^2 k}.
$$
To see the topological features of this theory, let's rewrite the BdG Hamiltonian (matrix) into Dirac Hamiltonian $\mathsf{H}(k)=\mathbf{d} (k) \cdot \boldsymbol{\sigma}$ with
$$
\textbf{d}(k)=-\big(0,\Delta_0\sin k, t\cos k+\mu \big),
$$
running in a $(\sigma_y,\sigma_z)$-plane $\mathbb{R}^2$ which is regard as the space of Hamiltonian as $k$ sweeps over the entire BZ. The trajectory of $\textbf{d}$ is an ellipse centered at $d_z=-\mu$. Together with fact that the energy spectrum of this parametric theory provides the clues for finding physically distinct phases and one can see, for $\Delta_0\neq0$, the gap closes at
$$
t\cos k +\mu=\sin k=0,
$$
which is valid at "$k=0$ with $\mu=-t$" and "$k=\pi$ with $\mu=t$".
So far so good, here is the point where weirdness starts:
Therefore, for $k$ goes from $0$ to $2\pi$, there are three phases:
-
For $0<t<\mu$, $\Delta>0$ the origin is outside the loop which is a trivial phase.
-
For $\mu=0, t>0$ the origin is inside the loop which is topological and has winding number 1.
- For $\mu=0, t<0$ the origin is still inside the loop, but the winding number is -1.
Hmm…I get 1 trivial phase like Kitaev model, but 2 non-trivial phases corresponding to winding number $\pm 1$!
But normally (see p198 of Bernevig and Hughes's book) we say that Kitaev chain has the topological number/invariant of $\mathbb{Z}_2$, i.e. $0,1$ (mod $2$). So $w=1$ and $w=-1$ would be the same phase. That is, one can transform from one to the other without closing the gap(, which is true and I know it.)
It is known that, in general, topological classifications are done by finding an integer number called topological invariant. Thus, naively I would expect different topological numbers would correspond to distinct topological classes (or phases.)
My question would be:
Where did I do wrong? And how should I explain the topological number $w=\pm1$ I got correspond to the same phase, while the general statement is that—we can classify distinct topological phases with these topological numbers?
My main references are:
[1]Bernevig and Hughes's book.
[2]Kitaev's original paper.
*I've been searching for relevant questions and answers on SE for a while and didn't find much. If there is any worth digging, please let me know :p
Best Answer
You are correct in observing that $w=+1$ and $w=-1$ are labeling the same topological phase of the Majorana chain. This paradox is resolved by understanding that the winding number $w$ is not the correct topological index for the Majorana chain. In this paper https://arxiv.org/pdf/1403.4938.pdf, it was shown that different winding numbers can correspond to the same topological phase of the Majorana chain, so there is indeed a "degeneracy" in the winding number.
There are at least two reasons why the winding number fails to serve as a topological index of the fermionic symmetry protected topological (SPT) phases:
The winding number, as calculated from the free fermion topological band theory, is not expected to work once the fermion interaction is introduced. However the fermionic SPT phases are in general interacting, so one should not expect the winding number to label the SPT phases correctly.
Even if we restrict ourselves to the free fermion SPT phases, the winding number is still not a good idea, because it is only well defined in the presence of a specific type of the time-reversal symmetry in the symmetry class BDI, where the time-reversal $\mathcal{T}$ act on the fermion as $\mathcal{T}^2=+1$. (As a side comment, we must also understand that there are many versions of Majorana chains depending on the symmetry class. For example, the BDI class has a $\mathbb{Z}_8$ classification, while the D and DIII classes have $\mathbb{Z}_2$ classifications. So it is also wrong to say that the Majorana chain has a $\mathbb{Z}_2$ topological index without specifying the symmetry class in the first place.)
The correct classification of Majorana chains is either based on the K-theory in the free fermion limit (for more discussions, see Topological insulators: why K-theory classification rather than homotopy classification?), or based on the $\mathbb{Z}_2$-graded group cohomology theory (see Fidkowski-Kitaev https://arxiv.org/pdf/1008.4138.pdf) or the spin cobordism theory (see https://arxiv.org/pdf/1406.7329.pdf) for the interacting case. In any case, the winding number $w\in\mathbb{Z}$ (or the homotopy theory) is never used as a topological index to label/classify the SPT phases. Just think that we don't even have a $\mathbb{Z}$-classified Majorana chain, so how can the $\mathbb{Z}$-valued winding number possibly be correct?
But loosely speaking, it is still fine to think that the topological index is "related" to the winding number modulo some integer. For example, for the class DIII Majorana chain, the $\mathbb{Z}_2$ topological index $\nu_\text{DIII}$ is related to the winding number $w$ by $\nu_\text{DIII}=w\text{ mod }2$, such that $w=+1$ and $w=-1$ are labeling the same topological phase, which is uniquely identified by the correct topological index $\nu_\text{DIII}=1$ without the $\pm1$ discrepancy.