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
