Can anyone explain how to apply Bogolyubov-de Gennes equations to solve the band structure of the Kitaev model Unpaired Majorana fermions? I kind of understand the Bogolyubov-de Gennes approach that solves the problem but still have problems with mathematical/physical calculations that give the band structure E vs q. Thanks!
1 Answer
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Maybe you should take a look at Kitaev's original paper, where all these questions are clearly addressed.
You do not need to apply the BdG equation to solve the band structure, you can just work with Majorana fermions and transform to the momentum space to obtain the band structure. Starting with the real space Hamiltonian $$H=\sum_{i}(\mathrm{i}t_1\chi_{iA}\chi_{iB}+\mathrm{i}t_2\chi_{iB}\chi_{i+1,A}).$$ Pluging in the definition of momentum space fermion operator $\chi_{iX}=\sum_{k}\chi_{kX}e^{\mathrm{i}ki}$ (for $X=A,B$), one obtains $$H=\frac{\mathrm{i}}{2}\sum_{k}\chi_{-k}^\intercal h(k)\chi_{k},\quad h(k)=\left[\begin{matrix}0&t_1-t_2 e^{-\mathrm{i}k}\\-t_1+t_2 e^{\mathrm{i}k}&0\end{matrix}\right].$$ Diagonalizing the matrix $h(k)$, one finds the band structure of the fermion $$E(k)=\pm|t_1-t_2 e^{-\mathrm{i}k}|,$$ which is fully gapped when $t_1\neq t_2$. The topological-trivial transition happens at $t_1=t_2$ where the bulk gap closes.
To see that there is indeed a unpaired Majorana zero mode on the each edge of the open chain, one can use the transfer matrix method. The Schrodinger equation for zero-mode is given by $h_{ab}\chi_b=0$, where $h_{ab}$ is the real-space single-particle Hamiltonian in its matrix form and the indices $a,b$ labels both unit-cell and sublattice. The Schrodinger equation implies the following relation $$\left[\begin{matrix}\chi_{i+1,A}\\\chi_{i+1,B}\end{matrix}\right]=\left[\begin{matrix}t_1/t_2&0\\0&t_2/t_1\end{matrix}\right]\left[\begin{matrix}\chi_{iA}\\\chi_{iB}\end{matrix}\right].$$ So if $|t_1/t_2|<1$, then the mode wave function starting from the left-edge A-sublattice will decay exponentially with a localization length $\xi=-1/\ln(t_1/t_2)$. So this explicitly constructs the localized Majorana zero mode on the edge, thereby proving its existance.
