Kitaev chain

In condensed matter physics, the Kitaev chain is a one-dimensional lattice model for superconductors, with the particularly that it features Majorana bound states. The Kitaev chain have been used as a toy model of semiconductor nanowires for the development of topological quantum computers. The model was first proposed by Alexei Kitaev in 2000.

Description

Hamiltonian

The tight binding Hamiltonian in of a Kitaev chain considers a lattice with spinless particles with a even number of sites N is given by:

$H=-\mu \sum _{j=1}^{N}\left(c_{j}^{\dagger }c_{j}-{\frac {1}{2}}\right)+\sum _{j=1}^{N-1}\left[-t\left(c_{j+1}^{\dagger }c_{j}+c_{j}^{\dagger }c_{j+1}\right)+|\Delta |\left(c_{j+1}^{\dagger }c_{j}^{\dagger }+c_{j}c_{j+1}\right)\right]$

where $\mu$ is the chemical potential, $c_{j}^{\dagger },c_{j}$ are Creation and annihilation operators, $t>0$ the energy needed for a particle to hop from one location of the lattice to another, $\Delta =|\Delta |e^{i\theta }$ is the induced superconducting gap and $\theta$ is the coherent superconducting phase.

The Hamiltonian can be rewritten using Majorana operators, given by

$\left\{{\begin{matrix}\gamma _{j}^{A}=c_{j}^{\dagger }+c_{j}\\\gamma _{j}^{B}=i(c_{j}^{\dagger }-c_{j})\end{matrix}}\right.$,

which can be thought as the real and imaginary parts of the creation operator $c_{j}={\tfrac {1}{2}}(\gamma _{j}^{A}+i\gamma _{j}^{B})$. These Majorana operator are Hermitian operators, and anticommutate,

$\{\gamma _{j}^{A},\gamma _{k}^{B}\}=2\delta _{jk}$.

Using these operators the Hamiltonian can be rewritten as

$H=-{\frac {i\mu }{2}}\sum _{j=1}^{N}\gamma _{j}^{B}\gamma _{j}^{A}+{\frac {i}{2}}\sum _{j=1}^{N-1}\left(\omega _{+}\gamma _{j}^{B}\gamma _{j+1}^{A}+\omega _{-}\gamma _{j+1}^{B}\gamma _{j}^{A}\right)$

where $\omega _{\pm }=|\Delta |\pm t$.

Trivial phase

In the limit $t=|\Delta |\to 0$, we obtain the following Hamiltonian

$H=-{\frac {i\mu }{2}}\sum _{j=1}^{N}\gamma _{j}^{B}\gamma _{j}^{A}$

where the Majorana operators are coupled on the same site. This condition is considered a topologically trivial phase.

Non-trivial phase

In the limit $\mu \to 0$ and $|\Delta |\to t$, we obtain the following Hamiltonian

$H_{\rm {M}}=it\sum _{j=1}^{N-1}\gamma _{j}^{B}\gamma _{j+1}^{A}$

where every Majorana operator is coupled to a Majorana operator of a different kind in the next site. By assigning a new fermion operator ${\tilde {c}}_{j}={\tfrac {1}{2}}(\gamma _{j}^{B}+i\gamma _{j+1}^{A})$, the Hamiltonian is diagonalized, as

$H_{\rm {M}}=2t\sum _{j=1}^{N-1}\left({\tilde {c}}_{j}^{\dagger }{\tilde {c}}_{j}+{\frac {1}{2}}\right)$

which describes a new set of N-1 Bogoliubov quasiparticles with energy t. The missing mode given by ${\tilde {c}}_{\rm {M}}={\tfrac {1}{2}}(\gamma _{N}^{B}+i\gamma _{1}^{A})$ couples the Majorana operators from the two endpoints of the chain, as this mode does not appear in the Hamiltonian, it requires zero energy. This mode is called a Majorana zero mode and highly delocalized. As the presence of this mode does not change the total energy, the ground state is two-fold degenerate. This condition is a topological non-trivial phase.