In condensed matter physics, the Kitaev chain is a simplified model for a topological superconductor. It models a one dimensional lattice featuring Majorana bound states. The Kitaev chain have been used as a toy model of semiconductor nanowires for the development of topological quantum computers.[1] [2] The model was first proposed by Alexei Kitaev in 2000.[3]
The tight binding Hamiltonian in of a Kitaev chain considers a one dimensional lattice with N site and spinless particles at zero temperature, subjected to nearest neighbour hoping interactions, given in second quantization formalism as[4]
| N | |
| H=-\mu\sum | |
| j=1 |
| \dagger | |
| \left(c | |
| j |
| N-1 | ||||
| c | ||||
|
| \dagger | |
| \left[-t\left(c | |
| j+1 |
cj+c
| \dagger | |
| j |
cj+1
| \dagger | |
| \right)+|\Delta|\left(c | |
| j+1 |
| \dagger+c | |
| c | |
| j |
cj+1\right)\right]
\mu
| \dagger,c | |
| c | |
| j |
t\geq0
\Delta=|\Delta|ei\theta
\theta
The Hamiltonian can be rewritten using Majorana operators, given by
| A | |
| \left\{ \begin{matrix} \gamma | |
| j=c |
| B | |
| j=i(c |
| \dagger-c | |
| j) \end{matrix} \right. |
| B | |
| c | |
| j) |
| A | |
| \{\gamma | |
| j,\gamma |
| B\}=2 | |
| k |
\deltajk
| H=- | i\mu |
| 2 |
| N | |
| \sum | |
| j=1 |
| B | |
| \gamma | |
| j |
| A | ||||
| \gamma | ||||
|
| N-1 | |
| \sum | |
| j=1 |
| A | |
| \left(\omega | |
| j+1 |
| B | |
| +\omega | |
| j+1 |
| A | |
| \gamma | |
| j\right) |
\omega\pm=|\Delta|\pmt
In the limit
t=|\Delta|\to0
| H=- | i\mu |
| 2 |
| N | |
| \sum | |
| j=1 |
| B | |
| \gamma | |
| j |
| A | |
| \gamma | |
| j |
In the limit
\mu\to0
|\Delta|\tot
H\rm
| N-1 | |
| =it\sum | |
| j=1 |
| A | |
| \gamma | |
| j+1 |
| A | |
| \tilde{c} | |
| j+1 |
)
H\rm
| N-1 | |
| =2t\sum | |
| j=1 |
| \dagger | |
| \left(\tilde{c} | |
| j |
| \tilde{c} | ||||
|
\right)
\tilde{c}\rm
| A | |
| =\tfrac{1}{2}(\gamma | |
| 1 |
)
The existence of the Majorana zero mode is topologically protected from small perturbation due to symmetry considerations. For the Kitaev chain the Majorana zero mode persist as long as
\mu<2t
Using Bogoliubov-de Gennes formalism it can be shown that for the bulk case (infinite number of sites), that the energy yields
E(k)=\pm\sqrt{(2t\cosk+\mu)2+4|\Delta|2\sin2k}
\mu=2t
k=0
Q=sign\left\{pf[iH(k=0)]pf[iH(k=\pi)]\right\}
pf[x]
\mu>2t
Q=1
\mu<2t
Q=-1
One possible realization of Kitaev chains is using semiconductor nanowires with strong spin–orbit interaction to break spin-degeneracy,[8] like indium antimonide or indium arsenide.[9] A magnetic field can be applied to induce Zeeman coupling to spin polarize the wire and break Kramers degeneracy. The superconducting gap can be induced using Andreev reflection, by putting the wire in the proximity to a superconductor. Realizations using 3D topological insulators have also been proposed.
There is no single definitive way to test for Majorana zero modes. One proposal to experimentally observe these modes is using scanning tunneling microscopy. A zero bias peak in the conductance could be the signature of a topological phase. Josephson effect between two wires in superconducting phase could also help to demonstrate these modes.
In 2023 QuTech team from Delft University of Technology reported the realization of a "poor man's" Majorana that is a Majorana bound state that is not topologically protected and therefore only stable for a very small range of parameters. It was obtained in a Kitaev chain consisting of two quantum dots in a superconducting nanowire strongly coupled by normal tunneling and Andreev tunneling with the state arising when the rate of both. Some researches have raised concerns, suggesting that an alternative mechanism to that of Majorana bound states might explain the data obtained.