Kitaev chain

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]

Description

Hamiltonian

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]

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

where ${\displaystyle \mu }$ is the chemical potential, ${\displaystyle c_{j}^{\dagger },c_{j}}$ are creation and annihilation operators, ${\displaystyle t\geq 0}$ the energy needed for a particle to hop from one location of the lattice to another, ${\displaystyle \Delta =|\Delta |e^{i\theta }}$ is the induced superconducting gap (p-wave pairing) and ${\displaystyle \theta }$ is the coherent superconducting phase. This Hamiltonian has particle-hole symmetry, as well as time reversal symmetry. ^[5]

The Hamiltonian can be rewritten using Majorana operators, given by ^[4]

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

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

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

Using these operators the Hamiltonian can be rewritten as ^[4]

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

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

Trivial phase

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

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

where the Majorana operators are coupled on the same site. This condition is considered a topologically trivial phase. ^[5]

Non-trivial phase

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

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

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

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

which describes a new set of N-1 Bogoliubov quasiparticles with energy t. The missing mode given by ${\displaystyle {\tilde {c}}_{\rm {M}}={\tfrac {1}{2}}(\gamma _{N}^{B}+i\gamma _{1}^{A})}$ 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 is highly delocalized. As the presence of this mode does not change the total energy, the ground state is two-fold degenerate. ^[4] This condition is a topological superconducting non-trivial phase. ^[5]

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 ${\displaystyle \mu <2t}$ and the superconducting gap is finite. ^[6] The robustness of these modes makes it a candidate for qubits as a basis for topological quantum computer. ^[7]

Bulk case

Using Bogoliubov-de Gennes formalism it can be shown that for the bulk case (infinite number of sites), that the energy yields ^[6]

${\displaystyle E(k)=\pm {\sqrt {(2t\cos k+\mu )^{2}+4|\Delta |^{2}\sin ^{2}k}}}$,

and it is gapped, except for the case ${\displaystyle \mu =2t}$ and wave vector ${\displaystyle k=0}$. For the bulk case there are no zero modes. However a topological invariant exists given by

${\displaystyle Q=\mathrm {sign} \left\{\mathrm {pf} [iH(k=0)]\mathrm {pf} [iH(k=\pi )]\right\}}$,

where ${\displaystyle \mathrm {pf} [x]}$ is the Pfaffian operation. For ${\displaystyle \mu >2t}$, the invariant is strictly ${\displaystyle Q=1}$ and for ${\displaystyle \mu <2t}$, ${\displaystyle Q=-1}$ corresponding to the trivial and non-trivial phases from the finite chain, respectively. This relation between the topological invariant from the bulk case and the existence of Majorana zero modes in the finite case is called a bulk-edge correspondence. ^[6]

Experimental efforts

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. ^[8] The superconducting gap can be induced using Andreev reflection, by putting the wire in the proximity to a superconductor. ^{[8] [9]} Realizations using 3D topological insulators have also been proposed. ^[9]

There is no single definitive way to test for Majorana zero modes. One proposal to experimentally observe these modes is using scanning tunneling microscopy. ^[9] A zero bias peak in the conductance could be the signature of a topological phase. ^[9] Josephson effect between two wires in superconducting phase could also help to demonstrate these modes. ^[9]

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. ^{[1] [2]} 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. ^{[1] [2]} Some researches have raised concerns, suggesting that an alternative mechanism to that of Majorana bound states might explain the data obtained. ^{[2] [7]}

See also

• Su–Schrieffer–Heeger chain

References

1. Wright, Katherine (2023-02-15). "Evidence Found for a Majorana "Cousin"". Physics. 16: 24.
2. Author, No (2024). "'Poor man's Majoranas' offer testbed for studying possible qubits". Physics World. Retrieved 2024-09-10.{{cite web}}: |last= has generic name (help)
3. Kitaev, A Yu (2001-10-01). "Unpaired Majorana fermions in quantum wires". Physics-Uspekhi. 44 (10S): 131–136. arXiv: cond-mat/0010440 . doi:10.1070/1063-7869/44/10S/S29. ISSN   1468-4780.
4. Schäpers, Thomas (2021-05-10). Semiconductor Spintronics. Walter de Gruyter GmbH & Co KG. ISBN   978-3-11-063900-1.
5. Stanescu, Tudor D. (2024-07-02). Introduction to Topological Quantum Matter & Quantum Computation. CRC Press. ISBN   978-1-040-04198-7.
6. Topology course team (2021). "Bulk-edge correspondence in the Kitaev chain". Online course on topology in condensed matter - Delft University of Technology.
7. Ball, Philip (2021-09-29). "Major Quantum Computing Strategy Suffers Serious Setbacks". Quanta Magazine. Retrieved 2024-09-10.
8. Topology course team (2021). "From Kitaev chain to a nanowire". Online course on topology in condensed matter – University of Delft.
9. Chen, Fei; Matern, Stephanie (2014). "Kitaev Chain" (PDF). Oberseminar: Quantum Knots - Prof. Dr A. Rosch, Prof. Dr. S. Trebst - University of Cologne.