Majorana fermions

where to find them and how to use them

Tami Pereg-Barnea (McGill)

INRTIQ fall meeting, October 11-12 2022

Predicted in 1938

• Majorana found a real solution to the Dirac equation

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• Real solutions represent a particle which is its own antiparticle due to particle-hole symmetry in the equation

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• To this date there is no confirmation of such elementary particle

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Majorana Fermions�

Fermion

Majorana

.. 60+ years later … Kitaev

• Unpaired Majoranas
• One can write any fermion as a sum of two Majoranans
• Is it useful?
• Only if we can make them spatially separate

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• The Kitaev chain, p-wave superconductor

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Kitaev Chain

Spinful model

Two regimes

• If is dominated by the Zeeman field �= topological superconductor.

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• Otherwise – trivial superconductor

Projection on bands

• In the occupied band the spin winds

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• Project on the occupied band

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• Due to momentum-spin locking the projection gives an odd function of momentum

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• P-wave pairing, triplet pairing

Majoranas in 2D

• Triplet pairing is key
• Majorana branches on the perimeter of the sample
• Majorana zero modes at the vortex cores

Abelian? Non-abelian!

• Majoranas are fermions since:

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• But

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• However, a system with many Majoranas is degenerate and the exchange of any two switches states

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Alicea RMP 2012

Counting degeneracy

• A Majorana mode is neither full nor empty.
• But a fermion made of two Majoranas could be filled
• A many body system with N Majoranans has N/2 fermions
• Each fermion is either full or empty -> Degeneracy

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• An exchange of two Majoranas is given by the operator
• Acting with this on any state doesn’t change the energy but �might change the state while preserving the parity

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Majorana box qubit

• 4 Majoranas
• 2- fermions - 4 states (0,0) (1,1)

(1,0) (0,1)��

• Coulomb blockade – preserved pairity – only 2 states

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• Universal computing, topological protection, scalability

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Stephan Plugge et al 2017 New J. Phys. 19 012001

Majorana evidence

Zero bias peaks Quantized conductance

Other systems

Iron on lead, CrBr3 on NbSe2 – Yazdani group

Vortex Majorana - heterostructures

Iron based superconductors

Hong Ding et al.

LiFeAs 

Smoking gun?

• Zero bias peaks are not enough
• Need to see localization
• Magnetic field dependence
• Majorana statistics is key
• Josephson effect – pair tunneling produces 2π periodic signal

- Single electron tunneling gives 4π periodicity

Quasiparticle poisoning

• Andreev bound states may allow single particle tunneling
• Use a tunable device such as

Manipulation: Moving Majoranas

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Dynamics of Majorana-based qubits operated with an array of tunable gates�Bela Bauer et al.�SciPost Phys. 5, 004 (2018)

Diabatic error while moving Majoranas�Bill Truong, Kartiek Agarwal, TPB�Unpublished

Manipulation: Exchange

Universal Geometric Path to a Robust Majorana Magic Gate

Karzig, et al.

Phys. Rev. X 6, 031019 (2016)

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New directions in the pursuit of Majorana

fermions in solid state systems

Jason Alicea, RMP 2012

Manipulation: Floquet Majoranas

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Manipulation: Floquet

Jiang et al. PRL (2011)

Kundu and Seradjeh, PRL (2014)

Reducing error

• Draiding

• Implement using a quantum dot

Double Braiding Majoranas for Quantum Computing and Hamiltonian Engineering

Ivar Martin and Kartiek Agarwal

PRX Quantum 1, 020324 (2020)

Dynamical approach to improving Majorana qubits and distinguishing them from trivial bound states �Min, Fajardo, TPB, Agarwal, PRB 105, 155412 (2022)

Future directions

• Smoking gun experiments
• Qubits and gates
• Scalability
• Disorder, interactions