Post on 13-Jul-2020
Mariana Malard
SYNTHESIZING MAJORANA ZERO-ENERGY MODES IN A PERIODICALLY GATED QUANTUM WIRE
University of Gothenburgmariana.malard@physics.gu.se
Universidade de Brasília mmalard@unb.br
What is a Majorana zero-energy mode (MZM)?
Majorana zero-energy modes
Majorana quasi-particle with non-abelian statistics.
What is a Majorana zero-energy mode (MZM)?
Majorana zero-energy modes
A Majorana particle is a fermion which is
its own antiparticle.
What is a Majorana zero-energy mode (MZM)?
Majorana zero-energy modes
In solid state systems, MZM’s appear as in-gap states
at zero energy.
A Majorana particle is a fermion which is
its own antiparticle.
What is it good for?
Majorana zero-energy modes
braiding MZM’s
Low-decoherence, fault-tolerant topological quantum computer
Majorana zero-energy modes
Where should I look for MZM’s?
Fractional quantum Hall systems
Majorana zero-energy modes
1D and 2D p-wave (topological) superconductors
Fractional quantum Hall systems
Where should I look for MZM’s?
Optically trapped cold fermions
Fractional quantum Hall systems
1D and 2D p-wave (topological) superconductors
Majorana zero-energy modes
Where should I look for MZM’s?
Majorana zero-energy modes
Optically trapped cold fermions
Quantum dots
Carbon nanotubes
Fractional quantum Hall systems
1D and 2D p-wave (topological) superconductors
Where should I look for MZM’s?
How can these completely different systems all be topological superconductors?
Majorana zero-energy modes
How can these completely different systems all be topological superconductors?
Majorana zero-energy modes
Hilbert spaces with the same topology.
Topology is the unifying feature.
For p-wave pairing, the crucial ingredient is the spinless character of the superconducting pairing.
Recall the Kitaev chain!
1,1 N,2
Majorana zero-energy modes
spin-orbit coupled quantum wire + magnetic field ++ s-wave superconductor
J.D Sau, R. M. Lutchyn, S. Tewari, and S. D. Sarma, Phys. Rev. Lett. 104, 040502 (2010).Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett 105, 177002 (2010). J. Alicea, Phys. Rev. B 81, 125318 (2010).
One-dimensional topological superconductors
helical stateS
p
S
p
spin-orbit coupled quantum wire + magnetic field ++ s-wave superconductor
J.D Sau, R. M. Lutchyn, S. Tewari, and S. D. Sarma, Phys. Rev. Lett. 104, 040502 (2010).Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett 105, 177002 (2010). J. Alicea, Phys. Rev. B 81, 125318 (2010).
One-dimensional topological superconductors
spin-orbit coupled quantum wire + magnetic field ++ s-wave superconductor
J.D Sau, R. M. Lutchyn, S. Tewari, and S. D. Sarma, Phys. Rev. Lett. 104, 040502 (2010).Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett 105, 177002 (2010). J. Alicea, Phys. Rev. B 81, 125318 (2010).
helical stateS
p
S
p
s-wave pairing
One-dimensional topological superconductors
s-wave pairinghelical state
But the magnetic field creates some problems…
S
p
S
p
One-dimensional topological superconductors
Canting of the spins.
Reduced robustness against disorder.
B might be slow and hard do apply locally. B breaks time reversal symmetry explicitly!
One-dimensional topological superconductors
s-wave pairinghelical stateS
p
S
p
Doing without the magnetic field…
Proposal of an one-dimensional topological superconductor
Henrik Johannesson
University of Gothenburg
Giorgi (Gia) Japaridze
Andronikashvili Institute of Physics
o Proposal of a magnetic field-free 1D topological superconductor
o Microscopic model
o Helical liquid
o From the helical liquid to a 1D topological superconductor
o Effective low-energy bosonized theory and RG analysis
o Phase diagram and experimental regimes
o Final remarks
Outline
o Proposal of a magnetic field-free 1D topological superconductor
o Microscopic model
o Helical liquid
o From the helical liquid to a 1D topological superconductor
o Effective low-energy bosonized theory and RG analysis
o Phase diagram and experimental regimes
o Final remarks
Outline
Proposal of an one-dimensional topological superconductor
Replace the magnetic field by a spatially modulated electric field!
s-wave superconductor
top gates
quantum wire
o Proposal of a magnetic field-free 1D topological superconductor
o Microscopic model
o Helical liquid
o Effective low-energy bosonized theory and RG analysis
o Phase diagram and experimental regimes
o Final remarks
Outline
Microscopic model
kinetic energy + chemical potential + uniform Rashba and Dresselhaus spin-obit interactions:
modulated Rashba spin-orbit interaction:
modulated chemical potential:
s-wave superconducting pairing potential:
e-e interactions:
o Proposal of a magnetic field-free 1D topological superconductor
o Microscopic model
o Helical liquid
o Effective low-energy bosonized theory and RG analysis
o Phase diagram and experimental regimes
o Final remarks
Outline
top gates
quantum wire
Helical liquid
(a)
* * * *ener
gy
Helical liquid
Spin-split bands in the BZ
Without external modulation
(b)
ener
gy
(a)
* * * *ener
gyHelical liquid
modulated chemical potential +
“trivial” modulated spin-orbit terms
Reduced BZSpin-split bands in the BZ
(b)
ener
gy
“spin-flipping”modulated term
+e-e interactions
+ tuning
of the Fermi level
* *
ener
gy
(c)
Helical liquid
Reduced BZ
Gap opening at the
outer Fermi points
(b)
ener
gy
* *
ener
gy
(c)
Helical liquid
What about Kramers theorem?
“spin-flipping”modulated term
+e-e interactions
+ tuning
of the Fermi level
(b)
ener
gy
* *
ener
gy
(c)
Helical liquid
The systems breaks time reversal symmetry spontaneously.
“spin-flipping”modulated term
+e-e interactions
+ tuning
of the Fermi level
The outer branch develops
a spin density wave.
(b)
ener
gy
* *
ener
gy
(c)
Helical liquid
The gap opening is selective!
Only around the Fermi points
separated by Q.
“spin-flipping”modulated term
+e-e interactions
+ tuning
of the Fermi level
(b)
ener
gy
* *
ener
gy
(c)
Helical liquid
Gapless helical Luttinger liquid
in the inner branch!
“spin-flipping”modulated term
+e-e interactions
+ tuning
of the Fermi level
o Proposal of a magnetic field-free 1D topological superconductor
o Microscopic model
o Helical liquid
o From the helical liquid to a 1D topological superconductor
o Effective low-energy bosonized theory and RG analysis
o Phase diagram and experimental regimes
o Final remarks
Outline
top gates
quantum wire
From the helical liquid to a 1D topological superconductor
s-wave superconductor
top gates
quantum wire
Topological superconductor hosting Majorana zero-energy modes?
From the helical liquid to a 1D topological superconductor
Competition between superconducting and spin-orbit correlations
at the outer Fermi points.s-wave superconductor
top gates
quantum wire
Topological superconductor hosting Majorana zero modes? * *
ener
gy
(c)
From the helical liquid to a 1D topological superconductor
s-wave superconductor
top gates
quantum wire
Topological superconductor hosting Majorana zero modes? * *
ener
gy
(c)
Coupling between inner and outer Fermi points.
From the helical liquid to a 1D topological superconductor
o Proposal of a magnetic field-free 1D topological superconductor
o Microscopic model
o Helical liquid
o From the helical liquid to a 1D topological superconductor
o Effective low-energy bosonized theory and RG analysis
o Phase diagram and experimental regimes
o Final remarks
Outline
Effective low energy bosonized theory and RG analysis
Effective low energy bosonized theory and RG analysis
Effective low energy bosonized theory and RG analysis
Bunch of sine-Gordon potentials
intensity of the combined spin-orbit couplings
intensity of the superconducting pairing
intensity of e-e interaction (Luttinger parameter)
Effective low energy bosonized theory and RG analysis
competition between spin-orbit and superconducting trends
superconducting pairing
Dimensionless parametersFlow equations
Effective low energy bosonized theory and RG analysis
Renormalization group analysis of the outer branch
Effective low energy bosonized theory and RG analysis
Critical plane:
Effective low energy bosonized theory and RG analysis
Critical plane:
Above critical plane - “good regime”
➢ Spin-orbit is strongly (a) or marginally (b) relevant.
➢ Superconductivity is irrelevant.
Effective low energy bosonized theory and RG analysis
Critical plane:
Below critical plane - “bad regime”
➢ Spin-orbit is irrelevant.
➢ Superconductivity is marginally (a) or strongly (b) relevant.
Effective low energy bosonized theory and RG analysis
spin-orbit term in the outer branch:
Effective low energy bosonized theory and RG analysis
spin-orbit term in the outer branch:
spin-orbit is strongly relevant
1 gets pinned
branches decouple
Effective low energy bosonized theory and RG analysis
spin-orbit term in the outer branch:
spin-orbit is strongly relevant
1 gets pinned
branches decouple
Spin-orbit is marginally relevant.
❖ How is the interplay with the mixing term?
❖ Does the branch decoupling persists?
o Proposal of a magnetic field-free 1D topological superconductor
o Microscopic model
o Helical liquid
o From the helical liquid to a 1D topological superconductor
o Effective low-energy bosonized theory and RG analysis
o Phase diagram and experimental regimes
o Final remarks
Outline
Phase diagram – combining the regimes of the inner and outer branches
experimental regime
Phase diagram – combining the regimes of the inner and outer branches
opening of the insulating gap in the outer branch
opening of the superconducting gap in the inner branch
Experimental regime – analysis of the scaling lengths
Above this lenght thermal leakage destroys the correlations.
superconducting gap > thermal energy
Cutoff length: system’s size or localization length.
Lins and Lsc must fit into the system.
Threshold for Lins above which the p-wave state is lost.
insulating gap > r x superconducting gap
opening of the insulating gap in the outer branch
opening of the superconducting gap in the inner branch
Experimental regime – analysis of the scaling lengths
o Proposal of a magnetic field-free 1D topological superconductor
o Microscopic model
o Helical liquid
o From the helical liquid to a 1D topological superconductor
o Effective low-energy bosonized theory and RG analysis
o Phase diagram and experimental regimes
o Final remarks
Outline
Final remarks
We propose a scheme for engineering a topological superconductor hostingMajorana zero modes with no magnetic fields nor topological insulators.
The scheme relies on the interplay between a modulated Rashba interaction, anuniform Dresselhaus interaction, s-wave pairing and e-e interactions.
A topological superconducting phase arises within a finite region of theparameter space.
The experimental viability of the proposed scheme is analyzed. We found thatwhile a cold atoms realization is well within present experimental capabilities,a solid state setup looks more challenging.