Luis Gregório Dias da Silva - USPluisdias/Presentations/NRG... · Kondo-like zero-bias peak...
Transcript of Luis Gregório Dias da Silva - USPluisdias/Presentations/NRG... · Kondo-like zero-bias peak...
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Application of Wilson's NRG to Majorana-Kondo systems.
Instituto de Física, Universidade de São Paulo
Luis Gregório Dias da Silva
NRG UFU/2019 Advanced Studies SchoolUberlândia, Feb. 01, 2019.
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Physics @ USP – São Paulo.
Physics Institute-USP
6 departments
~130 active faculty
~250 grad students
~400 undergrad studentsS.Paulo-Uberlândia ~600 km
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Physics @ USP – São Paulo.
São Paulo
Physics Institute-USP
6 departments
~130 active faculty
~250 grad students
~400 undergrad students
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Marcos Medeiros
Doutorado
Raphael Levy
Doutorado
Luis Gregório Dias da Silva
Professor
Jesus Cifuentes
Mestrado
Rafael Magaldi
Mestrado
Group Members
João Victor Ferreira Alves
Mestrado
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Outline
Basics: Majorana bound states in condensed matter systems.
Detecting Majorana states with quantum dots.
Interacting quantum dots: Wilson’s NRG
Kondo-Majoranana co-existence.
Manipulating Majorana states with (double) quantum dots.
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What are Majorana fermions?
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Majorana Fermions
Majorana solution: Representarions of
Dirac matrices with only imaginary non-zero
elements while still satisfying
http://www.giornalettismo.com/archives/255332/il-ritorno-di-ettore-majorana/
A Dirac fermion can be “written” in terms of two Majorana fermions
or
Real solutions:
E. Majorana, Nuovo Cimento 5, 171 (1937)
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Where do we find Majorana fermions?
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Majorana quasiparticles in condensed matter systems?
Fractional Quantum Hall liquids (ν=5/2):
“non-Abelian anyons”.
Two-channel Kondo non-Fermi-liquid state.
Interface of topological insulators with BCS
superconductors
Spin-polarized (“spinless”) p-wave
superconductors.
Moore and Read, Nucl. Phys. B (1991).
Read and Green, Phys. Rev. B (2000).
Fu and Kane, Phys. Rev. Lett. (2008).
Kitaev, Phys. Usp. (2001).
Motivation: entanglement of particles with non-abelian statistics
(“Ising anyons”); topologically protected quantum computation.
Zhang, Hewson, Bulla, Solid State Comm. (1999).
Maldacena, Ludwig, Nucl. Phys. B. (1997).
Coleman, Ioffe, Tsvelik PRB (1995).
Emery, Kivelson, PRB (1992).
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1D p-wave superconductor (Kitaev model)
P-wave pairing term (spinless fermions)
J. Alicea, Rep. Prog. Phys. 75, 076501 (2012)
Energy spectrum:
Gappless modes (E=0) :
Gapped: topological (∆≠0)
Gapped (E+-E->0): trivial
or
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Majorana states in the Kitaev model.
Map into a “chain of Majorana modes” using:
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Majorana states in the Kitaev model.
Topological regime: Majorana modes (e=µ=0!!!) at the edges of the chain!
Gapped: trivial. Special case:
Gapped: topological. Special case:
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Can the Kitaev model be realized experimentally?
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How to realize a p-wave SC: Quantum wires.
Theory: Lutchyn et al. PRL, 105, 077001 (2010); Oreg et al. PRL,105, 077002 (2010);
Step 1:
create spinless 1D fermions.
Ingredients: spin-orbit, B field.
Step 2:
Introduce SC pairing.
Ingredients: proximity with
a BCS SC
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Experiment on InSb nanowires.
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A success story??
Inside joke:
“Majorana found at the end of a quantum wire”
Theory: Lutchyn et al. PRL, 105, 077001 (2010); Oreg et al. PRL,105, 077002 (2010);
Experiment: V. Mourik et al. Science 336 1003 (2012)
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Alternative explanations for the zero-bias peak.
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Magnetic chain: spiral angle
MBSs in magnetic chains on topological insulators
(See poster by Raphael Levy…)
Honeycomb lattices:
Silicene, Stanene…
Kane-Mele-type TIs
R. Teixeira et al. Phys Rev B 99 035127 (2019).
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MBSs in magnetic chains on topological insulators
(See poster by Raphael Levy…)
L=61
L=21
L=21
MBS or ABS?R. Teixeira et al. Phys Rev B 99 035127 (2019).
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Detecting MBS with quantum dots.
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Better way to measure?
Quantum dot coupled to metallic leads coupled with at the end of the nanowire.
Liu and Baranger, Phys Rev B 84 201308 (2011).
Vernek et al., Phys Rev B 89 165314 (2014).
M.T. Deng et al., Science 354 1557 (2016).
Theory Experiment (Marcus’ group)
Ruiz-Tijerina et al. Phys Rev B 91 115435 (2015).
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Better way to measure?
Connect a quantum dot + metallic leads at the end of the nanowire.
Measure conductance through the dot
0.5 e2/h = signature of the Majorana mode for U=0
What happens for the (common) case of non-zero U???
Ruiz-Tijerina et al. Phys Rev B 91 115435 (2015).
Liu and Baranger, Phys Rev B 84 201308 (2011).
Vernek et al., Phys Rev B 89 165314 (2014).U=0!!!
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Majoranas + interaction
Kondo impurity + Majorana edge states (NRG)
Quantum dot + Kitaev (NRG)
Quantum dot + Kitaev (DMRG)
Interacting Kitaev model (DMRG)Stoudenmire et al., Phys. Rev. B 84 014503 (2011).
Cheng et al., Phys. Rev. X 4, 031051 (2014).
Thomale et al., Phys. Rev. B 88 161103(R) (2013).
Chirla et al., Phys. Rev. B 90, 195108 (2014).
M. Lee, et al., Phys. Rev. B 87, 241402 (2013).
R. Zitko, Phys. Rev. B 83, 195137 (2011).
R. Zitko, P. Simon, Phys. Rev. B 84, 195310 (2011).
Ruiz-Tijerina et al., Phys. Rev. B 91, 115435 (2015).
Korytár and Schmitteckert, JPCM 25 475304 (2014).
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Kondo Effect in Quantum Dots: zero-bias transport.
Vg
Vg
Co
nd
ucta
nce
(G
) ~U
U
~U
TK
•T>TK: Coulomb blockade (low G)
•T<TK: Kondo singlet formation
•Kondo resonance at EF (width TK).
•New conduction channel at EF:
Zero-bias enhancement of G (→2e2/h!)
Kondo
resonanceΓ
Even N Odd N
van der Wiel et al.,
Science 289 2105
(2000).
Drain
T>TKT<TK
Source
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Kondo resonance with Wilson’s NRG
γn~Λ-n/2εd
εd+U t γ1...
γ2 γ3
Spectral density:
Single-particle peaks at εd
and εd+U.
Many-body peak at the
Fermi energy: Kondo
resonance (width ~TK).
NRG: very good resolution at low ω.
Γ Γ
εd εd+ U
~TK
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Quantum dot defined in InAs/InP quantum wires coupled to superconducting leads.
Kondo-like zero-bias peak emerges at a critical field Bc .
Kondo zero-bias peak in quantum wires coupled to SC leads.
E.J. Lee et al. PRL 109 186802 (2012)
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Model: Quantum dot + quantum wire + SC pairing.
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Iterative Green’s functions + mean field (Hubbard I).
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Iterative Green’s functions + mean field (Hubbard I).
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Iterative Green’s functions + mean field (Hubbard I).
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Shortcomings of the mean-field approximation.
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Effective low-energy Anderson model
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Effective low-energy Anderson model
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Effective low-energy Anderson model
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Effective model
D. A. Ruiz-Tijerina et al. Phys Rev B 91 115435 (2015).
creates a fermion in state α
Quantum dot (VZ: Zeeman ; U: e-e interaction)
Coupling to the metallic leads
Coupling to one Majorana
number operator (=0,1) Majorana operators
Metallic leads
NRG: spectral function and conductance
zero energy mode
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NRG formulation: quantum numbers
Majorana operators
not a good QN!
OK!
OK!
Build blocks such as: etc,
However:
Fermion operators
See also: M. Lee, et al., Phys. Rev. B 87, 241402 (2013).
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NRG formulation: quantum numbers
H-1 : block-diagonal:
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NFL behavior: ln(2)1/2 residual entropy
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Majorana-Kondo co-existence
Consistent with:Cheng et al., Phys. Rev. X 4, 031051 (2014).
M. Lee, et al., Phys. Rev. B 87, 241402 (2013).D. A. Ruiz-Tijerina et al. Phys Rev B 91 115435 (2015).
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Manipulating MBS with quantum dots.
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Marcos Medeiros
Doutorado
Raphael Levy
Doutorado
Luis Gregório Dias da Silva
Professor
Jesus Cifuentes
Mestrado
Rafael Magaldi
Mestrado
Group Members
João Victor Ferreira Alves
Mestrado
![Page 42: Luis Gregório Dias da Silva - USPluisdias/Presentations/NRG... · Kondo-like zero-bias peak emerges at a critical field B c. Kondo zero-bias peak in quantum wires coupled to SC leads.](https://reader033.fdocuments.us/reader033/viewer/2022050219/5f64a5d825dcaa39572c75cb/html5/thumbnails/42.jpg)
Manipulation of Majorana fermions in Double Quantum Dots
Dot-Lead
Dot-MajoranaEnergy level
Coulomb Interaction
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Green’s Function
Density of States (DOS)
Non-interacting case: spectral densities
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Symmetric coupling Non-Interacting U=0
Majorana signature
No Majorana
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Symmetric coupling Interacting U>0 (NRG)
Kondo-MajoranaSatelites
Kondo Peak
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Interference destroying Majorana signatureNon-Interacting
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Indirect Majorana CouplingNon-Interacting
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Manipulating with couplings/gate voltages.
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Summary
Coexistence of MZM and Kondo states in interacting
quantum dots
Detecting MZMs using quantum dots: signature in the
spin-resolved densitity of states → large (e2/2) reduction
in the conductance.
Manipulating MZMs using double quantum dots using
only gate voltages and couplings.
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Support: FAPESP (2016/18495-4); CNPq (308351/2017-7 and 449148/2014-9); CAPES, FAPEMIG, USP-PRP Q-Nano.
Edson VernekCarlos EguesDavid Ruiz-Tijerina
Collaborators in these works
Annica Black-Schaffer