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Kirill Shtengel, UC Riverside
Looking beyond Majorana zero modes:Engineering non-Abelian anyons
Looking beyond Majorana zero modes:Engineering non-Abelian anyons
Support:DARPA QuEST
Joint work with: Jason Alicea, CaltechDavid Clarke, UC Riverside → Caltech
Euler Symposium, St. Petersburg 2013
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1D quantum Ising chain:
Ising model (of all things)Ising model (of all things)
Jordan—Wigner transformation:
γ's are MajoranaMajorana operators:
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Hamiltonian after Jordan—Wigner transformation:
Ising model (of all things)Ising model (of all things)1D quantum Ising chain:
γ's are MajoranaMajorana operators:
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Hamiltonian after Jordan—Wigner transformation:
Ising model (of all things)Ising model (of all things)1D quantum Ising chain:
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Hamiltonian after Jordan—Wigner transformation:
Ising model (of all things)Ising model (of all things)1D quantum Ising chain:
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1D spinless p-wave superconductor(Kitaev 2001):
From Ising to Majorana wiresFrom Ising to Majorana wires
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1D spinless p-wave superconductor(Kitaev 2001):
Unpaired Majorana
fermions at the ends!
From Ising to Majorana wiresFrom Ising to Majorana wires
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Kane & Mele, 2005; Bernevig, Hughes, Zhang, 2006; Fu & Kane, 2008
Realization in topological insulator edges Realization in topological insulator edges
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Kane & Mele, 2005; Bernevig, Hughes, Zhang, 2006; Fu & Kane, 2008
Realization in topological insulator edges Realization in topological insulator edges
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HgTe
1D and effectively ‘spinless’! Just need superconductivity...
‘spin down’
‘spin up’
Fu & Kane, 2008
Realization in topological insulator edges Realization in topological insulator edges
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HgTe
s-wave SC ‘spin down’
‘spin up’
Fu & Kane, 2008
Realization in topological insulator edges Realization in topological insulator edges
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HgTe
s-wave SC
FM insulator
“Terminating” the SC wire by a magnetic gap: Majorana zero modes localised at the ends
Fu & Kane, 2008
Realization in topological insulator edges Realization in topological insulator edges
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(Lutchyn, Sau, Das Sarma 2010; Oreg, Refael, von Oppen 2010)
1D spin-orbit-coupled wire (e.g. InAs)
Realization in 1D wiresRealization in 1D wires
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(Lutchyn, Sau, Das Sarma 2010; Oreg, Refael, von Oppen 2010)
1D spin-orbit-coupled wire (e.g. InAs)
Realization in 1D wiresRealization in 1D wires
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(Lutchyn, Sau, Das Sarma 2010; Oreg, Refael, von Oppen 2010)
1D spin-orbit-coupled wire (e.g. InAs)
Realization in 1D wiresRealization in 1D wires
s-wave SC
Generates a1D ‘spinless’ SC state with Majorana fermions!
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Mourik et al., Science 2012(Kouwenhoven's group, Delft)following proposals byLutchyn, Sau & Das Sarma, 2010;Oreg, Refael & von Oppen, 2010.
First possible experimental realization First possible experimental realization
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First possible experimental realization First possible experimental realization
Mourik et al., Science 2012(Kouwenhoven's group, Delft)following proposals byLutchyn, Sau & Das Sarma, 2010;Oreg, Refael & von Oppen, 2010.
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Great! Anything else? Great! Anything else?
Can this idea be generalised to other types of exotic (non-Abelian) zero modes?
At first glance, this is very doubtful: – Z8 classification of phases of interacting fermions in 1D (Fidkowski & Kitaev, 2009; Turner, Pollmann & Berg, 2010)
Nevertheless, it is possible!
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1D quantum clock model (Fendley, arXiv:1209.0472):
N=2: quantum Ising chain
Taking a cue from Stat MechTaking a cue from Stat Mech
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N ≠ 2: quantum clock
Taking a cue from Stat MechTaking a cue from Stat Mech1D quantum clock model (Fendley, arXiv:1209.0472):
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Taking a cue from Stat MechTaking a cue from Stat Mech
Jordan—Wigner transformation:
α's are parafermionicparafermionic operators:
N=2: these are Majorana fermions
1D quantum clock model (Fendley, arXiv:1209.0472):
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Taking a cue from Stat MechTaking a cue from Stat Mech
Hamiltonian after Jordan—Wigner transformation:
N=2: these are Majorana fermions
1D quantum clock model (Fendley, arXiv:1209.0472):
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Taking a cue from Stat MechTaking a cue from Stat Mech
Hamiltonian after Jordan—Wigner transformation:
N=2: these are Majorana fermions
1D quantum clock model (Fendley, arXiv:1209.0472):
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Taking a cue from Stat MechTaking a cue from Stat Mech
Hamiltonian after Jordan—Wigner transformation:
1D quantum clock model (Fendley, arXiv:1209.0472):
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Taking a cue from Stat MechTaking a cue from Stat Mech
Hamiltonian after Jordan—Wigner transformation:
1D quantum clock model (Fendley, arXiv:1209.0472):
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2D TI (HgTe)
s-wave SC
FM insulator
Recall topological insulator edges Recall topological insulator edges
Fu & Kane, 2008
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2D FTI?
s-wave SC
FM insulator
We could envision playing the same game with 2D fractional topological insulators (à la Levin & Stern, 2009), but...
What about fractional TI edges? What about fractional TI edges?
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2D FTI?
s-wave SC
FM insulator
There are no known fractional topological insulators (yet).But could we 'fake' the same physics elsewhere?
What about fractional TI edges? What about fractional TI edges?
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Realization in quantum Hall edges Realization in quantum Hall edges
Counter-propagating edge modes at the boundary betweeng > 0 and g < 0.The sign of g can be changed by stress.
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Counter-propagating edge modes at the boundary betweeng > 0 and g < 0.The sign of g can be changed by stress.
Realization in quantum Hall edges Realization in quantum Hall edges
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Counter-propagating edge modes at the boundary betweeng > 0 and g < 0.The sign of g can be changed by stress.
Realization in quantum Hall edges Realization in quantum Hall edges
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Counter-propagating fractionalized edge modes at the boundary betweeng > 0 and g < 0.The sign of g can be changed by stress.
What about fractional quantum Hall edges? What about fractional quantum Hall edges?
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Counter-propagating fractionalized edge modes at the boundary betweeng > 0 and g < 0.The sign of g can be changed by stress.
What about fractional quantum Hall edges? What about fractional quantum Hall edges?
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Majorana Fermions:
Parafermions:
Parafermions 1D quantum Clock/Potts modelMajoranas 1D quantum Ising model
Paul Fendley, arXiv:1209.0472
Parafermions vs MajoranasParafermions vs Majoranas
Upshot:
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A Laughlin edge state at is a natural starting point since
Parafermions from quantum Hall edges Parafermions from quantum Hall edges
for chiral edge excitations of charge e/m. Now, we have two counter-propagating modes, , which obey
The electron fields are
and hence
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Change of variables:
Parafermions from quantum Hall edges Parafermions from quantum Hall edges
Just need to show that a zero mode is bound at a domain wall between
and
where
Free Hamiltonian:
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Assuming strong tunnelling and pairing,
under the superconductors
under the SO coupled insulators
m
nϕπϕ =
m
nθπθ =
2ϕ1ϕ θ
Parafermionic zero modeParafermionic zero mode
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Majorana zero modeMajorana zero mode
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Parafermionic zero modeParafermionic zero mode
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Reminder: in the strong backscattering and pairing limit,
in the SC regions,
in the 'magnetic' regions
Quantum DimensionQuantum Dimension
Each n has eignenvalues 0,... 2m-1, but they don't commute!
⇒ One can fix nφ's or nθ's
Each n has eignenvalues 0,... 2m-1, but they don't commute!Each n has eignenvalues 0,... 2m-1, but they don't commute!
degenerate states for 2N modes ⇒
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Fu & Kane (2009)
2D Topological Insulator
SCFMSC
Laughlin state
SCSC
Laughlin state
support fractionally charged ground states!
Majorana vs. parafermion zero-modesMajorana vs. parafermion zero-modesMajorana vs. parafermion zero-modesMajorana vs. parafermion zero-modes
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2D Topological Insulator
SCSC
Laughlin state
SCSC
Laughlin state
Kitaev (2001)
Fractional Josephson effect
Effect persists even with electronic relaxation processes
Fu & Kane (2009)
Majorana vs. parafermion zero-modesMajorana vs. parafermion zero-modesMajorana vs. parafermion zero-modesMajorana vs. parafermion zero-modes
support fractionally charged ground states!
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Is this zero-mode a (non-Abelian) anyon? Is this zero-mode a (non-Abelian) anyon?
“It depends upon what the meaning of the word 'is' is”Bill Clinton
First and foremost, it is not a low-energy quasiparticle! It is a zero mode bound to a very high-energy topological defect – c.f. Majorana mode bound to a SC vortex or 'genons'
(M. Barkeshli, X-L. Qi, arXiv:1112.3311, arXiv:1208.4834)
Instead of the usual braiding statistics, one should think of projective braiding statistics projective braiding statistics (statistics up to a phase)
“Metaplectic anyons” – M. Hastings, C. Nayak, Z. Wang, arXiv:1210.5477
What are braiding statistics in 1D anyway?
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d = 3Only bosons & fermions(no knots in 3+1 dimensions)
d = 2
d = 1Exchange not well defined...
...because particles inevitably “collide”
Anyons are now possible!
Braiding statistics in 1D? Braiding statistics in 1D?
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Exchanging end modes in 1D wires Exchanging end modes in 1D wires
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Exchanging end modes in 1D wires Exchanging end modes in 1D wires
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Exchanging end modes in 1D wires Exchanging end modes in 1D wires
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Exchanging end modes in 1D wires Exchanging end modes in 1D wires
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Exchanging end modes in 1D wires Exchanging end modes in 1D wires
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Exchanging end modes in 1D wires Exchanging end modes in 1D wires
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Exchanging end modes in 1D wires Exchanging end modes in 1D wires
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Exchanging end modes in 1D wires Exchanging end modes in 1D wires
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Exchanging end modes in 1D wires Exchanging end modes in 1D wires
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Exchanging end modes in 1D wires Exchanging end modes in 1D wires
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Exchanging end modes in 1D wires Exchanging end modes in 1D wires
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Exchanging end modes in 1D wires Exchanging end modes in 1D wires
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Exchanging end modes in 1D wires Exchanging end modes in 1D wires
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Apparent problem: We cannot have Y-junctions: our modes live on the domain walls.. We can still exchange them:
Exchanging end modes in our case Exchanging end modes in our case
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Exchanging end modes in our case Exchanging end modes in our case
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Exchanging end modes in our case Exchanging end modes in our case
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Exchanging end modes in our case Exchanging end modes in our case
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Exchanging end modes in our case Exchanging end modes in our case
Integral of motion:
Energy-minimizing condition:
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Parafermion ZM BraidingParafermion ZM Braiding
(Majorana zero modes):
Upshot:
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Important observation: If quasiparticles of both chiralities are allowed to tunnel, the
braiding is not universal ⇒ Potential problem for fractional TI!
Parafermion ZM BraidingParafermion ZM Braiding
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Is this feasible?
Experimental realizations?Experimental realizations?
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Experimental realizations?Experimental realizations?Bending the Quantum Hall Effect (Matt Grayson's group,TU München)
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Which Parts are Really Necessary?
Do we need opposite g-factors (or exotic materials)?
No! Use spin-unpolarized states, like 2/3 or 2/5
D. Clarke, J. Alicea, KS, in preparation
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Which Parts are Really Necessary?
So do we really need the insulator at all?
Not in this case!
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Which Parts are Really Necessary?
Ok, but we need the superconductor, right?
Not if we use a bilayer system, then particle<-> hole is replaced
by layer<->layerBarkeshli, Jian, Qi, arxiv:1208.4834
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Topological Quantum Computation(Kitaev, Preskill, Freedman, Larsen, Wang)
• Multidimensional Hilbert space where we can
encode information → Qubits
• Ability to initialise and read-out a qubit
• Unitary operations → Quantum gates
Things we need:
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↔0 ↔1
↔
eterInterferom ↔
c
a b
(Hormozi, Bonesteel, et. al.)
Topological Quantum Computation
Or, perhaps use measurements to generate brading! (Bonderson, Freedman, Nayak, 2009)
Bonderson, KS & Slingerland, PRL 2006, PRL 2007, Ann. Phys. 2008
d
a b
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↔0 ↔1
↔
c
a b
(Hormozi, Bonesteel, et. al.)
Topological Quantum Computation
Or, perhaps use measurements to generate brading! (Bonderson, Freedman, Nayak, 2009)
d
a b
Majorana zero modes are not universal!
No entangling gates with braiding alone No phase gate
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↔0 ↔1
↔
c
a b
(Hormozi, Bonesteel, et. al.)
Topological Quantum Computation
Or, perhaps use measurements to generate brading! (Bonderson, Freedman, Nayak, 2009)
d
a b
Parafermionic zero modes are still not universal...
Can do entangling gates! No phase gate?
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ConclusionsConclusions
D. Clarke, J. Alicea & KS, arXiv:1204.5479
Parallel work:
N. Lindner, E. Berg, G. Refael & A. Stern, arXiv:1204.5733
M. Cheng, arXiv:1204.6084
Parafermionic zero modes can be localised in systems with counter-propagating fractionalised edge modes (FQHE, or fractional topological insulators)
Fractional Josephson effect with periodicity 4mπ Zero-bias anomaly – similar to the Majorana case, but with fractionalised charge tunnelling
Potential utility for quantum computing?