Partial supersymmetry breaking anddispersive Majorana modes
in quantum wires
Pasquale Marra1,2 , Daisuke Inotani2 , Muneto Nitta2
pmarra@ms.u-tokyo.ac.jp

1

Graduate School of Mathematical Sciences, The University of Tokyo,
2
Department of Physics, Keio University, Hiyoshi, YokohamaComm. Phys. 5 149 (2022)
Phys. Rev. B 105, 214525 (2022)
arXiv:2207.10103

Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

Supersymmetry (SUSY)
Loosely speaking, supersymmetry is a symmetry between bosons and fermions.
In particle physics, supersymmetric extensions of the Standard Model predicts
a partner particle for each known particle.

Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

SUSY in Majorana chains• Interacting Majorana chain with emergent
space-time SUSY: tricritical Ising model(via Jordan-Wigner mapping)XX
H = ıtγj γj+1 + g
γj γj+1 γj+2 γj+3 ,jj

Rahmani, Zhu, Franz, Aﬄeck, PRL 115, 166401 (2015);
PRB 92, 235123 (2015)• Translational invariant Majorana models withemergent quantum-mechanical SUSYrXN
HH=Hi ,
Q=T (1 + P )i=12γEEΔHsieh, Halász, Grover PRL 117, 166802 (2016)• Closed Majorana chains (with closed PH gap) withemergent quantum-mechanical SUSY
H=i2N
X

tj γj γj+1 ,

j=1N
Y
j=1

t2j−1 =N
Y

t2j ,

j=1

√
QA,B = γA,B H
Huang, Shimasaki, Nitta, PRB 96, 220504(R) (2017)
Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wiresQuantum Mechanical Supersymmetry (QM SUSY)• QM SUSY: Symmetry of the many-body Hamiltonian whichconnects states with even and odd fermion parity(i.e., bosons and fermions)• Supercharge Q which satisﬁes the superalgebra

E{Q, Q† } = QQ† + Q† Q = 2HSUSY{P, Q} = 0evenoddwith P the parity operator.

• SUSY is a “fermionic” symmetry (unlike unitary symmetries)
Q{HSUSY , Q} = 02

† 2

Q = (Q ) = 0• Extended QM SUSY:
many supercharges Qi which satisﬁes the superalgebra{Qi , Q†j } = 2δij H,{Qi , Qj } = {Q†i , Q†j } = 0,
{P, Qi } = 0.

Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

Spontaneous breaking of QM SUSY
• Unbroken QM SUSY:†
Q |0⟩ = Q† |0⟩ = 0⇒ eı(ϵQ+ϵ̄Q ) |0⟩ = |0⟩• Spontaneously broken QM SUSY:†
Q |0⟩ ̸= 0 or Q† |0⟩ ̸= 0 ⇒ eı(ϵQ+ϵ̄Q ) |0⟩ ̸= |0⟩• Witten index:
diﬀerence between the # of bosons and # of fermions
W = Tr(−1)F = (# even − # odd eigenstates)
W =1even

W =0oddevenodd• Spontaneously broken unitary symmetry → Goldstone mode (boson)
Spontaneously broken supersymmetry → Goldstino (fermion)• Extended SUSY:
Witten’s no-go theorem prohibits partial SUSY breaking
Loophole: central charges
Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

Bipartite Majorana chain• Chain of Majorana modesPHeﬀ = ı j wγAj γBj + vγBj γAj+1• In momentum space
Heﬀ =X[c†k , c−k ] · Heﬀ (k) · τ ·kckc†−kwhere Heﬀ (k) = (0, v sin k, v cos k − w)

• Dispersion
Ek = |Heﬀ (k)| =pw2 + v 2 − 2wv cos k• Topological invariant Meff = |w| − |v|• In the continuum limit → 1D Dirac equation
v H = vk τy + mv 2 − k2 τz2• Hybridization of 0D Majorana modes ⇒ 1D Majorana mode in a 1D system
Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

Chiral and helical 1D Majorana fermions
EEkk

Chiral SCEQHEkkEEkk

Helical SCEQSHEkk

Qi, Hughes, Raghu, Zhang, PRL 102, 187001 (2009)
(a)

(b)

Eky(c)

Eky

E

(d)

ky

E

kykx

Sato, Fujimoto, J. Phys. Soc. Jpn. 85, 072001 (2016)
Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

QM SUSY in Majorana chainsE• The Majorana chain exhibits SUSY with superchargesrHSUSY
1QM =(eγA + ıeγB )2r
HSUSY
2QT =T (1 + P )2
2HSUSY = QQ† + Q† Q = {Q, Q† }evenodd

Q{P, Q} = 0, Q2 = (Q† )2 = 0
with HSUSY = Heﬀ + const > 0
1 Huang, Shimasaki, Nitta, PRB 96, 220504(R) (2017)
2 Hsieh, Halász, Grover PRL 117, 166802 (2016)• Spontaneously broken SUSY: Witten index is zero
W = Tr(−1)F = (#even − #odd eigenstates) = 0• Goldstino → 1D Majorana fermion

Pasquale Marrapmarra@ms.u-tokyo.ac.jp

TPartial SUSY breaking and dispersive Majorana modes in quantum wires

QM SUSY in Majorana chainsE• The Majorana chain exhibits SUSY with superchargesr
HSUSY
1QM =(eγA + ıeγB )
2r
HSUSY
2
QT =T (1 + P )22HSUSY = QQ† + Q† Q = {Q, Q† }

even

odd

Q

{P, Q} = 0, Q2 = (Q† )2 = 0
with HSUSY = Heﬀ + const > 0
1 Huang, Shimasaki, Nitta, PRB 96, 220504(R) (2017)
2 Hsieh, Halász, Grover PRL 117, 166802 (2016)• Spontaneously broken SUSY: Witten index is zero
W = Tr(−1)F = (#even − #odd eigenstates) = 0• Goldstino → 1D Majorana fermion

Pasquale Marrapmarra@ms.u-tokyo.ac.jp

TPartial SUSY breaking and dispersive Majorana modes in quantum wires

Extended QM SUSY and partial SUSY breaking• The two N = 2 superalgebras can be combined• The Majorana chain exhibits extended SUSY N = 4 with central charges. The N = 4superalgebra is given by{Qi , Q†j } = 2δij HSUSY + Zij ,{Qi , Qj } = {Q†i , Qj† } = 0,{P, Qi } = 0,
• Supercharges:r
Q1 =HSUSY
dM (1 + P ),2r
Q2 =• Central charges:"†P (−1)dM dM )Zij = HSUSY −(1 +†{dM (1 − P ), T }where dM = (eγA + ıe
γB )/2.
• Partially broken SUSYp
Q2 |0⟩ = 2HSUSY |1⟩ ,Q†2 |1⟩ =HSUSY
T (1 + P ),2#{dM (1 + P ), T † } ,0p

2HSUSY |0⟩ ,Q1 |0⟩ =Q1 |1⟩ = Q†1 |0⟩ = Q†1 |1⟩ = Q†2 |0⟩ = Q2 |1⟩ = 0.
• Witten’s no-go theorem prohibits partial SUSY breaking
Loophole: central charges
Marra, Inotani, Nitta, Comm. Phys. 5 149 (2022), Phys. Rev. B 105, 214525 (2022)
Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

Array of quasi-Majorana modes induced by modulated ﬁelds• Oreg-Lutchyn model with spatially-modulated ﬁelds
 2pαH=+ σy p − µ τz + b(x) · σ + ∆(x)τx2mh̄p• local Majorana mass M = µ(x)2 + ∆(x)2 − |b(x)|quasi-Majorana modes localize at the nodes M = 0• Eﬀective theory → Chain of Majorana modesPHeﬀ = ı j wγAj γBj + vγBj γAj+1ﬁnite overlaps:
w = −ı ⟨γAj | H |γBj ⟩ ∝ e−LAB /ξMv = −ı ⟨γBj | H |γAj+1 ⟩ ∝ e−LBA /ξMMarra, Inotani, Nitta, Comm. Phys. 5 149 (2022), Phys. Rev. B 105, 214525 (2022)
Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

Energy dispersion and LDOS (numerical calculations)• Magnetic ﬁeld calculated with FEM:
• The local Majorana mass

|bnm (x)|2 ≈ |bnm |2 (2 − 2 cos (2πx/λ))ppM(x) ≈ µ2 + ∆2 − |bnm | (2 − 2 cos (2πx/λ))• LDOS at zero energy: localized modes• Energy dispersion Ek : Dispersive subgap level

Marra, Inotani, Nitta, Comm. Phys. 5 149 (2022), Phys. Rev. B 105, 214525 (2022)
Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

Edge modes of edge modes• Open boundaries in the globally nontrivial phase• Global topological invariant

Meff = |w| − |v|• LDOS at zero energy: localized modes accumulating at the boundary

Marra, Inotani, Nitta, Comm. Phys. 5 149 (2022), Phys. Rev. B 105, 214525 (2022)
Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

Adiabatic pumping of 0D Majorana modes• By applying a rotating magnetic ﬁeld ba [cos θ x̂ + sin θ ẑ]. produces a sliding ﬁeldM(x) ∝ sin (2πx/λ + θ)
• θ → θ + π : adiabatic pumping of one 0D Majorana state• θ → θ + 2π : adiabatic pumping of two 0D Majorana states (one fermion)• Analogy to Thouless pumps of cold atoms: local Majorana mass ∼ optical lattice

Marra, Inotani, Nitta, Comm. Phys. 5 149 (2022), Phys. Rev. B 105, 214525 (2022)
Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

Experimental considerations• Hierarchy of length scales(length of the wire) ≫ (spatial periodicity of the ﬁeld) ≳ (Majorana localization length)• Conditions for SUSY and massless Majorana fermions
LAB ≈ LBA
• The SUSY point appears at a single point of parameter space. However, electronicinteractions may pin the Majorana modes to zero energy in a larger window[see Domínguez, Cayao, San-Jose, Aguado, Yeyati, Prada, npj Quantum Materials 2, 13 (2017)]• The transition between a dip G = 0 to a quantized peak G = 2e2 /h̄ in the zero-bias
conductance may signal the onset of SUSY [see also Flensberg, PRB 82, 180516 (2010)]

Marra, Inotani, Nitta, Comm. Phys. 5 149 (2022), Phys. Rev. B 105, 214525 (2022)
Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

Trijunctions• T-shaped or Y-shaped trijunctions with lattices of 2N Majorana modes on each branch• Nanomagnets, piano keyboard, or in planar Josephson junctions

Heﬀ = ıu (γ1,1 γ2,1 + γ2,1 γ3,1 + γ3,1 γ1,1 )+ıPasquale Marra

3XNX

m=1

n=1

pmarra@ms.u-tokyo.ac.jp

wγm,2n−1 γm,2n +N−1X!
vγm,2n γm,2n+1n=1

Partial SUSY breaking and dispersive Majorana modes in quantum wires,

Braiding

See also Hegde et al., Ann. Phys. 423, 168326 (2020).

Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

Topological phases and energy scaling• |w| > |v|, trivial phase, ∆E/w → E0 for N → ∞• |w| = |v|, SUSY, ∆E/w ∝ 1/N for N → ∞• |w| < |v|, nontrivial phase, ∆E/w ∝ e−2N /ξeﬀ for N → ∞Marra, Inotani, Nitta, arXiv:2207.10103
Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

Conclusions• Spatially-modulated ﬁelds can induce
dispersive 1D Majorana fermions in 1D nanowiresby the hybridization of 0D Majorana modes• Real-world realization of centrally-extended QM-SUSY
and partial breaking of the extended SUSY algebras• Dispersive 1D Majorana fermion is the Goldstino of thespontaneously, partially broken SUSY
• Zero-bias dip to peak transition as a signature of SUSY
• Adiabatic pumping of Majorana modesanalogous to Thouless pumps of (non-Majorana) fermions• Implementation of novel braiding protocols?
Marra, Inotani, Nitta,
Comm. Phys. 5 149 (2022)
Phys. Rev. B 105, 214525 (2022)
arXiv:2207.10103
You can ﬁnd this presentation here:
https://www.ms.u-tokyo.ac.jp/~pmarra/slides
email to:pmarra@ms.u-tokyo.ac.jp
Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires

Pseudohelicity• Let’s deﬁne the Majorana pseudospin operator
T = 12 τ = 12 (τx , τy , τz )• The expectation value of T on a state Ek > 0 isv sin k v cos k − w
H(k)⟨T⟩ = 0,,=2Ek
2Ek
2Ek• two Majorana modes with opposite pseudospins⟨T⟩ = 12 sgn (v sin k)ŷ at k → 0, π for v = ±w.
e.g., ⟨T⟩ = ± 12 ŷ at k → 0± for v = w > 0• Pseudohelical, but not necessarily helical(the eﬀective Hamiltonian does not act
on the spin degrees of freedom)• The expectation values ⟨Tx ⟩ and ⟨Ty ⟩ coincide withthe Majorana polarization PMx and PMy(Sticlet, Bena, Simon, PRL 108, 096802, 2012)
Marra, Inotani, Nitta, Comm. Phys. 5 149 (2022), Phys. Rev. B 105, 214525 (2022)
Pasquale Marrapmarra@ms.u-tokyo.ac.jp

Partial SUSY breaking and dispersive Majorana modes in quantum wires