TIME-REVERSAL-INVARIANT TOPOLOGICAL SUPERCONDUCTORS: PROPOSALS AND - - PowerPoint PPT Presentation

TIME-REVERSAL-INVARIANT TOPOLOGICAL SUPERCONDUCTORS: PROPOSALS AND SIGNATURES

Liliana Arrachea Universidad Nacional de San Martín Argentina

ICTP- 2019 -

COLLABORATORS

• Armando Aligia, Bariloche
• Alberto Camjayi, Buenos Aires
• Leonel Gruñeiro, San Martín
• Oscar Casas, Colombia
• William Herrera, Colombia
• Alfredo Levy

Yeyati, Madrid

• Felix von Oppen, Berlin

TRITOPS

BCS Hamiltonian- fermions with spin Particle-hole symmetry

Class DIII

HBCS = X

k

Ψ†

kHBdG(k)Ψk

⇣ ψk,↑ ψk,↓ ψ†

−k,↓ − ψ† −k,↑

⌘t

[HBdG, Θ] = 0

{HBdG, Ξ} = 0

Time-reversal symmetry

{HBdG, Π} = 0, Π = ΘΞ Θ2 = 0, ±1, Ξ2 = ±1

Ξ2 = 1 Θ2 = −1

Altland, Zirnbauer, Phys, Rev. B 55,1145 (1997)

• A. Schnyder, et al, Phys. Rev. B 78, 195125 (2008)

NORMAL SUPERCONDUCTIVITY VS TRITOPS (1D)

E

∆ −∆

E

Zero modes

Non topological

• f Sz. S

n or

= ±1/4 Zero modes have fractional spin!

Γ†

E,σ, σ =

Topological

Kramers pairs of Majoranas

Γ−E,−σ

Γ†

0,σ = ±iΓ0,−σ

HISTORICAL NOTE

ULTIMAS NOTICIAS EDICION IMPRESA SUPLEMENTOS TAPAS ROSARIO/12 FIERRO FUTBOL EN VIVO

NOTA DE TAPA

La pista argentina

Bajo un manto de dudas subyace la historia, la parábola sobre la biografía de Ettore Majorana; quizás (quizás, quizás, eso al menos decía Fermi) uno de los grandes científicos de nuestra época (se anticipó al esbozo de la Teoría del Núcleo Atómico de Heisenberg que dio lugar al descubrimiento del neutrón) y que un buen día se esfumó por completo. Y bueno, hay malas

• buenas lenguas que dicen que anduvo por aquí, allá por 1950.

Por Matías Alinovi Ettore Majorana siempre vuelve. En el suplemento Radar del 23 de marzo pasado, Juan Forn comentó la reedición de Tusquets de La desaparición de Majorana, libro de Leonardo Sciascia. Se refería también a la pista argentina sobre la desaparición del físico italiano, aunque de un modo lateral,

SÁBADO, 31 DE MAYO DE 2008

FOTOGRAFIA FECHADA EL 3 DE NOVIEMBRE DE 1923, TOMADA DE SU LIBRETA UNIVERSITARIA.

FUTURO MIS RECORTES: 0 [0%] INDICE NOTA DE TAPA> NOTA DE TAPA

La pista argentina Historia de la ciencia: la sombra de Majorana (1906-?) Por Matías Alinovi

QUEMA EN EL DELTA

No son solamente pastizales Por Susana Gallardo

LIBROS Y PUIBLICACIONES

Redes Por Adrián Pérez

LA IMAGEN DE LA SEMANA

Marte rojo shocking

AGENDA CIENTIFICA

Semana de la Fisica. Jornada de Reciclado

INGRESAR | REGISTRARSE EDICIONES ANTERIORES BUSQUEDA AVANZADA CORREO RADAR RADAR LIBROS CASH TURISMO LIBERO NO LAS12 FUTURO M2 SOY SATIRA12 ESPECIALES FOTOGALERIA

Lunes, 2 de junio de 2008 | Hoy

Gmail Para Tu Negocio

Lucí más profesional con el email personalizado de Google Apps.

Comenzá Ahora

PALAS CARGADORAS

MAJORANA IN ARGENTINA?

MAJORANA’S ROUTE (1938) ?

NAPOLES-GENOVA- BUENOS AIRES?

EXAMPLES OF TRIPTOPS

Time-reversal-invariant topological superconductivity

Arbel Haima, and Yuval Oregb

aWalter Burke Institute for Theoretical Physics and Institute for Quantum Information and Matter, California Institute of

Technology, Pasadena, CA 91125, USA

bDepartment of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel

Review Article, arXiv: 1809.06863

(a) (b)

MINIMAL INGREDIENTS

ν = sgn(∆+)sgn(∆−),

ν = −1 Topological ν = 1 Trivial

Topological invariant

C.Wong, K.T.Law, Phys, Rev. B 86, 184516 (2012)

• E. Gaidamauskas, J. Paaske, and K. Flensberg,

Phys, Rev. Lett 112, 126402 (2014) C.Reeg, C. Schrade, J. Klinovaja,

• D. Loss, Phys, Rev. B (2017)

1D Rashba wire 2D Rashba layer

(a)

s±

nodeless wave SC Kramers pair of Majoranas: γ1γ2

1D Rashba wire

!

s±

nodeless wave SC

(b)

• F. Zhang, C. L. Kane, and E. J. Mele,

Phys, Rev. Lett. 111, 056402 (2013)

FROM PROXIMITY EFFECT

Rashba

Rashba

PROXIMITY+ PHASE TUNING

• L. Fu, C. Kane, Phys. Rev. B 2009

C-X Liu, B. Trauzettel , PRB 83, 229510 (2011)

• F. Parhizgar, AM.Black-Schaffer,
• Sci. Rep. 7, 9817 (2017)
• Δ/

Δ/

• 1

1

• ↑

↓ ↓ ↑

• A. Keselman, L. Fu, A. Stern and E. Berg,
• Phys. Rev. Lett. 111, 116402 (2013)

Rashba + Rashba - TI

WITH MANY-BODY INTERACTIONS

• L. Fu and E. Berg, Phys. Rev.
• Lett. 105, 097001 (2010)
• CuxBi2Se3
• 2
• 1

1 2 0.2 0.4 0.6 0.8 1

U/V

m/µ

+

1 2

• J. Wang,
• Y. Xu, S-C Zhang, Phys. Rev. B 90, 054503 (2014)

S.Nakosai,

• Y. Tanaka, N.Nagaosa, PRL 108, 147003 (2012)

(a) Ag Sn I (b)

• 3

+ + +

• 0.5

U/V

Δ1 Δ3

spin-singlet spin-triplet

1.0 0.8 0.6 0.4 0.2 0.0 0.5 1.0 1.5 2.0

M0/μ

∆

∆

−|∆

∼

• A. Haim, A. Keselman,
• Y. Oreg,
• Phys. Rev. B. 89, 220504 (2014)

2D 1D

TRIPTOPS 2D WITHOUT PHASE-TUNING

Proximity induced time-reversal topological superconductivity in Bi2Se3 films without phase tuning

Oscar E. Casas,1, 2 Liliana Arrachea,3 William J. Herrera,1 and Alfredo Levy Yeyati2

• Phys. Rev. B (RC) 99, 161301 (2019)

arXiv:1812.00931

3D TOPOLOGICAL INSULATORS

ARTICLES

PUBLISHED ONLINE: 10 MAY 2009 | DOI: 10.1038/NPHYS1270

Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface

Haijun Zhang1, Chao-Xing Liu2, Xiao-Liang Qi3, Xi Dai1, Zhong Fang1 and Shou-Cheng Zhang3*

Model Hamiltonian for Topological Insulators

Chao-Xing Liu1, Xiao-Liang Qi2, HaiJun Zhang3, Xi Dai3, Zhong Fang3 and Shou-Cheng Zhang2

1

• Nat. Phys. 5,438 (2009)
• Phys. Rev. B 82, 045122 (2010)

Bi2Se3

Basis

Atomic Bonding Antibonding Crystal field splitting

Parity structure:

SURFACE STATES Bi2Se3

Well defined helicity

+ ↑ + ↓

− ↓ − ↑

+ ↑ + ↓

− ↑ − ↓

Helicity-degenerate Even Odd

z bottom surface top surface

THIN FILMS

Bi2Se3

ARPES Spectra

Y Zhang, et al, Nat. Phys. 6,584 (2010)

Helicity-degenerate A gap appears

• z

bottom surface top surface

Bi2Se3 THIN FILMS+ELECTRIC FIELD

Degeneracy is broken

• +

PROXIMITY EFFECT IN THIN FILMS

Bi2Se3 film

Discretized model

TOPOLOGICAL INVARIANT

Weak coupling Analytical result

PHASE DIAGRAM

• the r = ∆+/∆−,

panels) and at

• fixed

, d = Λ/∆− (lower

INDUCED GAP N=6

Helical Majorana modes

TRIPTOPS 1D FRACTIONAL SPIN PROJECTION

A.Keselman, L. Fu, A. Stern and E. Berg,

• B. Phys. Rev. Lett. 111, 116402 (2013)

Entangled end states with fractionalized spin projection in a time-reversal-invariant topological superconducting wire

Armando A. Aligia1 and Liliana Arrachea2

M

B

SZ = 1/4 SZ = 1/4

+

• +
• Phys. Rev. B 98, 174507 (2018) arXiv:1806.06104

50 100 150

i

0.01 0.02 0.03 0.04 0.05

Si

z

Odd number of electrons

1. ∆Z E

⇥ ⇤ E↑ = 2E2 ∆Z , E↓ = 2E2 ∆Z + ∆Z 2 .

2. ∆Z ⌧ E

⇠ ⇥ ⇤ E↑ = E " 1 + 1 2 ✓∆Z 4E ◆2# ∆Z 4 , E↓ = E " 1 + 1 2 ✓∆Z 4E ◆2# + ∆Z 4 .

Zeeman splitting

SO along z

TRIPTOPS 1D SIGNATURES IN JOSEPHSON JUNCTIONS WITH EMBEDDED Q-DOTS

S-D-S JUNCTIONS

superconductor superconductor

∆eiφ, µ ∆, µ

U

εd

Hybrid superconductor–quantum dot devices

Silvano De Franceschi1*, Leo Kouwenhoven2, Christian Schönenberger3 and Wolfgang Wernsdorfer4

REVIEW ARTICLE

a

with well broadening spacings, with well broadening spacings,

JOSEPHSON CURRENT

• a

Initial 2 4 3 1 Intermediate Final N = even, S = 0 Is = lc sin()

• 2

4 3 1 Initial Intermediate Final

• N = odd, S = 1/2 Is = lc sin(+ )
• 0.4

0.3 0.2 1 2 3 4 Ic (nA)

Singlet S=1/2 junction

π

kBTK ∆ kBTK ⌧ ∆

filling) kBTK ¼ ffiffiffiffiffiffiffiffiffiffiffiffi ffi U=2 p expðU=8Þ k T & k T

Kondo

0-PI TRANSITION S-D-S JUNCTIONS

E. Vecino, A. Martín-Rodero, and A. Levy Yeyati, Phys. Rev. B 68, 035105 (2003)

1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 10 11 12 1

a

0´

π´ π

• ε0/Δ

U/Δ

Phase diagram

0.0 0.5 1.0 1.5 2.0

• 0.44
• 0.43
• 0.42
• 0.41
• 0.40
• 0.39
• 0.38

E/10Δ

a

2.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

• 0.41
• 0.40
• 0.39
• 0.38
• 0.37

d

φ/π

Singlet GS (Kondo) Doublet GS (~isollated S=1/2)

1.0

φ/π

∆ < kBTK ∆ > kBTK

E. Vecino, A. Martín-Rodero, and A. Levy Yeyati, Phys. Rev. B 68, 035105 (2003) Perturbation theory

• F. Siano and R. Egger, Phys. Rev. Lett. 93, 047002 (2004) Hirsch-Fye QMC

M.-S. Choi, M. Lee, K. Kang, and W. Belzig, Phys, Rev. B 70, 020502 (R) (2004) NRG

• D. Luitz and F. F. Assaad, Phys. Rev. B 81, 024509 (2010);D. J. Luitz, et al , Phys. Rev. Lett. 108, 227001 (2012) CTQMC

• t, Hc,α = −t⇤ ⇤

σ

• c†

α,1,σdσ + H.c.

⇥ . sents the quantum dot

tached to the wires. The Hamilton is H = ⇤

α=L,R (Hα + Hc,α) + Hd. T

Hα =

N

⌅

σ,j=1

• −tc†

α,j+1,σcα,j,σ + iλsσc† α,j+1,σcα,j,σ

−µ nα,j,σ + ∆eiϕαsσc†

α,j+1,σc† α,jσ + H.c.

⇥ , (1)

Hd = εd ⌅

σ=,⇥

nd,σ + Undnd⇥.

= t,

U

εd

TRITOPS

densit h t0 densit h t0

t, ∆eiφ, λ, µ

t, ∆, λ, µ

TRITOPS

• Phys. Rev. Lett. 119, 046801 (2017)

Fractional spin and Josephson effect in time-reversal-invariant topological superconductors

Alberto Camjayi,1 Liliana Arrachea,2 Armando Aligia,3 and Felix von Oppen4

JOSEPHSON JUNCTION

∆, λR, µ ∆eiφ, λR, µ

topological superconductor topological superconductor

Δ1 −Δ1 E

π 2π

φ (b)

Andreev states

4-fold symmetry protected crossing: periodicity 4π

Odd parity Even parity

s |µ| < 2⇥R n in Fig. 3(a)

J = 2t⇤

σ

Im ⇥ ⌃c†

α,1,σdσ⌥

⇤ = 2t⇤2 β

• σ
• n

Im ⇥ g(12)

1α,σ(iωn)G(21) d,σ (iωn)

⇤

• FIG. 2.

(Color online) Josephson current for the quantum dot with U = 0, t0 = t, ε = 0, λ = t/2. The length of the superconducting wires is N = 100 sites. The inverse of the temperature is β = 400. Energies are expressed in units of t = 1.

topological trivial

Evaluated exactly

2 0 0.5 1 1.5 2

φ/π

0.5 1 1.5 2

φ/π

• 0.5

0.5

E0(φ)

s |µ| < 2⇥R n in Fig. 3(a)

TRITOPS-QD-TRITOPS. U=0

JOSEPHSON CURRENT U#0

Trivial topological sign inversion junction π No sign- inversion U = 0 U = 10 U = 0 U = 10

s |µ| < 2⇥R n in Fig. 3(a)

Calculation with CTQMC

EFFECTIVE HAMILTONIAN I

U

εd

ence of the quantum dot, the effectiv s Heff = t0eiφ/4 ⇧

σ

⇤ †

L,σdσ + d† σR,σ

⌅ +

⌅ +H.c. + Hd.

†

L = † L,↑ = iL,↓,

†

R = † R,↑ = iR,↓

Zero-energy states (Bogoliubov q-particles)

EFFECTIVE HAMILTONIAN II

GS is always singlet!!

U

εd

↑ = 1

• 2 (L + R)

, ↓ = i

• 2

⇤ †

L † R

⌅

⇧ Heff = ⌃

σ

• tc†

σdσ tsσdσ

⇥ + H.c + Hd,

• f Sz. S

n or

= ±1/4

Hlow =J{Sz

d

 (nL + nR − 1) + i sin φ 2 ⇣ γ†

LγR − γ† RγL

⌘ + i cos φ 2 ⇣ S−

d γ† Lγ† R − S+ d γRγL

⌘ }

VS MONTE CARLO

Heff

• FIG. 5. (Color online) Upper panel: Local density of states at

the quantum dot in the topological phase with t0 = t, ε = U/2, λ = t/2, ∆ = t/5 and µ = 0. On the left sub-panel φ = 0.3π and on the right φ = 0.8π, as indicated. Lower panel: evolution of the spectrum as a function of φ with the same values of the parameters as above. The dark lines correspond to the prediction of Heff with t0 = 0.3, U = 1.2 with an additional on-site energy ε0 = 0.04 at the non-interacting site, in order to simulate the coupling to the continuum at φ = 0.

ρσ(ω) = 2Im[GR

d,σ(ω)]

Symmetry protected crossing 4-fold level degeneracy at

φ = π

Effective Hamiltonian

TRIPTOPS 1D SIGNATURES OF ORIENTATION OF FRACTIONAL SPIN IN JOSEPHSON EFFECT

Catalogue of Andreev spectra and Josephson effects in structures with time-reversal-invariant topological superconductor wires

Liliana Arrachea,1 Alberto Camjayi,2 Armando A. Aligia,3 and Leonel Gru˜ neiro1

Physical Review B 99, 085431 (2019)

TRITOPS-D-TRITOPS

tached to the wires. The Hamilton is H = ⇤

α=L,R (Hα + Hc,α) + Hd. T

Hd = εd ⌅

σ=,⇥

nd,σ + Undnd⇥.

• t, Hc,α = −t⇤ ⇤

σ

• c†

α,1,σdσ + H.c.

⇥ . sents the quantum dot

= t,

U

εd

TRITOPS

densit h t0 densit h t0

TRITOPS

tonian H↵ = X

i,j

h †

↵,i h↵ ij ↵,j + † ↵,i ∆↵ i j † ↵,j

i + H.c., h↵

i j = − (t↵ + i↵n↵ · σ) j,i+1 − µ↵i,j

∆↵

i j =

⇣ ˜ ∆↵ j,i+1 + ∆↵ i,j ⌘ iy ⇣ spinor †

↵,j =

⇣ c†

↵, j,↑, c† ↵, j,↑

⌘ and

• matrices. The terms of

t, ∆eiφ, λ, nL, µ t, ∆, λ, nR, µ

EFFECTIVE HAMILTONIAN II

U

εd

Heff

J,dot = HL + tφd† "γ itφd† #γ† + H.c. + Hd

HL = X

s=",#

⇣ ts˜ γ†ds + δs˜ γds ⌘

2 2

• 2

2 t" = tφ cos θ 2, t# = tφeiϕ sin θ 2, tφ = tJeiφ/4, δ" = itφeiϕ sin θ 2, δ# = itφ cos θ 2.

⇣ ⌘ ˜ γL," = cos θ 2 ˜ γ ieiϕ sin θ 2 ˜ γ†, γR," = γ, ˜ γL,# = eiϕ sin θ 2 ˜ γ + i cos θ 2 ˜ γ†, γ†

R,# = iγ,

†

↵,+ = i sgn

⇣ ↵ ˜ ∆↵ ⌘ ↵,−, ˜ †

↵,+ = −i sgn

⇣ ↵ ˜ ∆↵ ⌘ ˜ ↵,−.

nR = ˆ z

nL = (θ, ϕ)

• f Sz. S

n or

= ±1/4

SnL = ±1/4

• 7
• 6
• 5

E

• 7
• 6
• 5

E

0.5 1 1.5 2

φ/π

• 7
• 6
• 5

E

0.5 1 1.5 2

φ/π

• 1
• 0.5

0.5 1

J

θ=0

θ=0.3π

θ=π

TRITOPS-QD-TRITOPS.

U 6= 0

No signatures of transition induced by U

0 − π

• 0.1

0.1 J L = 200 L = 100 L = 50 L = 30 L = 20

• 0.1

0.1 J 0.5 1 φ/2π

• 0.1

0.1 J

Exact, U=0 Effective Hamiltonian

θ = 0

θ = π

θ = 0.3π

TRITOPS-QD-S.

0.5 1 φ/2π

• 0.1

0.1 J L = 200 L = 100 L = 50 L = 30 L = 20

0.5 1 1.5 2

φ/π

• 8
• 6
• 4

E

0.5 1 1.5 2

φ/π

• 0.2

0.2

J

Exact, U=0 Effective Hamiltonian

U 6= 0

OUTLOOK

• TRITOPS phase induced in thin films of BiSe by proximity to

s-wave superconductors.

• Zero-energy states with Sz=1/4 at the ends combine at the

junction to form 1/2-spin that screen the localized spin of the quantum dot: No transition to pi-junction!

• Signatures in Josephson junctions TRITOPS-TRITOPS and

TRITOPS-TRS. Main features well described by low-energy effective Hamiltonians.

• To do: experimental setups to realize the TRIPTOPS phase.

Xul Solar, Argentina, 1937-1963

THANK YOU!