From Integer Quantum Hall effect to Majorana Fermions in ...

A.Tagliacozzo (Universita’ di Napoli “Federico II”)

From Integer Quantum Hall effect to Majorana Fermions in Topological Materials.

P.Lucignano, G.Campagnano, F. Trani, D.Giuliano (Cz), A.Mezzacapo (now IBM Yorktown),

exp. groups of F.Tafuri (Napoli) & F.Lombardi(Chalmers)

The main idea is to convey to the audience the feeling that trying to understand the basic mathematics of Physics

can be a way towards the understanding of a basic unity of the laws of Nature.

Beppe Morandi

Bologna, 24-25 novembre 2017

Il ricordo del mio austero, rigoroso,a volte iroso, sensibile, attento,sempre generoso correlatore di tesi resta immutato nel mio cuore. Ho imparato tantissimo da Beppe professionalmente ma il privilegio di averlo frequentato, anche se per un breve periodo, da studente fuori sede e quindi anche al di là delle mura universitarie è stato e resta tuttora un'esperienza preziosa. Mi dispiace di non essere con voi ma vi abbraccio tutti con affetto.

Michela di Stasio, SISSA phd diploma 1993:

From quantum Spins to correlated fermions: a new strong coupling method

Content:

• 1-2-D models for Topological systems:

• p-wave pairing in spinless systems allows for Majorana

real fermion, zero energy excitations

• 3D TRI Topological Insulators

• helicity of topological states at the boundaries

• d-wave superconductive proximity of a wire

• Majorana bound state at a vortex core

• Fermi arcs in Weyl semimetals

from two bands, Time Reversal Invariant (TRI) Topological Insulators to Majorana excitations

The correspondence bulk-edges has been taken over in the Quantum Spin Hall 2D-systems, in the 3D-Topological Insulators which have Dirac states in the bulk gap, crossing the Fermi energy, localized at the boundaries and in Weyl semimetal.

=

Integer Hall conductance as the integral of the Berry curvature over the BZ

A non zero Chern number as an obstruction to Stokes’ Theorem over the whole BZ

e2

hC1

BZ

�xy

=e2

h

1

2⇡

Z Z

BZ

dkx

dky

Fxy

(k)

Fxy

(k) =@A

y

(k)

@kx

� @Ax

(k)

@ky

Ai

= �iX

a2 filled bands

⌧a ~k

��@

@ki

�� a ~k�

0-π π

A B

B A

π

0

��uIa(�~k)

E= ⇥

��uIIa (~k)

E

��uIIa (�~k)

E= �⇥

��uIa(~k)

E

non trivial Z2 index must be an obstruction to defining the TR-smooth gauge

between Kramers degenerate states

for Time reversal invariant systems F⇣�~k

⌘= �F

⇣~k⌘

C = 0

in half the BZ QSH : HgTe quantum well

Bulk-edge correspondence: topologically protected edge states

H =

X

k

H(k) with H(k) = ~d(k) · ~⌧

dx

= t+ + t� cos k, dy

= t� sin k, dz

= 0

C1 =

1

4⇡

Z

S2

d✓d' ~g(n) · [ @✓

~g(n)⇥ @'

~g(n) ] =

Z

S2

d⌦/4⇡.~g =~d

|~d |

H 0(x) = [�ivF

�y

@x

+m(x)�x

] 0(x) = 0

m(x) (i �y

+ �x

)�0 = 0 :

✓0 20 0

◆�0 = 0, ! �0 =

✓10

◆.

zero energy real fermion at the boundary in the SSH model

spin reverses between ! �†⇥k0

, ��⇥k0

Note that in BCS:

u⇥�k = vk, v⇥�k = �uk

�k0 = uk ck⇥ � vk c†�k⇤

�k1 = uk c�k⇤ + vk c†k⇥ Time Reversal implies that:

so that

neutral fermion requires superconductivity

model has to be spinless ! �⇥k = u⇥kc⇥k � v⇥kc�⇥k†

�⇥k† = u⇥⇥kc⇥k

† � v⇥⇥kc�⇥k provided:

�†⇥k

= ��⇥kis:

u⇤�~k

= �v~k, v⇤�~k= �u~k

spinless requires p-wave pairing !

spinful singlet (s-wave) pairing �⇣c†�k#c

†k" � c†�k"c

†k#

⌘

�⇣c†k"c

†�k# � c†k#c

†�k"

⌘

k -k

k "! �k #

�[k]⇣c†kc

†�k + c†�kc

†k

⌘spinless (p-wave) pairing

Δ[-k] = -Δ[k]

Hk = h(k) · ~� hx,y

(�k) = �hx,y

(k), hz

(�k) = hz

(k)

H = �µX

x

c†x

cx

� 1

2

X

x

⇥t c†

x

cx+1 +� ei�c

x

cx+1 +H.c.

⇤

Hk = h(k) · ~�

h(0) = s0z, h(⇡) = s⇡ z

⌫ = s0 s⇡

Z2 topological invariance

Kitaev model (’97)

J. Alicea, Rev. Mod. Phys. 2012A. Kitaev, Annals of Physics 2003

�T = �� HMF

=

✓h ��† �hT

◆

�x

HMF

�x

=

✓�hT �†

� h

◆⌘ �H⇤

MF

HMF

n

= ✏n

n

, HMF

�x

⇤n

= �2x

HMF

�x

⇤n

= ��x

H⇤MF

⇤n

= �✏n

�x

⇤n

n

=

✓un

vn

◆! �

x

⇤n

=

✓v⇤n

u⇤n

◆

�x

0⇤ = 0 ! 0 =

✓u0 + v⇤0v0 + u⇤

0

◆⌘

✓w0

w0⇤

◆

�†0 = w0 a†0 + w⇤

0 a0 ⌘ �0

Bogolubov-de Gennes Hamiltonian

with

Majoranas come always in pairs and give two degenerate ground states of opposite parity

admits solution with provided the phase acquired in the loop is χ= π . This is indeed the case at surfaces of Topological Insulators

✏ = �0 py

bound Andreev state in 2D when proximity is p-ip

rhe

(✏, ~p) = ↵ e�i�(~p)

reh

(✏, ~p) = ↵ ei�(~p)

↵ = e�i arccos(✏/|�(~p)|) e�i�/2

� (~p) = � arctan py

/px

� (~p) = �0 (px � ipy

) ⌘ �0 · p · ei�(~p)

1 = reh

(✏, px

, py

) rhe

(✏,�px

, py

)

= e�2i arccos(✏/�(~p))�i�(�p

x

,p

y

)+i�(p

x

,p

y

) e�i�

Bohr-Sommerfeld quantization:.

N

bound state of two Majorana in a π-Josephson Junction (1D)

Capitolo 2

Proprietà elettroniche diisolanti topologici 3D

2.1 Modello per gli isolanti topologici 3D.Il modello Hamiltoniano presentato e discusso in tale paragrafo fa uso dei risultatiteorici circa l’effetto dell’accoppiamento Spin-Orbita sugli stati elettronici in mate-riali caratterizzati da un elevato numero atomico Z.Ricordando la procedura esposta nella costruzione di un modello efficace alla de-scrizione di stati di edge presenti in un isolante topologico bidimensionale, si evi-denzia come il fenomeno, centrale nella comprensione delle proprietà del sistema,dell’inversione tra bande ad opera del forte accoppiamento Spin-Orbita, rientras-se nell’ambito di una più generale descrizione quantistico-relativistica del sistemastesso. La dispersione di fermioni a massa nulla1 ottenuta nel risolvere l’equazionedi Dirac, suggerisce l’estensione del modello (ora facilmente generalizzabile) al casodi stati di conduzione in isolanti topologici tridimensionali. Se da un lato, infatti,l’inversione tra la banda di conduzione e quella di valenza, rende chiara la necessitàdi una base di stati ibridi in cui si tiene conto sia del possibile orientamento dellospin che della differente parità orbitale delle stesse, dall’altro è proprio il formalismospinoriale della meccanica quantistica relativistica a giustificare le dimensioni dellospazio in cui viene modellizzato il materiale.Dopo tali delucidazioni risulta chiaro che una base che soddisfi tali requisiti è deltipo :

( g" , u" , g# , u# ) (2.1)

in cui u� è uno stato dispari per inversione spaziale ( ~r ! �~r ) , mentre g�

rappresenta uno stato pari per la medesima trasformazione.E’ ora, in tale base, possibile esibire una rappresentazione dell’Hamiltoniana efficace[9], che sia adatta alla descrizione delle proprietà elettroniche del sistema :

H0[~r] = ~vF

0

BB@

�M� µ i@z 0 i(@x � i@y)i@z M� µ i(@x � i@y) 0

0 i(@x + i@y) �M� µ �i@zi(@x + i@y) 0 �i@z M� µ

1

CCA (2.2)

Dove vF rappresenta la velocità di Fermi2. del sistema .1D’ora in avanti per uniformarsi alla letteratura si utilizzerà la nomenclatura inglese Dirac

massless fermion.2La velocità di Fermi è la velocità dei costituenti dei un sistema fermionico sulla superficie, nello

spazio delle fasi, cui corrisponde la massima energia assumibile allo stato fondamentale (Energia

15

Two bands model Hamiltonian for TI

Capitolo 2



( g" , u" , g# , u# ) (2.1)



H0[~r] = ~vF

0

BB@



1

CCA (2.2)




15

3

Here g�

is even for ~k ! �~k, while u�

is odd for ~k ! �~k. Other important symmetries that could be presentare the particle-hole symmetry ⌅ and the chiral symmetry �. ⌅ = �i�y⌧xK is an antiunitary symmetry suchthat H(k) = ⌅H(�k)⌅�1. The chiral symmetry � = i · ⇥⌅ is a unitary transformation that changes the sign ofthe Hamiltonian without reversing the sign of k. In presence of a non zero chemical potential µ the Hamiltonianof (1) has neither the particle-hole symmetry ⌅ nor the chiral symmetry �. In fact, ⌅T H(µ)⌅ = �H(�µ) and�†H(µ)� = H(�µ). Hence it belongs to the class AII of the Altland � Zirnbauer classification[? ]: ⇥2 = �1,⌅2 = 0, �2 = 0.

Bulk eigenfunctions correspond to E = ±pM2 + k2 + k2

z

where k2 = k2x

+ k2y

= k+k� and k± = kx

± iky

. andwavefunctions

��E,~k, 1E

=1N

0BB@E + M

kz

0k+

1CCA eikzzeikk·rk ,��E,~k, 2

E=

1N

0BB@0

k�E + M�k

z

1CCA eikzzeikk·rk (2)

The relative sign between M and C1,2 qualifies the insulator as being topologically non trivial or trivial. In AppendixC we derive the boundary states for a flat surface orthogonal to the z�axis, with the simplifying assumption thatC1,2 = 0 and that M changes sign at the boundary. This provides localized states at the boundary which decayexponentially with the same decay length 1/M on both sides of the boundary. The simplifying assumption allows foran analytical treatment of the matching condition[? ]. The reduced Hamiltonian for the boundary states conservestheir helicity, ~� · ~k.

Localized states at the surface could be:

��1; kk↵

=1N

0BB@M + E

iM sign(z)0

k+

1CCA e�M |z|eikk·rk ,��2; kk

↵= 1

N

0BB@0

k�M + E

�iM sign(z)

1CCA e�M |z|eikk·rk eigenvalue E = ±k (3)

They are derived from eigenstates of the Hamiltonian in k�space, continued to imaginary kz

:

H[M ; z] ⇧R

=12

0BB@M i M sign(z) 0 k�

i M sign(z) �M k� 00 k+ M �i M sign(z)

k+ 0 �iM sign(z) �M

1CCA . (4)

Note that the parity g/u is respected by the given form. However, so as they stand, they are not continuous at z = 0.Hence they are not a physical state. Let us introduce the time reversed states where ⇥ = i�

y

⌦ I2⇥2K. Here K is thecomplex conjugation, �

i

are the Pauli matrices in the spin space and ⌧i

are the Pauli matrices acting in the orbitalspace. I2⇥2 is the 2⇥ 2 identity. The time reversed states ( z > 0) :

��⇥1; kk↵

=1N

0BB@0

k��(M + E)

iM

1CCA e�Mze�ikk·rk = � ��2;�kk↵

,��⇥2; kk

↵=

1N

0BB@M + E

iM0�k+

1CCA e�Mze�ikk·rk =��1;�kk

↵eigenvalue E = ±k (5)

Note that the Hamiltonian of Eq.(??) is non real so that these states are the eigenstates of the transformed Hamiltonian⇥H⇥† = H(�kk, z): The time reversed states have a factor e�Mze�ikk·rk . Let P be the space inversion operator~r ! �~r. The application of P changes rk ! �rk and z ! �⇣. The states P

��⇥1/2±; kk↵

have a factor eMzeikk·rk

which converges for z < 0. Note that P⇥H⇥†P † = �H(�M). Therefore it is a good Hamiltonian for matching thesurface wavefunction for z > 0 to that for z < 0 with the correct exponentially decaying behavior on the two sides.But, given i = 1, 2: ��P⇥i; kk

↵= P⇥H⇥†P † ��P⇥i; kk

↵= �H(�M)

��P⇥i; kk↵

(6)

so that if��i; kk↵ is eigenstate of H(M) with eigenvalue ±k,

��P⇥i; kk↵

is eigenstate of H(�M) with eigenvalue ⌥k.Il modello, inclusivo delle derivate seconde presenta stati di superficie che si annullano al bordo e decadono dentrol’TI . Il cono di Dirac prevede un’avvolgimento dello spin in modo chirale L/R per la particella/buca, che decade

Bulk bands dispersion

TR invariant

p-h symmetric (μ=0)

transforms p-h (μ=0) conserving helicity

� = i ⇥⌅

Zhang et al. 2009minimal model in the continuum limit:

at the Γ point_

is the chirality

⇥ = I ⌦ i⌧y

K, ⌅ = �i�y

⌦ ⌧x

K, � = �y

⌦ ⌧z

⇥H0(~k)⇥�1 = H0(�~k)

�H0(~k)��1 = �H0(�~k)

⌅H0(~k)⌅�1 = �H0(~k)

Projecting H onto the L,R

chiral space: ⇧L =

(1� �)2

⇧R =(1 + �)

2

H0[M ] ⇧L

=✓

0 (�x

� i �z

) k�(�

x

+ i �z

) k+ 0

◆

H0[�M ] ⇧R

=✓

0 (�x

+ i �z

) k�(�

x

� i �z

) k+ 0

◆

E = k

+

Capitolo 2



( g" , u" , g# , u# ) (2.1)



H0[~r] = ~vF

0

BB@



1

CCA (2.2)




15

chirality fixes an internal structure:

k± = kx

± i ky

g" � i u", u# + i g#

g" + i u", u# � i g#

L-chiral

R-chiral

H[M ; z = 0]⇧L = �H[�M ; z = 0]⇧R =

✓0 k�k+ 0

◆= ~� · ~k

states localised at surfaces appear in the insulating gap

bulk bands inversion with Sb

Rashba SO

in 2DEG

from metal to semiconductor

magnitude !1 that initially points in the xx1 direction,then tilts (arbitrarily slowly) slightly toward xx2, where xx1and xx2 are orthogonal in-plane directions, it follows fromthe linear response limit of Eq. (2) that

"hdn2dt

! nz!1 " !dnz=dt;

"hdnzdt

! #!1n2 # !dn2=dt" !2;(3)

where !2 ! ~!! $ xx2. Solving these inhomogeneouscoupled equations using a Green function technique, itfollows that to leading order in the slow-time depen-dences n2%t& ! !2%t&=!1, i.e., the xx2 component of thespin rotates to follow the direction of the field, and that

nz%t& !1

!21

"hd!2

dt: (4)

Our intrinsic spin Hall effect follows from Eq. (4).When a Bloch electron moves through momentum space,its spin orientation changes to follow the momentum-dependent effective field and also acquires a momentum-

dependent zz component. We now show that in the case ofRashba spin-orbit coupling, this effect leads to an intrin-sic spin-Hall conductivity that has a universal value in thelimit of zero quasiparticle spectral broadening.

For a given momentum ~pp, the spinor originally pointsin the azimuthal direction. An electric field in thexx direction ( _ppx ! #eEx) changes the y component ofthe ~pp-dependent effective field. Applying the adiabaticspin dynamics expressions explained above, identifyingthe azimuthal direction in momentum space with xx1 andthe radial direction with xx2 we find that the z componentof the spin direction for an electron in a state withmomentum ~pp is

nz; ~pp ! #e "h2pyEx

2"p3 : (5)

(Linear response theory applies for eExrs ' !1, where rsis the interparticle spacing in the 2DES.) Summing overall occupied states the linear response of the zz spin-polarization component vanishes because of the odd de-pendence of nz on py, as illustrated in Fig. 1, but the spincurrent in the yy direction is finite.

The Rashba Hamiltonian has two eigenstates for eachmomentum with eigenvalues E( ! p2=2m)!1=2; thediscussion above applies for the lower energy (labeled" for majority spin Rashba band) eigenstate while thehigher energy (labeled #) eigenstate has the oppositevalue of nz; ~pp. Since !1 is normally much smaller thanthe Fermi energy [17], only the annulus of momentumspace that is occupied by just the lower energy bandcontributes to the spin current. In this case we find thatthe spin current in the yy direction is [23]

js;y !Z

annulus

d2 ~pp%2# "h&2

"hnz; ~pp2

py

m! #eEx

16#"m%pF" # pF#&;

(6)

where pF" and pF# are the Fermi momenta of the ma-jority and minority spin Rashba bands. We find that whenboth bands are occupied, i.e., when n2D > m2"2=# "h4 *n+2D, pF" # pF# ! 2m"= "h and then the spin Hall (sH)conductivity is

$sH * # js;yEx

! e8#

; (7)

independent of both the Rashba coupling strength and ofthe 2DES density. For n2D < n+2D the upper Rashba bandis depopulated. In this limit pF# and pF" are the interiorand exterior Fermi radii of the lowest Rashba split band,and $sH vanishes linearly with the 2DES density:

$sH ! e8#

n2Dn+2D

: (8)

The intrinsic spin Hall conductivity of the Rashbamodel can also be evaluated by using transport theoriesthat are valid for systems with multiple spin-orbit spit

FIG. 1 (color online). (a) The 2D electronic eigenstates in aRashba spin-orbit coupled system are labeled by momentum(green or light gray arrows). For each momentum the twoeigenspinors point in the azimuthal direction (red or darkgray arrows). (b) In the presence of an electric field theFermi surface (circle) is displaced an amount jeExt0= "hj attime t0 (shorter than typical scattering times). While movingin momentum space, electrons experience an effective torquewhich tilts the spins up for py > 0 and down for py < 0,creating a spin current in the y direction.

P H Y S I C A L R E V I E W L E T T E R S week ending26 MARCH 2004VOLUME 92, NUMBER 12

126603-2 126603-2

Capitolo 2. Proprietà elettroniche di isolanti topologici 3D – 2.4. Evidenze sperimentali 29

Tale parametro è sperimentalmente identificabile nella concentrazione di bismu-to presente in lega. Aumentando la concentrazione dell’elemento più pesante, ilmaggior contributo dell’interazione Spin-Orbita, comporta una fenomenologia nondissimile all’inversione tra la banda di valenza e la banda di conduzione, osservatanelle strutture a confinamento quantistico del HgTe e CdTe.

Figura 2.7: Rappresentazione schematica della struttura a bande del Bi1�xSbx, che evolve da uncomportamento metallico per x < 0, 07, ad uno semiconduttivo per 0, 07 < x < 0, 22, tornando alprimo per x > 0, 22.

Alla concentrazione di Antimonio xSb ⇡ 0, 04 l’ampiezza della gap, si annulla,presentando caratteristici stati cui corrisponde una dispersione lineare (Cono di Di-rac).La complessa struttura delle bande [16] rende difficile la descrizione di tali stati me-diante il modello sin qui presentato. L’ampiezza ridotta della gap energetica in talileghe, ne ha comportato inoltre, un limitato sviluppo applicativo. Tali motivazionihanno portato alla ricerca di nuovi materiali [17], generalmente identificati comeIsolanti topologici di seconda generazione. Tale nuova classe di materiali includeuna variegata serie di composti cristallini, la cui struttura a bande viene analizzatamediante tecniche di spettroscopia ad effetto tunnel. L’ STM ( Scanning Tunne-ling Microscopy ) permette di acquisire informazioni sulla densità locale degli statilocalizzati ( LDOS ) sulla superfice del campione.

Figura 2.8: Dispersione energetica della LDOS per Sb2Se3(a), Sb2Te3(b), Bi2Se3(c), Bi2Te3(d).

Proprio mediante STM, sulle superfici di composti come Bi2Te3, Sb2Te3, si indi-vidua una struttura a bande facilmente identificabile con quella prevista dal modelloteorico analizzato in tale lavoro di tesi.

10

M

! M

-0.4

-0.2

0

0.2

0.4

-0.6

-0.4

-0.2

0.0

0.2

0.30.20.10.0-0.1-0.2-0.3

Deg

1.00.50.0-0.5

0.2

0.0

-0.2

-0.4

-0.6

0.2

0.0

-0.2

-0.5 0.0 0.5 1.0

k (Å��)x-1

k(Å

��)y

-1E

��(eV

)B

C

E

D

e��RS

! M

SS h�RS e�RS

h�RS

! M

highlow

BS

-0.4

-0.2

0

0.2

0.4

"

Band�structure�calculation

2

"10.0

-0.4

0.4

EF

n =�-1M

# #

SS

-0.1

-0.2

-0.3

0.0

E��(

eV)

B

T H L

!

M

!M

K

K

=�1�topology�(BiSb)$0

-0.5 0.0 0.5 1.0

k (Å��)x-1

k (Å��)x-1

T H L

-0.3

-0.2

-0.1

0.0

-0.2 -0.1 0.0 0.1 0.2

k (Å��)y-1 Sb�(111)�topology

%&'''('''!''')'''&

E��(

eV)

B

EF

! Mkx

! Mkx

EB

EF

EB

A

Ls

La

La

Ls

=�0�topology�(Au-like)$0BF G

TopologicalDirac�cone

FIG. 7: Topological character of parent compound revealed on the (111) surface states. Schematic of the bulk bandstructure (shaded areas) and surface band structure (red and blue lines) of Sb near EF for a (A) topologically non-trivial and(B) topological trivial (gold-like) case, together with their corresponding surface Fermi surfaces are shown. (C) Spin-integratedARPES spectrum of Sb(111) along the Γ-M direction. The surface states are denoted by SS, bulk states by BS, and the hole-likeresonance states and electron-like resonance states by h RS and e− RS respectively. (D) Calculated surface state band structureof Sb(111) based on the methods in [25, 40]. The continuum bulk energy bands are represented with pink shaded regions, andthe lines show the discrete bands of a 100 layer slab. The red and blue single bands, denoted Σ1 and Σ2, are the surface statesbands with spin polarization ⟨P ⟩ ∝ +y and ⟨P ⟩ ∝ −y respectively. (E) ARPES intensity map of Sb(111) at EF in the kx-kyplane. The only one FS encircling Γ seen in the data is formed by the inner V-shaped SS band seen in panel-(C) and (F).The outer V-shaped band bends back towards the bulk band best seen in data in panel-(F). (F) ARPES spectrum of Sb(111)along the Γ-K direction shows that the outer V-shaped SS band merges with the bulk band. (G) Schematic of the surface FSof Sb(111) showing the pockets formed by the surface states (unfilled) and the resonant states (blue and purple). The purely

surface state Fermi pocket encloses only one Kramers degenerate point (kT ), namely, Γ(=kT ), therefore consistent with the ν0= 1 topological classification of Sb which is different from Au (compare (B) and (G)). As discussed in the text, the hRS ande−RS count trivially. [Adapted from D. Hsieh et al., Science 323, 919 (2009) (Completed and Submitted in 2008) [16]].

be lifted by the spin-orbit interaction [19-21]. However,according to Kramers theorem [16], they must remainspin degenerate at four special time reversal invariantmomenta (kT = Γ, M) in the surface BZ [11], which forthe (111) surface of Bi1−xSbx are located at Γ and threeequivalent M points [see Fig.5(A)].

Depending on whether ν0 equals 0 or 1, the Fermi sur-face pockets formed by the surface bands will enclose thefour kT an even or odd number of times respectively. Ifa Fermi surface pocket does not enclose kT (= Γ, M), itis irrelevant for the topology [10, 40]. Because the wavefunction of a single electron spin acquires a geometricphase factor of π [43? ] as it evolves by 360◦ in mo-

mentum space along a Fermi contour enclosing a kT , anodd number of Fermi pockets enclosing kT in total im-plies a π geometrical (Berry’s) phase [10]. In order torealize a π Berry’s phase the surface bands must be spin-polarized and exhibit a partner switching [10] dispersionbehavior between a pair of kT . This means that any pairof spin-polarized surface bands that are degenerate at Γmust not re-connect at M, or must separately connectto the bulk valence and conduction band in between Γand M. The partner switching behavior is realized in Fig.

5(C) because the spin down band connects to and is de-generate with different spin up bands at Γ and M. Thepartner switching behavior is realized in Fig. 7(A) be-cause the spin up and spin down bands emerging fromΓ separately merge into the bulk valence and conductionbands respectively between Γ and M.We first investigate the spin properties of the topo-

logical insulator phase [16]. Spin-integrated ARPES [41]intensity maps of the (111) surface states of insulatingBi1−xSbx taken at the Fermi level (EF ) [Figs 5(D)&(E)]show that a hexagonal FS encloses Γ, while dumbbellshaped FS pockets that are much weaker in intensity en-close M. By examining the surface band dispersion be-low the Fermi level [Fig.5(F)] it is clear that the centralhexagonal FS is formed by a single band (Fermi crossing1) whereas the dumbbell shaped FSs are formed by themerger of two bands (Fermi crossings 4 and 5) [14].This band dispersion resembles the partner switching

dispersion behavior characteristic of topological insula-tors. To check this scenario and determine the topo-logical index ν0, we have carried out spin-resolved pho-toemission spectroscopy. Fig.5(G) shows a spin-resolvedmomentum distribution curve taken along the Γ-M direc-tion at a binding energy EB = −25 meV [Fig.5(G)]. The

12

-0.2 0.0 0.20.1-0.1

0.0

-0.1

-0.2

← −Μ Γ Μ→E

(e

V)

B

k (Å )x-1

B

A

l2 l1 r1 r2

-0.2 0.0 0.20.1-0.1-1

Inte

nsity

(arb

. uni

ts)

0.3

C D

k (Å )x

0.0

-0.2

-0.4

0.2

0.4

Spi

n po

lariz

atio

n

0-1 1 0 1

-1

0

1

-1

0

1

F

Px

Py Pz

Pin plane

Inte

nsity

(arb

. uni

ts)

Ek (Å )x

PP

y �

z �

-0.2 0.0 0.20.1-0.1-1

E = -30 meVBr2

r1

l1

l2

e beamaccelerating optics

hν

Au foil40 kV

e

yx

z

x� y�

z�

Mott spin detector

θ

sample

spin || y

spin ||- y

k (Å )x

-0.2 0.0 0.20.1-0.1-1

E = -30 meVB

0

10

20

0

2

4

Momentum distribution of spin

I

I

y

y

Γ

ΣΣ

2

1

r1l1

l2

l1

r1

r2

FIG. 9: Spin-texture of topological surface states and topological chirality. (A) Experimental geometry of the spin-resolved ARPES study. At normal emission (θ = 0◦), the sensitive y′-axis of the Mott detector is rotated by 45◦ from thesample Γ to −M (∥ −x) direction, and the sensitive z′-axis of the Mott detector is parallel to the sample normal (∥ z). (B)Spin-integrated ARPES spectrum of Sb(111) along the −M-Γ-M direction. The momentum splitting between the band minimais indicated by the black bar and is approximately 0.03 A−1. A schematic of the spin chirality of the central FS based onthe spin-resolved ARPES results is shown on the right. (C) Momentum distribution curve of the spin averaged spectrum atEB = −30 meV (shown in (B) by white line), together with the Lorentzian peaks of the fit. (D) Measured spin polarizationcurves (symbols) for the detector y′ and z′ components together with the fitted lines using the two-step fitting routine [45].(E) Spin-resolved spectra for the sample y component based on the fitted spin polarization curves shown in (D). Up (down)triangles represent a spin direction along the +(-)y direction. (F) The in-plane and out-of-plane spin polarization componentsin the sample coordinate frame obtained from the spin polarization fit. Overall spin-resolved data and the fact that the surfaceband that forms the central electron pocket has ⟨P ⟩ ∝ −y along the +kx direction, as in (E), suggest a left-handed chirality(schematic in (B) and see text for details). [Adapted from D. Hsieh et al., Science 323, 919 (2009) [16]].

Berry’s phase.In a conventional spin-orbit metal such as gold, a free-

electron like surface state is split into two parabolic spin-polarized sub-bands that are shifted in k-space relativeto each other [46]. Two concentric spin-polarized Fermisurfaces are created, one having an opposite sense of in-plane spin rotation from the other, that enclose Γ. Sucha Fermi surface arrangement, like the schematic shownin figure 7(B), does not support a non-zero Berry’s phase

because the kT are enclosed an even number of times (2for most known materials).However, for Sb, this is not the case. Figure 7(C)

shows a spin-integrated ARPES intensity map of Sb(111)from Γ to M. By performing a systematic incident pho-ton energy dependence study of such spectra, previouslyunavailable with He lamp sources [47], it is possible toidentify two V-shaped surface states (SS) centered atΓ, a bulk state located near kx = −0.25 A−1 and res-onance states centered about kx = 0.25 A−1 and M thatare hybrid states formed by surface and bulk states [41]

(SM). An examination of the ARPES intensity map ofthe Sb(111) surface and resonance states at EF [Fig.7(E)]reveals that the central surface FS enclosing Γ is formedby the inner V-shaped SS only. The outer V-shaped SSon the other hand forms part of a tear-drop shaped FSthat does not enclose Γ, unlike the case in gold. Thistear-drop shaped FS is formed partly by the outer V-shaped SS and partly by the hole-like resonance state.The electron-like resonance state FS enclosing M doesnot affect the determination of ν0 because it must be dou-bly spin degenerate (SM). Such a FS geometry [Fig.7(G)]suggests that the V-shaped SS pair may undergo a part-ner switching behavior expected in Fig.7(A). This behav-ior is most clearly seen in a cut taken along the Γ-K direc-tion since the top of the bulk valence band is well belowEF [Fig.7(F)] showing only the inner V-shaped SS cross-ing EF while the outer V-shaped SS bends back towardsthe bulk valence band near kx = 0.1 A−1 before reach-ing EF . The additional support for this band dispersionbehavior comes from tight binding surface calculations

!− 1"!k = #nk=1;nj!k=0,1

"i=!n1n2n3". !2.12"

!0 is clearly independent of the choice of primitivereciprocal-lattice vectors bk in Eq. !2.3". !!1!2!3" are not.However, they may be viewed as components of a mod 2reciprocal-lattice vector

G! = !1b1 + !2b2 + !3b3. !2.13"

This vector may be explicitly constructed from the "i’s asfollows. A gauge transformation of the form of Eq. !2.7" canchange the signs of any four "i for which the #i lie in thesame plane. Such transformations do not change the invari-ants !!1!2!3". By a sequence of these transformations, it isalways possible to find a gauge in which "i=−1 for at mostone nonzero #i. Define #*=#i if there is one such point. Ifthere is none, then set #*=0. In this gauge, the mod 2reciprocal-lattice vector is G!=2#*. The remaining invariant!0 is then determined by "i at #i=0.

As we will explain below in Sec. II C 2, the latter invari-ants !k are not robust in the presence of disorder. We refer tothem as “weak” topological invariants. On the other hand, !0is more fundamental and distinguishes the “strong” topologi-cal insulator.

Formulas !2.10"–!2.12" are a bit deceptive because theyappear to depend solely on a local property of the wavefunctions. Knowledge of the global structure of $unk%, how-ever, is necessary to construct the continuous gauge requiredto evaluate Eq. !2.4". The existence of globally continuouswave functions is mathematically guaranteed because theChern number for the occupied bands vanishes due to time-reversal symmetry. However, determining a continuousgauge is not always simple.

C. Surface states

The spectrum of surface !or edge" states as a function ofmomentum parallel to the surface !or edge" is equivalent tothe spectrum of discrete end states of the cylinder as a func-tion of flux. Figure 2 schematically shows two possible endstate spectra as a function of momentum !or equivalentlyflux" along a path connecting the surface time-reversal in-variant momenta $a and $b. Only end states localized at oneof the ends of the cylinder are shown. The shaded regiongives the bulk continuum states. Time-reversal symmetry re-quires the end states at $a and $b to be twofold degenerate.However, there are two possible ways these degenerate statescan connect up with each other. In Fig. 2!a", the Kramerspairs “switch partners” between $a and $b, while in Fig.2!b", they do not.

These two situations are distinguished by the Z2 invariantcharacterizing the change in the time-reversal polarization ofthe cylinder when the flux is changed between the valuescorresponding to $a and $b. Suppose that at the flux corre-sponding to $a the ground state is nondegenerate, and alllevels up to and including the doublet %a1 are occupied. If theflux is adiabatically changed to $b, then for Fig. 2!a", thedoublet %b1 is half filled, and the ground state has a twofoldKramers degeneracy associated with the end. For Fig. 2!b",

on the other hand, the ground state remains nondegenerate.This construction establishes the connection between the sur-face states and the bulk topological invariants. When &a&b=−1 !+1", the surface spectrum is like Fig. 2!a" !2b".

It follows that when &a&b=−1 !+1", a generic Fermi en-ergy inside the bulk gap will intersect an odd !even" numberof surface bands between $a and $b. Thus, when &a&b=−1, the existence of surface states is topologically protected.The details of the surface-state spectrum will depend on theHamiltonian in the vicinity of the surface. In Fig. 2, we haveassumed that surface bound states exist for all momenta. Thisneed not be the case, since it is possible that by varying thesurface Hamiltonian, the degenerate states at $a and $b canbe pulled out of the gap into the bulk continuum states. This,however, does not change our conclusions regarding thenumber of Fermi energy crossings. When &a&b=−1, therestill must exist surface band traversing the energy gap.

In the two-dimensional quantum spin-Hall phase, &1&2=−1, and there will be an odd number of pairs of Fermipoints.9,10 In the simplest case where there is a single pair,the states at the Fermi energy will be spin filtered in thesense that the expectation value of the spin in the right andleft moving states will have opposite sign. These states arerobust in the presence of weak disorder and interactions be-cause time-reversal symmetry forbids elastic backscattering.Strong interactions, however, can lead to an electronic insta-bility that opens a gap.31,32 The resulting ground state, how-ever, breaks time-reversal symmetry.

In three dimensions, the Kramers degenerate band cross-ings that occur at $a in the surface spectrum are two-dimensional Dirac points. While such Dirac points will oc-cur in any time-reversal invariant system with spin-orbitinteractions, the nontrivial structure here arises from the wayin which the Dirac points at different $a are connected toeach other. This is determined by the relative signs of thefour &a associated with any surface.

In Fig. 3, we depict four different topological classes forthree-dimensional band structures labeled according to !0;

FIG. 2. Schematic representations of the surface energy levels ofa crystal in either two or three dimensions as a function of surfacecrystal momentum on a path connecting $a and $b. The shadedregion shows the bulk continuum states, and the lines show discretesurface !or edge" bands localized near one of the surfaces. TheKramers degenerate surface states at $a and $b can be connected toeach other in two possible ways, shown in !a" and !b", which reflectthe change in time-reversal polarization &a&b of the cylinder be-tween those points. Case !a" occurs in topological insulators andguarantees that the surface bands cross any Fermi energy inside thebulk gap.

TOPOLOGICAL INSULATORS WITH INVERSION SYMMETRY PHYSICAL REVIEW B 76, 045302 !2007"

045302-5

L.Fu, C.L.Kane

⇧L

��a,+; kk↵/

2

664e�i✓~k

✓i1

◆

1

✓1i

◆

3

775 e�R z dz M(z); ⇧L

��a,⇥+; kk↵/ �i

2

664

✓i1

◆

ei✓~k✓

1i

◆

3

775 . e�R z dz M(z)

s-wave proximity 2D boundary TI

a pairing appears� (px � ipy)

eigenvalues are: ±�

(±v|p|� µ)2 + �2

�(z) = ��g��g⇥⇥ = ��u��u⇥⇥

Ψ =

HBdG =

✓h0(k)� µ �

�† �hT0 (�k) + µ

◆

[ k", k#,� †�k#,

†�k"]

U : [�kL,�kR,�[��k ✓L]†, [��k ✓R]

†]

�kL =� k" + ei✓ k#

�, �kR =

�e�i✓ k" � k#

�

[H�]~k = ��he�i✓k �†kL [��k ✓L]

† � ei✓k �†kR [��k ✓R]†i+ h.c.

c~k ⌘

c~k"c~k#

�! 1p

2

1

ei✓k

�; �c�~k

† ⌘

c�~k#†

�c�~k"†

�! 1p

2

e�i✓k

1

�.

Hnew =1

2

⇣c~k

† � c�~k

⌘

.

✓[vk � µ] �e�i✓k

�ei✓k � [vk � µ]

◆✓c~k

�c�~k†

◆

Read & Green 2000, Fu & Kane 2008

3 D exchange of helicities between the two surfaces:

Tchoosing one single helical boundary state

vk ⇠ µ

Capitolo 3. Trasporto elettronico in isolanti topologici. – 3.2. Backscattering 35

buti:

�n(t) =

Z ⌧

0i hn;R(t)| d

dt|n;R(t)i dt = �n(t) =

Z R(⌧)

R(0)ihn;R(t)| @

@Ri|n;R(t)i dRi

(3.8)Nel nostro caso considerando l’autostato normalizzato:

| (0)i = |L1i = 1p2

✓ie�i✓

1

◆(3.9)

Il termine geometrico in Eq.(3.8) si esprime, nel considerare una rotazione oraria,come :

�n(t) =

Z ✓(⌧)

✓(0)i hL1| @

@✓|L1i d✓ (3.10)

Dunque esplicitandone i termini :

�n(t) =1

2

Z ⇡

0i��iei✓ 1

� @

@✓

✓ie�i✓

1

◆d✓ =

1

2

Z ⇡

0i��iei✓ 1

�✓ei✓0

◆d✓ =

=

1

2

Z ⇡

0i(�i) d✓ =

1

2

[✓]⇡0 =

⇡

2

Da cui tornando alla Eq.3.6

| (t)i0 = |�i = ei⇡

2 | (0)i (3.11)

Considerando ora una rotazione antioraria, nello spazio dei parametri, risulta im-mediato calcolare il fattore geometrico in maniera analoga a quanto appena fattoper la rotazione nel senso opposto:

�n(t) =1

2

Z �⇡

0i hL1| @

@✓|L1i d✓ = �⇡

2

Da cui si ricava:| (t)i00 = | i = e�i⇡

2 | (0)i (3.12)

La probabilità che lo stato subisca backscattering, è data dal modulo quadro dellafunzione d’onda, ottenuta sommando entrambi i contributi :

| (t)i = | i+ |�i = �ei

⇡

2

+ e�i⇡

2

� | (0)i = ei⇡

2

�e�i⇡

2

+ 1

� | (0)i = 0

In tale proceduta non vengono fatte precisazioni sulla rappresentazione spazialedegli stati (la descrizione può essere identicamente riaffrontata in termini delle fun-zioni d’onda). Essa viene trascurata in quanto implicitamente considerata identicaa quella rappresentata nella base delle onde piane introdotta nell’ambito della pre-sentazione del modello.Gli stati in Eq.3.11 e Eq.3.12, propagandosi lungo una medesima direzione, pre-sentano in virtù della chiralità del sistema, un identico orientamento del vettore diSpin. E’ proprio questa la motivazione per cui tale variabile viene resa implicitanella trattazione.E’ stato finora reso esplicito quanto anticipato nel Capitolo 1, e cioè la presenza intali materiali, di una robustezza globale che rende gli stati di conduzione insensibilialle impurità localizzate nel sistema. Tale robustezza si traduce, infatti, nella di-struttività con cui le funzioni d’onda interferiscono dopo aver effettuato rotazioni,Figura 3.2, nello spazio dei momenti, tali da invertire la direzione di propagazione.


buti:

�n(t) =

Z ⌧

0i hn;R(t)| d


Z R(⌧)

R(0)ihn;R(t)| @

@Ri|n;R(t)i dRi


| (0)i = |L1i = 1p2

✓ie�i✓

1

◆(3.9)


�n(t) =

Z ✓(⌧)

✓(0)i hL1| @

@✓|L1i d✓ (3.10)


�n(t) =1

2

Z ⇡

0i��iei✓ 1

� @

@✓

✓ie�i✓

1

◆d✓ =

1

2

Z ⇡

0i��iei✓ 1

�✓ei✓0

◆d✓ =

=

1

2

Z ⇡

0i(�i) d✓ =

1

2

[✓]⇡0 =

⇡

2


| (t)i0 = |�i = ei⇡

2 | (0)i (3.11)


�n(t) =1

2

Z �⇡

0i hL1| @

@✓|L1i d✓ = �⇡

2


2 | (0)i (3.12)


| (t)i = | i+ |�i = �ei

⇡

2

+ e�i⇡

2

� | (0)i = ei⇡

2

�e�i⇡

2

+ 1

� | (0)i = 0



buti:

�n(t) =

Z ⌧

0i hn;R(t)| d


Z R(⌧)

R(0)ihn;R(t)| @

@Ri|n;R(t)i dRi


| (0)i = |L1i = 1p2

✓ie�i✓

1

◆(3.9)


�n(t) =

Z ✓(⌧)

✓(0)i hL1| @

@✓|L1i d✓ (3.10)


�n(t) =1

2

Z ⇡

0i��iei✓ 1

� @

@✓

✓ie�i✓

1

◆d✓ =

1

2

Z ⇡

0i��iei✓ 1

�✓ei✓0

◆d✓ =

=

1

2

Z ⇡

0i(�i) d✓ =

1

2

[✓]⇡0 =

⇡

2


| (t)i0 = |�i = ei⇡

2 | (0)i (3.11)


�n(t) =1

2

Z �⇡

0i hL1| @

@✓|L1i d✓ = �⇡

2


2 | (0)i (3.12)


| (t)i = | i+ |�i = �ei

⇡

2

+ e�i⇡

2

� | (0)i = ei⇡

2

�e�i⇡

2

+ 1

� | (0)i = 0


Absence of backscattering:

✓~k 2 (0,�⇡)✓~k 2 (0, ⇡)

'Berry = �i

Z T

0dt

�1 �i ei✓(t)

� 12

d

dt

✓1

i e�i✓(t)

◆= �

Z T

0dt ✓ = ±⇡

2


3.2 BackscatteringIn tale sezione vengono mostrate le proprietà di conduzione delle superfici di bordonegli isolanti topologici 3D. Come anticipato anche nel caso di modelli bidimensiona-li, l’accoppiamento Spin-Orbita, gioca un ruolo fondamentale nella caratterizzazionedegli stati elettronici in tali materiali. L’interazione S.O. viene esplicitamente codifi-cata in modelli Hamiltoniani (si veda Eq.(2.39)), che a differenza di quelli utilizzatinella la descrizione di siStemi QHE, ammettono simmetria T.R. Tale simmetriaviene generalmente espressa nella relazione:

⇥H(k)⇥†= H(�k) (3.7)

Viene ora giustificata, mediante i risultati acquisiti nel precedente paragrafo, la -robustezza di tali stati di conduzione garantita dalla simmetria T.R.Supponiamo di localizzare nel materiale delle impurezze non magnetiche (che noncomportino rottura della simmetria Time Reversal). Esattamente come accade nelcaso di un fotone riflesso da una superficie opportuna, un elettrone può essere riflessonell’interazione con l’impurezza. Tale riflessione , nel formalismo della meccanicaquantistica, è codificata dal sorgere, dopo un opportuno tempo ⌧ , di una funzioned’onda caratterizzata da un verso di propagazione opposto rispetto a quello iniziale.Nel voler calcolare la probabilità che dopo un certo tempo ⌧ ( sufficiente affinchélo stato iniziale inverta la direzione del moto ),è necessario considerare entrambi icammini corrispondenti alle due possibili rotazioni, nello spazio degli Spin. Al finedi semplificare la visualizzazione di tale proposizione si riporta la figura discussa nelCapitolo 1:

Figura 3.1: Possibili cammini di un elettrone sul bordo di un sistema QSHE quando incontraun’impurezza non-magnetica.

Come mostrato nel precedente capitolo è proprio lo spin a costituire un elementofondamentale nello studio della dinamica di tali sistemi.Data la parametrizzazione nello spazio degli impulsi3 :

8><

>:

kx = k cos ✓ sin�

ky = k sin ✓ sin�

kz = k cos�

(✓,�) 2 (0,⇡)⇥ (0, 2⇡)

E’ possibile, come mostrato, esprimere mediante tali parametri le componenti spi-noriali degli autostati di superficie. Tale operazione rende esplicita la struttura divarietà differenziabile cui è munito lo spazio dei parametri M4. La restrizione allasuperficie viene imposta mediante la relazione � = 0, mentre la trasformazione ci-clica è implementata dalla rotazione di ⇡ lungo la sfera.Utilizzando la definizione di fase geometrica calcoliamo separatamente i due contri-

3Si preferisce parametrizzare uno spazio tridimensionale per poi ridursi, mediante condizioniopportune, a superfici bidimensionali.

4Nel caso in analisi la varietà è proprio la sfera S2 immersa in R3


3.2 BackscatteringIn tale sezione vengono mostrate le proprietà di conduzione delle superfici di bordonegli isolanti topologici 3D. Come anticipato anche nel caso di modelli bidimensiona-li, l’accoppiamento Spin-Orbita, gioca un ruolo fondamentale nella caratterizzazionedegli stati elettronici in tali materiali. L’interazione S.O. viene esplicitamente codifi-cata in modelli Hamiltoniani (si veda Eq.(2.39)), che a differenza di quelli utilizzatinella la descrizione di siStemi QHE, ammettono simmetria T.R. Tale simmetriaviene generalmente espressa nella relazione:

⇥H(k)⇥†= H(�k) (3.7)

Viene ora giustificata, mediante i risultati acquisiti nel precedente paragrafo, la -robustezza di tali stati di conduzione garantita dalla simmetria T.R.Supponiamo di localizzare nel materiale delle impurezze non magnetiche (che noncomportino rottura della simmetria Time Reversal). Esattamente come accade nelcaso di un fotone riflesso da una superficie opportuna, un elettrone può essere riflessonell’interazione con l’impurezza. Tale riflessione , nel formalismo della meccanicaquantistica, è codificata dal sorgere, dopo un opportuno tempo ⌧ , di una funzioned’onda caratterizzata da un verso di propagazione opposto rispetto a quello iniziale.Nel voler calcolare la probabilità che dopo un certo tempo ⌧ ( sufficiente affinchélo stato iniziale inverta la direzione del moto ),è necessario considerare entrambi icammini corrispondenti alle due possibili rotazioni, nello spazio degli Spin. Al finedi semplificare la visualizzazione di tale proposizione si riporta la figura discussa nelCapitolo 1:

Figura 3.1: Possibili cammini di un elettrone sul bordo di un sistema QSHE quando incontraun’impurezza non-magnetica.

Come mostrato nel precedente capitolo è proprio lo spin a costituire un elementofondamentale nello studio della dinamica di tali sistemi.Data la parametrizzazione nello spazio degli impulsi3 :

8><

>:

kx = k cos ✓ sin�

ky = k sin ✓ sin�

kz = k cos�

(✓,�) 2 (0,⇡)⇥ (0, 2⇡)

E’ possibile, come mostrato, esprimere mediante tali parametri le componenti spi-noriali degli autostati di superficie. Tale operazione rende esplicita la struttura divarietà differenziabile cui è munito lo spazio dei parametri M4. La restrizione allasuperficie viene imposta mediante la relazione � = 0, mentre la trasformazione ci-clica è implementata dalla rotazione di ⇡ lungo la sfera.Utilizzando la definizione di fase geometrica calcoliamo separatamente i due contri-

3Si preferisce parametrizzare uno spazio tridimensionale per poi ridursi, mediante condizioniopportune, a superfici bidimensionali.

4Nel caso in analisi la varietà è proprio la sfera S2 immersa in R3

0-π

0π✓~k

0

Topologically protected boundary states

Proximity from a 2D d-wave conventional Superconductor to a 1D- wire with spin-orbit coupling

(Numerical: tight-binding)

t =0.3, α=0.1, μ=-1. T

Htot

= Hs

+Hw

+HT

Gw (i!, kx

) =hi! � Z(⇠w

k

x

� �µ) ⌧z

� ⌃ (i!, kx

)i�1

⌃ (i!, kx

) = �i!(Z�1 � 1)� �µ ⌧z

+�0(i!, kx)(1� Z)⌧x

Top-nontrivial !

F (w)(k) =

1

2

hdk"d�k# � dk#d�k"i

F (w)(`) =

1

Nk

X

k

F (w)(k) cos(k`)

H± (r > ⇠o

) =

but both Majoranas have total mJ = 1/2 !

En = n ⇥0 = n�2

�F

other bound qp levels:

n 6= 0

as if the vortex splits into two half vortices...

Two spin polarized zero energy Majorana fermions bound at the interfaces with the

superconductor

Vortex along z

orbital angular momentum m=0 on surface z=0, but m=1 on surface z=L , due to the opposite

chirality,

B

3D TI

z

s-wave superconductor

s-wave superconductor z=L

z=0

z=0 spin m=0j= 1/2

z=L spin m=1j= 1/2

j = m + sz

Weyl semimetals

Weyl nodes at EF

of cones with opposite chirality

3D-type IQHE without magnetic field

Breaking of inversion symmetry

P3 / ~E · ~B

Shuang Jia, Su-Yang Xu and M. Zahid Hasan

Nat. Mat. 15,1140 (2016) Wan , et al PRB83, 205101 (2011)

addendum:

• TI harbours one helical Dirac boundary state at each surface

• s-wave proximity provides an effective p-wave pairing

• spinless p-wave pairing localizes quasi-1D Majorana states at the edges

• a vortex binds a Majorana fermion at top/bottom edges

wires, flakes and HTc superconductor:

Summary:

• quasi 1D-Josephson Junction with dx2-y2 sup-proximity in Kitaev simplified model chain

• intrinsic magnetic flux in aTI on a HTc tricrystal

From Integer Quantum Hall effect to Majorana Fermions in ...

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