Interaction and Disorder in Topological InsulatorsInteraction and Disorder in Topological Insulators School on Topological Quantum Matter, Harish-Chandra Research Institute, Allahabad

KARLSRUHE INSTITUTE OF TECHNOLOGY

Interaction and Disorder in Topological InsulatorsSchool on Topological Quantum Matter, Harish-Chandra Research Institute, Allahabad

Igor Gornyi | 13.02.2015

KIT – University of the State of Baden-Wuerttemberg and

National Laboratory of the Helmholtz Association

www.kit.edu

http://www.kit.edu

Outline

Topology

Topological insulators

2D topological insulators: Quantum spin Hall effect

3D topological insulators

Scaling theory of localization: Symplectic class

Interaction-induced criticality

Generic helical liquids

Igor Gornyi – Interaction and Disorder in Topological Insulators 13.02.2015 2/65

Topology (Wikipedia)

Homeomorphism:A, R, B, C, G, I, J, L, M, N, S, U, V, W, Z,D, O, E, F, T, Y, H, K, P, Q, XHomotopy:A, R, D, O, P, Q, B,C, E, F, G, H, I, J, K, L, M, N, S, T, U, V, W, X, Y, Z


Topology (Wikipedia)

Homeomorphism:A, R, B, C, I, J, L, M, N, S, U, V, W, Z,D, O, E, F, G, T, Y, H, K, P, Q, XHomotopy:A, R, D, O, P, Q, B,C, E, F, G, H, I, J, K, L, M, N, S, T, U, V, W, X, Y, Z


Topology (cf. Ilya Gruzberg’s lecture)


Topological insulator vs. trivial insulator


Insulator


Topological insulator


What is a topological insulator?

Topological insulator bulk insulator with metallic non-localizable edge/surface

characterized by a certain topological invariant

Examples:Quantum Hall effect at the plateau

Materials with extreme spin-orbit coupling (inverted gap)2D: Quantum spin-Hall effect (HgTe/CdHgTe, InAs/GaSb)3D: Bix Sb1x , BiTe, BiSe, strained HgTe

Reviews:M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).

X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).


What is a topological insulator?

Topological insulator bulk insulator with metallic non-localizable edge/surface

Examples:Quantum Hall effect at the plateau

Materials with extreme spin-orbit coupling (inverted gap)2D: Quantum spin-Hall effect (HgTe/CdHgTe, InAs/GaSb)3D: Bix Sb1x , BiTe, BiSe, strained HgTe

Reviews:M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).

X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).


Topological invariants

Topological invariants

QHE: time-reversal symmetry broken by magnetic fieldChern number = # of edge states = ::: 2;1; 0; 1; 2; ::: (Z)

=) Z topological insulator

QSHE: time-reversal symmetry preserved, spin-rotational brokenBand-structure topological invariant (Z2): n = 0 or n = 1

() odd vs. even number of Kramers pairs of edge states=) Z2 topological insulator


We focus on symplectic class AII (strong spin-orbit interaction)


Classification of topological insulators


Topological insulators & Topology

Two ways to detect existence of TIs of class p in d dimensions:(i) by inspecting the topology of classifying spaces Rp:8<

:TI of type Z

TI of type Z2() 0(Rpd ) =

8<:ZZ2

(ii) by analyzing homotopy groups of the -model manifolds:8<:TI of type Z() d (Sp) = Z Wess-Zumino term

TI of type Z2 () d1(Sp) = Z2 = topological term

WZ and = terms make boundary excitations “non-localizable”TI in d () topological protection from localization in d 1

Bott periodicity: d (Rp) = 0(Rp+d ) , periodicity 8


Quantum Hall effectvon Klitzing, Dorda, Pepper ’80

Quantized resistance

Electron states

extended

localized

E

EF

Landau levelsZ topological insulator:topological invariant 2 N describes number of chiral edge modesHall resistance is quantized in units of RK = h=e2 26 kΩ


QHE insulator: Z - topological insulator

N N+1edge states edge states


Quantum spin Hall effectKane, Mele ’05, Sheng et al ’05, Bernevig, Zhang ’06

No magnetic field but strong spin-orbit interaction=) Electrons with opposite spins feel opposite effective magnetic field

Electric current leads tospin accumulation at the edges

=) spin-Hall effect

Extreme spin-orbit couplingopens a band gap

=) quantum spin-Hall effect?


QSHE: Cartoon

two QH states with opposite magnetic field! effective magnetic field due to spin-orbit coupling

RL

momentum

energy

Z2 TI: one time reversal pair! helical edge state

Naively: quantized conductance G0 = 2e2=h


Quantum spin-Hall effect: Clean case

Kane and Mele ’05


Quantum (spin-)Hall effect with disorder

Impurities do not destroy the edge (spin) current:

QHE

due to chirality of carriersany disorder

QSHE

due to time-inversion symmetryno magnetic impurities


QSHE: theory proposalBernevig, Hughes, Zhang ’06

HgTe/CdTe quantum well band structure

2D Dirac Hamiltonian with tunable mass: m ? 0 when d 7 dc


HgTe quantum wells: HamiltonianJz -symmetric Hamiltonian in basis E1+;H1+;E1;H1(Bernevig-Hughes-Zhang Hamiltonian):

HBHZ =

h(k) 0

0 h(k)

!; h(k) =

(k) + m(k) Ak+

Ak (k)m(k)

!

k = kx iky ; (k) = C + Dk2; m(k) = M + Bk2:


HgTe quantum wells: HamiltonianJz -symmetric Hamiltonian in basis E1+;H1+;E1;H1(Bernevig-Hughes-Zhang Hamiltonian):

HBHZ =

h(k) 0

0 h(k)

!; h(k) =

(k) + m(k) Ak+

Ak (k)m(k)

!

k = kx iky ; (k) = C + Dk2; m(k) = M + Bk2:

Spin-orbit interaction (block mixing) due to inversion asymmetry:

HSO =

0BBB@

0 0 2ek+ ∆0

0 0 ∆0 2hk2ek ∆0 0 0∆0 2hk+ 0 0

1CCCA+

0BBB@

0 0 2ir0k 00 0 0 0

2ir0k+ 0 0 00 0 0 0

1CCCA


QSHE: experimentMolenkamp group ’07

I — d = 5:5nm: normal insulatorII, III, IV — d = 7:3nm: inverted band gap — topological insulator


3D topological insulatorsHasan group ’08


3D topological insulator: spectroscopy

ARPES measurement on Bi0:9Sb0:1

Odd number of surface modes =) nontrivial topology


Phenomenological descriptioncf. Volkov & Pankratov ’85

Bulk Hamiltonian (Dirac): Hb =

M pp M

!E

M

−M

k

Domain wall:

0 x

M<0M>0

Edge state

Decays into the bulk: Ψ = ejMxj

!

Surface Hamiltonian

Hs =rn2|z

curvature

+12

n[p ] + [p ]n

| z

Rashba

) n[p ]

E

M

−M

k


Surface of 3D topological insulators ofsymmetry class AII

3D Topological Insulators have 2D Dirac modes at the surface

surface of a 3D TI = single-valley spin-polarized graphene

2D disordered Dirac fermions of symmetry class AII:topological protection against localization


Scaling theory of localizationAbrahams, Anderson, Licciardello, Ramakrishnan ’79

Dimensionless conductance [in units e2=h]

Metallic sample (Ohm’s law): g Ld2

Insulating sample (tunneling): g eL=

Universal scaling function

d ln gd ln L

= (g) =

8<:d 2; g 1; (metal);

ln g; g 1; (insulator):

ln g

0

dln

gd

lnL

3D

2D

1D


Weak antilocalization (g 1)

Without spin-orbit interaction: enhanced backscattering= interference suppresses conductivity

weak localization

Spin-orbit interaction: additional phase factor (Berry phase)interference suppresses backscattering = enhances conductivity

weak antilocalization


Weak localization correction in 2DGor’kov, Larkin, Khmelnitskii ’79; Hikami, Larkin, Nagaoka ’80

Scaling of conductivity (no e-e interaction)

d ln gd ln L

=

8>><>>:1=g; orthogonal (TR preserved);

1=2g2; unitary (TR broken);

+1=2g; symplectic (TR preserved, spin-orbit) we are here!

ln Σ

0

dlnΣd

lnL

ΣSp*

»1.4

Sp U O

2D Dirac electrons:Metal or insulator?

MIT in symplectic class at Sp 1:4e2=h


One-dimensional symplectic wire

ΨL

out

Ψin

LΨout

R

Ψin

R

Scattering matrix of a symplectic system

ΨLout

ΨRout

!=

r t 0

t r 0

! ΨL

in

ΨRin

!TI symmetry =)

r = rT

r 0 = r 0T

t = t 0T

For N channels:

det r = (1)N det rT =) no localization if N is odd ! ! !


Topological insulator: reduction to 1D

Hollow cylinder threaded with magnetic flux Φ

Surface states: En(k) = s

k2 +

n +

12 eΦ

hc

2

Time-inversion symmetry is preserved ifeΦ

hcis integer or half-integer

no 1D localization =) no 2D localization


Single copy of Dirac fermionsAbsence of localization (cf. Nomura, Koshino, Ryu ’07):

Energy spectra with changing twist angle (boundary conditions)

TR symmetry holds at = 0 and = ) Kramers degeneracy

Single-valley massless Dirac model: Kramers pairs change partners

Localized states must be insensitive to boundary conditions) no localization of massless Dirac fermions


Dirac fermions in symplectic class:sigma model

Fendley ’01, Altland ’06, Ostrovsky, IG, Mirlin ’07, Ryu, Mudry, Obuse, Furusaki ’07

Random potential: symplectic time-reversal symmetry H = y HTy

Symplectic sigma model: topological -term with =

S[Q] =xx

16Str(rQ)2 + iN[Q] N[Q] = 0; 1

Similar to Pruisken sigma model for IQHE (instantons suppress localization)

No localization! Criticality?

Minimal conductivity: = 4Sp e2=h, or

Absolute antilocalization: !1


Topology

2

O (2N)

O (N) O (N)

= Z2

Local expression of theta term:

S = SWZW jO(x)=OT (x) =

(0

)for

(trivial configuration

non-trivial configuration

Upon inclusion of vector potential S ! SChernSimons


Scaling of conductance: numerics

Bardarson, Tworzydło, Brouwer, Beenakker ’07 Nomura, Koshino, Ryu ’07

Absence of localization confirmed

Supermetallic behaviour for microscopic models considered


Beta functions for symplectic system

(g) =d log gd log L

β(g)

log g0

g∗ ≈ 1.4

supermetal

insulator

Usual spin-orbit metal

β(g)

log g0

supermetal

Dirac fermions


2D surface states of a 3D TI:Disorder and interaction

Disorder:Topological protection from localization, RG flow towards supermetal

What is the effect of Coulomb interaction?

assume not too strong interaction rs =p

2e2=~vF . 1

=) no instabilities, no symmetry-breaking

=) topological protection from localization persists

But interaction may destroy the supermetal phase!


Effect of Coulomb interactionAltshuler, Aronov ’79; Finkelstein ’83

Any 2D metallic sample g 1Diffusion + Coulomb repulsion) Altshuler-Aronov correction:

∆gee =h 1 + (N2 1)F

ilog LT

N = number of independent equivalent species (spin, valleys etc.)

Include correction into symplectic beta function

(g) =d log gd log L

=1g

N2 1 + (N2 1)F

Surface of a 3D topological insulator: N = 1

(g) ) 1=2g

Coulomb repulsion destroys supermetallic phase!


Interaction-induced criticalityOstrovsky, IG, Mirlin, PRL’10

Interaction =) tendency to localization at g 1

Topology =) prevents strong localization (g 1 forbidden)

Result: Interaction induces a novel quantum critical state with someuniversal conductivity g 1 on the surface of a 3D topological insulator.

β(g)

log g0

critical state

“Self-organized” criticality: no adjustable parameters


Beta functions for 2D spin-orbit systemsno interaction with interaction

usual

spin-orbit

β(g)

log g0

g∗ ≈ 1.4

supermetalinsulator

β(g)

log g0

insulator

Dirac

ferm

ions

β(g)

log g0

supermetal

β(g)

log g0

critical state


QSHE: Phase diagram

Obuse, Furusaki, Ryu, Mudry ’07

In the presence of disorder, normal and topological insulating phasesare separated by the supermetal phase

Transitions between them are conventional symplectic MIT

No quantum spin-Hall transition


QSHE + Coulomb: Phase diagram

(a) no interaction (b) with interaction

0

disorder

band gap0inverted normal

supermetal

normalinsulator

QSHinsulator

0

disorder

band gap0inverted normal

critical

normalinsulator

QSHinsulator

Interaction restores direct quantum spin-Hall transition viaa novel critical state

Ostrovsky, IG, Mirlin, PRL’10


2D: Summary

Two critical stateson the surface of 3D topological insulator

at the quantum spin-Hall transition

Common features:symplectic symmetry

topological protection

interaction-induced criticality

conductivity of order e2=h

Further developments:3D TI in slab geometry (intersurface interaction, E. Konig et al. ’13)

Quantum Hall Effect in 3D TI (E. Konig et al. ’14)


Transport experiments on Bi2Se3

At present most conventional 3D TI: Bi2Se3

Advantage: huge bulk gap 0:3 eV

Disadvantages:Fermi-Energy not in bulk gap) requires chemical or electrostatic gating

experiment reveal large bulk contribution) thin films and thickness dependent measurements



DSa

dl

l

1

2

3D TI

substrate

VSD

ξ

coatsurface 1

surface 2

top gate

back gate

VTG

VBG



“weaklocalization” in (T )

Proof of electron-electron interaction!



weak anti-localization in (B)





weak anti-localization in (B)





Evidence for combinedeffects of

weak antilocalization

electron-electroninteraction


Conductivity of a generic helical liquid1D quantum liquids

spinlessR

L

spinful

R, spin up L, spin down

L, spin up R, spin down

chiral

helical

time reversal symmetry broken

time reversal symmetry preserved

R

R, spin up L, spin down


Physics in one dimension

Many body problem, Fermi sea, Fermi surface

1D : Fermi surface consists of two pointsparticles are moving to the right (R) or to the left (L)

currentJ = e vF (NR NL)


Physics in 1D: strong correlations

Electrons in 1D have no way “around” each other

Arbitrarily weak interaction qualitatively changes the ground state

Fundamental excitations: collective density waves - plasmons


Tomonaga-Luttinger liquid

Interacting model (quartic in fermions):

H0 = ivF

Zdx

X=R;L

y(x)@x (x)

Hint =gZdx (x)(x); (x) =

X

y(x) (x)

Bosonization : Exact mapping of fermionic to bosonic Hilbert space

(x) = (2)1=2 exp (iph'

i)

Nonlocal commutation relations: ['(x); (y)] = iΘ(x y)


Operator Bosonization

Interacting model remains quadratic in bosons

H = H0 + Hint =u2

hK (@x )2 + K1(@x ')2

icharacterized by two parameters

Luttinger liquid parameter

K = (1 + g=vF )1=2

Plasmon velocity

u = vF=K

Luttinger liquid parameter describes fermionic interaction strength:K = 1 noninteracting, K < 1 repulsive, K > 1 attractive


Disordered Luttinger liquidGiamarchi & Schulz ’88; IG, Mirlin, Polyakov ’05, ’07

Single-channel infinite wire: right(left) movers ; =

Spinless (spin-polarized, = +) or spinful ( = ) electrons

Linear dispersion, k = kvF

Short-range weak e-e interaction, V (0)=2vF 1

No e-e backscattering; g-ology with g2 and g4

White-noise weak ( EF0 1 ) disorder,

hU(x)U(x 0)i = (x x 0)=200:


Bosonization and disorder averaging

Giamarchi & Schulz ’88

Bosonization: given realization of disorder,

Disorder averaging. Quenched disorder: replicas n

Bosonized replicated action (no spin):

S[] =1

2vF

Xn

Zdx d

n[@n(x ; )]2 u2[@xn(x ; )]2

o

vF k2F

20

Xn;m

Zdx d d 0 cos[2n(x ; ) 2m(x ; 0)]


Two steps: virtual & real processesIG, Mirlin, Polyakov ’05, ’07; Bagrets, IG, Mirlin, Polyakov ’09

Step 1: Integrate out T < < EF (RG, virtual processes)Giamarchi & Schulz ’88 ! T -dependent static disorder

(T ) = 0(T=EF )2;

all power-law (Luttinger) terms / (EF=T ) in renormalized couplings

Step 2: Refermionizesolve kinetic equation: classical (Drude) conductivity

D(T ) / (T ) / T 2;

inelastic relaxation processes;interference and dephasing



Linear spectrum with Dirac pointElastic backscattering from nonmagnetic impurity forbiddenInelastic (two-particle) backscattering allowed

Xu & Moore ’06, Wu, Bernevig, Zhang ’06

Broken Sz symmetry: Inelastic backscattering allowedSchmidt, Rachel, von Oppen, Glazman ’12

RL

momentum

ener

gy

RL

momentum

ener

gy

Dirac point


Transport in a generic helical liquid

Previous work

Correction to conductance of a short edge for weakly interactingelectrons1;2

Luttinger liquid renormalization2;3;4;5

1 T. L. Schmidt, S. Rachel, F. von Oppen and L. I. Glazman, Phys. Rev. Lett. 108, 156402 (2012).2 F. Crepin, J. C. Budich, F. Dolcini, P. Recher, and B. Trauzettel, Phys. Rev. B 86, 121106(R) (2012).3 A. Strom, H. Johannesson, and G. I. Japaridze, Phys. Rev. Lett. 104, 256804 (2010).4 N. Lezmy, Y. Oreg, and M. Berkooz, Phys. Rev. B 85, 235304 (2012).5 F. Geissler, F. Crepin, and B. Trauzettel, Phys. Rev. B 89, 235136 (2014)

Poster by Niko Kainaris:

Conductivity of long edge channels including LL renormalizationN. Kainaris et al. ’14


Model

Generic helical liquidwith SO interaction – block mixing = Sz broken, TR respected.Rotation of basis (spins vs. chirality/helicity) 1

k ;"

k ;#

!= Bk

k ;R

k ;L

!; By

k Bk = 1; Bk = Bk

General form of Bk for k k0 (here k10 – strength of SO interaction):

Bk =

0@1 k4

2k40

k2

k20

k2

k20

1 k4

2k40

1A ; h

Byk Bp

i;0

= ;0 + ;0k2 p2

k20

;

1Schmidt, Rachel, von Oppen, Glazman (PRL 2012)


Model

Microscopic model for 1D time-reversal invariant quantum liquid2

Spinless fermions withmomentum k and chirality :

H0 = vFXk ;

k y;k ;k

Linear spectrum

RL

momentum

ener

gy Dirac point

2Schmidt, Rachel, von Oppen, Glazman ’12


Modelscreened two-particle interaction

H2 =V

L

Xk;p;q;

y;k

y;p ;p+q ;kq

H3 =V

k40 L

Xk;p;q;

(k2 (k q)2)

p2 (p + q)2 y;k

y;p ;p+q ;kq

H5 = V

k20 L

Xk;p;q;

(k2 p2) y;k+q

y;pq ;p ;k + h:c:

L

R

L

R

g2

forward

R

L

L

R

g3

umklapp

R

LR

g5

umklapp

R


Model: disorderShort range, non-magnetic impurities

Himp =UL

Xk ;p;

( y;k ;p +

k2 p2

k20

y;k ;p)

R R R L

Magnetic impurities & conducting islands (Kondo, spin glass...):

J. Maciejko, C. Liu, Y. Oreg, X.-L. Qi, C. Wu, and S.-C. Zhang ’09

V. Cheianov, L. Glazman ’13; J.I. Vayrynen, M. Goldstein, L.I. Glazman ’13; J.I. Vayrynen, M. Goldstein, Y. Gefen, L.I. Glazman ’14

B.L. Altshuler, I.L. Aleiner, V.I. Yudson ’13

Not included here; might be important for experiments


Model

Microscopic model for 1D TR invariant quantum liquid

momentum factor ensures TRI

k10 measures spin-orbit

coupling

R,k L,p


Dominant scattering mechanisms

process has to changechirality of incomingparticles

j = e vF (NR NL)

Disorder backscatteringreduced by TRI) forward scatteringdominant in combinedprocesses

Transport: combinedeffects of interactionand disorder

RL L R

R

R

L L

L

R L

RR

L R

R

L

R

R

inelastic single-particle (1P) inelastic two-particle (2P) g5

L

(a)

(b)




j = e vF (NR NL)



RL L R

R

R

L L

L

R L

RR

L R

R

L

R

R


L

(a)

(b)




j = e vF (NR NL)



RL L R

R

R

L L

L

R L

RR

L R

R

L

R

R


L

(a)

(b)




j = e vF (NR NL)



RL L R

R

R

L L

L

R L

RR

L R

R

L

R

R


L

(a)

(b)


Experiment

Konig et al. (Science 2007)

HgTe/CdTe: short ( 1m) edges

Knez et al. (PRL 2014)

InAs/GaSb: longer ( 10m) edges


Experiment

Gusev, Kvon, et al. ’12, ’13:

long edges (5-50 m),resistance much higher than quantum resistance,

temperature dependence saturates


Summary

Topology

Topological insulators

2D topological insulators: Quantum spin Hall effect

3D topological insulators

Scaling theory of localization: Symplectic class

Interaction-induced criticality



Documents

Interaction and Disorder in Topological InsulatorsInteraction and Disorder in Topological Insulators School on Topological Quantum Matter, Harish-Chandra Research Institute, Allahabad