Hideo AokiDept Physics, Univ Tokyo

Quantum Hall effect and real-space wavefunctions- multifractality in 2DEG & lattice systems

Newton Institute Workshop“Delocalization Transitions and

Multifractality”, Wales, 4 Nov 2008

Tsuneyoshi Nakayama, Takamichi Terao Hokkaido Univ

Mikito Koshino Tokyo Institute of Tech

Yasuhiro Hatsugai, Akihiro Arikawa Univ TsukubaTakahiro Fukui Ibaraki Univ

Toru Kawarabayashi Toho Univ

Outline

Introduction

Localisation in QHE systems--- Landau index dependence

QHE in graphene

QHE in 3D

Early papers

Wegner, Z Phys B 1980:Participation ratio p =(Ld∫|Y|4dr)-1 EEc 0 only an infinitesimal fraction of the volume occupied by Y

Aoki, JPC 1980:Real-space renormalisation-group for Anderson localisation

scale invariant

Aoki, JPC 1983, PRB 1986:Y = fractal

Castellani & Peliti, JPA 1986; Schreiber & Grussbach, PRL 1991; Huckestein, RMP 1995; Avishai et al 1995, …Y = multifractal

Scaling theory of localisation for QHE systems

Aoki & Ando, PRL 1985Finite-size scaling critical exponent

Khmel’nitskii, JETPL 1983; Pruisken, PRB 1985; PRL 1988

Two-parameter scalingHuckestein 1990’sIrrelevant scaling length to recover universality

If Anderson transition ～ phase transition

no particular length scale at the criticality

self-similar (fractal)

Localisation transition = quantum phase tr ?

(Aoki, JPC 1983)

D(E

)

localis

ation length

localised delocalised

Integer quantum Hall effect

(Aoki & Ando SSC 1981； Aoki, Rep Prog Phys 1987)

Finite-size scaling study for the quantum criticality(Aoki & Ando, PRL 1985)

s model (Pruisken 1985)Numerical result (Aoki & Ando 1986)

sxx-sxy flow

|Y|2

a

b

c

d

e

a b c d e

D(E)

Ｌocalisation length

(Aoki, JPC 19７７)

maximal hopping between cyclotron orbits drift of cyclotron orbit centres

l

Short-range vs long-range random pontetials

(Aoki, Rep Prog Phys, 1987)

Outline

Introduction

Localisation in QHE systems--- Landau index dependence

QHE in 3D

QHE in graphene

N = 0

N = 1

N = 1, long-range randomness

Localisation transition in QHE － multifractal analysis

[Terao, Nakayama & Aoki, Phys. Rev. B 54, 10350 (1996)

with Lanczos diag with no re-orthogonalisation]

Why Landau index N dependence interesting?loc. length x ～ l exp(s2

SCBA), sSCBA ∝ (N+1/2)explodes with N

1. parabolic f(a) log-normal dist for Y

(analytic: Zirnbauer 1999; Evers et al 2001;this workshop)

Result for the multifractal spectrum

2. f(a) depends on N

3. for finite-range random potential universality in f(a) recovered

(Terao et al, PRB 1996)

Parabolic f(a) = normal dist for log |y| )

|Y| ln |Y|

(Terao, Nakayama & Aoki, 1996)

□ Conformal mapping (Cardy 1984) scaling invariance extends to local scale inv (conformal tr)

□ Scaling amplitude (Janssen 1994)limM∞lM/M = Lc = 1/p(a0 - d)

Can we relate multifractality with scaling

a0●： present result

---: 1/ln(1+√2) Lee-Wang-Kivelson 1993

recovery of 1-parameter scaling in agreement with Huckestein 1992

N-dependence of multifractality

<|Y(r1) Y(r2)|2 > ����～ 1/ |r1-r2|

a0 range ↑

d

D(2)

N 210

range ↑

1

1.5

Anomalous diffusion

Fractal wf’s anomalous diffusion (Chalker & Daniell, 1988)

Normally: s -- e2 N(EF)D0 : Einstein’s relation (D0 : diffusion const)

At criticality:s(q, w) ～ e2 N(EF)D(q/√w): dynamical scaling

D(q/√w) -- D0 / (q/√w):

q, w0

q, w0 = 2 - D(2) ～ 0.38

(Gammel & Brenig, PRB 1996)

[Morgenstern et al, PRL 90, 056804 (2003)]

Real-space imaging of disorder potential

STS study for 0.8% Fe deposited on cleaved n-InAs(110)[Morgenstern et al, PRL 90, 056804 (2003)]

Experimentally obtained multifractal spectrum

(Hashimoto et al, arXiv:0807.3784)

Real-space imaging experiments

STS study for 0.01 monolayer of Cs on cleaved n-InSb(110), T = 0.3 K

Turbulence(Kida et al, 1989)

Quantum “turbulence”

N=0, 300l x300l

Outline

Introduction

Localisation in QHE systems--- Landau index dependence

QHE in graphene

QHE in 3D

Massless Dirac

Graphene

QHE in graphene

rxx

sxyKK

K

K’

K’

K’

(Novoselov et al, Nature 2005; Zhang et al, Nature 2005)

sxy = (2N+1)(-e2/h)

2(N+1/2)(-e2/h)

(half-integer)

for zero-mass Dirac

particles

sxy /(e2/h) = 1

3

5

7

-3

-1

-5

-7

(Thouless, Kohmoto, Nightingale & den Nijs, 1982)

Linear response

clean, periodic systems

= ∑nband (Chern #)n

Berry’s “curvature”

“Gauss-Bonnet”

Hall conductivity = a topological number

disordered systems(Aoki & Ando, 1986)

(Avron et al, 2003)

distribution of topological #in disordered systems(Aoki & Ando, 1986;Huo & Bhatt, 1992;

Yang & Bhatt, 1999)

Edge states in graphene QHE systems

Hatsugai, Fukui & Aoki (2006);Arikawa, Aoki & Hatsugai (2008)

real space Momentum along the edge

E=0 mode has to exist topologically# of edge mode (2q-1) is odd

for magnetic field = p/q

Edge states in graphene QHE

Hatsugai, Fukui & Aoki (2006);Arikawa, Aoki & Hatsugai (2008)

© J Meyer, UC Berkeley

“Ripple” in graphene

Graphene has a mobility higher than any known semiconductor i.e., graphene is the cleanest semiconductor ever studied [Bolotin, Sikes, Hone, Stormer & Kim, PRL 101, 096802 (2008)]

But the graphene sheet has “ripples”

(Meyer, Geim et al, Nature 2007)

E=0 Landau level is robust against disorder? How universal ?

Sensitivity to the spatial correlation ? Castro Neto et al. (arXiv:0709.1163);Guinea et al., PRB77 (2008) 205421

B

2D Honeycomb Lattice in disordered magnetic fields

'

',

',

uni rr

rr

iccteH rr †

><

Disordered components 0uni', /)(2 p - rr eh /0

= uni

)2/exp(2

1)(

22

2

sps

-P

)4/||exp(222

s jiji --

12 sW effective width

: correlation of random flux

Density of states

rrr EGE )i(Im1

)( , p

r -

Green’s function methodSchweitzer, Kramer, MacKinnon (1984)

t01.0a

250

P.B.C.

Numerical study (Kawarabayashi et al, in prep)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-0.8 -0.4 0 0.4 0.8

DOS

DOS

DOS

DOS

E/t

D(E)

uni/0=1/41 (l =2.4|a|)

2.0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-0.8 -0.4 0 0.4 0.8

DOSEE

cf. Random site energy

12/2 W

W/t = 0, 1, 2

00 < EE

Random magnetic field

W /0=0

0.5

1.0

(Kawarabayashi et al, in prep)

0

0.1

0.2

0.3

0.4

0.5

0.6

-0.8 -0.4 0 0.4 0.8

DOSEE

uni/0=1/41 (l =2.4a)W /0=2.0

Correlation Length

||/3 a

0.0

E/t

Spatially correlated random magnetic field

3.0

5.0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

-0.8 -0.4 0 0.4 0.8

DOSE

cf. sensitivity to the mean value uni

E/t

W /0=2.0 uni/0=0, 1/41>>

Cf. Anomaly in DOS at E=0 for the random flux model with zero mean (Furusaki PRL (1999), etc)

uni/0=

1/41

0

(Kawarabayashi et al, in prep)

Uncorrelated random t

uni/0=1/41 (l =2.4a)

Wt /t =0

D(E)

E/t

0.2

0.40.50.6

E/t

Random hopping energy

Correlated random t

Wt /t = 0.4, uni/0=1/41 (l =2.4a)

||/3 at

5.0

3.0

2.0

1.0

Broadening is smaller than the numerical resolution.i.e., no difference

from Wt=0 case

0

Class AIII[Evers & Mirlin, Rev Mod Phys

80, 1355 (2008)]

(Kawarabayashi et al, in prep)

Vg (V)

rxx(k

)

-6

2

4

-60 -30 0 6030

sxy (e

2/h)

0

10

20

30

-4

-2

0

B = 29 T

T = 300 K

Novoselov et al, Science 2007

T ↑(Room-T QHE)

Dirty massless Dirac(Nomura et al, arXiv0801.3121)

Singularly robust n=0 Landau level

B 0

t’

t’=-1: p-flux lattice t’=0: honeycomb t’=+1: square

t

Is the “massless Dirac” specific to honeycomb?

Hatsugai, Fukui & Aoki (2006)

Dirac cones in other systems ?---d-wave superconductors

Dispersion of the quasi-particlein the Bogoliubov Hamiltonian 2D Dirac Hamiltonian

Edge states in anisotropicsuperconductors(Ryu & Hatsugai, PRL 2002)

Outline

Introduction

QHE in graphene

QHE in 3D

Prospects

QHE specific to 2D+B

3D+B something like QHE ?

If, for some reason, gaps IQHE (Montambaux & Kohmoto 1990;

Kohmoto, Halperin & Wu 1992)

3D2D

usually

B

Integer QHE in 3D

Hall current in 3Dˆ( ) - j σ E

Koshino et al, PRL 86 (2001);

Koshino and Aoki, PRB 67 (2003)

Hofstader’s butterfly in 2D

(exp: Geisler et al, PRL 2004)

(sxy, szx) in units of (e2/ah)

BandE

LandauE

2D butterfly Bragg’s gap + Landau’s quantisation

2

Landau

Band /

E Ba

E h e

Origin of the butterfly

Landau’s q on xy-plane + xz plane

tanyxz

xy z

BE

E B

3D butterfly

interference

k-space picture

ky kx

kz

doubly-periodic 1D Harper problem

(ratio of periods: By / Bz = tan q) quantum tunnelling

between semiclassical orbits

Koshino et al, PRL 86 (2001);

Koshino and Aoki, PRB 67 (2003)

cf. 2D butterfly: single-period Harper’s eqncf. 2D butterfly:

～h/e B ～ 4000T for a = 10Å

Hall current = Thouless pumping

Kx’

k-space hopping

2

( , , ) ( , , )yz zx xy x y z

em m m

has s s -

ˆ 'eB

- r z Kh

Real space hopping

* * *ˆ ˆ ˆ( )x y zm m m K x y z

Hall conductivity

--- Alternative interpretation for the 3D IQHE

Ky’

Ez’ hkz’ = -eE

.

cf. 1D adiabatic pumping

(Thouless 1983)

Koshino and Aoki, PRB 67 (2003)

N

M

B

Maxwell’s relation

loop

31 2, ,ne n n

h a b c

-

j

‘Quantised surface-wrapping current’

Streda’s

formula

loop j j n

n

bulk( , , )yz zx xy

e N

Vs s s

-

B

surfacebulk( , , )yz zx xy

e N

Vs s s

-

B

bulk

=

Koshino, Aoki & Halperin, PRB 66, 081301R (2002)

Surface wrapping current and its quantisation

3D continuous system

(just an ordinary metal) in B

G1, G2: modulations

(e.g., acoustic waves)

(Koshino & Aoki, PRB 2004)

Acoustic waves for the 3D QHE

wavelength: 1/G > 10nm

(for B ~ 10T)

w < 100 GHz for Bi

(Arita et al,PRB 2004)

zeolite + metal

Band structure: Tight-binding model of superatoms (“supercrystal”)

～10A

Plateaux in disordered butterflies

(Aoki, Surf. Sci. 1992)

Future problems

(Kagalovsky, Horovitz & Avishai, 2004)

● QHE vs layered disordered superconductors

● Many-body effects

rxy (h/e2)

(Kivelson et al, 1992)

a Localisation in QHE systems--- Landau index dependence

a QHE in grapheneedge states, randomness

a QHE in 3D

Summary