quantum hall effect and real-space...
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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
top related