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Integer Quantum Hall effect• basics
• theories for the quantization
• disorder in QHS
• Berry phase in QHS
• topology in QHS
• effect of lattice
• effect of spin and electron interaction
M.C. Chang
Dept of Phys
Hall effect (1879), a classical analysis
* *
ˆ; / 0 at steady state
dv v vm eE e B mdt c
B Bz dv dtτ
= − − × −
= =
*
*
/ // /
x x
y y
v Em eB ce
v EeB c mτ
τ⎛ ⎞ ⎛ ⎞⎛ ⎞
= −⎜ ⎟ ⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠
*
2
0*
2
11
x x xc
y y yc
m BE j jne necE j jB m
nec ne
ω ττ ρω τ
τ
⎛ ⎞⎜ ⎟⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟= =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟−⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠−⎜ ⎟⎝ ⎠
ρxy
B
j env= −*
0 *2 , cm
neBm ce
ρτ
ω ==
( )1 0
2
10
1
111
1
1
0 // 0
c
c
c
cc
c
c
nec Bnec B
ω τ
ω τ
ω τσω τω τ
ω τσ
ω τ
−
>
−⎛ ⎞= = ⎜ ⎟
+ ⎝ ⎠
−⎛ ⎞⎯⎯⎯→ ⎜ ⎟
⎝ ⎠−⎛ ⎞
⎯⎯⎯→ ⎜ ⎟⎝ ⎠
σ ρ2
0 *
nemτσ =
• Hall conductivity
• Hall resistivity
Resistance and conductance
,x xx xy x x xx xy xy yx yy y y yx yy y
V R R I I VV R R I I V
Σ Σ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟Σ Σ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠
Note:
detyy
xxRΣ
=Σ
So it’s possible to have Rxx and Σxxsimultaneously be zero (provided Rxy and Σxyare nonzero).
0
3 :
2 :
y
y yxx xx yx yx
x xI
yxx xx yx yx
x
V E WL WD R RA I J A A
E WLD R RW J W
ρ ρ
ρ ρ
=
= ≡ = =
= = =
L
Wx
y
Quantum Hall effect 0, .
1 1det det
xx yx
yx yxyx yx
yx yx
const
RR R
ρ ρ
ρσ
ρ ρ
= =
→ Σ = = = = =
Measurement of Hall resistance
2-dim electron gas (2DEG)
GaAs/AlGaAs heterojunction
(broadened) Landau levels in a magnetic field
Ener
gy
μ
subband
Dynamics along z-direction is frozen in the ground state
Ando, Matsumoto, and Uemura JPSJ 1975
Effect of disorder on σxy (theoretical prediction before 1980)
Kawaji et al, Supp PTP 1975
Si(100) MOS inversion layer
9.8 T, 1.6 K
1985
ρxy deviates from (h/e2)/n by less than 3 ppm on the very first report.• This result is independent of the shape/size of sample. • Different materials lead to the same effect (Si MOSFET, GaAsheterojunction…)
→ a very accurate way to measure α-1 = h/e2c = 137.036 (no unit)→ a very convenient resistance standard.
Quantum Hall effect (von Klitzing, 1980)
An accurate and stable resistance standard (1990)
Kinoshita, Phys. Rev. Lett. 1995• theory
• experiment
Condensed matter physics is physics of dirt - Pauli
dirty clean
• Flux quantization
0 2he
φ =
• Quantum Hall effect
• …
Often protected by topology, but not vice versa.
The triangle of quantum metrology
QCP
e
(to be realized)
I
QHEV h / e 2
Josephsoneffect
f
e / h
Quantum Hall effect requires • Two-dimensional electron gas• strong magnetic field • low temperature Note: Room Temp QHE in graphene (Novoselov et al, Science 2007)
Plateau and the importance of disorder
Broadened LL due to disorder
Why RH has to be exactly (h/e2)/n ?
• see Laughlin’s argument below
( )B ck T ω<
Filling factor
Aoki, CMST 2011
The importance of localized states
Width of extended states?
256 states in the LLL. ε(Φ) periodic in Φ0Aoki 1983
• Finite-Size Scaling
~ , 1/ 2xE N x ν−Δ =
ΔE ΔE
Huo
and Bhatt P
RL 1992
Exponent for correlation length
Li et al PR
L 2005
• experiment Ensemble average over 100-2000 disorder configurations
States that can carry Hall current (with non-zero Chern number)
Quantization of Hall conductance, Laughlin’s gauge argument (1981)21 ( ) ( )
2 i i eiei
eH p A r V Vrm c⎛ ⎞= + +⎜ ⎟⎝
+⎠
∑
1 ( )
x x iix y
x y x y
e ej A rm L L i x c
c H cL L A L
HΦ
− ∂⎡ ⎤= +⎢ ⎥∂⎣ ⎦
∂∂= − = −
∂ ∂Φ
∑
x xA LΦ =
x
y
solve | |H Eψ ψΦ Φ Φ Φ>= >By the Hellman-Feynman theorem, one has
| | | |
xy
H EH
EcjL
ψ ψ ψ ψΦ ΦΦ Φ Φ Φ Φ
Φ
∂ ∂∂< >= < >=
∂Φ ∂Φ ∂Φ∂
∴ = −∂Φ
• EF at localized states, no charge transfer whatever Φ is.
• EF at extended states, only integer charges may transfer along y when Φ is changed by one Φ0. 0
2( ) yx y
y
Vn ej c e EL
nh
−= − =
Φ
• Due to gauge symmetry, the system needs to be invariant under Φ→ Φ+ Φ0,
• Simulate a longitudinal EMF by a fictitious time-dependent flux Φ
1
Edge state in quantum Hall system
• Bending of LLsGapless excitations at the edges
• Robust against disorder (no back-scattering)
• Classical pictureChiral edge state (skipping orbit)
• number of edge modes = n
Inclusion of lattice (more details later)
• Bulk states: En(kx,ky) (projected to ky); Edge states: En(ky)
• when the flux is changed by 1 Φ0, the states should come back.
→ Only integer charges can be transported.
Figs from Hatsugai’s ppt
2Streda formula (1982)
Giuliani and Vignale, Sec 10.3.3
• If ν bands are filled, then the number of electrons per unit area is n=νeB/hc
∴ σH=νe2/h
L
R
Nonzero along edgeˆc M z= ∇ ×
Degeneracy of a LL: D=BA/Φ0
Current response: conductivity
• Vector potential of an uniform electric field
22
00 0
0
1 ( )2
'
latt
i t
e eH p A V H A p O Am c m ceH p A e
m cω
ω−
⎛ ⎞= + + = + ⋅ +⎜ ⎟⎝ ⎠
= ⋅
1 ( )( )
( ) , then ( ) ;i t i t
A tE tc t
iE t E e A t A e E Ac
ω ωω ω ω ω
ω− −
∂= −
∂
= = =
2
,
( ) m m m
m m m
m m m
m
f f v veiV
v v
α
α
β
α
β
ασ ω ω ω
ω ψ
ω
ω ω ψ
−=
+
≡ − ≡
∑
• 1st order perturbation in E →
Kubo-Greenwood formula
j Eα αβ βσ=
00
1=
=
m
m m
m
m
pm
u
u k um
uk k
i
α
α
α
α
α
ε δ ω∂ ∂+∂ ∂
∂ +
Quantization of Hall conductanceThouless et al’s argument (1982)
3
2
2 20
2
1
2 nk nk nk nk
DC m m m m
m m
nknk
u u u ui
p p p pe fim V
e fV k k k k
α β
α β
α
α β
β
β
α
σω≠−
=
⎛ ⎞= ⎜ ⎟⎜ ⎟
∂ ∂ ∂ ∂−
∂ ∂ ∂ ∂⎝ ⎠
∑
∑
ℓ, m = (n, k)
• Berry curvature
( ) ( )2
2
( , , are
(
( )
)
cyclic)
1
2
nk nk nk nk
n zH nBZ
n
e
u u u ukk k
ik
d k k
k
h
γα β β α
α β γ
σπ
⎛ ⎞∂ ∂ ∂ ∂Ω ≡ −⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝
Ω⎦
⎠
⎡ ⎤= ⎢ ⎥
⎣∫
• Hall conductivity for the n-th band
cell-periodic function um
an integer for a filled band
( )
( )
n n nk k
nk
k i u u
A k
Ω = ∇ × ∇
= ∇ ×
( )n n nkA k i u u≡ ∇
• Berry curvature (for n-th band)
• Berry connection
2
( ,0) ( , ) ( , ) (0, )
BZb c d a
a b c d
x x x x x y y y x y y y
d k
dk A dk A dk A dk A
dk A k A k g dk A g k A
A
k→ ↑
= ⋅ + ⋅ + ⋅ + ⋅
⎡ ⎤ ⎡ ⎤= − + −⎣ ⎦
∇
⎦
×
⎣
∫
∫ ∫ ∫ ∫∫ ∫
1 2( ) ( )ˆ ˆ
2 2
,
( ,0) ( , ) ( ) ( )
etc
y x
x y
i k i kk k g x k k g y
x x x x x y
u e u u e u
dk A k A k g a b
θ θ
θ θ
+ +
→
= =
⎡ ⎤− = −⎣ ⎦∫
2
2 2 1 1( ) ( ) ( ) ( )2
BZ
d k
a b d a
A
nθ θ θ θπ
= −
∇×
+ −=
∫
Pf:a b
cd
Brillouinzone
• Niu-Thouless-Wu generalization to system with disorder and electron interaction (PRB 1985).
BZ
Zeros and vortices
total vorticity in the BZ
Czerwinski and Brown, PRS (London) 1991
gx
gy
21 intege( r )2 B
zZ
k kd nπ
Ω =∫
[ ]
1
2
1
2
1 2 1 2
( )
( )
( )
( )
( ) ( ) ( ) ( )
i aa b
i bb c
i dc d
i ad a
i a b d aa a
u e u
u e u
u e u
u e u
u e u
θ
θ
θ
θ
θ θ θ θ
−
−
+ − −
=
=
=
=
∴ =
Connection with localization in disordered system (Anderson, 1958)
• one-parameter scaling hypothesis(Abrahams et al, 1979 < Thouless, Landauer…): assumeβ(g) depends only on g
Localized
extended
Quasi-extended
20( ) , ( ) 2
dg L L g dσ β−= = −
• For large g (good conductor)
• For small g (insulator)/( ) , ( ) lnLc
c
gg L g e gg
ξ β−= =
Lagendijk et al, Phys Today 2009
• All wave functions of disordered systems in 1D and 2D are localized.
• QHE belongs to a new class of disordered systems.
This analysis does not apply to the QHS, since the extended states are crucial there.
Flow follows the increase of L
MIT
conductance
Fig from Altshuler’s ppt
Spectral distribution of random matrix (rank N>>1)• eigenvalues Ei• mean level spacing d1= (taking ensemble average)
• spacing between NN s=(Ei+1-Ei)/d1• P(s): distribution function of s
• spectral rigidity: P(0)=0
• level repulsion: P(s
Fig from Altshuler’s ppt
Wigner-Dyson classes
GOE
GUE
GSE
AI
A
AII
Altland-Zirnbauerclasses
Quantization ofmagnetic monopole (see Sakurai Sec 2.6)
• Vector potential (use 2 “atlas” to avoid Dirac string)
• gauge transformation between 2 atlas
→ monopole charge is quantized
YM
Shn
ir, M
agne
tic m
onop
oles
Note:
2 2
2 /
N S ig ig
N S ieg c
A A ie ee
ϕ ϕ
ϕψ ψ
−
⋅
− = − ∇
=
2eg nc=
Analogy in QH system
• Gauge transformation
Kohmoto, Ann. Phys, 1985
• Two atlases
, , ,( , ; ) ( ) ( )n n nH r p x E xλ λ λλ ψ ψ=
• Fast variable and slow variable
• “Slow variables Ri” are treated as parameters λ(t)
(Kinetic energies from Pi are neglected)
• solve time-independent Schroedinger eq.
“snapshot” solution
{ }( , ; , )i iH r p R P
electron; {nuclei}
Born-Oppenheimer approximation
e-
H+2 molecule
nuclei move thousands of times slower than the electron
Instead of solving time-dependent Schroedinger eq., one uses
Connection with Berry phase First, a brief review of Berry phase:
• After a cyclic evolution
0
, ( ) , (
'
0
(
)
' )T
ni dt E
n
t
n T eλ λψ ψ− ∫=
Dynamical phase
Adiabatic evolution of a quantum system
0 λ(t)
E(λ(t)) ( ) (0)Tλ λ=
xx
n
n+1
n-1
• Phases of the snapshot states at different λ’sare independent and can be arbitrarily assigned
(, ( , ( )
))
nn t n t
ieλ λγ λψ ψ→
• Do we need to worry about this phase?
• Energy spectrum:
( , ; )H r p λ
, , 0n n ni λ λγ ψ ψ λλ∂
= ⋅ ≠∂
≣An(λ)
• Fock, Z. Phys 1928• Schiff, Quantum Mechanics (3rd ed.) p.290
No!
Pf :
( ) ( )H t i ttλ λ∂
Ψ = Ψ∂
0' ( ')( )
,( )t
nni dt E ti
nt e eγ λ
λ λψ−
Ψ = ∫
Consider the n-th level,
Stationary, snapshot state
, ,nn nH Eλ λψ ψ=
( ), ,'
nn n
ie φ λλ λψ ψ=
nφλ
∂= −
∂Choose a φ (λ) such that,
Redefine the phase,
Thus removing the extra phase
An’(λ) An(λ)
An’(λ)=0
• One problem: ( )Aλφ λ∇ =does not always have a well-defined (global) solution.
0C
A d λ⋅ =∫ 0C A d λ⋅ ≠∫
Vector flow A
Contour of φ
C
Vector flow
Contour of φ
A φ is not defined here
C
0' ( ')
( ) (0)C
Ti dt t
Ti Ee eγλ λψ ψ
− ∫=
0C C i dλ λγ ψ ψ λλ∂
= ⋅ ≠∂∫
• Berry phase (path dependent)
M. Berry, 1984 : • Parameter-dependent phase NOT always removable!
Index nneglected
• Berry connection (or Berry potential)
• Berry curvature (or Berry field)
( )A i λλ λλ ψ ψ≡ ∇
( ) ( )F A iλ λ λλ λλ λ ψ ψ≡ ∇ × = ∇ × ∇
C C SA d A daλγ λ= ⋅ = ∇ × ⋅∫ ∫
• Stokes theorem (3-dim here, can be higher)
• Gauge transformation
( )
( ) ( )
( ) ( )
i
C C
e
A A
F F
φ λλ λ
λ
ψ ψ
λ λ φ
λ λγ γ
→
→ −∇
→→
i
i
ii
Redefine the phases of the snapshot states
Berry curvature and Berry phase are gauge invariant
2λ
3λ
1λ
( )tλ
C
S
Some terminology
λ→ k in QHS
spin × solid angle
Example: spin-1/2 particle in slowly changing B field
BBH Bλ μ σ= = ⋅
y
z
x( )B t
CS
• Real space • Parameter space
Berry curvature
a monopole at the origin
Berry phase
S
1= ( )2
F da Cγ ± ± ⋅ = Ω∫ ∓
yB
zB
xB( )B t
C
E(B)
B
Level crossing at B=0
−
+
2, ,
ˆ1( )2B BB B
BF B iB
ψ ψ± ± ±= ∇ × ∇ = ∓
Examples of the Berry phase:
Magnetic monopole / Berry phase / fiber bundle
( )A k i λ λ λψ ψ≡ ∇
in parameter space
Berry connection
Berry curvature (in 3D)
( ) ( )F Aλλ λ≡ ∇ ×
S
( )
=
C CA d
F da
γ λ λ= ⋅
⋅
∫∫
Berry phase
Total curvature
1 ( ) integer2
F daλπ
⋅ =∫
in real space
Vector potential
Magnetic field
( ) ( )B r A r≡ ∇×
( )A r
( )
=C
S
A r dr
B da
Φ = ⋅
⋅
∫∫
Magnetic flux
Monopole charge
1 ( ) integer4
B r daπ
⋅ =∫
connection
curvature
1st Chern number
horizontal lift (along a U(1) fiber)
U(1) fiber bundle
A
F
γ
C1
(QHE: λ→ k in BZ)
Connection with geometry First, a brief review of topology:
K≠0
G≠0K≠0
G=0
• extrinsic curvature K vs• intrinsic (Gaussian) curvature G
G>0
G=0
G
Euler characteristic 歐拉特徵數
2 ( ), 2(1 )M
da G M gπχ χ= = −∫
• Gauss-Bonnet theorem (for a 2-dim closed surface)
0g = 1g = 2g =
The most beautiful theorem in differential topology
• Gauss-Bonnet theorem (for a surface with boundary)
( ),2gM M M Mda G ds k π χ∂ ∂+ =∫ ∫
Marder, Phys Today, Feb 2007
• Can be generalized to higher dimension.
• Nontrivial fiber bundleMöbius band
• Trivial fiber bundle(a product space R1 x R1)
Simplest examples:
R1
R1base
fiber
• Fiber bundle ~ base space × fiber space
Fiber bundle: a generalization of product space
• In physics, a fiber bundle ~ Physical space × Inner space
• In QHS, we have T2 x U(1)
• The topology of a fiber bundle is classified by Chern numbers ~the topology of a closed surface is classified by Euler characteristics
(spin, gauge field…)
base space
fiber space
Lattice electron in a magnetic field: magnetic translation symmetry
consider a uniform B field
Indep of r• Magnetic translation operator
Commute if this is 1
Xiao et al, RMP 2010
B
a1
a2
Simultaneous eigenstates: magnetic Bloch states
• If Φ=(p/q)Φ0 per plaquette, then Magnetic Brillouin zone = BZ/q.
e.g., p/q=1/3
Hofstadter spectrum
Band structure of a 2DEG subjects to both a periodic potential V(x,y) and a magnetic field B.
Can be studied using the tight-binding model (TBM).
B The tricky part:
q=3 → q=29 upon a small change of B!Also, when B → 0, q can be very large.
1
3 110
1029
13
187−
= = +
Surprisingly complex spectrum!Split of energy band depends on flux/plaquette.If Φplaq/Φ0= p/q, where p, q are co-prime integers, then a Bloch band splits to q subbands (for TBM).
Hofstadter’s butterfly (Hofstadter, PRB 1976)
• A fractal spectrum with self-similarity structure
Self-similarity (heirarchy)
B → 0 near band button, evenly-spaced LLs
• The total band width for an irrational q is of measure zero (as in a Cantor set).
Wid
th o
f a B
loch
ban
d w
hen
B=0
Landau subband
集異璧
著作:Douglas R. Hofstadter翻譯:郭維德
Pulitzer 1980M
IT: h
ttp://
ocw
.mit.
edu/
high
-sch
ool/c
ours
es/g
odel
-esc
her-b
ach/
C1 = 1
C2 = −2
C3 = 1
Bloch energy E(k) Berry curvature Ω(k) p/q=1/3
Distribution of Hall conductance among subbands(Thouless et al PRL 1982)
r rr pt qs= +• Diophantine equation
• for rectangular lattice: sr should be as small as possible
• for triangular lattice: sr and tr cannot both be odd(Thouless, Surf Sci 1984)
e.g., / 2 / 52 5
5 2(0) 5(1)4 2(2) 5(0)3 2( 1) 5(1)2 2(1) 5(0)1 2( 2) 5(1)0 2(0) 5(0)
r r
p qr t s
== += += += − += += − += +
See Xiao et al RMP 2010 for another derivation
1
/
H
tot
tot plaq
plaq
H r
necB
N r rnA q A q
BApq hc e
t
σ
σ
∂=
∂
= =
=
∴ =
• Streda formula
• for weak magnetic field: (σH)r = tr - tr-1• for strong magnetic field: (σH)r = sr - sr-1
Jump of Hall conductance induced by band-crossingLee, Chang, and Hong, PRB 1998
Φ=2/5
Lattice with edges
• Energy dispersion of edge states
Hatusgai, J Phys 1997