Download Topology in Condensed Matter Physics: Quantum Hall Effect, Chern

Topology in Condensed Matter Physics:
Quantum Hall Effect, Chern Number, Topological Insulators,
and All That Jazz
Kwon Park
Colloquium
Pyeong-Chang Summer Institute (PSI)
July 29, 2015
The quantum Hall effect (QHE) is one of the most fascinating phenomena in
physics, induced by a topological quantum state of matter.
Written by Edwin A. Abbott in 1884
“Electrons in Flatland” under a high magnetic field: QHE
Pan et al., PRL 88, 176802 (02)
Classical Hall effect
Lorentz force due to magnetic field
e
j
F= v⇥B=
⇥B
c
⇢c
Electric force due to charge accumulation
F = eE
The Hall resistance is given by the steady-state condition balancing the two forces.
Rxy =
Ey
B
=
jx ρ ec
Integer quantum Hall effect (IQHE)
h
Rxy = 2
ne
Von Klitzing, Dorda, Pepper (1980)
Figure: Nobel prize press release (1988)
•  With help of the disorder-induced Anderson localization, the incompressibility of
completely filled Landau levels at integer filling factors can explain the IQHE.
Magnetic algebra
Ÿ There is a close similarity between the Hamiltonian of a particle moving in 2D under a
uniform magnetic field and that of an 1D harmonic oscillator.
1 ⇣
H=
p
2m
e ⌘2
1
A =
(⇡x2 + ⇡y2 )
c
2m
i~e
i~2
[⇡x , ⇡y ] =
r⇥A= 2
c
lB
2
lB
~c
=
eB
H = ~!C
1
a† a +
2
◆
[x, p] = i~
r
lB /~
a = p (⇡x + i⇡y )
2
lB /~
a† = p (⇡x i⇡y )
2
✓
⇤
1 ⇥ 2
2 2
H=
p + (m!) x
2m
m!
a=
2~
r
m!
a† =
2~
!C =
eB
mc
✓
⇣
p ⌘
x+i
m!
⇣
p ⌘
x i
m!
H = ~! a† a +
1
2
◆
Landau levels in the Landau gauge
Ÿ In the Landau gauge, the Hamiltonian reduces to that of an
1D harmonic oscillator with its center location proportional to
the perpendicular momentum, kylB2.
"
✓
1
H=
p2x + py
2m
)
eB
x
c
1 2 1
2
px + m!C
(x
2m
2
y
◆2 #
x
A = (0, Bx, 0)
2 2
k y lB
)
E
(x, y) = (x)eiky y
···
Ÿ  Each Landau level has a macroscopic degeneracy!
2
ky,max lB
= Lx
2⇡nmax 2
lB = L x
Ly
BA
Lx Ly
=
nmax =
=
2
2⇡lB
2⇡~c/e
···
Ly
0
Lx
x
Laughlin’s gauge argument for the quantized Hall resistance
Laughlin, PRB 23, 5632 (1981)
External Magnetic Field B
Eigenstate wave function in the Landau gauge:
• Gaussian wave packet in the x-direction
• Plane wave in the y-direction
• x0=kyl2 with l a magnetic length
y
Test flux
δΦ
x
Current
Applied Voltage V
• Magnetic moment: µ = IA/c
• Energy change:
δE = µ δB = µ δΦ/A = I δΦ/c
• Therefore, I = c δE/δΦ
Rxy =
V
h
= 2
I
ne
x
Effect of adding test flux δΦ:
• δΦ increases a vector potential shifting ky.
• The center of wave packets moves.
• When δΦ=hc/e, the physics must be invariant.
• Therefore, the wave packets move by one unit.
• When n Landau level is filled, n electrons are
transferred with the energy gain δE = neV.
The Hall resistance is so precisely quantized since only
the extended states can respond to the global flux change.
“Landauer” approach for the quantized Hall resistance
Edge states
µ+
µ = eVx
Edge states
dpy
hvy i
2⇡~
⌧
Z
dpy @H
=e
2⇡~ @py
X Z dpy @En
=e
2⇡~ @py
n
X Z µ+ dEn
=e
2⇡~
µ
n
ne
=
(µ+ µ )
h
ne2
=
Vx
h
Iy = e
µ+
µ
Ly
···
RH =
Lx
Z
Vx
h
= 2
Iy
ne
~v =
~⇡
1 ⇣
=
p~
m
m
µ+
e ~⌘
A
c
µ = eVx
Linear response theory for the quantized Hall resistance
Ÿ The fluctuation-dissipation theorem is a general result of statistical mechanics
stating that the fluctuation in a given system at equilibrium is related with the response
of the system to a small applied perturbation.
Fluctuation/Correlation
Dissipation/Response function
Density-density
Dielectric function
Spin-spin
Spin susceptibility
Current-current along the same direction
Conductivity
Current-current between orthogonal directions
Hall conductivity
Ÿ The Kubo formula provides the precise mathematical formulation of the fluctuationdissipation theorem.
xy
/
Z
1
0
dt hJx (t)Jy (0)i
Connection between the Chern number and the Hall conductance
Ÿ  Kubo formula:
↵

i
n0 e 2
(q, !) =
⇧↵ (q, !) +
!
m
⇧↵ (q, !) =
i
V
Z
1
1
↵
dtei!t ✓(t)h |[j↵† (q, t), j (q, 0)]| i
Ÿ  Current operator in the long wavelength limit:
j↵ (q ! 0) = ev↵ = e
X
v̂↵ (k) = e
k
X
k
c†k
@Hk
ck
@k↵
Ÿ  Hall conductivity in the long wavelength limit:

ie2 X
h✏n |vx |✏m ih✏m |vy |✏n i h✏n |vy |✏m ih✏m |vx |✏n i
(!
+
i⌘)
=
f
(✏
)
+
xy
n
V (! + i⌘) n,m
! + i⌘ + ✏n ✏m
(! + i⌘) + ✏n ✏m
Chern number as a topological order parameter:
TKNN formula
xy
/
=
Z
Z
X
2
d k
k2BZ
✏µ (k)<✏F <✏⌫ (k)
2
d k
k2BZ
X
✏µ (k)<✏F
huµ (k)|rk H(k)|u⌫ (k)i ⇥ hu⌫ (k)|rk H(k)|uµ (k)i
· ẑ
[✏µ (k) ✏⌫ (k)]2
hrk uµ (k)| ⇥ |rk uµ (k)i · ẑ
where uµk(r) is the periodic part of a Bloch wave function:

⌘2
1 ⇣
e
i~r + ~k
A(r) + U (r) uµk (r) = ✏µk uµk (r)
2m
c
• This is the famous Thouless-Kohmoto-Nightingale-Nijs (TKNN) formula relating the
the topologically invariant “order parameter” called the Chern number with the Hall
conductivity via σxy=ne2/h. n=
X
✏µ (k)<✏F
Cµ
i
Cµ =
2⇡
Z
k2BZ
d2 khrk uµ (k)| ⇥ |rk uµ (k)i · ẑ
Berry curvature flux piercing through the Brillouin zone
(BZ) for the µ-th energy band QHE in the lattice: Hofstadter’s Butterfly
Energy band
Color: Hall conductance
0
Magnetic flux through the unit cell
in units of magnetic flux quantum
1
Topology plays an intriguing role in quantum physics under the name of
the Berry phase.
Adiabatic evolution and geometrical phase
Ÿ  Adiabatic theorem (originally by Born and Fock, 1928):
A physical system remains in its instantaneous energy eigenstate if a given perturbation is
acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the
spectrum.
@
i~ | (t)i = H(t)| (t)i
@t
H(t)|
|
n (t)i
'|
n⇤ (t)i
= En⇤ (t)|
n⇤ (t)ie
i
~
Rt
0
n⇤ (t)i
dt0 En⇤ (t0 ) i
e
n (t)
Berry realized that there was an additional
geometrical phase that can have a physical effect.
Berry, “Quantal Phase Factors Accompanying Adiabatic Changes,”
Proc. R. Soc. A 392, 45 (1984)
Formal theory of the Berry phase
↵z
h
|
↵
n (~
|
↵y
↵x
+
↵)i
n (~
↵
~ )i
↵)| n (~
↵
n (~
=1+
=e
↵
~h
+
↵
~ )i
↵)| r↵~
n (~
~n (~
i ↵
~ ·A
↵)
|
↵)i
n (~
Ÿ Berry connection: vector potential
~ n (~
A
↵) = i h
↵)| r↵~
n (~
|
↵)i
n (~
Ÿ Berry curvature: magnetic field
~n (~
~ n (~
B
↵) = r↵~ ⇥ A
↵) = i hr↵~
n
↵)|
n (~
⇥ |r↵~
↵)i
n (~
Ÿ Berry phase: Aharonov-Bohm phase
I
Z
Z
~ n (~
~ ·B
~n (~
~ · hr↵~ n (~
=
d~
↵·A
↵) =
dS
↵) = i
dS
↵)| ⇥ |r↵~
C
A
A
↵)i
n (~
Formal theory of the Berry phase
n
=i
=i
Z
ZA
A
=i
Z
A
~ · hr↵~
dS
~·
dS
X
m6=n
↵)|
n (~
hr↵~
X h
~·
dS
⇥ |r↵~
↵)i
n (~
↵)| m (~
↵)i
n (~
↵)|r↵~
m (~
↵)i
n (~
⇥ h m (~
↵)|r↵~ H(~
↵)|
[En (~
↵) Em (~
↵)]2
↵)|r↵~ H(~
↵)| m (~
↵)i
n (~
m6=n
Note
⇥h
H(~
↵)|
↵)i
n (~
= En (~
↵)|
↵)i
n (~
↵)|r↵~
Multiplying both sides by h m (~
h
↵)|r↵~
m (~
↵)i =
n (~
h
↵)|r↵~ H(~
↵)| n (~
↵)i
m (~
En (~
↵)
Em (~
↵)
↵)i
n (~
TKNN formula revisited
Ÿ  The TKNN formula tells us that the Hall conductivity is proportional to the Berry phase
of a closed path encompassing the entire Brillouin zone.
n
=i
Z
A
~ · hr↵~
dS
↵)|
n (~
⇥ |r↵~
↵)i
n (~
↵
~ $k
xy
/
Z
2
d k
k2BZ
X
✏µ (k)<✏F
hrk uµ (k)| ⇥ |rk uµ (k)i · ẑ
When does the topology become non-trivial?
An answer to this question reveals that there is a profound connection between
the Rabi oscillation and topological insulators.
Rabi oscillation
H(t) =
!t
⌦ = |g|B0
B
i~
~⌦
ˆz + (cos !tˆx + sin !tˆy )
2
@
| (t)i = H(t)| (t)i =
@t
✓
~⌦/2
ei!t
= ~|g|B1 /2
i!t
e
~⌦/2
◆
| (t)i
Ÿ Exact solution:
| (t)i =
✓
e
˜ =
~⌦
i !t
2
0
0
e
i !t
2
p
~2 (⌦
◆✓
˜
˜
cos ⌦t
i cos ✓ sin ⌦t
˜
i sin ✓ sin ⌦t
!)2 /4 +
2
sin ✓ =
˜
~⌦
˜
i sin ✓ sin ⌦t
˜ + i cos ✓ sin ⌦t
˜
cos ⌦t
cos ✓ =
~(⌦
◆
!)/2
˜
~⌦
| (0)i
Geometrical meaning of the Berry phase
Ÿ  In the adiabatic limit, i.e., ω << Ω, the exact solution says:
p
E±⇤ (t) = ± ~2 ⌦2 /4 +
|
+ (t)i
'
!t
'
✓⇤
B
✓
✓
2
= ±~⌦⇤ /2
✓⇤
i !t
cos 2 e 2
✓⇤ i !t
sin 2 e 2
✓⇤
i !t
cos 2 e 2
!t
sin ✓2⇤ ei 2
◆
◆
˜
e
i ⌦t
2
e
i ⌦2 t
⇤
e
i ⌦
2 ⌦⇤
!t
1 ⌦
1
!t = cos ✓⇤ !t
+ (t) =
2 ⌦⇤
2
S(✓⇤ )
h
+ (t
= 2⇡/!)|
+ (t
= 0)i = e
⇤
i ⇡⌦
!
e
i
2 S(✓⇤ )
Solid angle enclosed by
a circular path
Magnetic monopole in the Rabi oscillation
~ = |g|S
~ ·B
~
H(B)
h
~+ (~
B
↵) = i
Note
H(~
↵) = ~ · ↵
~
(~
↵)i ⇥ h (~
↵)|r↵~ H(~
↵)|
[E+ (~
↵) E (~
↵)]2
↵)|r↵~ H(~
↵)|
+ (~
|
r↵~ H(~
↵) = ~
E± (~
↵) = ±|~
↵| = ±↵
~+ (~
B
↵) =
+
=
Z
A
↵
ˆ
2↵2
|
↵)i
+ (~
=
(~
↵)i =
cos ✓2 e i 2
sin ✓2 ei 2
1
(Solid angle)
2
!
sin ✓2 e i 2
cos ✓2 ei 2
“Magnetic field” exerted by a Dirac monopole
at the center with the strength -1/2
~ ·B
~+ (~
dS
↵) =
↵)i
+ (~
!
B
q=-1/2
Topological insulators (TIs) are generated by a magnetic monopole in
the pseudospin space.
Graphene
0
K, K = (
Ÿ Graphene Hamiltonian:
H=
t
X
hi,ji
H=
=
t
Z
Z
2
d k (e
BZ
d2 k
BZ
⇥
ik·
|k, Ai
1
+e
ik·
|k, Bi
2
2⇡
2⇡
,± p )
3a
3 3a
(|ri ihrj | + |rj ihri |)
ik·
+e
✓
0
fk⇤
)|k, Aihk, B| + H.c.
◆✓
◆
fk
hk, A|
0
hk, B|
3
⇤
Graphene
Hk =
✓
0
fk⇤
fk
0
◆
fk =
t(eik·
1
+ eik·
2
+ eik· 3 )
Ÿ Energy dispersion: Ek = ±|fk |
Hk=K+q ' ~vF q ·
⇤
Hk=K0 +q ' ~vF q ·
~vF = 3at/2
Massless Dirac Hamiltonian, or the Hamiltonian for “Zeeman coupling”
with B replaced by q
Haldane model
ai / bi: vectors connecting between nearest/next nearest neighbors
H(k) =
✓
gk
fk⇤
fk
gk
dk = (Refk , Imfk , gk )
X
f k = t1
eik·ai
i
gk = M + 2t2
X
i
◆
= dk ·
cos (k · bi + )
Chern number from the Landau level structure
Ÿ Energy of the Landau levels:
En = sgn(n)
q
(m↵ vF2 )2 + 2e~vF2 |nB| (n 6= 0)
E0 = ↵m↵ vF2 e sgn(B)
m↵ vF2 = M
ν
E
p
3 3↵t2 sin
ν
−9
−7
−5
−3
+9
+7
+5
+3
−1
+1
+1
−1
+3
+5
+7
+9
−3
−5
−7
−9
ν
B
E
ν
−9
−7
−5
−3
+9
+7
+5
+3
−1
+1
+1
−1
+3
+5
+7
+9
−3
−5
−7
−9
Graphene Landau levels
B
Chern number from the Berry phase:
Connection between the Rabi oscillation and TI
H(k) =
✓
gk
fk⇤
fk
gk
◆
= dk ·
dk,z
ky
dk,y
dk,x
kx
Momentum space
Bloch sphere
Chern number, or the winding number:
Z
1
C=
d2 k d̂k · (@kx d̂k ⇥ @ky d̂k )
4⇡
Kane-Mele model: spin-orbit-coupled graphene
Kane, Mele, PRL 95, 226801 (2005)
( )
( )
HKM = Hnn + Hnnn
⌘1
i
1
t
3
i
t
Hnn =
t
SO
i
(")
HKM =
fk =
i
SO
⌘3
⌘4
c†i cj
(")
Hnnn
=
SO
X
ei
ij
c†i cj
hhi|jii
d2 k
BZ
t
SO
SO
hi|ji
Z
i
SO
⌘5
2
X
⌘2
⌘6
i
t
SO
3
X
i=1
⇣
e
c†k,A
ik·
m
c†k,B
gk =
⌘✓ g
k
fk⇤
i
SO
fk
gk
6
X
i=1
◆✓
ck,A
ck,B
( 1)m eik·⌘m
◆
Low-energy effective Hamiltonian for the Kane-Mele model
(")
HKM (k) =
✓
gk
fk⇤
fk
gk
◆
= dk ·
H (#) (k) = H (")⇤ ( k)
(1) The energy spectrum is gapped:
dk = (Refk , Imfk , gk )
Time-reversal symmetry
E±,k = ±|dk | = ±
q
|fk |2 + gk2
(2) The low-energy effective Hamiltonian:
• Around k = K + q,
fq = ~vF (qx + iqy )
gq =
p
3 3
p
9 3
SO +
4
0
• Around k = K + q,
fq = ~vF (qx
iqy )
p
gq = 3 3
p
SO
9 3
4
SO a
SO a
2
2
(qx2 + qy2 )
(qx2 + qy2 )
Condition for topological non-triviality
(")
HKM (k) = dk ·
H (#) (k) = H (")⇤ ( k)
dk=K+q ' (Aqx , Aqy , M + B(qx2 + qy2 ))
dq,z = M + B(qx2 + qy2 )
dq,y =
dq,x = Aqx
dq,z = M + B(qx2 + qy2 )
Aqy
dq,y =
dq,x = Aqx
M/B > 0
M/B < 0
Topologically trivial
Topologically non-trivial
Aqy
How to measure: quantum spin Hall effect (QSHE)
Interface between a quantum Hall insulator (in the
Haldane model) and an ordinary insulator
1D energy bands for a strip of spinorbit coupled graphene as described
by the Kane-Mele model
Interface between a quantum spin Hall insulator (in
the Kane-Mele model) and an ordinary insulator
Hasan, Kane, RMP 82, 3045 (2010)
Kane, Mele, PRL 95, 226801 (2005)
Bernevig-Hughes-Zhang (BHZ) model: HgTe quantum well
!Schematically speaking, a half of the Kane-Mele model
0
M + B(kx2 + ky2 )
B
A(kx iky )
H(k) = ✏k I + B
@
0
0
Spin-up electrons in the slike E1 conduction and the
p-like H1 valence bands
A(kx + iky )
[M + B(kx2 + ky2 )]
0
0
Bernevig, Hughes, Zhang, Science 314, 1757 (2006)
0
0
M + B(kx2 + ky2 )
A(kx + iky )
0
0
A(kx iky )
[M + B(kx2 + ky2 )]
1
C
C
A
Spin-down electrons in the slike E1 conduction and the
p-like H1 valence bands
Two-terminal charge
conductance, not the
spin-filtered Hall
conductance!
M/B > 0
M/B < 0
König et al., Science 318, 766 (2007), adapted by Qi, Zhang, Phys. Today 63, 33 (2010)
How to promote TI from 2D to 3D?
3D TI as a system of stacked 2D TI layers: weak TI
2D TIs
Ÿ Unfortunately, unlike the 2D helical edge states, the time-reversal symmetry
does not protect the surface states in a weak TI. Here, the surface states may be
localized in the presence of disorder.
Strong 3D TI: BiSe-family materials
0
2
M + B1 k?
+ B2 kz2
B
A1 (kx iky )
H(k) = ✏k I + B
@
0
A2 k z
Ÿ  Lattice regularization:
A1 (kx + iky )
2
(M + B1 k?
+ B2 kz2 )
A2 k z
0
k↵ ! sin (k↵ a)/a
0
A2 k z
2
M + B1 k?
+ B2 kz2
A1 (kx + iky )
k↵2 ! 2(1
1
A2 k z
C
0
C
A
A1 (kx iky )
2
2
(M + B1 k? + B2 kz )
cos (k↵ a))/a2
Ÿ  Strong TI:
kz
⇡
0
⇡
kx
kz = ⇡ plane
⇡
ky
kz = 0 plane
A 3D TI becomes a strong TI if
band topology is opposite between
two 2D subsystems in the k-space
containing one set of time-reversal
invariant momenta (TRIM) and
the other.
Helical surface states: spin-momentum locking
Bi2Te3
Bi2Se3
Chen et al., Science 325, 178 (2009), adapted by Qi,
Zhang, Phys. Today 63, 33 (2010)
Xia et al., Nature Phys. 5, 398 (2009), Hsieh et al., Nature 460,
1101 (2009), adapted by Qi, Zhang, RMP 83, 1057 (2011)
How to measure topology directly in the bulk without reference to boundaries?
A quotation from Lord Kelvin, “To measure is to know.”
W.-R. Lee & KP, arXiv:1503.01870
Bloch oscillation
Experiments
Waschke et al., PRL (1993): Semiconductor superlattice
Dahan et al., PRL (1996): Optical lattice
Band
width
e
εk
k
•  The electron motion in the lattice is
bounded and oscillatory due to the fact that
no states are available outside the energy
band.
•  Another way of viewing this is that the
group velocity becomes negative once the
crystal momentum crosses the zone
boundary.
Quantized Bloch oscillation: Wannier-Stark ladder (WSL)
W.-R. Lee & KP, arXiv:1503.01870
eEak
En (k? ) = Ē(k? ) + neEak
Zak phase: the WSL revisited
W.-R. Lee & KP, arXiv:1503.01870
~
eEa /4D
0.3
=
(a)
(b)
0.2
0.1
0
-1
✓
En (k? ) = Ē(k? ) + n +
Zak (k? )
2⇡
◆
eEak
0
- ~
(ħω-Ɛ)/4D
1
-1
0
- ~
(ħω-Ɛ)/4D
1
Winding number of the WSL: BHZ model
W.-R. Lee & KP, arXiv:1503.01870
Trivial topology
Disorder strength (self-consistent Born approx.)
M/B > 0
Non-trivial topology
M/B < 0
Winding number of the WSL: Kane-Mele model
W.-R. Lee & KP, arXiv:1503.01870
Non-trivial topology
SO
6= 0
M/B < 0
Critical topology:
Usual graphene
SO
=0
M =B=0
Winding
becomes
discontinuous!
Winding number of the WSL: Strong 3D TI model
W.-R. Lee & KP, arXiv:1503.01870
! 2D TI
! 3D TI
(a)
2D subspace
containing TRIM:
Abelian
Berry phase
General 2D subspace:
Non-Abelian
Berry phase
Winding number of the WSL: Strong 3D TI model
W.-R. Lee & KP, arXiv:1503.01870
Non-trivial
topology
2D subspace
containing TRIM:
Abelian Berry
connection
General 2D subspace:
Non-Abelian Berry
connection
Trivial
topology
Topological insulators provide one of the most dramatic physical examples
accentuating an intriguing role of the geometrical phase in quantum physics.