Download Topological phases of matter

Topological phases of matter:
From the quantum Hall effect to topological insulators to
Weyl semimetals
Kwon Park
Colloquium
POSTECH
November 30, 2016
2016 Nobel Prize in physics
David J. Thouless (1/2), F. Duncan M. Haldane (1/4), and J. Michael Kosterlitz (1/4)
“for theoretical discoveries of topological phase transitions and topological phases of
matter”
The Nobel prize in 2016 was awarded to two different topics: (i)
topological phase transitions, which is exemplified by the so-called
Kosterlitz-Thouless (KT) transition, and (ii) topological phases of
matter, which include quantum Hall states, topological insulators,
topological conductors (i.e., Weyl semimetals), and so on.
Spontaneous symmetry breaking
Disordered
Ordered
No order in the two-dimensional world?
•  Mermin-Wagner theorem: Thermal fluctuations destroy all orders (obtained by
spontaneously breaking the continuous symmetry) in a flat, two-dimensional
world even in the limit of absolute zero temperature.
•  In particular, this would mean that superconductivity and superfluidity cannot
exist in a thin film since both are long-range ordered states, where the phases of
Cooper pairs and superfluid bosons, respectively, are all aligned in the same way
as the magnetic moments in a ferromagnet.
Kosterlitz-Thouless transition
Kosterlitz and Thouless, Long-range order and metastability in two-dimensional solids and superfluids, J.
Phys. C: Solid State Phys. 5, L124 (1972); Ordering, Metastability, and phase transitions in two-dimensional
systems, ibid. 6, 1181 (1973).
•  The Kosterlitz-Thouless transition theory tells us that there can be a quasiordered state, which undergoes a topological phase transition at a critical
temperature.
2016 Nobel Prize press release
The correlation function
shows a power-law decay.
The correlation function
shows a exponential decay
Are there any topological phases of matter as stable phases, i.e., ground
states rather than just topological fluctuations?
Topological phases of matter are new phases of matter, which cannot be
characterized by conventional order parameters, but rather by
topological invariants.
What is the topological invariant?
A 3D topological object in real space: Magnetic Skyrmion
Winding number:
1
C=
4⇡
Z
d2 r n · (@x n ⇥ @y n)
Topological objects in momentum space?
Integer quantum Hall effect (IQHE)
h
Rxy = 2
ne
Von Klitzing, Dorda, Pepper (1980)
Figure courtesy: Nobel prize press release in1998
The incompressibility of the filled Landau level at integer fillings, combined with the
disorder-induced Anderson localization, explains the IQHE.
Nobel Prize in physics for the IQHE
Klaus von Klitzing (1980)
Rxy
h
= 2
ne
Nobel prize awarded in 1985 “for the discovery of the quantized Hall effect”
Quantized Hall conductance: TKNN formula
Thouless, Kohmoto, Nightingale, den Nijs, PRL 49, 405 (1982)
•  The Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula relates the topological
invariant called the Chern number with the Hall conductivity. xy
n=
e2
=n
h
X
✏µ (k)<✏F
i
Cµ =
2⇡
Z
k2BZ
Cµ
d2 khrk uµ (k)| ⇥ |rk uµ (k)i · ẑ
Berry curvature flux piercing through the Brillouin zone
(BZ) for the µ-th energy band 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)
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 (~
Interpretation of the TKNN formula via the Berry phase
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
A cartoon picture for the meaning of the TKNN formula
2016 Nobel Prize press release
Why is the longitudinal resistance zero?
•  The longitudinal resistance becomes zero since all electrons can flow freely at the
edge of system without any back-scattering.
•  The edge state is protected since it is the boundary between two phases, i.e.,
inside and outside the system, with different topological numbers.
Hasan, Kane, RMP 82, 3045 (2010)
Is it possible to make the Chern number non-trivial without the magnetic field?
Graphene
Hk =
✓
0 fk
fk⇤ 0
fk = t(eik·
1
◆
+ eik·
Hk=K+q ' ~vF q ·
2
+ eik· 3 )
⇤
Hk=K0 +q ' ~vF q ·
~vF = 3at/2
Massless Dirac Hamiltonian, which is equivalent to the Hamiltonian for the
Rabi oscillation with B replaced by q, which is known to have a magnetic
monopole!
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 + )
Magnetic monopole in the pseudospin space
H(~
↵) = ~ · ↵
~
~n (~
~ n (~
B
↵) = r↵~ ⇥ A
↵) = i hr↵~
~+ (~
B
↵) =
↵
ˆ
2↵2
↵)|
n (~
⇥ |r↵~
↵)i
n (~
•  This is precisely equal to the “magnetic field” exerted by a
Dirac monopole at the center with monopole strength −1/2
B
q=−1/2
Equivalence between the Berry flux and the Chern number
H(k) =
✓
gk
fk⇤
fk
gk
◆
= dk ·
dk,z
ky
dk,y
dk,x
kx
Momentum space
Berry flux = Chern number, or the winding number
Z
1
C=
d2 k d̂k · (@kx d̂k ⇥ @ky d̂k )
4⇡
Bloch sphere
A condition for the non-trivial topological invariant
H(k) =
✓
gk
fk⇤
fk
gk
◆
= dk ·
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 does the topological invariant of topological insulators manifest
itself?
One manifestation is the quantum spin Hall effect.
A manifestation of the non-trivial topological invariant:
Quantum spin Hall effect (QSHE)
Interface between a quantum Hall insulator (in the
IQHE or 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)
2D TI: HgTe quantum well
ŸSchematically 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 topological insulators 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
: Spin-mixing terms
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
0
A2 k z
2
M + B1 k?
+ B2 kz2
A1 (kx + iky )
k↵ ! sin (k↵ a)/a
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
⇡
kz = ⇡ plane
0
⇡
kx
⇡
ky
kz = 0 plane
A 3D TI becomes a strong TI if the
band topology is opposite between two
inversion-symmetric 2D subspaces in
the k-space containing one set of timereversal 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)
Weyl semimetal is a topological conductor extending the topological classification
of matter beyond topological insulator.
What does it mean that Weyl semimetal is topological?
A hallmark of Weyl semimetal is the existence of surface Fermi arcs.
Surface Fermi arcs
H =k·
Balents, “Weyl electrons kiss,” Physics 4, 36 (2011)
Xu et al., “Discovery of a Weyl fermion semimetal and topological Fermi arcs,” Science 349, 613 (2015)
A simple model Hamiltonian for Weyl semimetal
Yang, Lu, & Ran, PRB 84, 075129 (2011)
kz
k0
k0
ky
kx
Topologically
non-trivial since M/B < 0
H(k) = [ 2t(cos kx
cos k0 ) + m(2
+ 2t sin ky y + 2t sin kz z
h
i
m
' t(kx2 k02 ) + (ky2 + kz2 )
2
M
B
cos ky
x
+ 2tky
cos kz )]
y
+ 2tkz
x
z
While the existence of surface Fermi arcs is important, it is a surface property.
Is there a bulk property that can directly tell us about the non-trivial topological
structure of Weyl semimetal?
Our answer
Bloch oscillation
Experiments
Mendez & Bastard, Phys. Today 46, 34 (1993): Semiconductor superlattice
Raizen, Salomon, & Niu, Phys. Today 50, 30 (1997): 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, PRB 92, 195144 (2015)
En (k? ) = Ē(k? ) + neEak
Bohr-Sommerfeld quantization
eEak
Experimental observation of the Wannier-Stark ladder:
Semiconductor superlattice
Photocurrent spectra and interband transition energies of 40-20 Å GaAsGa0.65Al0.35As superlattice, obtained from photocurrent spectra at 5K
Mendez, Bastard, Phys. Today 46, 34 (1993)
Zak phase: Topological shift of the WSL
W.-R. Lee & KP, PRB 92, 195144 (2015)
Zak (k? )
=
✓
I
~
eEa /4D
0.3
=
C
dkk · Ak
Zak (k? )
En (k? ) = Ē(k? ) + n +
2⇡
(a)
(b)
0.2
0.1
◆
eEak
0
-1
0
- ~
(ħω-Ɛ)/4D
k? a? = ±⇡
1
-1
0
- ~
(ħω-Ɛ)/4D
k? a? = 0
1
Winding number of the WSL: BHZ model
W.-R. Lee & KP, PRB 92, 195144 (2015)
Trivial topology
Disorder strength (self-consistent Born approx.)
M/B > 0
Non-trivial topology
M/B < 0
Weyl nodes as topological defects of the WSL
K. W. Kim, W.-R. Lee, Y. B. Kim & KP, Nat. Commun. 7, 13489 (2016)
ŸApplied to the Yang-Lu-Ran model [PRB 84, 075129 (2011)]
Surface Fermi arc
Zak phase: I
Zak (k? )
=
dkk · Ak
Bulk Fermi arc!
3D plot of the
Zak phase
showing screw
dislocations
Adiabatic WSL:
EnWSL (k? )

= Ē(k? ) + eaE n +
Zak (k? )
2⇡
Weyl nodes, which are responsible for the non-trivial topological structure of Weyl
semimetal, can be directly manifested as topological defects of the WSL.
γ zak
kz
kx
This opens up the possibility of a novel spectroscopic method to characterize Weyl
semimetal.