Topological insulators and
superconductors
Rok Žitko
Ljubljana, 22. 7. 2011
• States of matter
– insulators
– quantum Hall effect
• Topological insulators (TI)
– 2D TI and helical edge states
– 3D TI and helical surface states
• Proximity effect and topological
superconductors
– Majorana edge states
– Detections schemes
States of matter
• Characterized by
– broken symmetries (long range correlations)
– topological order
• Quantified by
– order parameter
– topological quantum number
• Described by
– Landau theory of phase transitions
– topological field theories
Solid-liquid phase transition
Broken translation invariance
Order parameter: FT of <r(r)r(0)>,
Bragg peaks
FLandau=ay2+by4 a=a0(T-Tc)
Insulators
• Anderson insulators
disorder  electrons become localized
• Mott insulators
Coulomb interaction (repulsion) between
electrons  motion suppressed
• Band insulators
absence of conduction states at the Fermi
level  forbidden band
Band insulators
• vacuum (“Dirac sea” model):
Egap=2mc2=106 eV
• atomic insulators (solid argon):
Egap=10eV
• covalent-bond semiconductors and
insulators:
Egap=1eV
Bloch, 1928
2D electron gas in strong magnetic field
wc=eB/mc
Landau levels
Egap=ħwc
Quantum Hall Effect
von Klitzing et al. (1980)
Jy=xyEx
2
e
 xy  n
h
chiral
edge states
Gauss-Bonnet theorem
1

dS
K

2
Gaussian curvature, K=1/R1R2
Sphere =2
Torus =0
Topological insulators
• “Topological”: topological properties of the
band structure in the reciprocal space
• “Insulators”: well, not really. They have
gap, but they are conducting (on edges)!
• Quantum Hall effect: in high magnetic field, broken
time-reversal symmetry (von Klitzing, 1980)
• Time-reversal-invariant topological insulators
(Kane, Mele, Fu, Zhang, Qi, Bernevig, Molenkamp,
Hasan and others, from 2006 and still on-going)
Smooth transformations and
topology
Band structure: mapping from the Brillouin
zone (k) to the Hilbert space (y): k 
|y(k)
Bloch theorem: y(k)=eikr uk(r)
uk(r)=uk(r+R)
Smooth transformations: changes of the
Hamiltonian such that the gap remains
open at all times
See Fig.
TKNN (Chern) invariant
Thouless-Kohmoto-Nightingale-den Nijs, PRL 1982
1
2
n
d
k
F

2
F (k )  k  A(k )
Berry curvature
Integer
number!
N
A(k )  i  um  k um
m 1
Same n as in xy=ne2/h.
An integer within an accuracy of at least 10-9!
New resistance standard: RK=h/e2=25812.807557(18) W
Spin-orbit coupling



vE
B 2
c
H SO
eNucleus
B
1 U (r )  

L

S
2
mec r r
Stronger effect for heavy elements (Pb, Bi, etc.)
from the bottom of the periodic system
Reinterpretation:
Quantum Spin Hall effect (QSHE)
(“2D topological insulators”)
• Two copies of QHE, one for each spin, each
seeing the opposite effective magnetic field
induced by spin-orbit coupling.
• Insulating in the bulk, conducting helical edge
states.
• Theoretically predicted (Bernevig, Hughes and
Zhang, Science 2006) and experimentally
observed (Koenig et al, Science 2007) in
HgTe/CdTe quantum wells.
Edge states in 2D TIs
Helical modes: on each edge one pair of 1D modes related by the
TR symmetry. Propagate in opposite directions for opposite spin.
3D topological insulators
• Generalization of QSHE to 3D.
• Insulating in the bulk, conducting helical
surface states.
• Theoretically predicted in 2006,
experimentally discovered in
BiSb alloys (Hsieh et al., Nature 2008)
and in Bi2Se3 and similar layered materials
(Xia et al., Nature Phys. 2009).
Surface states on
3D topological insulators
• Conducting surface states must exist on the
interface between two topologically different
insulators, because the gap must close
somewhere near the interface!
• Single Dirac cone = ¼ of graphene.
In graphene, there is spin and valley
degeneracy, i.e., fourfold degeneracy.
Experimental detection in Bi2Te3
Chen et al. Science (2009)
Spin-momentum locking
Spin-resolved ARPES
Hsieh, Science (2009)
Topological field theory
q=0, topologically trivial, q=, topological insulator
Qi, Hughes, Zhang, PRB (2008), Wang, Qi, Zhang, NJP (2010).
Z2 invariants
Fu, Kane, Mele (2007)
Equivalence shown by Wang, Qi, Zhang (2010)
Time-reversal symmetry, t  -t
k
T
s
-k
Time-derivatives
-s (momenta) are reversed!
• Time-reversal operator: T=K exp(iy)
• Half-integer spin: rotation by 2 reverses the
sign of the state.
• Kramer’s theorem: T2=-1  degeneracy!
• Spin-orbit coupling does not break TR.
• Magnetic field breaks TR: Zeeman splitting!
Suppression of backreflection
Kramers doublet: |k↑=T|-k↓
k↑|U|-k↓=0
for any time-reversal-invariant operator U
Semiclassical picture: destructive interference.
Quantum picture: spin-flip would break TRI.
Magnetic impurities can open gap
Chen et al., Science (2010)
Kondo effect
in helical electron liquids
• Broken SU(2) symmetry for spin, but total
angular momentum (orbital+spin) still
conserved
• Previous work: incomplete Kondo
screening, residual degrees of freedom
leading to anomalies in low-temperature
thermodynamics
• My little contribution: complete screening,
no anomalous features
R. Žitko, Phys. Rev. B 81, 241414(R) (2010)
The problem has time-reversal symmetry,
so the persistance of Kondo screening seems
likely. The Kramers symmetry, not the spin
SU(2) symmetry, is essential for the Kondo
effect.
General approach: reduce the problem to a one-dimensional
tight-binding Hamiltonian (Wilson chain Hamiltonian) with the
impurity attached to one edge
K. G. Wilson, RMP (1975)
H. R. Krisnamurthy et al., PRB (1980)
R. Žitko, Phys. Rev. B 81, 241414(R) (2010)
Quantum anomalous Hall (QAH) state
= QHE without external magnetic field.
Proposal: magnetically doped HgTe quantum wells, Liu et al. (2008)
See also Qi, Wu, Zhang, PRB (2006), Qi, Hughes, Zhang, PRB (2010)
Chiral topological superconductor
= QAH + proximity induced superconductivity
One has to tune both the magnetization, m, and the
induced superconducting gap, D.
Qi, Hughes, Zhang, PRB (2010)
chiral Majorana mode
Review: Qi, Zhang (2010), Hasan, Kane, RMP (2010)
Majorana fermions
Two-state system: 0, 1
Complex “Dirac” fermionic operators y and y† defined as:
y† 0= 1, y 1= 0, y 0=0, y† 1=0
Canonical anticommutation relations: {y,y}=0, {y†,y†}=0, {y,y†}=1.
We “decompose” complex operator y into its “real parts”:
y=(h1+ih2)/2, y † =(h1-ih2)/2
Inverse transformation: h1=(y+y † )/2, h2=(y-y † )/(2i)
Real operators: hi † =hi
Canonical anticommutation relations: {h1,h1}=1, {h2,h2}=1, {h1,h2}=0.
Thus hi2=1/2.
Is this merely a change of basis?
• Not if a single Majorana mode is considered!
(Or several spatially separated ones.)
• Two separated Majorana fermions correspond to
a two-state system (i.e., a qubit, cf. Kitaev 2001)
where information is encoded non-locally.
• Many-particle systems may have elementary
excitations which behave as Majorana fermions.
• Single Majorana fermion has half the degrees of
freedom of a complex fermion → (1/2)ln2 entropy
Majorana excitations in
superconductors
• Solutions of the Bogoliubov-de Gennes
equation come in pairs:
y†(E) at energy E  y(E) at energy –E.
• At E=0, a solution with y†=y is possible.
•  Majorana fermion level at zero energy
inside the vortex in a p-wave
superconductor.
Reed, Green, PRB (2000), Ivanov, PRL (2001), Volovik
Non-Abelian states of matter
• In 2D, excitations with unusual statistics,
anyons (= particles which are neither
fermions nor bosons):
y1y2=eiqy2y1 with q0,
Wilczek, PRL 1982
• Zero-energy Majorana modes 
degenerate ground state
• Non-Abelian statistics:
y1y2=y2y1U
unitary transformation within
the ground state multiplet
Majorana fermions
in condensed-matter systems
• p-wave superconductors (Sr2RuO4, cold atom
systems)
• n=5/2 fractional quantum Hall state
• topological superconductors
• superconductor-topological insulator-magnet
heterostructures
Building blocks for topological quantum computers?
For a review, see Nayak, Simon, Stern, Freedman, Das Sarma,
RMP 80, 1083 (2008).
Detection of Majorana fermions
• Problem: Majorana excitations in a
superconductor have zero charge.
• Proposals:
– electrical transport measurements in
interferometric setups (Akhmerov et al, 2009;
Fu, Kane, 2009; Law, Lee, Ng 2009)
– “teleportation” (Fu, 2010)
– Josephson currents (Tanaka et al. 2009)
– non-Fermi-liquid kind of the Kondo effect
Interferometric detection
Electron can either be transmitted as an electron or as a hole (Andreev
process), depending on the number of flux quanta enclosed.
Akhmerov, Nilsson, Beenakker, PRL (2009); Fu, Kane, PRL (2009)
2-ch Kondo effect – experimental
detection in a quantum-dot system
Potok, Rau, Shtrikman, Oreg,
Goldhaber-Gordon (2007)
Two-channel Kondo model
TD
Can be solved by the numerical
renormalization group (NRG), etc.
TK
Bosonisation and refermionisation
One Majorana mode
decouples!
Emery, Kivelson (1992)
Majorana detection via induced
non-Fermi-liquid effects
Chiral TSC:
single
Majorana
edge mode
Source-drain linear
conductance:
R. Žitko, Phys. Rev. B 83,
195137 (2011)
Impurity decoupled from one
of the Majorana modes (a=0)
Standard Anderson
impurity (a=45º)
Parametrization:
Conclusion
• Spin-orbit coupling leads to non-trivial
topological properties of insulators
containing heavy elements.
• More surprises at the bottom of the
periodic system?
• Great news for surface physicists: the
interesting things happen at the surface.