Transcript Slide 1

Effective Topological Field Theories in
Condensed Matter Physics
Theoretical prediction: Bernevig, Hughes and Zhang, Science 314, 1757 (2006)
Experimental observation: Koenig et al, Science 318, 766 (2007)
New Developments: Qi et al, Nature Physics 4, 273, 08’, Phy Rev B78, 195424, 08’, Science 323, 1184, 09’
The search for new states of matter
The search for new elements led to a golden age of chemistry.
The search for new particles led to the golden age of particle physics.
In condensed matter physics, we ask what are the fundamental states of matter?
In the classical world we have solid, liquid and gas. The same H2O molecules can
condense into ice, water or vapor.
In the quantum world we have metals, insulators, superconductors, magnets etc.
Most of these states are differentiated by the broken symmetry.
Crystal: Broken
translational symmetry
Magnet: Broken
rotational symmetry
Superconductor: Broken
gauge symmetry
The quantum Hall state, a topologically non-trivial
state of matter
 xy
e2
n
h
• TKNN integer=the first Chern number.
d 2 k 
n
 F (k )
2
(2 )
• Topological states of matter are
defined and described by topological
field theory:
S eff 
 xy
2
2

d
xdt

A  A

• Physically measurable topological
properties are all contained in the
topological field theory, e.g. QHE,
fractional charge, fractional statistics
etc…
Chiral (QHE) and helical (QSHE) liquids in D=1


k
k
kF
-kF
The QHE state spatially separates the two
chiral states of a spinless 1D liquid
kF
-kF
The QSHE state spatially separates the
four chiral states of a spinful 1D liquid
x
2=1+1
4=2+2
x
No go theorems: chiral and helical states can never be constructed microscopically
from a purely 1D model. (Wu, Bernevig, Zhang, 2006) Helical liquid=1/2 of 1D fermi liquid!
Time reversal symmetry in quantum mechanics
• Wave function of a particle
with integer spin changes by 1
under 2 rotation.
Spin=1
• Wave function of a half-integer
spin changes by -1 under 2
rotation.
• Kramers theorem, in a time
reversal invariant system with
half-integer spins, T2=-1, all
states for degenerate doublets.
• Application in condensed
matter physics: Anderson’s
theorem. BCS pair=(k,up)+(k,down). General pairing
between Kramers doublets.
y> y
Spin=1/2
y>-y
The topological distinction between a conventional
insulator and a QSH insulator
Kane and Mele PRL, (2005); Wu, Bernevig and Zhang, PRL (2006); Xu and Moore, PRB (2006)
• Band diagram of a conventional insulator, a conventional insulator with accidental
surface states (with animation), a QSH insulator (with animation). Blue and red
color code for up and down spins.

k
Trivial
k=0 or 
Trivial
Non-trivial
From topology to chemistry: the search for the QSH state
• Graphene – spin-orbit coupling only about 10-3meV. Not realizable in experiments.
(Kane and Mele, 2005, Yao et al, 2006, MacDonald group 2006)
• Quantum spin Hall with Landau levels – spin-orbit coupling in GaAs too small.
(Bernevig and Zhang, PRL, 2006)
Bandgap vs. lattice constant
(at room temperature in zinc blende structure)
6.0
5.5
• Type III quantum
wells work. HgTe has a
negative band gap!
• Tuning
4.5
Bandgap energy (eV)
(Bernevig, Hughes and
Zhang, Science 2006)
5.0
the thickness of
the HgTe/CdTe quantum
well leads to a
topological quantum
phase transition into the
QSH state.
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Band Structure of HgTe
S
P
P3/2
S
S
S
P1/2
P
P3/2
P1/2
Quantum Well Sub-bands
Let us focus on E1, H1
bands close to
crossing point
HgTe
HgTe
H1
E1
CdTe
H1
normal
CdTe
CdTe
CdTe
E1
inverted
Effective tight-binding model
Square lattice with 4-orbitals per site:
s,  , s,  , ( p x  ip y ,  ,  ( p x  ip y ), 
Nearest neighbor hopping integrals. Mixing matrix
elements between the s and the p states must be
odd in k.
0 
 h(k )

H eff (k x , k y )  

h ( k ) 
 0
m( k )
A(sin k x  i sin k y ) 

  d a (k ) a
h(k )  

 m( k )
 A(sin k x  i sin k y )

m
A(k x  ik y ) 


 

A
(
k

ik
)

m
x
y


Relativistic Dirac equation in 2+1 dimensions, with a mass term tunable by the
sample thickness d! m<0 for d>dc.
Mass domain wall
Cutting the Hall bar along the y-direction we see a domain-wall structure in the
band structure mass term. This leads to states localized on the domain wall
which still disperse along the x-direction.
y
y
m>0
x
m<0
0
x
m>0
E
E
kx
Bulk
0
Bulk
m
x
Experimental Predictions
x


k
k
Experimental evidence for the QSH state in HgTe
Fractional charge in the QSH state, E&M duality!
• Since the mass is proportional to the magnetization, a magnetization
domain wall leads to a mass domain wall on the edge.
mx
e/2
x
x
x
• The fractional charge e/2 can be measured by a Coulomb blockade
experiment, one at the time! Jackiw+Rebbie, Qi, Hughes & Zhang
E
 V=e/C
G
Vg
E
G
Vg
3D insulators with a single Dirac cone on the surface
z
(a)
(b)
y
y
x
Quintuple
layer
x
(c)
C
t1
t2
t3
Bi
Se1
Se2
A
B
C
A
B
C
Relevant orbitals of Bi2Se3 and the band inversion
(a)
(b)
0.6
E (eV)
Bi
Se
0.2
c
-0.2
0
(I)
(II)
(III)

0.2
(eV)
0.4
Bulk and surface states from first principle calculations
(a) Sb2Se3
(b) Sb2Te3
(c) Bi2Se3
(d) Bi2Te3
Effective model for Bi2Se3, Zhang et al
Pz+, up, Pz-, up, Pz+, down, Pz-, down
Minimal Dirac model on the surface of Bi2Se3,
Surface of Bi2Se3 = ¼ Graphene !
Zhang et al
Arpes experiment on Be2Te3 surface states,
Shen group
Doping evolution of the FS and band structure
Undoped
Under-doped
Optimallydoped
Over-doped
EF(undoped)
BCB bottom
BVB bottom
Dirac point position
General definition of a topological insulator
• Z2 topological band invariant
in momentum space based on
single particle states.
(Fu, Kane and Mele, Moore and Balents,
Roy)
• Topological field theory term in
the effective action. Generally
valid for interacting and
disordered systems. Directly
measurable physically. Relates to
axion physics! (Qi, Hughes and
Zhang)
• For a periodic system, the system is time
reversal symmetric only when
q=0 => trivial insulator
q= => non-trivial insulator
• Arpes experiments
(Hasan group)
q term with open boundaries
• q= implies QHE on the boundary with
1 e2
 xy 
2 h
• For a sample with boundary, it is only insulating when a
small T-breaking field is applied to the boundary. The
surface theory is a CS term, describing the half QH.
• Each Dirac cone contributes xy=1/2e2/h to the QH.
Therefore, q= implies an odd number of Dirac cones on
the surface!
T breaking
M
E
j//
• Surface of a TI = ¼ graphene
The Topological Magneto-Electric (TME) effect
• Equations of axion electrodynamics predict the robust TME effect.
4πM=a q/2 E
4πP=a q/2 B
Wilzcek, axion electrodynamics
• P3=q/2 is the electro-magnetic polarization, microscopically given by the CS term
over the momentum space. Change of P3=2nd Chern number!
Low frequency Faraday/Kerr rotation
(Qi, Hughes and Zhang, PRB78, 195424, 2008)
Adiabatic
Eg
Requirement:
(surface gap)
Topological contribution
normal contribution
qtopo» 3.6x 10-3 rad
Seeing the magnetic monopole thru the mirror of a TME
insulator, (Qi et al, Science 323, 1184, 2009)
q
TME insulator
(for =’, =’)
higher order
feed back
similar to Witten’s dyon effect
Magnitude of B:
An electron-monopole dyon becomes an anyon!
q  2a 2 P3
New topological states of quantum matter
QH insulator (U(1) integer), QSH insulator (Z2 number), chiral (U(1) integer) and
helical (Z2 number) superconductors.
Chiral Majorana fermions
Chiral fermions
massless Majorana fermions
massless Dirac fermions
Taking the square root in math and physics
1  i
Klein Gordon  Dirac
Dirac  Chiral fermion
SpaceTimeSymmetry Supersymme
try
Gravity  Supergravity
1D spinlessliquid  Chiral edgestateof QHE
1D spinfulliquid  Helicaledgestateof QSHE
Summary: the search for new states of matter
Crystal
Quantum Hall
Magnet
s-wave superconductor
Quantum Spin Hall
Recurrence of effective field theories
Dirac at MeV
Schroedinger at eV
Dirac at meV
Theta vacuum and axion of QCD
Topological insulators in CM
Monopoles in cosmology
table top experiments in CM
To see the world in a
grain of sand,
To hold infinities in an
hour!