Effective Topological Field Theories in Condensed Matter Physics

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
2
Quantum spin Hall effect and topological
insulators
Theoretical prediction Bernevig, Hughes and
Zhang, Science 314, 1757 (2006) Experimental
observation Koenig et al, Science 318, 766
(2007)New Developments Nature Physics 4, 273,
08, Phy Rev B78, 195424, 08, Science 323, 1184,
09 Theoretical prediction Zhang et al
cond-mat/0812.1622 Experimental observation
Chen et al, submitted
3
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.
Superconductor Broken gauge symmetry
Magnet Broken rotational symmetry
Crystal Broken translational symmetry
4
The quantum Hall state, a topologically
non-trivial state of matter

• TKNN integerthe first Chern number.

• Topological states of matter are defined and
  described by topological field theory

• Physically measurable topological properties are
  all contained in the topological field theory,
  e.g. QHE, fractional charge, fractional
  statistics etc

5
The Generalizations of the Hall Effect

• Theoretical predictions of the spin Hall effect
  (Dyakonov, Murakami, Nagaosa and Zhang, Science
  2003, Sinova et al PRL 2004)
• The spin Hall effect has now been experimentally
  observed. (Kato et al, Science 2004, Wunderlich
  et al PRL 2004)

What about the quantum spin Hall effect?
6
Quantum Spin Hall Effect

• The QSH state can be thought of as two copies of
  QH states, one for each spin component, each
  seeing the opposite magnetic field. (Bernevig and
  Zhang, PRL, 2006)
• The QSH state does not break the time reversal
  symmetry, and can exist without any external
  magnetic field.

7
Chiral (QHE) and helical (QSHE) liquids in D1
e
The QHE state spatially separates the two chiral
states of a spinless 1D liquid
The QSHE state spatially separates the four
chiral states of a spinful 1D liquid
211
422
No go theorems chiral and helical states can
never be constructed microscopically from a
purely 1D model. (Wu, Bernevig, Zhang, 2006)
Helical liquid1/2 of 1D fermi liquid!
8
Taking the square root in math and physics
9
Time reversal symmetry in quantum mechanics

• Wave function of a half-integer spin changes by
  -1 under 2p rotation.

• Kramers theorem, in a time reversal invariant
  system with half-integer spins, T2-1, all states
  for degenerate doublets.

ygt y

• Application in condensed matter physics
  Andersons theorem. BCS pair(k,up)(-k,down).
  General pairing between Kramers doublets.

Spin1/2
ygt-y
10
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.

e
k
k0 or p
Trivial
Trivial
Non-trivial
11
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)

• Type III quantum wells work. HgTe has a negative
  band gap! (Bernevig, Hughes and Zhang, Science
  2006)
• Tuning the thickness of the HgTe/CdTe quantum
  well leads to a topological quantum phase
  transition into the QSH state.

12
Band Structure of HgTe
13
Quantum Well Sub-bands
Let us focus on E1, H1 bands close to crossing
point
HgTe
HgTe
H1
E1
CdTe
CdTe
CdTe
CdTe
E1
H1
normal
inverted
14
Effective tight-binding model
Square lattice with 4-orbitals per site
Nearest neighbor hopping integrals. Mixing matrix
elements between the s and the p states must be
odd in k.
Relativistic Dirac equation in 21 dimensions,
with a mass term tunable by the sample thickness
d! mlt0 for dgtdc.
15
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
mgt0
x
mlt0
m
0
x
mgt0
16
Experimental setup

• High mobility samples of HgTe/CdTe quantum wells
  have been fabricated.
• Because of the small band gap, about several
  meV, one can gate dope this system from n to p
  doped regimes.
• Two tuning parameters, the thickness d of the
  quantum well, and the gate voltage.

• (Koenig et al, Science 2007)

17
Experimental Predictions
18
Smoking gun for the helical edge state
Magneto-Conductance
The crossing of the helical edge states is
protected by the TR symmetry. TR breaking term
such as the Zeeman magnetic field causes a
singular perturbation and will open up a full
insulating gap
e
B-Field
k
19
Experimental evidence for the QSH state in HgTe
20
Magnetic field dependence of the residual
conductance
21
Nonlocal transport in the QSH regime
R14,143/4 h/e2
I 1-4
V 2-3
R14,231/4 h/e2
22
QSH state in InAs/GaSb type II quantum wells

• HgTe is not a material that can be easily
  fabricated. We are searching for new
  semiconductor materials which can lead to QSH.
• In HgTe, the band inversion occurs intrinsically
  in the material. However, in InAs/GaSb quantum
  wells, a similar inversion can occur, since the
  valance band edge of GaSb lies above the
  conduction band edge of InAs.
• Our theoretical work show that the QSH can occur
  in InAs/Gab quantum wells. This material can be
  fabricated commercially in many places around the
  world.

23
Fractional charge in the QSH state, EM duality!

• Since the mass is proportional to the
  magnetization, a magnetization domain wall leads
  to a mass domain wall on the edge.

• The fractional charge e/2 can be measured by a
  Coulomb blockade experiment, one at the time!
  JackiwRebbie, Qi, Hughes Zhang

24
Electromagnetic response of an insulator

• Electromagnetic response of an insulator is
  described by an effective action

• However, another quadratic term is also allowed

4pP(?-1)E
4pM(1-1/?)B

• Physically, this term describes the
  magneto-electric effect. Under time reversal

4pPa q/2p B
4pMa q/2p E
25
q periodicity and time reversal

• Consider an analog system of a period ring. The
  flux enters the partition function as

• Therefore, the physics is completely invariant
  under the shift of

• Under time reversal, fgt-f, therefore, time
  reversal is recovered for two special values of
  f, f0 and fp.

• The ME term is a total derivative, independent
  of the bulk values of the fields

• Integrated over a spatially and temporally
  periodic system,

• Its contribution to the partition function is
  given by . Therefore, the partition
  function is invariant under the shift

Time reversal symmetry is recovered at
26
3D insulators with a single Dirac cone on the
surface
(b)
z
y
(a)
y
x
x
Quintuple layer
(c)
C
A
B
t2
t3
t1
C
A
B
C
27
Relevant orbitals of Bi2Se3 and the band inversion
(a)
(b)
0.6
Bi
0.2
E (eV)
Se
?c
-0.2
0
0.2
0.4
?
(eV)
(I)
(II)
(III)
28
Bulk and surface states from first principle
calculations
(a) Sb2Se3
(b) Sb2Te3
(c) Bi2Se3
(d) Bi2Te3
29
Effective model for Bi2Se3, Zhang et al
Pz, up, Pz-, up, Pz, down, Pz-, down
Minimal Dirac model on the surface of Bi2Se3,
Zhang et al
Surface of Bi2Se3 ¼ Graphene !
30
Arpes experiment on Be2Te3 surface states, Shen
group
Doping evolution of the FS and band structure
31
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
• q0 gt trivial insulator
• qp gt non-trivial insulator

• Arpes experiments (Hasan group)

32
q term with open boundaries

• qp implies QHE on the boundary with

• 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 sxy1/2e2/h to the
  QH. Therefore, qp implies an odd number of Dirac
  cones on the surface!

T breaking

• Surface of a TI ¼ graphene

33
Topological stability of the surface states

• No-go theorem it is not possible to construct a
  2D model with an odd number of Dirac cones, in a
  system with T2-1 TR symmetry. Surface states of
  a TI with qp is a holographic liquid! Wu,
  Bernevig Zhang, Holographical principle
• TI surface states can not rust away by surface
  chemistry.

• For a sample with boundary, physics is not
  periodic in q. However, T-invariant
  perturbations, like disorder, can induce plateau
  transitions with Dsxy1 e2/h, or Dq2p. For TI
  with qp, the surface QH can never disappear, no
  matter how strong the disorder! sxy1/2 e2/h gt
  sxy-1/2 e2/h.
• States related by interger plateau transition
  defines an equivalence class. There are only two
  classes!

34
q periodicity and time reversal

• Consider an analog system of a period ring. The
  flux enters the partition function as

• Therefore, the physics is completely invariant
  under the shift of

• Under time reversal, fgt-f, therefore, time
  reversal is recovered for two special values of
  f, f0 and fp.

• The ME term is a total derivative, independent
  of the bulk values of the fields

• Integrated over a spatially and temporally
  periodic system T4,

• Its contribution to the partition function is
  given by . Therefore, the partition
  function is invariant under the shift

Time reversal symmetry is recovered at
35
The Topological Magneto-Electric (TME) effect

• Equations of axion electrodynamics predict the
  robust TME effect.

Wilzcek, axion electrodynamics
4pPa q/2p B
4pMa q/2p E

• P3q/2p is the electro-magnetic polarization,
  microscopically given by the CS term over the
  momentum space. Change of P32nd Chern number!

36
Low frequency Faraday/Kerr rotation (Qi, Hughes
and Zhang, PRB78, 195424, 2008)
Adiabatic Requirement (surface gap)
Eg
Topological contribution ?topo 3.6x 10-3 rad
normal contribution
37
STM probe of the topological surface states (Liu
et al cond-mat/0808.2224)
38
Seeing the magnetic monopole thru the mirror of a
TME insulator, (Qi et al, Science 323, 1184, 2009)
higher order feed back
(for ??, ??) similar to Wittens dyon effect
Magnitude of B
39
An electron-monopole dyon becomes an anyon!
40
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 fermions
Chiral Majorana fermions
massless Dirac fermions
massless Majorana fermions
41
Taking the square root in math and physics
42
Topological superconductors and superfluids
The BCS-BdG Hamiltonian for equal spin pairing
where ppxipy. The edge Hamiltonian is given by
forming a pair of Majorana fermions. Mass term
breaks T symmetrygt topological protection! He3 B
phase provides a physical realization!
See also Roy Schneider et al Kitaev
43
Summary the search for new states of matter
s-wave superconductor
Magnet
Crystal
Quantum Hall
Quantum Spin Hall
44
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45
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!
46
Summary
Semiconductors HgTe/CdTe, InAs/GaSb, Bi1-xSbx,
Bi2Se3,
High energy physics ? vacuum, anomalies axion,
dyon
Magnetism fractional charge, spin charge
separation Topological Magneto-electric effect
Topological insulators
Thanks!
And what else?
Strongly correlated electron systems Topological
Mott insulators, Na2IrO3,
Superconductivity Nonabelian statistics, Majorana
fermion
47
Completing the table of Hall effects
Hall 1879 Anomalous Hall 1889 Spin Hall 2004
QHE 1980 QAHE 2008? QSHE 2007
48
Momentum space topology of the tight-binding model
Critical points
Ferromagnetic
Ferromagnetic
Skyrmion
Skyrmion
(p, p)
(0, p)
(p,0)
(0, 0)
X
49
Topological quantum phase transition
Meron in continuum picture
50
Inversion symmetry breaking in zincblend lattices
Inversion breaking term comes in the form
-spin 3/2 matrices
,
which couples E1, H1- and E1-,H1 states and is
a constant in quasi-2d systems
E/t
Gap closes at nodes away from k0, gap reopens at
non-zero value of M/2B. In the inverted regime,
the helical edge state crossing is still
robust. Tight-binding model by X Dai, Z Fang,
k
51
Quantum control of the electron spin

• The electron spin can be rotated by a pure AB
  flux, without any interaction with the
  electromagnetic field.