A particle physicist’s perspective
on topological insulators.
Andreas Karch
Seminar at IPMU, June 15, 2010
based on: “Fractional topological insulators in three dimensions”,
by J.Maciejko, X.-L. Qi, AK, S. Zhang, arxiv:1004.3628
Part I: Motivation and Summary.
What is a topological insulator?
Preview of fractional topological insulator.
2
Basic Tool: Effective Theory
Standard strategy to determine low energy
physics of a system with unknown or too
complicated microscopic physics:
1. Guess correct light degrees of freedom.
2. Write down most general Lagrangian with least
number of derivatives consistent with symmetries.
3. Determine unknown constants by “matching” or
experiment.
3
Basic Tool: Effective Theory
Examples:
QCD:
SU(3) YM → Mesons → chiral Lagrangian→ fπ.
Any substance close to equlibrium:
?→ Energy density → Navier-Stokes → P(E), η
Note: One Way Street! UV → IR
4
Effective theory on insulators.
What is the low energy description of a
generic, time reversal invariant insulator?
Insulator = gapped spectrum
Low energy DOFs: only Maxwell field.
Task: Write down the most general action for
E and B, with up to two derivatives, consistent with
symmetries.
5
Low energy effective action.
Low energy DOFs: only Maxwell field.
Permittivity and Permeability.
6
Rotations allow one extra term.

3

d
xdt ( E  B )


But: Under time reversal E is even, B odd
So naively the most general description of
a time reversal invariant insulator does not
allow for a theta term.
7
Flux Quantization.
Dirac Quantization
of magnetic charge:
e
gn
2
Implies quantization of magnetic flux!

S
F g
On any Euclidean closed 4-manifold M:
8
Flux Quantization w/o monopoles.
 
 
   B  dS   A  dl
e
As the electron encircles the flux, it picks up
a phase Φ. Single valued wavefunction requires
quantized flux!
On any 4-manifold M:
9
Flux Quantization.





Partition function
is periodic in +2 (Abelian version of the “
vacuum” (Callan, Dashen, Gross 1976, Jackiw&Rebbi, 1976))
 is time-reversal odd
time-reversal invariant insulator can have =0 or 
Z2 classification
10
Topological Insulators
Low energy description of a T-invariant insulator
described by 3 parameters: ε, μ, and:
11
Physical Consequences.
Theta term is total derivative. In the absence of interface
can be absorbed by redefining E and B. But:
Boundary of topological insulator
= domain wall of 
12
Physical Consequences.
E
4P=(-1)E
B
4P=(/2)B
B
A topological term
in the action
4M=(1-1/)B
E
4M=(/2)E
13
Magnetic Monopoles in TI
prediction: mirror charge of an electron
is a magnetic monopole
first pointed out by Sikivie, re-obtained
in the TI context by Qi et. al.
Compact expression from requiring
E&M duality covariance (AK).
(for =’, =’)
14
Faraday and Kerr Effect
change in θ
B independent contribution to
Kerr/Faraday (Qi et al)
(Polarization of transmitted and
reflected wave rotated by angle θ)
15
Permittivity and Permeability.


Mirror charge and Kerr/Faraday angle depend
on θ as well as ε and μ.
While θ is quantized, ε and μ are not.
Quantum Hall physics (2+1 dimensions):
Chern Simons dominates, Maxwell irrelevant
Topological Insulator (3+1 dimensions):
Maxwell term and the topological term marginal!
16
Faraday and Kerr Effect
Kerr and Faraday
together can give
quantization!
(Maciejko et al)
17
What’s coming up:
A simple microscopic model mapping
the properties of θ to the chiral anomaly.
Review of experimental results on
topological insulators.
18
Fractional Topological Insulators.
This discussion will be modified if there are
additional light degrees of freedom.
As long as all charge carriers are gapped we still
describe an insulator (in fact even conventional
insulators have light phonons).
I’ll give examples of effective theories that give
rise to theta being π/(odd integer).
19
Part II: Anomalies and fTI’s


Microscopic Example in a continuum field
theory and the ABJ anomaly.
Fractional Topological insulators in 3+1
dimensions.
20
A Microscopic Model
A microscopic model: Massive Dirac Fermion.
Time Reversal:
M
 M
Time reversal system has real mass.
Two options: positive or negative.
*
21
Chiral rotation and ABJ anomaly.
Massless theory invariant under chiral rotations:
Symmetry of massive theory if mass transforms:
Phase can be rotated away! Chose M positive.
22
Chiral rotation and ABJ anomaly.
But in the quantum theory chiral rotation
is anomalous. Measure transforms.
C   fields q  1 1  1
2
2
Single field with unit charge.
23
Chiral rotation and ABJ anomaly.
Axial rotation with Φ=π:
 Rotates real negative mass into
positive mass.
 Generates θ=π!
Positive mass = Trivial Insulator.
Negative mass = Topological Insulator.
24
Localized Zero Mode on Interface.
Domain Wall has localized zero mode!
Kaplan’s Domain Wall Fermion
Domain Wall = TI/non-TI Interface
25
The boundary point of view.
Surface state stable against perturbations:
* T-invariance does not allow for mass term
* “Dirty” surface layer can only add/remove
fermions in pairs! (Nielsen/Ninomiya)
* Single 2+1 fermion has parity anomaly
* Hall conductivity can be shifted by integers;
half-integer contribution robust
Surface zero mode =experimental signature!
26
Generalizations:
Boundary theory
With anomaly
Bulk topological insulator
4+1
Callan-Harvey, Kaplan
Z
axial anomaly
3+1
3+1
TRI topological insulator
Z2
parity anomaly
2+1
2+1
Quantum Hall
Z
chiral anomaly
1+1
27
A Lattice Realization.
How to get θ =π from non-interacting electrons
in periodic potential (Band-Insulator)?
Topology of Band Structure!
Define Z2 valued topological invariant of
bandstructure to distinguish trivial (“positive mass”)
from topologically non-trivial (“negative mass”).
28
TKKN theory of Quantum Hall.
(Thouless, Kohmoto, Nightingale, den Nijs)
“Berry Connection” has
quantized flux.
 xy
2
e
m
h
29
TKKN for topological insulator.
Multi-Band-Berry-Connection.
(Qi, Hughes, Zhang)
θ=0
Vacuum, …
θ=π
Bi1-xSbx, Bi2Se3, Bi2Te3, Sb2Te3
30
predicted: TlBi(Sb)Te(Se, S)2 , LaPtBi etc (Heusler compounds)
Experimentally found Zero Modes.
Hasan group, Nature 2008
H. J. Zhang, et al, Nat Phys 2009
Hasan group,
Bi2Se3
Nat Phys 2009
Bi2Te3
Chen et al
Science 2009
Summary of Strategy:
Low Energy Effective Theory:
Dirac Quantization
θ= Integer ∙ π
IR
Microscopic Model:
ABJ anomaly
θ/π=  (charge)2
Connection to Experiment:
Band Topology
θ= QHZ-invariant
UV
32
Fractional Topological Insulators?
Recall from Quantum Hall physics:
electron
e- interactions
 xy
e2
n
h
Quantum Hall
fractionalizes into
m partons
 xy
(m odd for fermions)
n e2

m h
Fractional
Quantum Hall
33
Fractional Topological Insulators?
TI = half of an integer quantum hall state on
the surface
expect: fractional TI = half a fractional QHS
Hall quantum = half of 1/odd integer.
Can we get this from charge fractionalization?
34
Partons.
Microscopic Model:
θ/π=  (charge)2
ABJ anomaly
e-
electron breaks
up into m partons.
=
2
1
1
 /    (charge )  m    
m
m
2
(m odd so e- is fermion)
35
(if partons form a TI = have negative mass)
Trial Wavefunction.
Recall fractional QHE (“projective construction”):
Parton wavefunction = Slater determinant of integer QHE:
Electron wavefunction = all partons on top of each other:
36
Trial Wavefunction for FTI.
Parton wavefunction = Slater determinant
(e.g. from lattice Dirac model):
Electron wavefunction :
37
Partons.
e-
=
To ensure that partons
confine into electron add
“statistical” SU(m) gauge field
Simplest model: SU(3) with Nf=1
38
Relativistic Partons.
In a relativistic field theory the electron=baryon
has spin m/2 in the SU(m) model.
Not a problem in non-relativistic context.
Alternative: Quiver Model
Still need m partons to
be gauge invariant.
Baryons of any spin possible.
39
General Parton model.
Total number of partons.
40
General Parton model.
41
Fractional θ and Dirac Quantization
Why is fractional θ allowed from the point
of view of Dirac quantization?
Basic electric charge: e/m
Basic magnetic charge: m g
42
Important dynamical question.
Is the gauge theory in a confining or
deconfining phase?
We need: deconfined! Favors abelian models.
(or N=4 SYM with N=2 massive hyper)
(similar to Takayanagi and Ryu)
Gapless modes present; charged fields all gapped.
43
How to make a fractional TI?
Need: Strong electron/electron interactions
(so electrons can potentially fractionalize)
Strong spin/orbit coupling
(so partons can form topological insulator)
How can one tell if a given material is a
fractional TI (in theory/in practice)?
44
Transport Measurements
Transport sees only sum of Hall currents.
Add to zero – can be aligned by external B.
45
Faraday and Kerr Effect
Kerr rotation sees
only one interface.
Directly measures
change in θ.
change in θ
46
Fractional TI or Surface FQHE?
Can be in principle be distinguished.
47
Witten Effect for fractional TI.
Experimental Protocol:.
 Catch a monopole from
cosmic rays
 Insert it into bulk
material
Measure electric flux to
determine charge
48
Ground State Degeneracy.
On non-trivial 3 manifold groundstate can be
degenerate.
Different for fractional TI and TI + fractional QH
49
Ground State Degeneracy in QH.
Review of QH:
 xy
1 e2

m h
Effective Description: CS of level m
Equation of motion:
F  0
Degrees of freedom: flat connections.
50
Ground State Degeneracy in QH.
Degrees of freedom: flat connections.
Trivial on R2
On the torus: Wilson lines
x and y
Wilson lines periodic: x and y= positions on a torus.
51
Ground State Degeneracy in QH.
S CS  m  A  dA  m  yx
CS terms = Magnetic field
Groundstates: Particle on a torus with m
units of magnetic flux.
dGS  m
52
Ground State Degeneracy in QH.
genus g Riemann surface:
2g non-contractible cycles
→ 2g Wilson lines
→ particle on 2g-torus with
m units of magnetic flux.
dGS  m
g
53
FTI versus TI+FQHE.
TI+FQHE:
FTI:
Two independent
CS theories.
 )
d GS  m
g 2
   
Linked by
F2 = 0
constraint for
vacuum.
dGS  m
g
54
T3.
Groundstate trivial on T3.
But: Swingle, Barkeshli, McGreevy, Senthil:
No fractional θ unless groundstate
on T3 is degenerate!
Assumes gap for all excitations (not just charged).
Avoided in deconfined phase due to color magnetic
flux!
55
fTI in N=4 SYM and AdS/CFT
0
D3
D7
1
2
3
4
5
6
7
8
X
X
X
X
-
-
-
-
X
X
X
X
X
X
X
X
D7
D3
mass
(8,9) space
9
-
-
T-invariant =real mass
= D7 at x9=0, any x8
(here this is a choice to preserve T,
Takayanagi and Ryu impose orientifold that
makes T-violating mass inconsistent) 56
holographic fTI (with Carlos Hoyos and Kristan Jensen)
AdS4 x S4
D7
D3
AdS5 x S3
(3,8) space
Find:
Smooth embedding. Approaches
step at r=∞ (boundary of AdS)
57
holographic fTI


other mass profiles possible. expect Hall
current with filling fraction 1/(2m) for any
zero crossing profile.
this can easily be verified from AdS.
Independent of details of embedding, the Hall
current is uniquely determined by WZ term.
Was expected: WZ term = anomalies
Explicit Realization of a non-abelian fTI
58
To Do:


Construct explicit embeddings numerically.
Can we learn about fTI’s properties beyond
anomalies? Anything generic?
Like Janus, construction can be
supersymmetrized by adding localized scalar
mass term. Localize SUSY Chern-Simons. Is
this useful? Can we find analytic
embeddings?
59
Summary.
• Effective field theory for fractional TI can be
constructed
• Basic ingredient: fractionalization
• Effective θ follows from anomaly/Dirac
quantization
• Requires strong LS coupling and strong
interactions.
• Experimental signatures: transport +
Kerr/Faraday
60