A particle physicist’s perspective
on Topological Insulators.
Andreas Karch

University of Washington“Electric-Magnetic Duality and Topological Insulators”,
by AK, Phys.Rev.Lett. 103:171601“Fractional topological insulators in three dimensions”,
with J.Maciejko, X.-L. Qi, S. Zhang, Phys.Rev.Lett. 105:246809
and more recent work with Hoyos, Jensen, Maciejko and Takayanagi

A particle physicist’s perspective
on Topological Insulators.

Description of Insulators:

3

Where does this come from?

Why replace ε0 with ε?

4

ε from ε0: microscopic picture+Q------++++free (macroscopic)
charge(appears explicitly as sourcein Maxwell’s equation)++-Q5

ε from ε0+Q------++++“bound” charges++implies polarization:-Q
6

ε from ε0
For small electric field expand:7

ε from ε0
For small electric field expand:typically vanishes. no built in polarization
non-zero Po: ferroelectric[[Po=0 for unbroken rotation invariance]]8

ε from ε0
For small electric field expand:negligible for “small” electric fields(small compared to intrinsic scales)9ε from ε0
For small electric field expand:negligible for “small” electric fields(small compared to intrinsic scales)“Low energy effective field theory”
10

ε from ε0
For small electric field expand:negligible for “small” electric fields(small compared to intrinsic scales)“To a physicist, everything is aharmonic oscillator”
11ε from ε0
For small electric field expand:electric susceptibility
parametrize complete response by a single number;in principle computable

12

Reorganize:

withSimilar for the magnetic field (non-ferromagnetic insulator):

13

Reorganize:with
Similar for the magnetic field (non-ferromagnetic insulator):

Question:

Why not include in D a term linear in B?
Why not include in H a term linear in E?

14

In particle physics language:2-derivative effective
action:

Question:

Why not include in D a term linear in B?
Why not include in H a term linear in E?
That is, why not include a θ-term?
15

Time reversal symmetry.
Time Reversal:tcharge density:

t
even

current density:

odd

even
odd16

Time reversal.
Time Reversal:

evenodd17

Time Reversal Invariant Insulators
Generalized Constitutive Relation:
even

even

odd

only time reversal
invariant for18

A particle physicist’s perspective
on Topological Insulators.

Time Reversal Invariant Insulators
Generalized Constitutive Relation:
Quantization of Magnetic
Flux ensures T invariance ator

20

(Qi, Hughes, Zhang)Topological Insulators
Need 3 parameters to describe media: ε, μ, and:21

Topological Insulators
Need 3 parameters to describe media: ε, μ, and:vacuum, air, NaCl,MgF2 …..

22

Topological Insulators
Need 3 parameters to describe media: ε, μ, and:vacuum, air, NaCl,MgF2 …..
Bi1-xSbx, Bi2Se3, Bi2Te3, Sb2Te3,TlBi(Sb)Te(Se, S)2 , LaPtBi, ...

θ uniquely determined by bandstructure.
23

Flux Quantization:
Modified ConstitutiveRelation from:
Study theory on Euclidean 4-manifold. iSθ/θ integer quantized(due to magnetic flux quantization).

In Path Integrals only

θ is 2 π periodic!

matters24

Consequence of Flux Quantization:
This is the abelian version of “θ-vacuaa” of QCD.
(Callan, Dashen, Gross 1976, Jackiw&Rebbi, 1976))

θ = - π equivalent toθ= π

θ = π is time reversal invariant.
Relation to Band Structure? Topology?

25

Topology= classifying different geometries without introducing
an explicit notion of distance.

(movie from wikipedia)

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Topology= classifying different geometries without introducing
an explicit notion of distance.

{A,R} {B} {C,G,I,J,L,M,N,S,U,V,W,Z}{D,O} {E,F,T,Y} {H} {P,Q} {K,X}
Equivalence classes of the English alphabet
in uppercase sans-serif font (Calibri)
27

Topology= classifying different geometries without introducing
an explicit notion of distance.
2,10,22,0{A,R} {B} {C,G,I,J,L,M,N,S,U,V,W,Z}{D,O} {E,F,T,Y} {H} {P,Q} {K,X}
0,1

3,0

4,0

1,1

4,0

Different topologies typically classified byintegers (= Topological Quantum Numbers)
For example: number of boundaries.
number of loops28Topology= classifying different geometries without introducing
an explicit notion of distance.

{A,R} {B} {C,G,I,J,L,M,N,S,U,V,W,Z}{D,O} {E,F,T,Y} {H} {P,Q} {K,X}

topologically H

topologically X(singular limit)

topologically 2copies of V2929

Bandstructure and Topology
Beautiful Math. But why physics?

Topology of what? The sample?

30

Bandstructure and Topology
Non-interacting electrons in periodic potential:

BANDSTRUCTURE

31

Bandstructure and Topology
Band structure can have
non-trivial topology.
Topological Quantum
Numbers are strictly integer.
Protected under deformations
that do not close the gap!
32

TKNN theory of Quantum Hall.

33

TKNN theory of Quantum Hall.

34

TKNN theory of Quantum Hall.
(Thouless, Kohmoto, Nightingale, den Nijs)“Berry Connection” hasquantized flux.
35

TKNN for topological insulator.
Multi-Band-Berry-Connection.

Note:(Qi, Hughes, Zhang)(see also Fu, Kane, Mele) QHZ invariant is Z2 (not Z) valued.
 Non-zero θ indicatesstrong L · S coupling

Next: Implications of generalized constitutive relation?

Implications of Generalized
Constitutive Relation.

37

Implications of Generalized
Constitutive Relation.

38

Implications of Generalized
Constitutive Relation.

39

Boundary Conditions
Maxwell unmodified → BC unmodified(in the absence ofmacroscopic surfacecharge or current densities)
40

E=?

qε,μ41

Method of Images:
qEε,μq2

Discontinuity in E┴:
microscopic surface charge density

42

Method of Images:
q

Now:

E
ε,μq2

non-zero43

Method of Images:qNow:E,Bε,μq2
g2: mirror magnetic
monopole charge

Discontinuity in B║:
microscopic surface current density

non-zero44

Magnetic Monopoles in TI
Mirror charge of an electron
is a magnetic monopole(Qi, Li, Zang, Zhang Science)

first pointed out by Lee and Sikivie for“axion domain walls”

45Maxwell has E/q ↔ B/g symmetry. So why so complicated?

Duality Covariant Mirror Charges (AK)Duality:

Constitutive Relation:
Mirror Charges:
46

Faraday Rotation
Plane waves reflected off/transmitted throughthe interface experience rotation in their polarization
proportional to αθ = Kerr/Faraday effect.
Typically happens in the presence of magnetic fields
but here it survives in the limit of zero (time reversalbreaking) external field.
47

Faraday/Kerr effect:
θK48

Faraday/Kerr effect:49

A Microscopic model?
Low Energy Effective Theory:

Flux Quantization

θ= Integer · πIRIs there something
in between?
Full Band Structure:

Band Topology

θ= TKNN-like-invariant

UV50

A Microscopic model?
Low Energy Effective Theory

IRFree Dirac Equation• θ from chiral anomaly• experimental signatures• generalization to fractional TI

Full Band Structure

UV
51

Microscopic Model:
E

kSpectrum of free massive
Dirac fermion.

52

A Microscopic Model
A microscopic model: Massive Dirac Fermion.

M
 M

Time Reversal:*Time reversal system has real mass.
positive or negative.
for energies << M

θ=0 or θ=π53

A Microscopic Model
A microscopic model: Massive Dirac Fermion.

M
 M

Time Reversal:*Time reversal system has real mass.
positive or negative.
for energies << M

θ=0 or θ=πchiral anomaly – robust
against interactions.
54

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.

55

Chiral rotation and ABJ anomaly.
But in the quantum theory chiral rotation
is anomalous. Measure transforms.

C   fields q  1 1  122Single field with unit charge.
56

Microscopic Model of Interface.
M>0ε,μM<057Microscopic Model of Interface.
M>0ε,μMass has to cross zero.

M<058Microscopic Model of Interface.
M>0Mass has to cross zero.

Localized Zero Mode.

M<0Domain WallFermion.

59Microscopic Model of Interface.
M>0ε,μM<0

Mass has to cross zero.

Localized 2+1 d Zero Mode.
EE~pk

Single Dirac Cone!
Impossible in pure 2+1.
E.g. graphene: 4 Dirac Cones.

60

Excellent Signature!
Hasan group, Nature 2008

H. J. Zhang, et al, Nat Phys 2009

Hasan group,Bi2Se3Nat Phys 2009

Bi2Te3 Chen et al
Science 2009

TlBiTe2 Kuroda et al
Phys.Rev.Lett 2010

Fractional Topological Insulators?
Recall from Quantum Hall physics:
electrone- interactions

fractionalizes intom partons(m odd for fermions)

Quantum Hall

Fractional
Quantum Hall

62

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?
63

Partons.
Microscopic Model:

θ/π=  (charge) 2

chiral anomaly

e-electron breaks
up into m partons.

=21
1
 /    (charge)  m    mm
2(m odd so e- is fermion)64(if partons form a TI = have negative mass)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 afractional TI (in theory/in practice)?
Transport!

Fractional Hall + Kerr.

65

Generalizations.

66Quantum Spin Hall Effect in HgTe
(Kane, Mele; Bernevig Zhang)

2+1 dimensional
HgTe quantum wellelectric field 0xy

spinxye2

67

Fractional Quantum Spin Hall.
• Quantum Spin Hall is also well described bytime reversal invariant free Dirac equation• Transport again determined by anomaly; robust
against interactions. “Quantum R-Hall-Effect”.
• Fractionalization of charge gives fractionaltransport coefficients.
(AK, Maciejko, Takayanagi)

68holographic fTI(Jensen, Hoyos, AK)

D3 branes
D7 brane

Basic phenomenon indeed robust against
strong interactions (in the effective theory).

Summary.
Particle
TheoryTOPOLOGICALINSULATORS
Condensed
Matter Physics

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