A particle physicist’s perspective on topological insulators.

Andreas Karch

based on: “Fractional topological insulators in three dimensions”, by J.Maciejko, X.-L. Qi, AK, S. Zhang, arxiv:1004.3628

Seminar at IPMU, June 15, 2010

Part I: Motivation and Summary.

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What is a topological insulator?

Preview of fractional topological insulator.

Basic Tool: Effective Theory

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Standard strategy to determine low energyphysics 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.

Basic Tool: Effective Theory

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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

Effective theory on insulators.

What is the low energy description of ageneric, time reversal invariant insulator?

Insulator = gapped spectrum

Low energy DOFs: only Maxwell field.

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Task: Write down the most general action forE and B, with up to two derivatives, consistent withsymmetries.

Low energy effective action.

Low energy DOFs: only Maxwell field.

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Permittivity and Permeability.

Rotations allow one extra term.

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)(3 BExdtd

But: Under time reversal E is even, B odd

So naively the most general description ofa time reversal invariant insulator does notallow for a theta term.

Flux Quantization.

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On any Euclidean closed 4-manifold M:

2

eng Dirac Quantization

of magnetic charge:

Implies quantization of magnetic flux!

S

gF

Flux Quantization w/o monopoles.

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On any 4-manifold M:

ldASdB

e

As the electron encircles the flux, it picks upa phase Φ. Single valued wavefunction requires quantized flux!

Flux Quantization.

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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

Topological Insulators

Low energy description of a T-invariant insulatordescribed by 3 parameters: ε, μ, and:

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Physical Consequences.

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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

Physical Consequences.

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E B

B E

4P=(-1)E 4M=(1-1/)B

4P=(/2B 4M=(/2E

A topological term in the action

Magnetic Monopoles in TIprediction: mirror charge of an electronis a magnetic monopole

first pointed out by Sikivie, re-obtainedin the TI context by Qi et. al.

Compact expression from requiring E&M duality covariance (AK).

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(for =’, =’)

Faraday and Kerr Effect

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B independent contribution toKerr/Faraday(Polarization of transmitted and reflected wave rotated by angle θ)

(Qi et al)

change in θ

Permittivity and Permeability.

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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!

Faraday and Kerr Effect

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(Maciejko et al)

Kerr and Faradaytogether can givequantization!

What’s coming up:

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A simple microscopic model mappingthe properties of θ to the chiral anomaly.

Review of experimental results on topological insulators.

Fractional Topological Insulators.

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This discussion will be modified if there are additional light degrees of freedom.

As long as all charge carriers are gapped we stilldescribe an insulator (in fact even conventionalinsulators have light phonons).

I’ll give examples of effective theories that giverise to theta being π/(odd integer).

Part II: Anomalies and fTI’s

Microscopic Example in a continuum field

theory and the ABJ anomaly. Fractional Topological insulators in 3+1

dimensions.

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A Microscopic Model

A microscopic model: Massive Dirac Fermion.

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Time Reversal:*MM

Time reversal system has real mass. Two options: positive or negative.

Chiral rotation and ABJ anomaly.

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Massless theory invariant under chiral rotations:

Symmetry of massive theory if mass transforms:

Phase can be rotated away! Chose M positive.

Chiral rotation and ABJ anomaly.

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But in the quantum theory chiral rotationis anomalous. Measure transforms.

111 22 fieldsqC

Single field with unit charge.

Chiral rotation and ABJ anomaly.

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Axial rotation with Φ=π:

Rotates real negative mass into positive mass. Generates θ=π!

Positive mass = Trivial Insulator.Negative mass = Topological Insulator.

Localized Zero Mode on Interface.

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Domain Wall has localized zero mode!

Domain Wall = TI/non-TI Interface

Kaplan’s Domain Wall Fermion

The boundary point of view.

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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!

Generalizations:

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Bulk topological insulatorBoundary theory

With anomaly

2+1 Quantum Hall chiral anomaly 1+1Z

Callan-Harvey, Kaplan axial anomaly4+1 3+1Z

TRI topological insulator parity anomaly3+1 2+1Z2

A Lattice Realization.

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How to get θ =π from non-interacting electronsin periodic potential (Band-Insulator)?

Topology of Band Structure!

Define Z2 valued topological invariant ofbandstructure to distinguish trivial (“positive mass”)from topologically non-trivial (“negative mass”).

TKKN theory of Quantum Hall.

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(Thouless, Kohmoto, Nightingale, den Nijs)

h

emxy

2

“Berry Connection” hasquantized flux.

TKKN for topological insulator.

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(Qi, Hughes, Zhang)Multi-Band-Berry-Connection.

θ=0

θ=π

Vacuum, …

Bi1-xSbx, Bi2Se3, Bi2Te3, Sb2Te3

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:

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Low Energy Effective Theory:

Microscopic Model:

Connection to Experiment:

Dirac Quantization θ= Integer ∙ π

ABJ anomaly θ/π=

Band Topology θ= QHZ-invariant

2)charge(

IR

UV

Fractional Topological Insulators?

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Recall from Quantum Hall physics:

e- interactions

h

enxy

2

h

e

m

nxy

2

electron fractionalizes intom partons

Quantum Hall FractionalQuantum Hall

(m odd for fermions)

Fractional Topological Insulators?

TI = half of an integer quantum hall state on the surface

expect: fractional TI = half a fractional QHSHall quantum = half of 1/odd integer.

Can we get this from charge fractionalization?

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Partons.

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Microscopic Model:

ABJ anomaly θ/π= 2)charge(

=e-

mmm

11)charge(/

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electron breaksup into m partons.

(m odd so e- is fermion)

(if partons form a TI = have negative mass)

Trial Wavefunction.

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Recall fractional QHE (“projective construction”):

Parton wavefunction = Slater determinant of integer QHE:

Electron wavefunction = all partons on top of each other:

Trial Wavefunction for FTI.

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Parton wavefunction = Slater determinant (e.g. from lattice Dirac model):

Electron wavefunction :

Partons.

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=e-

To ensure that partonsconfine into electron add“statistical” SU(m) gauge field

Simplest model: SU(3) with Nf=1

Relativistic Partons.

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In a relativistic field theory the electron=baryonhas spin m/2 in the SU(m) model.

Not a problem in non-relativistic context.

Alternative: Quiver Model

Still need m partons tobe gauge invariant.Baryons of any spin possible.

General Parton model.

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Total number of partons.

General Parton model.

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Fractional θ and Dirac Quantization

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Why is fractional θ allowed from the pointof view of Dirac quantization?

Basic electric charge: e/m

Basic magnetic charge: m g

Important dynamical question.

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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)Gapless modes present; charged fields all gapped.

(similar to Takayanagi and Ryu)

How to make a fractional TI?

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Need: Strong electron/electron interactions

Strong spin/orbit coupling

(so electrons can potentially fractionalize)

(so partons can form topological insulator)

How can one tell if a given material is afractional TI (in theory/in practice)?

Transport Measurements

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Transport sees only sum of Hall currents. Add to zero – can be aligned by external B.

Faraday and Kerr Effect

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change in θ

Kerr rotation sees only one interface.

Directly measureschange in θ.

Fractional TI or Surface FQHE?

47Can be in principle be distinguished.

Witten Effect for fractional TI.

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Experimental Protocol:.

Catch a monopole from cosmic rays Insert it into bulk materialMeasure electric flux to determine charge

Ground State Degeneracy.

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On non-trivial 3 manifold groundstate can be degenerate.

Different for fractional TI and TI + fractional QH

Ground State Degeneracy in QH.

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Review of QH:

Effective Description: CS of level m

0FEquation of motion:

Degrees of freedom: flat connections.

h

e

mxy

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Ground State Degeneracy in QH.

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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.

Ground State Degeneracy in QH.

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xymdAAmSCS

CS terms = Magnetic field

Groundstates: Particle on a torus with m units of magnetic flux.

mdGS

Ground State Degeneracy in QH.

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gGS md

genus g Riemann surface:

2g non-contractible cycles→ 2g Wilson lines→ particle on 2g-torus with m units of magnetic flux.

FTI versus TI+FQHE.

54 2g

GS md gGS md

TI+FQHE: FTI:

Two independentCS theories.

Linked by F2 = 0constraint forvacuum.

T3.

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Groundstate trivial on T3.

But: Swingle, Barkeshli, McGreevy, Senthil:

No fractional θ unless groundstateon T3 is degenerate!

Assumes gap for all excitations (not just charged).Avoided in deconfined phase due to color magneticflux!

fTI in N=4 SYM and AdS/CFT

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0 1 2 3 4 5 6 7 8 9

X X X X - - - - - -

X X X X X X X X

D3D7

D3

(8,9) space

D7

T-invariant =real mass= D7 at x9=0, any x8

mass (here this is a choice to preserve T, Takayanagi and Ryu impose orientifold that makes T-violating mass inconsistent)

holographic fTI (with Carlos Hoyos and Kristan Jensen)

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D3

D7

(3,8) space

AdS5 x S3

AdS4 x S4

Find:Smooth embedding. Approachesstep at r=∞ (boundary of AdS)

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.

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Was expected: WZ term = anomalies

Explicit Realization of a non-abelian fTI

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?

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Summary.

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• 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