Topological Physics inTopological Physics inBand InsulatorsBand Insulators

Gene MeleUniversity of Pennsylvania

Lecture Topics:Lecture Topics:

1. Topological Band Insulators (mostly 1D)

2. Topological Insulators (2D and 3D)

3. Low energy models for real materials

4. Wannier representations and band projectors

Collaborators:Collaborators:Charlie Kane

David DiVincenzoLiang FuPetr Kral

Andrew RappeMichael RiceSaad ZaheerFan Zhang

Electronic States of MatterElectronic States of MatterBenjamin Franklin (University of Pennsylvania)

That the Electrical Fire freely removes from Place to Place in and thro' the Substance of a Non-Electric, but not so thro' the Substance of Glass. If you offer a Quantity to one End of a long rod of Metal, it receives it, and when it enters, every Particle that was before in the Rod pushes it's Neighbour and so on quite to the farther End where the Overplus is discharg'd… But Glass from the Smalness of it's Pores, or stronger Attraction of what it contains, refuses to admit so free a Motion. A Glass Rod will not conduct a Shock, nor will the thinnest Glass suffer any Particle entringone of it's Surfaces to pass thro' to the other.

Band Insulators (orthodoxy)Band Insulators (orthodoxy)

Examples: Si, GaAs, SiO2, etc

Because of the “smallness of its pores”

Band Insulators (atomic limit)Band Insulators (atomic limit)

Examples: atoms, molecular crystals, etc.

“Attraction for what it contains”

Transition from covalent to atomic limitsTransition from covalent to atomic limits

Weaire & Thorpe (1971)

Tetrahedral semiconductors

Modern view: Gapped electronic states Modern view: Gapped electronic states are equivalentare equivalent

Kohn (1964): insulator is exponentially insensitivity to boundary conditions

weak coupling strong coupling “nearsighted”, local

Postmodern: Gapped electronic states are distinguished by topological invariants

Topological StatesTopological StatesTopological classification of valence manifold

Examples in 1D lattices: variants on primer

† †1 1

2 cos

n n n nn

k

H t c c c c

t ka

empty

saturated

?

Cell Doubling (Cell Doubling (PeierlsPeierls, SSH), SSH)

21 2

21 2

0( )

0

ika

ika

t t eH k

t t e

2( ) ( )2

H k H ka

smooth gauge

( ) ( )H k h k 1 2

2

cos 2sin 2

0

x

y

z

h t t kah t kah

Su, Schrieffer, Heeger (1979)

Project onto Bloch SphereProject onto Bloch Sphere

2 21 2 1 2( ) 2 cos2h k t t t t ka

( ) ( )H h k d k

Closed loop Retraced path

2 1t t 1 2t t

Formulation as a BerryFormulation as a Berry’’s Phases Phase

1 11 1;2 2i ie e

Im |

ie

d

: A phase0 : B phase Closed loop

Retraced path

Formulation using Electric PolarizationFormulation using Electric Polarization

( ) ( , )h k h k

1 2

( , )

t t

k

Im |A B dkk

connection:kA

Stokes Integral of Berry CurvatureStokes Integral of Berry Curvature

( ) ( , )h k h k

1 2

( , )

t t

k

2Im | |A BS

dk k

Berry curvature,kF

Polarization from adiabatic currentPolarization from adiabatic current

( ) ( , )h k h k

1 2

( , )

t t

k

A B Solid angle swept out :

J

Ground state current (via Kubo)

22 2 ( )A Bdt J P Pe eL

King-Smith and Vanderbilt (1993)

Example: Example: dimerizeddimerized 1D lattice1D lattice

2 1t t

1 2t t

2eP

00 ( )P ne

(spinless)

42

P P e

displaces a single charge across each unit cell

*

2eQ (spinless)

Topological Domain WallsTopological Domain Walls

1 †; ( 1)nn n

nC H C H C c c

State CountingState Counting

Particle-hole symmetry

on odd numbered chain

self conjugate state on one sublattice

……with spinwith spin

charge-less spinsspin-less charge spin-less charge

……for strong couplingfor strong coupling

charge-less spinsspin-less charge spin-less charge

Continuum ModelContinuum Model

4 4

4 4

0 0ˆ ˆ

20 0

i i

F x yi i

e eH q iv m

a xe e

1 2m t t “relativistic bands” back-scattered by mass

2 2 2E m v q

Mass inversion at domain wallMass inversion at domain wall

0

0( )( ) exp1

x m xx A dxv

Sublattice-polarized E=0 bound state

vm

Jackiw-Rebbi (1976)

Backflow of filled Fermi seaBackflow of filled Fermi sea

Fractional depletion of filled Fermi sea

Occurs in pairs(topologically connected)

† †1 1

† † †1 1( 1) ( ) ( 1)

n n n nn

n nn n n n n n

n n

H t c c c c

c c c c c c

HeteropolarHeteropolar LatticeLattice

modulate bond strength staggered potential

Rice and GM (1982)

On Bloch SphereOn Bloch Sphere

( ) ( )H k h k

( ) cos 2( ) sin 2

x

y

z

h t t kah t kah

000

000

High symmetry casesHigh symmetry cases0, 0

0, 0

0, 0

Continuously tunes the domain wall charge

* ( , )vQ ne Q

HeteropolarHeteropolar NTNT’’s of Boron Nitrides of Boron Nitride

BN is the III-V variantof graphene. The B and N occupy different sublattices-- this lowers the symmetry and leads to new physical effects

Conducting v. Semiconducting Conducting v. Semiconducting NanotubesNanotubes

Conducting v. Semiconducting Conducting v. Semiconducting NanotubesNanotubes

m=n mod(m-n,3) = ±1 mod(m-n,3) = 0, mn

NanotubeNanotube PolarizationPolarization

† ( )( , , )

( )xFx

xF

v q iH q

v q i

ionicity

chirality

,2 22

ee eP

NanotubeNanotube PolarizationPolarization

GM and P. Kral (2002)

Polarization and elastic strainPolarization and elastic strain(twist is a gauge field that modulates )

Heteropolar NT’s are molecular piezoelectrics

GM and P. Kral (2002)

One Parameter CycleOne Parameter Cyclee.g. periodic tube torsion modulates P

Bond strengths are modulated periodicallyin a reversible cycle

Two Parameter CycleTwo Parameter Cycleionicity and chirality are nonreciprocal potentials

Nonreciprocal cycle enclosing a point of degeneracy withChern number: 1 ( , )

4 k tS

n dk dt d k t d d

ThoulessThouless Charge PumpCharge Pump

( , ) ( , )H k t T H k t

,

2

2

k kt T t

k tS

eP A dk A dk

e F dk dt ne

with no gap closure on path

n=0 if potentials commuteThouless (1983)

Quantized Hall ConductanceQuantized Hall Conductance

2

2 2 xy

xy

xy

R dt j R dt E Q

h Q nee

enh

Adiabatic flux insertion through 2DEG on a cylinder:

TKNN invariantis the Chern number

Laughlin (1981), Thouless et al. (1983)

Some References:

Berry Phases: Di Xiao, M-C Chang, Q. NiuRev. Mod. Phys. 82, 1959 (2010)

SSH Model: A.J. Heeger. S. Kivelson, J.R. Schriefferand W-P Su, Rev. Mod. Phys. 60, 781 (1988)

Heteropolar One Dimensional Lattice: M. J. Riceand E.J. Mele, Phys. Rev. Lett. 49, 1455 (1982)

Heteropolar Nanotubes: E.J. Mele and P. KralPhys. Rev. Lett. 88, 05603 (2002)