Topological Physics inTopological Physics inBand InsulatorsBand Insulators

Gene MeleDRL 2N17a

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.

Electronic States of MatterElectronic States of Matter

Conductors & Insulators…

This taxonomy is incomplete…

…“Superconductors” (1911)

Electronic States of MatterElectronic States of Matter

Conductors, Insulators & Superconductors

(Onnes 1911)

from Physics Today, September 2010

Electronic States of MatterElectronic States of Matter

Conductors, Insulators & Superconductors

BCS mean field theory ofthe superconducting state (1957)

Univ. of Penn. (1962-1980)

Soliton theory of doped polyacetylene (1978)

Electronic States of MatterElectronic States of Matter

Conductors, Insulators & Superconductors…

This taxonomy is still incomplete…

…“Topological Insulators” (2005)

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

1 †; ( 1)nn n

nC H C H C c c

Domain WallsDomain Walls

Particle-hole symmetry

on odd numbered chain

self conjugate state on one sublattice

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

One Parameter CycleOne Parameter Cyclee.g. modulate the staggered on-site potential

Charge is shifted 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)

Electronic States of MatterElectronic States of Matter

Topological Insulators

This novel electronic state of matter is gapped in the bulk and supports the transport of spin and charge in gapless edge states that propagate at the sample boundaries. The edge states are …insensitive to disorder because their directionality is correlated with spin.

2005 Charlie Kane and GMUniversity of Pennsylvania

Electron spin admits a topologicallydistinct insulating state

Electronic States of MatterElectronic States of Matter

This state is realized in three dimensional materials where spin orbit coupling produces a bandgap “inversion.”

It has boundary modes (surface states) with a 2D Dirac singularity protected by time reversal symmetry.

Bi2Se3 is a prototype.

.Hasan/Cava (2009)

Topological Insulators

GrapheneGraphene: the Parent Phase: the Parent Phase

The dispersion of a free particle in 2D..

…is replacedby an unconventional E(k) relation on thegraphene lattice

……. it has a critical electronic state. it has a critical electronic state

The low energy theory is described byThe low energy theory is described byan effective mass theory for an effective mass theory for masslessmassless electronselectrons

( ) • ( )Bloch Wavefunction Wavefunction s at K r

eff FH r iv r ( ) ( )

NOTE: Here the “spin” degree of freedom describes the sublatticepolarization of the state, called pseudospin. In addition electrons carrya physical spin ½ and an isospin ½ describing the valley degeneracy.

It is a massless Dirac Theory in 2+1 Dimensions

D.P. DiVincenzo and GM (1984)

A continuum of structures all with √3 x √3 period hybridizes the two valleys

Gapping the Dirac PointGapping the Dirac PointValley mixing from brokentranslational symmetry

Gapping the Dirac PointGapping the Dirac PointValley mixing from brokentranslational symmetry

Kekule0

'0

ix

ix

eH

e

BN

0'

0z

z

H

Gapping the Dirac PointGapping the Dirac PointCharge transfer from broken

inversion symmetry

Gapping the Dirac PointGapping the Dirac PointOrbital currents from modulated flux

(Broken T-symmetry)

Gauged second neighbor hopping breaks T. “Chern insulator” with Hall conductance e2/h

FDM Haldane “Quantum Hall Effect without Landau Levels” (1988)

FDMH

0'

0z

z

H

Topological ClassificationTopological Classification

1 2

21 2

1 ( , ) 04 k k

S

n d k d k k d d

FDMH

0'

0z

z

H

1 2

21 2

1 ( , ) 14 k k

S

n d k d k k d d

Topological ClassificationTopological Classification

2

xyeh

“Chern Insulator” with (has equal contributionsfrom two valleys)

2 ' 0 ''

( ) cos sin

2 cos cos sin sin

n x n y zn

n n zn

H k t k a k a M

t k b k b

Orthodoxy: Spectrum Gapped only for Broken Symmetry States

Crucially, this ignores the electron spin

:na triad of nearest neighbor bond vectors

' :nb triad of directed “left turn”

second neighbor bond vectors

Breaks P

Breaks TBreaks e-h symmetry

Coupling orbital motion to the electron spin

SOH s V p

( ) ( )V r V r T Microscopic

Lattice model † † ( )SO m n n m m nH i r r

Spin orbit field Bond vector

Intersite hopping with spin precession

0, 0xy z ˆ ˆ( )SO z z effH s n p p s n p a

0effa d

/ 2

0effa d

† †2

† †

2 † †

cos

sin

i in m m n

n m m n

n m m n

t e c c e c c

c c c ct

i c c c c

Preserve mirror symmetry with a parallel spin orbit field

Generates a spin-dependent Haldane-type mass (two copies)

SO SO z z zs

Coupling orbital motion to the electron spin

Mass Terms (amended)Mass Terms (amended)

z

z z

z z zs x z y y xs s

,x x x y Kekule: valley mixing

Heteropolar (breaks P)

Modulated flux (breaks T)

Spin orbit (Rashba, broken z→-z)

Spin orbit (parallel)**This term respects all symmetries and is therefore present, though possibly weak

spinless

For carbon definitely weak, but still important

Topologically different statesTopologically different states

1 2

21 2

1 ( , )4 k k

S

n d k d k k d d

Charge transfer insulator Spin orbit coupled insulator

0n 1 ( 1) 0n

Topology of Chern insulatorin a T-invariant state

Boundary ModesBoundary Modes

Ballistic propagation through one-way edge state

Counter propagating spinpolarized edge statesIntrinsic SO-Graphene

model on a ribbon

Quantum Spin Hall EffectQuantum Spin Hall EffectIts boundary modes are spin filtered

propagating surface states (edge states)

Symmetry ClassificationSymmetry Classification

Conductors: unbroken state1

Insulators: broken translational symmetry:bandgap from Bragg reflection2

Superconductor: broken gauge symmetry

Topological Insulator ?

1possibly with mass anisotropy

2band insulators

Prospects

Late 1950’s: The band theory of solids introduces novel quantum kinematics (“light” electrons, “holes” etc.) Blount, Luttinger, Kane, Dresselhaus, Bassani…

Mid 1980’s: Identification of pseudo-relativisticphysics at low energy (graphene and its variants)DiVincenzo, Mele, Semenoff, Fradkin, Haldane…

Post 2000: Topological insulators and topological classification of gapped electronic states. Kane, Mele, Zhang, Moore, Balents, Fu, Roy, Teo, Ryu, Kitaev, Ludwig…

Looking forward: Topological band theory as a new materials design principle.(developing)

Study “all things practical and ornamental”