Effective Topological Field Theories in Condensed Matter Physics

Theoretical prediction: Bernevig, Hughes and Zhang, Science 314, 1757 (2006) Experimental observation: Koenig et al, Science 318, 766 (2007)New Developments: Qi et al, Nature Physics 4, 273, 08’, Phy Rev B78, 195424, 08’, Science 323, 1184, 09’

The search for new states of matter

Crystal: Broken translational symmetry

Magnet: Broken rotational symmetry

Superconductor: Broken gauge symmetry

The search for new elements led to a golden age of chemistry.

The search for new particles led to the golden age of particle physics.

In condensed matter physics, we ask what are the fundamental states of matter?In the classical world we have solid, liquid and gas. The same H2O molecules can condense into ice, water or vapor.

In the quantum world we have metals, insulators, superconductors, magnets etc.Most of these states are differentiated by the broken symmetry.

The quantum Hall state, a topologically non-trivial state of matter

h

enxy

2

• TKNN integer=the first Chern number.

)()2( 2

2

kFkd

n

• Topological states of matter are defined and described by topological field theory:

AAxdtdS xy

eff 2

2

• Physically measurable topological properties are all contained in the topological field theory, e.g. QHE, fractional charge, fractional statistics etc…

Chiral (QHE) and helical (QSHE) liquids in D=1

k

kF-kF

The QHE state spatially separates the two chiral states of a spinless 1D liquid

No go theorems: chiral and helical states can never be constructed microscopicallyfrom a purely 1D model. (Wu, Bernevig, Zhang, 2006) Helical liquid=1/2 of 1D fermi liquid!

k

kF-kF

The QSHE state spatially separates the four chiral states of a spinful 1D liquid

2=1+1 4=2+2

x

x

Time reversal symmetry in quantum mechanics

• Wave function of a half-integer spin changes by -1 under 2 rotation.

• Kramers theorem, in a time reversal invariant system with half-integer spins, T2=-1, all states for degenerate doublets.• Application in condensed matter physics: Anderson’s theorem. BCS pair=(k,up)+(-k,down). General pairing between Kramers doublets.

Spin=1/2

• Wave function of a particle with integer spin changes by 1 under 2 rotation. Spin=1

-

The topological distinction between a conventional insulator and a QSH insulatorKane and Mele PRL, (2005); Wu, Bernevig and Zhang, PRL (2006); Xu and Moore, PRB (2006)

• Band diagram of a conventional insulator, a conventional insulator with accidental surface states (with animation), a QSH insulator (with animation). Blue and red color code for up and down spins.

k

Trivial TrivialNon-trivial

k=0 or

From topology to chemistry: the search for the QSH state

• Graphene – spin-orbit coupling only about 10-3meV. Not realizable in experiments. (Kane and Mele, 2005, Yao et al, 2006, MacDonald group 2006)

• Quantum spin Hall with Landau levels – spin-orbit coupling in GaAs too small. (Bernevig and Zhang, PRL, 2006)

5 .1 5 .2 5 .3 5 .4 5 .5 5 .6 5 .7 5 .8 5 .9 6 .0 6 .1 6 .2 6 .3 6 .4 6 .5 6 .76 .6

1 .0

1 .5

0 .5

0 .0

-0 .5

2 .0

2 .5

3 .0

3 .5

4 .0

4 .5

5 .0

6 .0

5 .5

B a n d g a p v s . la ttic e co n sta n t(a t ro o m te m p e ra tu re in z in c b le n d e s t ru c tu re )

Ban

dga

p e

nerg

y (e

V)

la ttice con sta n t a [Å ]0

• Type III quantum wells work. HgTe has a negative band gap! (Bernevig, Hughes and Zhang, Science 2006)• Tuning the thickness of the HgTe/CdTe quantum well leads to a topological quantum phase transition into the QSH state.

Band Structure of HgTe

S

P

S

P3/2

P1/2

S

P SP1/2

P3/2

CdTe

HgTe

CdTe

E1

H1

normal

HgTe

CdTe CdTe

H1

E1

inverted

Quantum Well Sub-bands

Let us focus on E1, H1 bands close to crossing point

Effective tight-binding model

),(,,(,,,, yxyx ippippss

Square lattice with 4-orbitals per site:

Nearest neighbor hopping integrals. Mixing matrix elements between the s and the p states must be odd in k.

)(0

0)(),(

kh

khkkH yxeff

mikkA

ikkAm

kdkmkikA

kikAkmkh

yx

yx

aa

yx

yx

)(

)(

)()()sin(sin

)sin(sin)()(

Relativistic Dirac equation in 2+1 dimensions, with a mass term tunable by the sample thickness d! m<0 for d>dc.

kx

E

Mass domain wall

Cutting the Hall bar along the y-direction we see a domain-wall structure in the band structure mass term. This leads to states localized on the domain wall which still disperse along the x-direction.

m>0

m<0 m

y

0

m>0

y

x

x

Bulk

Bulk

E

0

Experimental Predictions

k

k

x

x

Experimental evidence for the QSH state in HgTe

• Since the mass is proportional to the magnetization, a magnetization domain wall leads to a mass domain wall on the edge.

Fractional charge in the QSH state, E&M duality!

mxx

e/2

• The fractional charge e/2 can be measured by a Coulomb blockade experiment, one at the time! Jackiw+Rebbie, Qi, Hughes & Zhang

E G

Vg

G

Vg

E V=e/C

x

x

(b)

Se2

BiSe1

Quintuplelayer

B

B

CA

C

A

C

(a)

t1

t2 t3

z

x

y

(c)x

y

3D insulators with a single Dirac cone on the surface

(I) (II) (III)

(a)

Bi

Se

(eV)

E (

eV)

-0.2

0.2

0.6

0 0.2 0.4

(b)

c

Relevant orbitals of Bi2Se3 and the band inversion

(a) Sb2Se3 (b) Sb2Te3

(c) Bi2Se3 (d) Bi2Te3

Bulk and surface states from first principle calculations

Effective model for Bi2Se3, Zhang et al

Pz+, up, Pz-, up, Pz+, down, Pz-, down

Minimal Dirac model on the surface of Bi2Se3, Zhang et al

Surface of Bi2Se3 = ¼ Graphene !

Doping evolution of the FS and band structureDoping evolution of the FS and band structure

EF(undoped)BCB bottom

Dirac point position

Undoped Under-doped Optimally-doped Over-doped

BVB bottom

Arpes experiment on Be2Te3 surface states, Shen group

General definition of a topological insulator

• Z2 topological band invariant in momentum space based on single particle states.(Fu, Kane and Mele, Moore and Balents, Roy)• Topological field theory term in the effective action. Generally valid for interacting and disordered systems. Directly measurable physically. Relates to axion physics! (Qi, Hughes and Zhang)

• For a periodic system, the system is time reversal symmetric only when=0 => trivial insulator= => non-trivial insulator

• Arpes experiments (Hasan group)

term with open boundaries• = implies QHE on the boundary with

EM

j//

T breaking

• For a sample with boundary, it is only insulating when a small T-breaking field is applied to the boundary. The surface theory is a CS term, describing the half QH.• Each Dirac cone contributes xy=1/2e2/h to the QH. Therefore, = implies an odd number of Dirac cones on the surface!

h

exy

2

2

1

• Surface of a TI = ¼ graphene

The Topological Magneto-Electric (TME) effect• Equations of axion electrodynamics predict the robust TME effect.

4πP= /2B4πM= /2E

• P3=/2 is the electro-magnetic polarization, microscopically given by the CS term over the momentum space. Change of P3=2nd Chern number!

Wilzcek, axion electrodynamics

Low frequency Faraday/Kerr rotation(Qi, Hughes and Zhang, PRB78, 195424, 2008)

Topological contribution

topo» 3.6x 10-3 radnormal contribution

Adiabatic

Requirement:

(surface gap)

Eg

Seeing the magnetic monopole thru the mirror of a TME insulator, (Qi et al, Science 323, 1184, 2009)

TME insulator

q

(for =’, =’)similar to Witten’s dyon effect

higher order feed back Magnitude of B:

An electron-monopole dyon becomes an anyon!

322 P

New topological states of quantum matter

QH insulator (U(1) integer), QSH insulator (Z2 number), chiral (U(1) integer) and helical (Z2 number) superconductors.

Chiral Majorana fermions

Chiral fermions

massless Majorana fermions

massless Dirac fermions

Taking the square root in math and physics

i 1

DiracGordonKlein

trySupersymmeSymmetryTimeSpace

tySupergraviGravity

QHEofstateedgeChiralliquidspinlessD 1

QSHEofstateedgeHelicalliquidspinfulD 1

fermionChiralDirac

Summary: the search for new states of matter

Magnet s-wave superconductor

Crystal

Quantum Hall

Quantum Spin Hall

Recurrence of effective field theories

Dirac at MeV Schroedinger at eV Dirac at meV

Theta vacuum and axion of QCDTopological insulators in CM

Monopoles in cosmologytable top experiments in CM

To see the world in a grain of sand,To hold infinities in an hour!