Inducing a Magnetic Monopole with Topological Surface StatesXiao-Liang Qi1, Rundong Li1, Jiadong Zang2, Shou-Cheng Zhang1

1. Department of Physics, Stanford University, Stanford, CA 94305–4045, USA.2. Department of Physics, Fudan University, Shanghai, 200433, China

Quantum Hall State

2

2 ( )(2 )d k

n F k

2

2xy

effS d xdt A A

2

xyenh

• TKNN integer = the first Chern number

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

Z classes, distinguishing conventional band insulator and quantum Hall states in 2D .

• Anomalous transverse Hall conductance

Econduction

valenceGap

k

Chiral liquid

3D Topological Insulators

kx

ky

valence

Conduction

E

2D massless Dirac point

Promising candidates:BiSb alloy, Bi2Se3, Bi2Te3 , Sb2Te3PbTe/SnTe superlattice.

nontrivial trivial

Z2 Topology

Electromagnetic response of an insulator

2 231 1( )8effS d xdt E B

E

P=(-0)E

B

M=(1/0-1/)B

B

P=qB

E

M=qE

• Electromagnetic response of an insulator is described by an effective action:

• However another quadratic term is also allowed:

3

2 2S d xdtE Bq

q

• physically, this term describes the magneto-electric effect. Under time reversal:

E E

B B

q q

q Term• Integrate over a spatially and temporally periodic system:

3 20z t zcdtd xE B dxdyB cdtdz A n

• q term’s contribution to the partition function is given by exp(iqn). Therefore the partition function is invariant under the discrete U(1) translation:

2 nq q 0q q

• q term is actually a total derivative, contributing to the equation of motion only when the system has open boundary.

3 3 ( )2 16 2 4

S d xdt F F d xdt A A q

q q

Chern Simons term

Conventional insulator :Topological insulator :

Topological Magneto-Electric Effect

• Equation of motion is given by:

Electric and magnetic fields are linearly coupled together.

3 2P q

3

3

40

1

4 1

4 24 2

c t

c c tPP

DB

BE

DH J

D E P BH B M E

2nd Chern number in the bulk.

Magnetic Monopole

A charge just above the surface of a TI induces a mirror charge attaching a magnetic monopole on it, namely a dyon.

for =’, =’

Analogous to Witten’s dyon

( / 2 )q gq

Experimental Proposals

MFMTip

Magnetic layer

3min 1/f r for topological insulator

6min 1/f r for conventional insulator

Magnitude of B: SQUID

Anyon Gas

Conclusion We have shown theoretically that the topological surface

states of a 3D topological insulator can act as amirrorthat images an electron as a magnetic monopole.

A 2D electron gas in the neighborhood of the surface will become a dyon gas with fractional statistics.

We have presented realistic experimental setups to observe the field of the image magnetic monopole and the statistical angle of the dyon.

Science 323, 1184 (2009)