Inducing a Magnetic Monopole with Topological Surface States Xiao-Liang Qi 1 , Rundong Li 1 , Jiadong Zang 2 , Shou-Cheng Zhang 1 1. Department of Physics, Stanford University, Stanford, CA 94305–4045, USA. 2. Department of Physics, Fudan University, Shanghai, 200433, China Q uantum H allState 2 2 () (2 ) dk n F k 2 2 xy eff S d xdt A A 2 xy e n h •TKNN integer = the firstChern num ber •Topologicalstates ofm atter are defined and described by topological field theory: Z classes,distinguishing conventional band insulator and quantum H all states in 2D . •Anom alous transverse Hallconductance E conduction valence Gap k Chiralliquid 3D TopologicalInsulators k x k y valence Conduction E 2D m assless D irac point Prom ising candidates: BiSballoy ,Bi 2 Se 3 ,Bi 2 Te 3 ,Sb 2 Te 3 PbTe/SnTesuperlattice. nontrivial trivial Z2 Topology Electrom agnetic response of an insulator 2 2 3 1 1 ( ) 8 eff S d xdt E B Z Z E P=( - 0 ) E B M =(1/ 0 -1/ ) B B P=qB E M =qE •Electrom agnetic response ofan insulatoris described by an effective action: •H ow ever another quadratic term is also allow ed: 3 2 2 S d xdtE B q q Z Z •physically,this term describes the m agneto-electric effect. Undertim e reversal: E E Z Z B B q q q Term • Integrate overa spatially and tem porally periodic system : 3 2 0 z t z cdtd xE B dxdyB cdtdz A n Z Z • q term ’s contribution to the partition function is given by exp(i qn).Therefore the partition function is invariantunderthe discrete U(1) translation: 2 n q q 0 q q •q term is actually a totalderivative,contributing to the equation of m otion only w hen the system has open boundary. 3 3 ( ) 2 16 2 4 S d xdt F F d xdt A A q q q Chern Sim ons term Conventional insulator: Topological insulator: TopologicalM agneto-Electric Effect • Equation ofm otion is given by: Electric and m agnetic fields are linearly coupled together. 3 2 P q 3 3 4 0 1 4 1 4 2 4 2 c t c c t P P D B B E D H J D E P B H B M E 2 nd Chern num ber in the bulk. M agnetic M onopole A charge justabove the surface ofa TI induces a m irrorcharge attaching a m agnetic m onopole on it,nam ely a dyon. for = ’, = ’ Analogous to W itten’s dyon (/2) q g q Experim entalProposals M FM Tip M agnetic layer 3 min 1/ f r for topologicalinsulator 6 min 1/ f r for conventionalinsulator M agnitude ofB: SQ U ID Anyon Gas C onclusion W e have show n theoretically thatthe topologicalsurface states ofa 3D topologicalinsulator can actas am irror thatim ages an electron as a m agnetic m onopole. A 2D electron gas in the neighborhood ofthe surface w ill becom e a dyon gas w ith fractionalstatistics. W e have presented realistic experim entalsetups to observe the field ofthe im age m agnetic m onopole and the statisticalangle ofthe dyon. Science 323 , 1184 (2009)