Witten effect in a crystalline topological insulator arXiv:1001.3179v3 [cond-mat.mes-hall] 21 May 2010 G. Rosenberg and M. Franz Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada V6T 1Z1 (Dated: October 25, 2018) It has been noted a long time ago that a term of the form θ ( e $^{2}$/ 2 πh ) B · E may be added to the standard Maxwell Lagrangian without modifying the familiar laws of electricity and magnetism. θ is known to particle physicists as the ‘axion’ field and whether or not it has a nonzero expectation value in vacuum remains a fundamental open question of the Standard Model. A key manifestation of the axion term is the Witten effect: a unit magnetic monopole placed inside a medium with θ = 0 is predicted to bind a (generally fractional) electric charge − e ( θ/ 2 π + n ) with n integer. Here we conduct a first test of the Witten effect based on the recently established fact that the axion term with θ = π emerges naturally in the description of the electromagnetic response of a new class of crystalline solids called topological insulators – materials distinguished by strong spin- orbit coupling and non-trivial band structures. Using a simple physical model for a topological insulator we demonstrate the existence of a fractional charge bound to a monopole by an explicit numerical calculation. We also propose a scheme for generating an ‘artificial’ magnetic monopole in a topological insulator film, that may be used to facilitate the first experimental test of Witten’s prediction. sponse of a STI is characterized by the axion term The idea of the axion was introduced in 1977 by Pec- cei and Quinn 1 as a means to solve what is known as the ‘strong CP problem’ in the physics of strong inter- actions. The strong CP problem, the details of which are quite subtle, has to do with the vacuum structure of Quantum Chromodynamics. In simple physical terms it can be stated as a question: Why is the electric dipole moment of the neutron (currently unobserved) so small? The Standard Model predicts a value for the neutron dipole moment | d$_{n}$ | ∼ 10 − $^{16}$θ e cm, with θ of order unity, that should be readily measurable. Peccei-Quinn’s solu- tion promotes θ to a dynamical field describing a new elementary particle, the axion, whose vacuum expecta- tion value has relaxed to a very small value, explaining the smallness of | d$_{n}$ | . The actual value of θ , and the validity of the Peccei-Quinn solution and its variants 2,3 remain open questions of considerable importance to fun- damental physics. The axion is also believed to be a vi- able candidate for the elusive dark matter that comprises the majority of matter in our universe 4 and is subject to active experimental searches. 5,6 In a remarkable development axion electrodynamics has recently emerged as a key tool in the description of crystalline solids called strong topological insulators (STIs). These three-dimensional time-reversal invariant (TRI) materials possess anomalous band structures char- acterized by a Z$_{2}$-valued topological invariant. 7,8 This in- variant, called ν$_{0}$ , counts the number of topologically pro- tected gapless surface states (modulo 2). A non-zero in- variant means that the surface of such an insulator will be metallic. This behavior has been predicted to occur 9,10 and subsequently experimentally discovered 11–13 in sev- eral 3-dimensional solids such as Bi$_{1}$$_{−}$$_{x}$Sb$_{x}$ alloys and Bi$_{2}$Se$_{3}$, Bi$_{2}$Te$_{3}$ crystals. More recently it has been realized, 14,15 remarkably, that the electromagnetic re- with θ = π , the only non-zero value permitted by the time-reversal symmetry. When the time-reversal symme- try is broken, e.g. in a crystal showing weak magnetism, θ can acquire an arbitrary value. Fluctuations in the magnetic order parameter then act as a dynamical axion field and can be thought of as emergent axion particles. 17 Thus, aside from possible practical applications, crys- talline solids with topologically non-trivial band struc- tures have the potential to provide tabletop laboratories for the testing and exploration of fundamental physical paradigms. A fundamental property of the axion medium is the Witten effect: 18 in the quantum theory, a magnetic monopole of unit strength (i.e. projecting magnetic flux Φ$_{0}$ = hc/e ) immersed in an axion medium must carry electric charge − e ( θ/ 2 π + n ) with n integer. This ef- fect, although theoretically well established, has never been experimentally tested because until now both a suitable axion medium and the means to produce a magnetic monopole have been lacking. In this study we demonstrate how the connection between the axion response 14,15 and strong topological insulators 7–13 may serve to overcome both obstacles. We remark that a 1D realization of the Witten effect in antiferromagnetic spin chains was proposed a long time ago. 19 Here we furnish the first concrete physical example of the Witten effect in 3D by modeling a STI with a magnetic monopole inserted in its bulk. We show that the monopole binds a fractional charge ± e/ 2 consistent with Witten’s prediction. 18 We then discuss possible ways to overcome the second ob- stacle by creating an emergent magnetic monopole in a topological insulator. This can be achieved by exploiting the degrees of freedom associated with a vortex in the exciton condensate that may emerge in a thin film topo- Δ L axion = θ$^{(}$ e 2 2 πh$^{)}$ B · E (1) I. AXIONS