ï  7 ï  5 ï  3 ï  1 1 3 5 7 ï  5 ï  3 ï  1 1 3 5 7 x y ï  0.01 0.00  0.01  0.02  0.03  0.04  ï  7 ï  5 ï  3 ï  1 1 3 5 7 ï ï 5 ï 3 ï 1 1 3 5 7 x y ï  0.06 ï  0.04 ï  0.02 0.00  0.02  0.04  0.06  ï  7 ï  5 ï  3 ï  1 1 3 5 7 ï ï 5 ï 3 ï 1 1 3 5 7 x y ï  0.06 ï  0.04 ï  0.02 0.00  0.02  0.04  0.06  ï  7 ï  5 ï  3 ï  1 1 3 5 7 ï  5 ï  3 ï  1 1 3 5 7 x y ï  0.01 0.00  0.01  0.02  0.03  0.04  0 2 4 6 8 10 12 0.0 0.1 0.2 0.3 0.4 0.5 r δQ ( r ) µ =0.00 µ =0.02 µ =0.05 µ =0.20 4 5 6 0.494 0.496 0.498 0.500 ï  7 ï  5 ï  3 ï  1 1 3 5 7 ï  7 ï  5 ï  3 ï  1 1 3 5 7 ï  0.01 0.00  0.01  0.02  0.03  0.04  x y 2 − ρ 1 b c a d e f FIG. 2: (Color online) A cubic sample of a TI including disorder with a planar unit monopole at its center, size L = 14 and parameters as in Fig. 1. In all cases shown we use the same disorder realization but vary its overall strength parametrized by µ . Panels (a) and (b) show charge density 2 − ρ$_{1}$ for the layer just below the planar monopole for weak disorder µ = 0 . 05Δ, and (c) larger disorder µ = 0 . 20Δ. Panels (e) and (f) show the difference in charge density δρ = ρ$_{0}$ − ρ$_{1}$ for µ/ Δ = 0 . 05 , 0 . 20, respectively, for the same layer. (d) The excess charge δQ ( r ) (in units of e ) for different values of disorder strength µ . The inset shows a close up of the saturation. At this scale a small deviation from the expected asymptotic value 1 / 2 that increases with the disorder strength becomes visible. We attribute this deviation to the finite-size effect in our numerical calculation. This identification is supported by the fact that the deviations grow more pronounced for smaller system sizes and close to the surface. Also, it is consistent with the notion that the bound charge is localized on the length scale ξ ∼ 1 / Δ which increases as the disorder reduces the spectral gap. field with a soliton profile 28,29 as realized in dimerized polyacetylene. 30 In the more general case when the Zee- man term or weak disorder are present, there exist no ex- act zero modes in the spectrum of electrons yet the frac- tional charge remains precisely quantized. This reflects the more subtle topological order that underlies the axion response of a STI, which is robust against any weak per- turbation that respects time reversal symmetry. 7,8,14,15 der optimal conditions it should be higher than it is in bilayer graphene, where the occurrence of this effect is hotly debated. Once the exciton condensate is formed, vortices can be nucleated by applying an in-plane mag- netic field. Since the exciton condensate is itself insulat- ing, the main conduction channel in this situation will be through vortices, each carrying − e/ 2 charge. Frac- tional charge of the carriers then can be detected using established techniques. 35,36 An interesting open question is the fate of the Witten effect in the presence of magnetic disorder. Experimen- tally, this can be implemented by adding a small con- centration of magnetic ions (such as Fe or Mn) into a topological insulator. Although at the microscopic level θ is no longer quantized in the presence of T -breaking perturbations, we expect its effective value, relevant to the physics at long lengthscales, to remain pinned at π as long as the magnetic moments stay disordered. This is because the net magnetic moment in a macroscopic region containing many impurities will effectively vanish. Qualitatively, this suggests that the Witten effect may survive inclusion of a moderate concentration of mag- netic dopants with randomly oriented moments. At low temperatures moments may order ferromagnetically. 37 In this case both T and the inversion symmetry are bro- ken (the latter due to the random position of magnetic dopants) and the effective θ can acquire an arbitrary Predicted more than 30 years ago in the context of high-energy physics, but never before observed in a real or numerical experiment, the Witten effect is realized in a strong topological insulator. A unit magnetic monopole inserted in a model STI binds electric charge − e/ 2 in accordance with the prediction 18 and furnishes a rare ex- ample of charge fractionalization and spin-charge sepa- ration in 3 spatial dimensions. In the special case when the underlying system possesses particle-hole symmetry and when the Zeeman term (15) can be neglected, the ap- pearance of fractional charge follows from the same ‘zero- mode’ arguments that underlie charge fractionalization in a one-dimensional system of fermions coupled to a scalar VI. OUTLOOK AND OPEN QUESTIONS 6