5 pointed out in Ref. 20 that to electrons in a STI such a vortex is indistinguishable from a ‘planar monopole’, i.e. a monopole with magnetic field radiating in the plane of the surface. A planar monopole can be viewed as an adiabatic de- formation of an ordinary monopole achieved by flattening the field lines in a cylindrically symmetric fashion. One expects that the total charge bound to the monopole via the Witten effect should be insensitive to such an adia- batic deformation and therefore a vortex in the exciton condensate should bind fractional charge ± e/ 2. This in- deed has been argued to happen in Ref. 20 based on the Dirac equation describing the low-energy physics of the surface states in the presence of the exciton condensate. Here, taking a more general point of view, we establish the existence of the fractional charge in such a condensate by studying a planar monopole embedded inside a STI. Our calculation below does not rely on the low-energy approximation for the surface states and is insensitive to the detailed microscopic structure of the condensate. Rather, it exploits only the most fundamental property of the STI given by its nontrivial axion response. In general, the fractional charge is expected to be ro- bust against weak disorder that does not break TRI. Such a disorder will be present in a real sample and we model it here by adding a term Although there is no known theoretical principle that prohibits the existence of fundamental magnetic monopoles in nature, 26 none have been observed to date despite extensive searches. 27 This null observation has led to a consensus that fundamental monopoles either do not exist for some heretofore unknown reason or they are very rare in our part of the universe. In either case the observed absence of fundamental monopoles poses a challenge to the idea of experimental verification of the Witten effect using a STI. At best, one could conceive of a new scheme for possible detection of magnetic monopoles using a STI in the role of a sensor if a convenient way to detect the fractional charge could be found. A much more promising avenue for the verification of the Witten effect is suggested by exploiting emergent instead of fundamental monopoles. A classic example of such an emergent behavior in a crystalline solid is the 2007 theoretical prediction 31 and the subsequent ex- perimental observation 32–34 of monopoles in frustrated magnetic systems called ‘spin ice’, realized in certain magnetic pyrochlore compounds such as Dy$_{2}$Ti$_{2}$O$_{7}$ or Ho$_{2}$Ti$_{2}$O$_{7}$. Magnetic monopoles in these systems arise as elementary excitations above the collective ground state of spins and the monopole-like magnetic field configura- tion originates from the magnetic moments of the con- stituent spins. In principle, the emergent monopoles in the spin ice could be used to test the Witten effect if a compound that is simultaneously a STI and a spin ice could be identified. Unfortunately no such material is known at present although we note that STI behavior has been theoretically predicted to occur in crystals with the same pyrochlore structure 23,24 that underlies the spin ice behavior. It is thus possible that a suitable material will be discovered in the future. to the Hamiltonian. The first term represents a parity- preserving on-site disorder (independent of the orbital), while the second term is a parity breaking disorder. Here we focus on a different type of emergent magnetic monopole that can arise in a thin film STI placed in a uniform external electric field. The basic idea and the feasibility of its experimental realization have been dis- cussed in Ref. 20. Following that work we envision the simplest STI with just one gapless Dirac state per surface and the chemical potential initially tuned to the neutral point. When a strong enough electric field is applied per- pendicular to the plane of the film the chemical poten- tial undergoes a shift that is opposite in the two surfaces. This creates a small electron Fermi surface in one surface and a small hole Fermi surface in the other. The essence of the proposal 20 lies in the observation that an arbitrar- ily weak Coulomb interaction between the surface states produces an exciton condensate, which may be viewed as a coherent fluid of electron-hole pairs drawn from the opposite surfaces. Such an exciton condensate is char- acterized by a complex scalar order parameter Φ, which can fluctuate in space and time. In particular Φ = Φ$_{0}$ e iχ can contain vortices – point-like topological defects with the phase χ winding by ± 2 π around a vortex. It has been In the framework of the current proposal the key ingre- dient required to produce a monopole-like configuration is the exciton condensate. As explained in Ref. 20 it is difficult to reliably estimate the critical temperature T$_{EC}$ for the formation of the exciton condensate, but un- As before we solve the Hamiltonian H = H$_{SO}$ + H$_{cd}$ + H$_{D}$ , in a cube containing L 3 sites, by exact numerical diagonalization. The planar monopole projects an effective magnetic field B$_{eff}$ = (Φ$_{0}$ / 2 πr ) δ ( z )ˆ r (in cylindrical coordinates) and the vector potential can be chosen as A = (Φ$_{0}$ / 2 π ) ϕδ ( z )ˆ z . The effective field does not couple to electron spin, 20 so there is no Zeeman term in this case. The disorder coefficients µ ± j are chosen from a Gaussian distribution with standard deviation µ . Note that the disorder breaks the four-fold rotational symmetry of the system, so we cannot exploit this symmetry in this case to efficiently diagonalize the Hamiltonian. Consequently we are limited to system sizes up to L = 14. For weak disorder µ ≪ Δ the charge bound to the planar monopole remains − e/ 2 (see Fig. 2), and for strong disorder µ ≫ Δ the charge bound is zero. Remarkably, even for fairly significant disorder (such that it generates charge density fluctuations comparable to the charge density induced by the monopole) the difference in charge density δρ shown in Fig. 2f is only weakly affected. H$_{D}$ = ∑ j µ + j Ψ † $_{j}$Ψ$_{j}$ + ∑ j µ − j Ψ † $_{j}$τ$_{z}$ Ψ$_{j}$ , (16) V. PROPOSAL FOR EXPERIMENTAL REALIZATION