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value. In this situation we expect a monopole to still bind fractional charge according to Eq. (6) but we leave a detailed study of this case to future investigation. 

Can the Witten effect be observed experimentally in the near future? We believe that the answer is affir- mative. One essential ingredient, the axion medium, is now widely available in any of the recently discovered STIs. 11–13 If an emergent monopole can be realized, ex- ploiting the proposed exciton condensate, 20 the spin ice- type physics, 31–34 or by some other means, then the ex- perimental challenge is reduced to designing a suitable method for the detection of fractional charge bound to the monopole. The fractional charge of elementary exci- tations in fractional quantum Hall fluids has been previ- ously detected 35,36 and it should be possible to adapt these methods to topological insulators. In this way, studies of crystalline quantum matter with non-trivial topological properties could help settle one of the endur- ing challenges of fundamental physics and provide new insights into the behavior of electrons placed in unusual situations. 

The authors are indebted to I. Affleck, A.M. Essin, I. Garate, H.-M. Guo, J.E. Moore, C. Weeks, X.-L. Qi, and S.-C. Zhang for helpful discussions and correspon- dence. This work was supported by NSERC and CI- fAR and was performed, in part, at the Aspen Center for Physics. 

The feature that makes our model Hamiltonian (9) easy to analyze, is its matrix structure in the combined orbital/spin space, which consists of four anticommut- ing 4 × 4 matrices. Such matrices are by convention de- noted γ$_{µ}$ , µ = 0 , 1 , 2 , 3 and form a representation of the Clifford algebra defined by the anticommutation relation { γ$_{µ}$, γ$_{ν}$ } = 2 δ$_{µν}$ . In terms of these matrices Eq. (9) can be written as 

with d$_{µ}$ ( k ) = ( m$_{k}$, − 2 λ sin k$_{x}$, − 2 λ sin k$_{y}$, − 2 λ sin k$_{z}$ ). The actual form of our γ matrices is apparent by com- paring Eq.(A1) to (9) but our result for θ is independent of any particular representation, as long as the matrices satisfy the requisite commutation relation. In fact in the subsequent calculations it will be advantageous to use a 

1 R.D. Peccei and H.R. Quinn, Phys. Rev. Lett. 38 , 1440 (1977); Phys. Rev. D 16 , 1791 (1977). 

## Acknowledgment 

## Appendix A: Evaluation of θ 

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different representation of the Clifford algebra, obtained by a uniform rotation by angle π/ 2 around the τ$_{1}$ axis: we use γ$_{0}$ = τ$_{1}$ ⊗ ✶ and γ$_{i}$ = τ$_{2}$ ⊗ σ$_{i}$ , i = 1 , 2 , 3. In this representation the two normalized negative- 

energy eigenstates of H k can be written as 

where E = ( ∑ µ d 2 $_{µ}$) 1 / $^{2}$, and from now on we suppress the momentum dependence of all quantities. The eigenstates ψ$_{1}$ and ψ$_{2}$ above are degenerate and we are thus free to choose any (orthogonal) linear combination of these. Such a change of basis corresponds to a gauge transfor- mation on A . When evaluating θ using Eq. (13) one must keep in mind that the integrand is not gauge invariant while the integral taken over the BZ is gauge invariant modulo 4 π $^{2}$. This property reflects the Z$_{2}$ nature of the topological invariant ν$_{0}$ that underlies the physics of a STI. 

Using the eigenstates given in Eq. (A2) we find the Berry connection to be of the form A i = n$_{i}$ · σ , where σ is a vector of Pauli matrices and the components of vector n$_{i}$ read 

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with D$_{µ}$ = d$_{µ}$$_{/}$√$_{2}$$_{E}$$_{.}$ After substituting these into Eq. (13) a tedious but ultimately straightforward calculation leads to the expression 

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Using the values of d$_{µ}$ for our model given below Eq. (A1), one obtains a complicated integrand in terms of trigonometric functions. Although the value of the inte- gral is guaranteed to be either 0 or π in practice it is not obvious how to perform the required three-dimensional integration. However, it is clear from the structure of the integrand in (A3) that in the limit | m$_{k}$ | ≪ λ contri- butions to the integral come only from the vicinity of the 8 Dirac points. We evaluate these 8 contributions sepa- rately by linearizing d ’s as d ℓ $_{µ}$( k ) = ( m$_{ℓ}$, v ℓ $_{x}$k$_{x}$ , v ℓ $_{y}$k$_{y}$ , v ℓ $_{z}$k$_{z}$ ) and obtain 

An elementary evaluation then yields the result quoted in Eq. (14). 

2 S. Weinberg, Phys. Rev. Lett. 40 , 223 (1978). 3 

F. Wilczek, Phys. Rev. Lett. 40 , 279 (1978). 

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