2 III. TOPOLOGICAL INSULATOR AS AN AXION MEDIUM II. MONOPOLE AND THE WITTEN EFFECT Although the model specified by Eqs. (7) and (8) prob- ably does not describe any real solid, it is physical in that it is local in space and preserves time reversal and inversion symmetries. We show below that for a range of parameters ( λ, ϵ, t ) it represents a topological insula- tor, and therefore can be adiabatically deformed into any of the more realistic models 9,10,23,24 characterized by the same topological invariants. Any physical property that depends only on the topological invariants, such as the electrical charge bound to a monopole, can thus be cal- culated in the present model and the result will remain applicable to any topological insulator in the same topo- logical class. We note that a similar model was used in Refs. 10,17,25. Our Hamiltonian has a simple representation in mo- mentum space, H = $^{∑}$$_{k}$Ψ † k H $_{k}$Ψ$_{k}$ with and m$_{k}$ = ϵ − 2 t $^{∑}$$_{µ}$cos k$_{µ}$ . The spectrum of excitations has two doubly degenerate bands, In the limit ϵ, t → 0 the bands touch at 8 non-equivalent Dirac points located at Γ$_{ℓ}$$_{=(}$$_{n}$$_{x}$$_{n}$$_{y}$$_{n}$$_{z}$$_{)}$ = π ( n$_{x}$, n$_{y}$, n$_{z}$ ) with n$_{µ}$ = 0 , 1. These Γ$_{ℓ}$ also coincide with the 8 time-reversal invariant momenta 9 (TRIM). When ϵ, t are small but non-zero, the low-energy excitations of the system can be described in terms of 8 massive Dirac Hamiltonians obtained by a straightforward expansion of H k to lin- ear order in momentum in the vicinity of Γ$_{ℓ}$ . Here The integer n accounts for the possibility of binding extra electrons, which can always occur – only the fractional part of Q is non-trivial. where Δ θ is the net change in θ and we assumed that there was no initial electrical charge bound to the monopole, as should be the case for a charge-conjugation and parity (CP) invariant theory 18 with θ = 0. In a topological insulator Δ θ = π , thus one expects a mag- netic monopole to bind fractional charge We see that an electric field is generated in this process. Integrating Eq. (4) over space and time, we find that this field can be thought of as originating from a point electric charge Q located at the origin with magnitude Now consider a unit monopole, ∇ · B = Φ$_{0}$ δ ( r ), placed at the origin, in a medium initially characterized by θ = 0. We wish to understand what happens when we turn on θ as a function of time (but keep it uniform in space). To this end we set ∇ θ = 0 and j = 0 (no currents in vacuum) and take the divergence of Eq. (3) to obtain where α = e $^{2}$/ ℏ c is the fine structure constant and ρ , j are the electric charge density and current, respectively. We observe that for uniform, constant θ , Eqs. (2) and (3) revert to the familiar Maxwell’s equations, consistent with the notion that Δ L axion can be written as a total derivative in this case. An important related property 21 is the periodicity under θ → θ + 2 πn of the axion action S $_{axion}$, implying that θ can be chosen from the interval [0 , 2 π ). We start with a brief overview of the Witten effect. Although the effect is quantum-mechanical in nature its essence can be understood by studying the classical Maxwell’s equations modified in the presence of Δ L $_{axion}$. The axion term revises both Gauss’ law and Amp` ere’s law by adding extra source terms, 21 We now specify our model for a topological insulator and show that it indeed possesses the axion term. In order to minimize computational difficulties we consider a very simple model, inspired by Ref. 22, with electrons hopping on the cubic lattice with two orbitals per site, denoted as c and d . The Hamiltonian H = H$_{SO}$ + H$_{cd}$ , consists of a spin dependent part, with hopping between neighboring sites of the lattice, where Ψ$_{j}$ = ( c$_{j}$$_{↑}$, c$_{j}$$_{↓}$, d$_{j}$$_{↑}$, d$_{j}$$_{↓}$ ) $^{T}$, j labels sites of the cubic lattice, τ$_{µ}$ and σ$_{µ}$ are Pauli matrices in orbital and spin space, respectively, µ = x, y, z , and the spin-independent terms that connect the two orbitals, logical insulator under external bias. 20 We conclude that the prospects for experimental verification of the Wit- ten effect in a tabletop experiment using a STI appear promising. H$_{SO}$ = iλ ∑ j,µ Ψ † $_{j}$τ$_{z}$ σ$_{µ}$ Ψ$_{j}$$_{+}$$_{µ}$ + h . c ., (7) H$_{cd}$ = ϵ ∑ j Ψ † $_{j}$τ$_{x}$ Ψ$_{j}$ − t ∑ 〈 ij 〉 Ψ † $_{i}$τ$_{x}$ Ψ$_{j}$ + h . c . (8) ∇ · E = ρ − α 4 π 2 ∇ θ · B , (2) ∇ × B = ∂ E ∂t + j + α 4 π 2 ( ∇ θ × E + ∂θ ∂t B$^{)}$ , (3) ∇ · ∂ E ∂t + α 4 π 2 ∂θ ∂t ∇ · B = 0 . (4) H k = − 2 λ ∑ µ τ$_{z}$σ$_{µ}$ sin k$_{µ}$ + τ$_{x}$m$_{k}$, (9) Q = − α 4 π $_{2}$Φ$_{0}$ Δ θ = − Δ θ 2 π e, (5) E$_{k}$ = ± $^{√}$4 λ $^{2}$(sin 2 k$_{x}$ + sin 2 k$_{y}$ + sin 2 k$_{z}$ ) + m 2 $_{k}$. (10) H ℓ k = ∑ µ τ$_{z}$σ$_{µ}$v ℓ $_{µ}$k$_{µ}$ + τ$_{x}$m$_{ℓ}$ (11) Q = − e$^{(}$ 1 2 + n$^{)}$ . (6)