Emergent Momentum-Space Skyrmion Texture on the Surface of Topological Insulators

The quantum anomalous Hall effect has been theoretically predicted and experimentally verified in magnetic topological insulators. In addition, the surface states of these materials exhibit a hedgehoglike “spin” texture in momentum space. Here, we apply the previously formulated low-energy model for Bi2Se3, a parent compound for magnetic topological insulators, to a slab geometry in which an exchange field acts only within one of the surface layers. In this sample set up, the hedgehog transforms into a skyrmion texture beyond a critical exchange field. This critical field marks a transition between two topologically distinct phases. The topological phase transition takes place without energy gap closing at the Fermi level and leaves the transverse Hall conductance unchanged and quantized to e2/2h. The momentum-space skyrmion texture persists in a finite field range. It may find its realization in hybrid heterostructures with an interface between a three-dimensional topological insulator and a ferromagnetic insulator.

which was worked out in Ref. [2] for the orbital basis functions given in the main text. Here, k 3 ≡ (k x , k y , k z ), The momenta along and perpendicular to the (111) direction are denoted by k z and k ⊥ = k 2 x + k 2 y , respectively. The lattice generalization of the above Hamiltonian is obtained by substituting . The parameters are modified by powers of the lattice constants a and c, i.e.
The finite thickness of the slab is accounted for by performing the partial reverse Fourier transformation of the fermionic operators c k3 = (1/N z ) lz e ikzlz c kx,ky,lz . The Hamiltonian for the slab then reads With the exponential forms for the sine and cosine functions and the identities kz e ikz(lz−l z −1) = N z δ lz,l z +1 , kz e ikz(lz−l z +1) = N z δ lz,l z −1 , the Hamiltonian (S3) reduces to the Hamiltonian (1) in the main text.

S2. Complementary "spin" textures
The hedgehog and skyrmion textures appear mainly in the occupied part of the exchange split surface states. A more precise statement is hampered by the fact that the two surface bands hybridize around the avoided levelcrossing momenta R ks = k 2 x + k 2 y 0.3/a. Near theΓ point, the surface states are truly confined to either the top or the bottom surface layer. Yet, upon moving away from theΓ point, the surface states spatially extend continuously more towards the interior of the slab. The concomitant increasing overlap of the surface states' wave functions in the central layer of the slab is the origin of the hybridization of the exchange split and the unsplit band which are truly surface bands only at and near theΓ point. For momenta beyond R ks , the spatial character of the surface bands is interchanged (see Fig. 3(c) in the main text). The hybridization and the avoided level crossing at R ks result in a complementarity of the associated "spin" textures. This is demonstrated in Fig. S1. This figure shows the "spin" texture of both surface bands within the surface layer in which the exchange field is applied. For the exchange-split surface band (the green band in Fig. 3(a), (b), (c) in the main text), the "spin" expectation values sharply drop to nearly zero when the magnitude of the momentum exceeds the ring with radius R ks . The origin of this drop is the significantly reduced amplitude of the wave function in the selected surface layer due to the interchange of the surface states and their associated spatial character at the avoided level crossing momenta with magnitude R ks .
The "spin" expectation values for the unsplit surface band (the red band in Fig. 3(a), (b), (c) in the main text) display the complementary behavior. Within the selected surface layer, the "spin" expectation values drop to nearly zero when the magnitude of the momenta is smaller than the ring radius R ks -for the same reason as outlined above, i.e. the interchange of the character of the two surface bands at R ks .  Fig. 3(a), (b), (c)). At and near theΓ point, the eigenstates of this band are spatially confined in the surface layer in which no field is applied. (a) and (c) show the corresponding textures in the second-to-highest energy band (the green band in Fig. 3(a)

S3. Finite-size effect on the Hall conductance
As studied previously in Ref. [3][4][5][6][7], the energy gap in thin slabs of three dimensional topological insulator is not truly closed at the Dirac point even in the absence of time-reversal symmetry breaking magnetic field. This gap decreases in an oscillatory manner with increasing the layer number N z . As discussed in Ref. [3][4][5][6][7], there is a phase transition from a band insulator to a topologically non-trivial insulating phase at N z = 3, made evident by an increase in the energy gap at theΓ point as shown in Fig. S2. More transitions follow with increasing N z involving parity changes of the bands closest to the Fermi level [6]. The oscillations in the energy gap with increasing N z are shown in the inset of Fig. S2. A non-zero energy gap prevails for all slab thicknesses even though it shrinks to tiny values with increasing N z . This energy gap causes a finite contribution to σ xy even in the absence of any exchange field. As shown in Fig. S2, σ xy increases linearly with N z beyond N z = 4. The variation of σ xy with N z is connected to the Berry curvature of the surface-band states. The Berry curvature is largest near theΓ point and increases with decreasing energy gap. The rise in the energy gap at N z = 3 is also reflected as a cusp in σ xy . σ xy is calculated for the full slab, and, therefore, it is the conductance rather than the conductivity. This explains the linear rise of σ xy with the slab thickness for n z > 4.