Supplementary Figure 3: Interaction effects in the proposed state preparation with Bloch oscillations. The numerical results are obtained by

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1 Supplementary Figure : Bandstructure of the spin-dependent hexagonal lattice. The lattice depth used here is V 0 = E rec, E rec the single photon recoil energy. In a and b, we choose the spin dependence parameter α to be 0 and 0.03, respectively. Switching the sign of α does not change the bandstructure. Supplementary Figure : Interaction effects in spin Hall response. The position vectors r and r are charge centers of the two spin components. The spatial coordinates r and r are parallel and perpendicular to the force F applied to the system, respectively. The numerical results are obtained by time-dependent Gross Pitaevskii simulations based upon the two-band Bose Hubbard model (see Supplementary Note 3). In our simulations, we choose D/t = 3, F/t = 0.5. When the interactions are weak, the spin Hall response of the chiral spin superfluid is only slightly modified compared to the non-interacting case.

2 Supplementary Figure 3: Interaction effects in the proposed state preparation Bloch oscillations. The numerical results are obtained by time-dependent Gross Pitaevskii simulations based upon the two-band Bose Hubbard model (see Supplementary Note 3) a, the occupation fractions of the ground and excited bands, n g and n e, obtained by projecting the time-dependent condensate wavefunction to the Bloch function basis. The time unit is T Dirac 4π. b and c, the difference of momentum 3λ distributions of two spins n (k) n (k) in the excited band at time T Dirac for interaction strengths U band /t = 0. and 0., respectively. The momentum distributions are normalized such that dkn σ(k) =. In our simulation we choose D = 3t and λ = 0.t.

3 3 Supplementary Note. DETAILS OF THE BOGOLIUBOV ANALYSIS In this section, we give the details of our Bogoliubov analysis. The ground state energy corrections are explicitly calculated for the () and ( ) states. Symmetry guarantees the same result for the ( ) and ( ) states. Fluctuations are included as φ r = ρ e ik r φ (k)e ik r k φ r = ρ e ipk r φ (pk)e ipk r, () p = and for chiral charge and spin states, respectively. The effective Bogoliubov Hamiltonian controlling these fluctuations is H eff = Ψ (k)h(k)ψ(k) const, () where Ψ(k) = [φ (k ), φ (k ), φ (k ), φ (k ) T and H(k) = k k M (k, k ) M (k), (3) M (k) M (k, k ) (k, k, k, k ) defined to be (k, K k, k, K k) and (k, K k, K k, k) for chiral charge and spin states, respectively. These M σσ matrices are = = M σσ (k σ, k σ ) ɛ(kσ ) ρ σ U σσ (K k) ρ σ U σσ (k K) ρ σ Uσσ(k K), ɛ(k σ ) ρ σ U σσ (K k) M (k) [ U (k K) ρ ρ U (k K) ρ ρ U (k K) ρ ρ U (k K). (5) ρ ρ (4) From Bogoliubov Hamiltonian in Eq. ( 3), we treat the spin-mixing part M as a perturbation which is well justified in the weakly interacting limit (see Fig. 3a in the main text). We thus write H eff = H (0) eff H() eff, H(0) eff block diagonal in the spin space. The leading part is readily diagonalized in terms of Ψ(k) = [ φ (k ), φ (k ), φ (k ), φ (k ) T, The coefficients are determined to be The Bogoliubov spectra are φ σ (k,σ ) = u σ (k σ, k σ )φ σ (k σ ) v σ (k σ, k σ )φ σ(k σ ), φ σ (k,σ ) = v σ (k σ, k σ )φ σ (k σ ) u σ (k σ, k σ )φ σ(k σ ). (6) vσ(k σ, k σ ) = u σ(k σ, k σ ) = [ ɛ(kσ, k σ ) ρ σ U σσ (k K) ε σ (k σ, k σ ), ε σ(k σ, k σ ) = ɛ(k σ, k σ ) [ɛ(k σ, k σ ) ρ σ U σσ (K k) ɛ(k σ, k σ ) = (ɛ(k σ ) ɛ(k σ )) /. ξ ±,σ = ε σ (k σ, k σ ) ± ɛ(k σ, k σ ) (7) ɛ(k σ, k σ ) = (ɛ(k σ ) ɛ(k σ ))/.

4 Under the condition U σσ (k) > 0 already assumed, we have ξ ±,σ > 0, which means the system is stable [,. Then H (0) eff takes a diagonal form H(0) eff = k p ξ φ p,σ σ(k p,σ ) φ σ (k p,σ ) E (0), E (0), E (0) /N s = k ɛ(k, K k) σ ε σ(k, K k) ρ σ U σσ (K k) (8) the same for chiral charge and spin states. Treating the spin mixing terms perturbatively, (see Sec. Supplementary Note ) the ground state receives an energy correction Introducing E () /N s = ρ ρ k Γ (k, k, k, k ), (9) ξ ξ ξ ξ Γ(k, k, k 3, k 4 ) = U (k K) (u (k, k ) v (k, k )) (u (k 3, k 4 ) v (k 3, k 4 )). (0) for the chiral charge case we have g(k) = Γ(k, K k, k, K k), () E () χ c /N s = ρ ρ k g (k) Q = K, while for the chiral spin case we have ε (k,q k)ε (k,q k) 4, () E () χ s /N s = ρ ρ k g (k) ε (k,q k)ε ( Qk, k) ɛ(k,q k) ɛ( Qk, k) ε ( Qk, k)ε (k,q k) ɛ( Qk, k) ɛ(k,q k). (3) We also calculate the energy correction by numerically diagonalizing the full Bogoliubov Hamiltonian (Eq. ( 3)), finding excellent agreement our analytic results when the inter-species interactions are weak (see Fig.3a in the main text). Supplementary Note. PERTURBATION THEORY FOR BOGOLIUBOV GROUND STATES In this section, we discuss the perturbative method to calculate the ground state energy of a Bogoliubov problem H Bog = Ψ H Bog Ψ H Bog (, ) H Bog (4, 4), Ψ a column vector of bosonic operators [φ, φ, φ, φ T. This Bogoliubov Hamiltonian is one momentum slice of Eq. ( ) and the momentum k index is suppressed for brevity. The 4 4 matrix H Bog can be rewritten as M G H Bog = G, (4) M where the matrices can be expanded in terms of Pauli matrices, M σ = c 0σ c xσ σ x c zσ σ z, and G takes a special form g( σ x ). The terms c 0σ, c xσ and c zσ can be read off from Eq. ( 3). Here we will treat the off-diagonal part G perturbatively. The leading part is readily diagonalized as H (0) = σ [ φ σ, φ [ φσ σ D σ φ σ D σ (, ) M σ (, ) D σ = ɛ σ c zσ σ z (5) ɛ σ = c 0σ c xσ

5 5 and [ φσ φ σ φσ = T σ φ, σ and u σ = vσ = reads uσ v T σ = σ v σ u σ c 0σ ɛ σ. The Bogoliubov spectra are ξ σ,± = ɛ σ ± c zσ. In terms of φ, the perturbative part H () = g(u v )(u v ) φ φ φ φ φ φ φ φ h.c. (6) Then standard perturbation theory applies, and only the third and fourth terms in Eq. ( 6) contribute at second order. The ground state energy is thus obtained to be E = (ɛ σ c 0,σ ) g(u v )(u v ). (7) ξ σ ξ ξ ξ Supplementary Note 3. TWO BAND BOSE HUBBARD MODEL To study interaction effects in the spin Hall effect and the dynamical population of excited band, we introduce a two band Bose Hubbard model H = H band H linear H int, single-particle terms H band and H linear already given in Methods, and the interaction term, H int = [U σσ A φ Aσ,r φ Aσ,r φ Aσ,rφ Aσ,r Uσσ B φ Bσ φ φ Bσ,rê,rê Bσ,rê φ Bσ,rê. (8) r With the combined spin-space inversion symmetry as in the spin-dependent hexagonal lattice, we have U A = U B U () band, U B = U A U () band and U A = U A = U B = U B V band. With weak spin dependence, these three independent interactions U () band, U () band and V band are approximately equal for 87 Rb atoms as used in the experiments [3. We take U () band = U () band = V band U band in the simulations shown in Figs.,3. The case difference in these interactions is also studied and no significant effect is found. Supplementary Note 4. LARGE U LIMIT OF THE HEXAGONAL LATTICE MODEL In the large U limit, we can project out double occupancy, and the Gutzwiller state is G = ( ) ) f,r,0 f,r, φ,r (f,r,0 f,rê,φ,rê vac, r a normalization condition f σ,r,0 f σ,r, =. To minimize kinetic energy we take f /,r, = f /, e ±ik r and f σ,r,0 = fσ,, where f σ, is a real number. Then the energy cost of the Gutzwiller state is which after minimization leads to E/N s = [ µ 3t eff ( f f ) 3t eff 3V/ ( f f ) 3t eff 3V/ ( f f ), f f = 0, if t eff > V/ f f = µ6t eff 6t eff 3V, otherwise. (0) The transition from the unpolarized superfluid to the fully polarized state is at V c = t eff in this large U limit, where the transition is first order. (9)

6 6 Supplementary References [ Stamper-Kurn, D. M. & Ueda, M. Spinor bose gases: Symmetries, magnetism, and quantum dynamics. Rev. Mod. Phys. 85, 9 44 (03). [ Pethick, C. & Smith, H. Bose-Einstein Condensation in dilute gases (cambridge university press, 008). [3 Soltan-Panahi, P., Luhmann, D.-S., Struck, J., Windpassinger, P. & Sengstock, K. Quantum phase transition to unconventional multi-orbital superfluidity in optical lattices. Nat Phys 8, 7 75 (0).