3 γ band in Figs. 4a (since the folded γ ′ is very weak) has happened above T$_{N}$ . The fact that the bands shift smoothly across the SDW transition temperature can only be interpreted into two completely opposite ways: either the magnetism is coupled to the electronic struc- ture very weakly; or the electronic structure has been already strongly coupled to magnetic fluctuations above T$_{N}$ . However, the former case is unlikely. Since if such a large change in electronic structure were not related to the magnetic ordering, it must be caused by the lattice distortion or some parasitical secondary orbital/charge order. Neutron and x-ray scattering experiments have found that the lattice distortion in NaFeAs takes place at T$_{S}$ , and saturates at T$_{N}$ with 0.36% total lattice dis- placement. [11, 18]. It would cause about 1% of change to the hopping parameters among various d orbitals [19]. Since 1% of the measured occupied bandwidth is just about 1 meV, the observed 12 meV shift between T$_{S}$ and T$_{N}$ is well beyond what the lattice distortion could cause. The strong coupling between magnetism and elec- tronic structure is further supported by the observation of band folding above the SDW transition in Fig. 5. At k$_{‖}$ = 0 . 52 ˚ A − $^{1}$, we expect that the γ ′ band becomes visi- ble in EDC when there is a band folding. Indeed, there is an abrupt change on the spectrum taken at 55 K; and the evolution below 55 K is rather smooth. Therefore, band folding occurs already at the structural transition temperature. Since the structural transition itself does not cause zone folding, this band folding indicates the not show any sign of bending due to the opening of any gap. We note the intensity of certain band may be weak at the Fermi level, but this is rather due to the matrix element effects, which has been the case for the normal state spectra in Fig. 1 already. To understand the band structure in the SDW state, Figure 3 plots band structures in the normal and SDW states together. As shown in Fig. 3b, the band folding mainly occurs on the γ band in the SDW state, caus- ing the additional γ ′ band. Both γ and γ ′ slightly push α and β near Γ, and they hybridize with δ , generat- ing a complicated band structure around M . Therefore they are slightly asymmetric with respect to the SDW zone boundary (the dash-dotted line). By comparing the bands in the normal and SDW states in Fig. 3c, pro- nounced band shifts could be identified. The β and δ bands shift downward, while the γ band shifts upward near M. Such a large shift, especially the shift of a band well below Fermi surface, indicates that the band recon- struction is not due to any Fermi surface instability. To study the band reconstruction in detail, Figs. 4a-d show the temperature dependence of photoemission spec- tra at four representative momenta. It is clear that posi- tions of various bands start to shift rapidly at the struc- tural transition temperature T$_{S}$ , and continue smoothly into the SDW state (Fig. 4e). For example, the β band shifts downward by about 16 meV (Fig. 4c), in which a 12 meV shift has already happened above the SDW transition temperature T$_{N}$ , while half of the shift for the FIG. 4: (Color online) Temperature dependence of EDCs at selected momenta: (a) k x = − 0 . 28 ˚ A − $^{1}$, (b) k x = 0 (Γ), (c) k x = 0 . 92 ˚ A − $^{1}$, and (d) k x = 1 . 1 ˚ A − 1 (M). (e) The summary of feature shifts in panels a-d. FIG. 3: (Color online) Band structure of NaFeAs determined experimentally at (a) the normal state, and (b) the SDW state. (c) The normal and SDW state bands are compared. Momentum positions for EDCs in Figs. 4 and 5 are marked on the top. E-E F  (meV) -0.5 0.5 0 1.0 k$_{//}$$_{ }$ ( Å -1 ) 0 -60 -120 0 -60 -120 α β γ δ α 0 -60 -120 β γ γ ' δ Γ Μ 4(a) 4(b) 5 4(c) 4(d) (a) (b) (c) γ ' γ Figs. E - E$_{F}$ (meV) Intenstiy (arb. units) k$_{//}$ = -0.28 Å -1 (a) 0 -50 -100 γ 75 55 49 43 37 30 10 65 70 60 52 46 40 34 20 T (K) Γ  point β (b) 0 -50 -100 75 55 49 43 37 30 10 65 70 60 52 46 40 34 20 T (K) (c) k$_{//}$ =0.92 Å -1 β 0 -50 -100 75 70 65 60 55 52 49 46 43 40 37 34 30 20 10 T (K) δ  ( β )  Μ point (d) 0 -50 -100 75 70 65 60 55 52 49 46 43 40 37 34 30 20 10 T (K) γ  in (a) β  in (b) β  in (c) δ ( β)  in ( d ) (e) T$_{N}$ T$_{S}$ -20 -30 -40 -50 -60 Peak Position (meV) 20 40 60 Temperature (K)