20 peak intensity (a) peak intensity (b) x=0.4, SC x=0.45, NSC (c) (d) x=0.45, NSC x=0.45, NSC 4 two bands as shown in Ref.[34], where the renormaliza- tion factors around M point are smaller and the α 3 band is much flatter, we obtain the velocity of α 3, v$_{F,α}$$_{3}$ ≈ 22 meV ˚ A. In summary, we observe the hourglass dispersion in both SC and NSC FeSe$_{x}$Te$_{1}$$_{−}$$_{x}$ samples, phenomenologi- cally similar to the case of cuprates and pure chromium metals. Whether the hourglass behavior of spin excita- tions is a very common feature in the metallic states of these magnetically-fluctuating systems needs to be fur- ther investigated both experimentally and theoretically. This work is supported by Chinese Academy of Sci- ence, 973 Program (2010CB833102), and by the US DOE, BES, through DOE DE-FG02-05ER46202 and Di- vision of Scientific User Facilities. system, where the two hole pockets ( α 2 and α 3) at Γ point are nested to the electron pockets at M point as shown in Figs. 1c-1e, the low energy excitations which disperse inward to the commensurate wavevector Q = (0 . 5 , 0 . 5) start at a incommensurate wavevector about Q = (0 . 5 − 0 . 15 , 0 . 5 + 0 . 15), which is different from the incommensurate scattering at Q = (0 . 5 − 0 . 09 , 0 . 5+0 . 09) defined by the high energy dispersion [26, 27]. The reason why such nesting conditions are favored may be related to orbital characters[28, 33, 34]. Although this nesting scheme is appealing, extremely flat band is required to produce the flat dispersion of spin excitations observed at low energies. The bare dispersion calculated by local- density approximation (LDA) [27] is not flat enough to produce the observed excitations. However, we note that recent angle resolved photoemission data [33, 34] have shown that compared with LDA calculations, certain bands in the FeSe$_{x}$Te$_{1}$$_{−}$$_{x}$ have very large renormalization factors and become much flatter, possibly due to strong electron-electron correlation effects [35]. These ARPES results can provide an consistent understanding of our experimental results based on the Fermi surface nesting between α 3 and β bands as shown in Fig. 1. Based on the proposed nesting picture in Fig. 1c-1d, we can de- rive the slope of the lower-energy spin excitation in Fig. 1f-1g as dE/dδ ≈ 2 Qcos $^{2}$θ (1 /v$_{F,α}$$_{3}$ + 1 /v$_{F,β}$ ) $^{$_{−}$1}$, where θ is the angle between the incommensurate wavevector Q =(0.5- δ , 0.5+ δ ) and the AF wavevector (0.5,0.5). We obtain dE/dδ ≈ 50 meV/r.l.u., fitting to our experimen- tal data. 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