2 3 4 5 6 7 0.05 K (a) 1.7 K, MARI 1.4 1.5 1.6 1.7 1.8 1.9 2 S ( Q , ω ) [a. u.] 0.5 1 1.5 0 0.75 0 0.75 0 0.75 0 0.75 0 0.75 1.5 Q [ ˚ A − $^{1}$] In t egrated Intensity [a. u.] 0.05 K 0.5 K 1.7 K 5 K 50 K (b) 0 2 4 0.4 0.8 1.2 1.6 2 T [K] Peak Intensity [a. u.] T ∗ (c) 2 3 4 5 6 7 5 K 50 K 1 2 3 2.4 2.6 2.8 3 3.2 Q [ ˚ A − $^{1}$] Integra t ed Intensity [a. u.] (d) ω [meV] 1 1.5 2 2.5 3 Q [ ˚ A − $^{1}$] ω [meV] 1 1.5 2 2.5 3 Q [ ˚ A − $^{1}$] 3 FIG. 3: (a) S ( Q,ω ) from IN4 at T = 0 . 05 , 5 , 50 K, and MARI at 1 . 7 K. The integration ranges used for the cuts in panels (b) and (d) are shown as black rectangles. (b) Q -dependence of intensity over energy ranges and 3-4 meV for 0.05-5 K (fit by two resolution convoluted Lorentzians) and 1-6 meV for 50 K (fit by Eq. 1). (c) Temperature variation of intensities of the Q 1 (open circles) and Q 2 (closed circles) modes. T ∗ = 1 K is indicated by a red vertical line. (d) Q -dependence of the flat mode integrated over 4.5-5.5 meV. S ( Q ) for a singlet-triplet excitation is shown in red. at Q = 1 . 08(2) ˚ A − $^{1}$, which coincides with the Q$_{2}$ peak in S ( Q ). Both of these features sharpen as temperature is reduced towards 0 . 05 K, with a second bar of scattering at Q = 0 . 68(4) ˚ A − 1 ∼ Q$_{1}$ growing below 1 . 7 K. Q -cuts through the Q$_{1}$ and Q$_{2}$ modes are shown in the lower 4 panels of Fig. 3(b). While the peaks narrow somewhat with decreasing T , a more dramatic change is observed in their respective intensities, I ( Q$_{1}$ ) and I ( Q$_{2}$ ). On cool- ing, I ( Q$_{2}$ ) remains constant, while I ( Q$_{1}$ ) increases to a final ratio I ( Q$_{1}$ ) /I ( Q$_{2}$ ) = 1 . 6 at 0 . 05 K. The buildup of dynamical correlations at Q$_{1}$ position thus coincides with the transition at T ∗ observed in NMR and µ SR. contrary to observations. To understand the low T short range ordered state, it is therefore necessary to look the- oretically further afield. From structural considerations outlined in the introduction, the Hamiltonian of volbor- thite can include three exchange terms ( J$_{1}$, J ′ $_{1}$, and J$_{2}$ ) as well as a Dzyaloshinskii-Moriya (DM) interac- tion, for which the dominant component is considered to be D$_{z}$ . The subsets of this model which have been treated semi-classically or quantum mechanically are: J$_{1}$ = J ′ 1 > 0 ,J$_{2}$ = 0 ,D$_{z}$ = 0, the isotropic kagome DM model (IKDM) [7, 8, 18] and J$_{1}$ = J ′ 1 > 0 ,J$_{2}$ = 0 ,D$_{z}$ = 0, the anisotropic kagome model (AK) [5, 6]. In addition, recent L(S)DA+U calculations have suggested a model where − J ′ $_{1}$/J$_{1}$ ∼ 1 . 2 − 2, − J$_{2}$/J$_{1}$ = 1 . 1 − 1 . 6, and J ′ 1 = 8 . 6 meV. This model, which we call the coupled chain model (CC), has only been treated classically so far [19]. The 5 possible ordered states which emerge from these models are: q = 0 order (IKDM for D$_{z}$ > 0 . 1, AK for 1 / 2 < J ′ $_{1}$/J$_{1}$ < 1), q = √$_{3}$ × √$_{3 (AK for 1}$$_{< J}$$_{1}$$_{/J}$ ′ 1 < 1 . 3), chirality stripe (AK for J$_{1}$/J ′ 1 > 1 . 3), ferrimagnetic (AK for J ′ $_{1}$/J$_{1}$ < 1 / 2, CC for J$_{2}$ < | J$_{1}$ | / 4 + J ′ $_{1}$/ 8), and spiral (CC for J$_{2}$ > | J$_{1}$ | / 4+ J ′ $_{1}$/ 8). The position of Bragg peaks for these structures are compared with our exper- imental Q$_{1}$ and Q$_{2}$ in figure 4(a). None of the proposed structures yield strong scattering at Q$_{1}$ . The lineshape and ampliude of the flat mode, on the other hand, show little temperature dependence, with only slight narrowing to become resolution limited be- tween 5 K and 1 . 7 K. Such narrow flat modes are often associated with two level excitations, eg. between a sin- glet and triplet. Indeed, such excitations are expected in the site diluted QKHAF [4, 16]. While the predom- inantly nearest neighbour correlations observed in S ( Q ) are consistent with such a state, the Q -dependence of the flat mode does not match the singlet-triplet S ( Q ) [3, 4] [Fig. 3(d)]. Another possible explanation for the flat band is thus that it is associated with the short range order observed at the Q$_{1}$ and Q$_{2}$ positions. This leads to two scenarios: i.) That the ground state possesses a degeneracy which results in a flat band, as is the case in the ground state of the classical kagome systems [17] or ii.) That the flat band arises as a consequence of powder averaging at the zone boundary of a spin wave dispersion. In the pure QKHAF, no short range order is expected, Therefore, instead of attempting to describe the ob- served S ( Q,ω ) in terms of the microscopic model above, we take the phenomenological approach of constructing a generic dispersion emanating from antiferromagnetic zone centers at high symmetry positions on the circles defined by Q$_{1}$ and Q$_{2}$ (Fig. 4a). One example of such a