1 1.5 2 2.5 2 3 4 5 6 7 Spin Wave Q [ ˚ A − $^{1}$] (b) ω [meV] 1 1.5 2 2.5 0.05 K, IN4 Q [ ˚ A − $^{1}$] J e a = 5 . 1 meV J e b = 15 . 3 meV 4 dispersion among the scenarios considered assumes zone centers close to the (10) and (01) positions, yielding the dispersion: where | F ( τ ) | 2 is the structure factor at the chosen po- sitions, 1 /ω describes antiferromangetic spin wave in- tensity, and the numerator is a geometric term yielding zero intensity at ferromagnetic zone centers. To yield a smooth continuous function, S ( q , ω ) was interpolated between adjacent Brillouin zones. Finally, the spectrum was powder averaged and convoluted with the experi- mental resolution. The result of this procedure using J e a = 5 . 1 meV and J e b = 15 . 3 meV closely resembles the experimental data (Fig. 4(b)). One important con- clusion from this analysis is that the steepness of the Q$_{2}$ mode implies that the 5 meV band cannot be the the global zone boundary energy. Instead the flat band is found to be a saddle point, which requires sizeable anisotropy between J e a and J e $_{b}$. This result is consistent with LS(D)A+U calculations, which also suggest a sig- nificant anisotropy in exchange along a and b . FIG. 4: (a) Reciprocal space of volborthite: the structural unit cell is indicated by the rectangle, and the extended Bril- louin zone of the kagome lattice by the dotted hexagon. Sym- bols represent Bragg peaks of the orders listed in the text. (b) The experimental S ( Q, ω ) measured at 0 . 05 K compared with the powder averaged S ( Q, ω ) derived from our empirical spin wave model. The dashed line in the left panel indicates the ( Q, ω ) window of the experiment. where J e a,b are effective exchanges, giving the amplitudes of the dispersion along a and b . Then, ω ( q ) = √ (2 J e a + 2 J e $_{b}$) 2 − [2 J e a cos( q$_{x}$a ) + 2 J e b cos( q$_{y}$b )] 2 (2) S ( q , ω ) = | F ( τ ) | $_{2}$2 − cos( q$_{x}$a ) − cos( q$_{y}$b ) ω ( q ) (3) In summary, we have reported polarised and inelastic neutron scattering results on the quasi-kagome s = 1 / 2 antiferromagnet volborthite. These reveal three impor- tant features: i) Buildup of nearest neighbor pair correla- tions from 50 K to 10 K; ii) Short range order indicated by peaks at Q$_{1}$ = 0 . 65(3) ˚ A − 1 and Q$_{2}$ = 1 . 15(5) ˚ A − 1 in the diffuse and inelastic scattering below 5 K; and iii) The excitation spectrum displays dispersive modes em- anating from both Q$_{1}$ and Q$_{2}$ along with a flat mode at ω$_{f}$ = 5 meV. The inelastic intensity at Q$_{1}$ becomes dominant below 1 . 7 K, identifying the nature of the low- T state reported from $^{51}$V NMR and C$_{p}$ . Most mod- els proposed for volborthite however inaccurately predict no scattering at Q$_{1}$ , and could therefore be ruled out. We provide an empirical dispersion model which repro- duce the experimental observations for an anisotropic set of exchanges which are in rough agreement with recent LS(D)A+U calculations. A quantum treatment of this model could well yield a correct description of both the correlations and excitation spectrum that we have un- covered experimentally. We thank Z. Hiroi, B. F˚ ak, P. Mendels, F. Bert, and O. Janson for stimulating discussions, G. Ehlers and J. Gardner for assistance during initial experiments, and SNF for support. [9] L. Messio, C. Lhuillier, and G. Misguich, arxiv:1101.1212 [8] I. Rousochatzakis, S. R. Manmana, A. M. Laeuchli, B. Normand, and F. Mila, Phys. Rev. B 79 , 214415 (2009). [7] O. Cepas, C. M. Fong, P. W. Leung, and C. Lhuillier, Phys. Rev. B 78 , 140405 (2008). [6] F. Wang, A. Vishwanath, and Y. B. Kim, Phys. Rev. B 76 , 094421 (2007). [5] T. Yavors’kii, W. Apel, and H.-U. Everts, Phys. Rev. B 76 , 064430 (2007). [4] R. R. P. Singh, Phys. Rev. Lett. 104 , 177203 (2010). [3] M. A. de Vries, J. R. Stewart, P. P. Deen, J. O. Piatek, G. J. Nilsen, H. M. Rønnow, and A. Harrison, Phys. Rev. Lett. 103 , 237201 (2009). [2] Z. Hiroi, M. Hanawa, N. Kobayashi, M. Nohara, H. Tak- agi, Y. Kato, and M. Takigawa, J. Phys. Soc. Japan 70 , 3377 (2001). [1] M. Shores, E. Nytko, B. Bartlett, and D. Nocera, J. Am. Chem. Soc. 127 , 13462 (2005). † Current Adress: Dep. of Pure and Applied Chemistry, University of Strathclyde, Glasgow, G1 1XL UK ∗ Email address: gorannilsen@gmail.com