0 1 2 3 4 Omega 0 0.5 1 1.5 2 2.5 3 Structure Factor Pinwheel Spectra  Different qs, Broadening=0.01 0 1 2 3 4 omega 0 0.1 0.2 0.3 0.4 0.5 Structure Factor Pinwheel Spectra Different qs, Broadening=0.1 FIG. 2: Kink-antikink or 2-spinon states for the pinwheel. A triplet excitation is created by breaking a singlet bond in the ground state. The Kink spinon lies on the inner hexagon, while the more mobile antikink spinon can move on the out- side vertices of the pinwheel through states (a) through (f) shown in the figure. When the two spinons are together as in Fig. (a), the kink spin can also move by going through higher energy intermediate states. FIG. 3: Angle averaged dynamic structure factor versus fre- quency of the Pinwheel state at T = 0 with small broadening. dimerization.[28] In the VBC state, their is significant quantum fluctuations. In particular, the dimerization in- side the resonating hexagons is strongly reduced. But, the pinwheel region is geometrically protected against quantum fluctuations and hence remains essentially fully dimerized.[29] Different Valence Bond phases have energy difference of only 0 . 001 J per site.[16] Thus one expects any phase transition to such a state to occur at a very low tem- perature. However, short-range Valence Bond order is set by J and can develop at significantly higher tem- peratures. Once this short-range order is a few lattice constants, pin-wheel like structures should begin to form locally. Because they have extremely low local energies, they should be stable up to higher temperatures. The triplet excitations of the pinwheel are kink- antikink pairs.[21, 26, 27] In an infinite Delta chain, the kinks are immobile, where as the antikink can hop from one triangle to another. The kink has zero excitation energy, whereas the energy of the antikink can be ap- proximated by Note that this means that excitations extend over the energy range J/ 4 < ϵ < 9 J/ 4. For the infinite sys- tem more detailed analytical and numerical calculations show[26, 27] that the lowest energy antikink is roughly at 0 . 219 J , whereas the upper energy may extend up to as much as 3 J . For the finite system, spin excitation from the ground state creates a pair of parallel spins on one of the singlet bonds of the ground state. While the kink an- tikink description is roughly valid, because both the kink and antikink remain in each other’s vicinity, both free spins or spinons become mobile. Fig. 2, shows the con- figurations that correspond to the mobile antikink. On the other hand the kink can also move by going through higher energy intermediate configuration, when the two spins are next to each other. In the S z = 1 sector, assum- ing two free spins and the rest of the system in the ground state leads to 66 states, of which 36 states are of the kink- antikink type, where one spin is in the inner hexagon, whereas the other spin is on the outside vertices of the pinwheel. One expects the spectral weight to be primar- ily spread over these states, giving rise to spectral weight spread roughly over the energy range J/ 4 < ϵ < 9 J/ 4. FIG. 4: Angle averaged dynamic structure factor versus fre- quency of the Pinwheel state at T = 0 with larger broadening. ϵ ( k ) = 5 / 4 − cos k. (1) 2