We study the low temperature static and dynamical properties of the classical bond-disordered antiferromagnetic Heisenberg model on the kagome lattice. This model has recently been shown to host a new type of spin liquid exhibiting an exponentially large number of discrete ground states. Surprisingly, despite the rigidity of the groundstates, we establish the vanishing of the corresponding spin stiffness. Locally, the low-lying eigenvectors of the Hessian appear to exhibit a fractal inverse participation ratio. Its spin dynamics resembles that of Coulomb Heisenberg spin liquids, but exhibits a new low-temperature dynamically arrested regime, which however gets squeezed out with increasing system size. We also probe the properties of the energy landscape underpinning this behaviour, and find energy barriers between distinct ground states vanishing with system size. In turn the local minima appear highly connected and the system tends to lose memory of its inital state in an accumulation of soft directions.
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