Classical roughening of dislocations and the effect of shear modulus softening in solid $^4$He.*

We propose that shear modulus $\mu(T)$ softening with increasing temperature $T$ observed by Day and Beamish [1] is due to a crossover experienced by dislocations from quantum smooth to classically rough state in the Peierls potential. Quantum dislocation is described by the Sine-Gordon model in dimensions $d=1+1$ with long-range interactions between kinks (induced by exchanging bulk phonons). Monte Carlo simulations of this model show that finite $T$ response on external stress can fit well the data $\mu(T)$ [1] for the parameters typical for $^4$He. We compare this model with the one proposed in Ref. [1]: the $^3$He impurities boiling off from the dislocations. Good fit of $\mu(T)$ cannot be achieved within this model for realistic values of the dislocation densities and relative fractions of $^3$He atoms. [1] J. Day and J. Beamish, Nature {\bf 450}, 853(2007)