content_image/407232.jpg content_image/407238.jpg content_image/407240.jpg content_image/407242.jpg content_image/407236.jpg content_image/407239.jpg content_image/407241.jpg content_image/407237.jpg content_image/407244.jpg content_image/407233.jpg content_image/407234.jpg content_image/407235.jpg content_image/407243.jpg
<img src='content_image/407232.jpg'>

In particular, when ρ$_{s}$$_{0}$ = 0 (normal solids), c$_{2}$ vanishes, and we only have 

which agrees with the longitudinal sound speed obtained by Zippelius et al. 44 once we identify ˜ λ = λ $^{ZHN}$+2 γ $^{ZHN}$+ 1 /χ and γ = ( γ $^{ZHN}$+1 /χ ) /ρ$_{0}$ . Moreover, when the Lam´ e coefficients and the coupling constant γ vanish we recover the sound speed of a normal fluid. As discussed earlier, there is one pair of transverse sound modes with speed 

Finally, we can calculate the correlation functions from Eq. (45): 

<img src='content_image/407238.jpg'>

and 

Since the density fluctuation is related to the defect den- sity fluctuation and the strain tensor by Eq. (34), the density-density correlation function becomes 

<img src='content_image/407240.jpg'>

where 

Then, by performing the analytic continuation iω$_{n}$ = ω + iδ , the density-density response function can be ob- tained from the imaginary part of the density-density correlation function: 

<img src='content_image/407242.jpg'>

<img src='content_image/407236.jpg'>

<img src='content_image/407239.jpg'>

<img src='content_image/407241.jpg'>

<img src='content_image/407237.jpg'>

where we have used the identity 

It is easy to show that the response function satisfies the thermodynamic sum rule (for the derivation of the static correlation function see Appendix B) 

and the f-sum rule 

## B. Dynamics of supersolid with dissipation 

We continue our discussion of the density correlation and response functions by including dissipative terms in the equations of motion. As mentioned above, the dis- sipative terms will broaden the δ -function peaks in the density response function. In addition, as noted by Mar- tin et al. $^{38}$, the dissipation is necessary to identify the “missing” defect diffusion mode in normal solids. The dissipative hydrodynamic equations of motion for a su- persolid were first obtained by Andreev and Lifshitz, 1 who used standard entropy-production arguments to gen- erate the dissipative terms. For an isotropic supersolid (including the nonlinear term neglected by Andreev and Lifshitz) we have (with the new dissipative terms on the right hand side) 

<img src='content_image/407244.jpg'>

where j = ρ$_{n}$ v$_{n}$ + ρ$_{s}$ v$_{s}$ is the total mass current, Σ and Λ coefficients of viscosity, ζ the bulk viscosity coefficient, η the shear viscosity coefficient, Γ the diffusion coefficient for defects. 

We next linearize the dissipative hydrodynamic equa- tions by considering small fluctuations from the equilib- rium values. Writing δµ and λ$_{ij}$ in terms of δρ and δw$_{ij}$ , 

<img src='content_image/407233.jpg'>

<img src='content_image/407234.jpg'>

<img src='content_image/407235.jpg'>

<img src='content_image/407243.jpg'>

