a pair of propagating sound modes and a thermal diffu- sion mode. The density fluctuations important for light scattering only couple to the longitudinal modes, so three modes are observed: the diffusion mode appears as the Rayleigh peak ω = 0 and the pair of sound modes as the Brillouin doublet at ω = ± cq (with a sound speed c ). In the absence of dissipation these peaks are δ -functions; dissipation turns each δ -function into a Lorentzian of width Dq $^{2}$, with D being an attenuation coefficient. What happens in a superfluid? In addition to the five conserved densities that exist in a normal fluid there is a broken gauge symmetry, so from mode counting we conclude there are six collective modes. Two of these are transverse momentum diffusion modes (just as for the normal fluid), leaving four longitudinal modes for the superfluid: a pair of propagating first sound modes, and a new pair of propagating second sound modes. In essence, the central Rayleigh peak in the normal fluid splits into a new Brillouin doublet upon passing into the superfluid phase. This remarkable phenomenon has been observed in light scattering experiments on $^{4}$He.48,49 We show below that an analogous splitting occurs in a su- persolid, and should be observable in a light scattering experiment. fying assumptions: the solid is isotropic, and two dimen- sional. The isotropy causes the transverse and longitu- dinal modes to neatly decouple; the two dimensionality results in only one pair of propagating transverse modes, rather than two pair. Since we’re interested in longi- tudinal fluctuations, the latter simplification is of little consequence to the main results of this section. With these assumptions, the thermodynamic relations are 6 To facilitate the calculation of the density-density cor- relation function for a supersolid we’ll make two simpli- where χ is the isothermal compressibility at constant strain, γ is a phenomenological coupling constant be- tween the strain and the density, and ˜ λ and ˜ µ are the bare Lam´ e coefficients at constant density. Then in Fourier space the Lagrangian density, Eq. (35), reduces to where ω$_{n}$ = iω , Q = ( q , ω$_{n}$ ), u$_{L}$ = ( q · u ) /q with q = | q | , u$_{T}$ = u − ( u$_{L}$/q ) q , and where λ ≡ ˜ λ + 2˜ µ . The collective modes are determined from the determinant Δ$_{A}$ of A : Setting Δ$_{A}$ = 0, we find the longitudinal first sound speed c$_{L}$ and second sound speed c$_{2}$ : Δ$_{A}$ = ρ$_{n}$$_{0}$ω 4 n + [ λ + ρ$_{n}$$_{0}$ ( 1 ρ$_{0}$χ − 2 γ )] q $^{2}$ω2 n − ρ$_{s}$$_{0}$ ( γ 2 − λ χρ 2 0 ) q $^{4}$. (47) c 2 L = λ 2 ρ$_{n}$$_{0}$ + 1 2 ρ$_{0}$χ − γ + 1 2 √ ( λ ρ$_{n}$$_{0}$ + 1 ρ$_{0}$χ − 2 γ ) 2 − 4 ρ$_{s}$$_{0}$ ρ$_{n}$$_{0}$ [ λ χρ 2 0 − γ 2 ] , (48) A =    1 ρ 2 $_{0}$χ − ω$_{n}$ − iq ( γ − 1 ρ 0 χ ) ω$_{n}$ q $^{2}$ρ$_{s}$$_{0}$ iω$_{n}$qρ$_{s}$$_{0}$ iq ( γ − 1 ρ 0 χ ) iω$_{n}$qρ$_{s}$$_{0}$ ρ$_{n}$$_{0}$ω 2 n + q 2 λ + 1 χ − 2 ρ$_{o}$γ )    , (46) L SS = 1 2 ( δρ$_{Δ}$ ( Q ) θ ( Q ) u$_{L}$ ( Q ) ) A   δρ$_{Δ}$ ( − Q ) θ ( Q ) u$_{L}$ ( − Q )   + 1 2 ( ρ$_{n}$$_{0}$ω 2 n + ˜ µq $^{2}$) u$_{T}$ ( Q ) u$_{T}$ ( − Q ) , (45) ∂λ$_{ji}$ ∂w$_{lk}$ ∣ ρ = ˜ λδ$_{ji}$δ$_{lk}$ + ˜ µ ( δ$_{il}$δ$_{jk}$ + δ$_{ik}$δ$_{jl}$ ) , (42) ∂µ ∂w$_{ij}$ ∣ ρ = γδ$_{ij}$ , (43) ∂µ ∂ρ$_{Δ}$ ∣ w$_{ij}$ = ∂µ ∂ρ ∣ w$_{ij}$ = 1 ρ 2 $_{0}$χ , (44) A. Dynamics of supersolid without dissipation