Taking ψ = √$_{ne}$ i ˜ φ with the number density n , the Gross- Pitaevskii Lagrangian density becomes The first term contributes i ℏ N/ 2 to the action (with N the number of particles) and does not contribute to the dynamics. Identifying ρ = mn and φ = ( ℏ /m ) ˜ φ , we see that the Gross-Pitaevskii Lagrangian density, Eq. (A12), is identical to the ideal fluid Lagrangian den- sity, Eq. (A9), with In this appendix we calculate the thermodynamic re- lation for the potential energy density in Eq. (1). Given the Lagrangian density, Eq. (1), the total energy density for a supersolid is defined as the sum of the kinetic energy densities and the internal energy density Following Andreev and Lifshitz $^{1}$, this total energy den- sity can be related to the energy density ϵ measured in the frame where the super-component is at rest as where ϵ has a thermodynamic relation We can obtain the thermodynamic relation for the total energy E$_{SS}$ by differentiating Eq. (B2) and using Eq. (B3) for dϵ , with the result This thermodynamic relation agrees with Eq. (2.18) of Saslow 4 and Eq. (2.1) of Liu 5 after identifying where j, k = ( ρ, u$_{L}$ ). Then, ˜ A$_{jk}$ , ˜ B$_{jk}$ , ˜ C$_{jk}$ and ˜ D$_{jk}$ can be written in terms of a$_{jk}$ , b$_{jk}$ , d$_{jk}$ and e$_{jk}$ along with the sound velocities ( c$_{L}$ and c$_{2}$ ) and the attenuation co- efficients ( D$_{L}$ and D$_{2}$ ): Each term in the Kubo function given in Eq. (80) can be separated into the first sound part and the second sound part by performing a partial fraction expansion, Then the static correlation functions can be easily read off from Eq. (B6): Using Eqs. (42) - (44) for an isotropic supersolid, the free energy (in Fourier space) can be written as For a supersolid at rest we can expand the free energy F$_{SS}$ = E$_{SS}$ − T S up to the second order in the density fluctuations δρ and the strains w$_{ij}$ = ∂$_{i}$u$_{j}$ : ϵ Saslow, Liu = E$_{SS}$ , and µ Saslow, Liu = µ − v$_{n}$$_{i}$v$_{s}$$_{i}$ + v$_{s}$ 2 $_{i}$/ 2. Then the differentiation of Eq. (B1) and the use of Eq. (B4) give the thermodynamic relation for U$_{SS}$ , Eq. (3). L GP = i ℏ 2 ∂$_{t}$n − ℏ n∂$_{t}$ ˜ φ − ℏ 2 8 mn ( ∂$_{i}$n ) $^{2}$− ℏ 2 2 m n ( ∂$_{i}$ ˜ φ ) $^{2}$− g 2 n $^{2}$. (A12) U$_{IF}$ ( ρ ) = ℏ 2 2 m $_{2}$( ∇ √ ρ ) 2 + g 2 m $_{2}$ρ$^{2}$. (A13) F$_{SS}$ = 1 2 ∂µ ∂ρ ∣ w$_{ij}$ ( δρ ) 2 + ∂µ ∂w$_{ij}$ ∣ ρ δρ w$_{ij}$ + 1 2 ∂λ$_{ij}$ ∂w$_{lk}$ ∣ ρ w$_{ij}$w$_{lk}$. (B5) F$_{SS}$ = 1 2 ˜ µq $^{2}$u2 T + 1 2 ( δρ ( q ) u$_{L}$ ( q ) ) B ( δρ ( − q ) u$_{L}$ ( q ) ) , (B6) B = ( 1 ρ 2 $_{0}$χ − iqγ iqγ q $^{2}$λ ) . (B7) χ$_{ρρ}$ ( q ) = β 〈 δρ ( q ) δρ ( − q ) 〉 = ρ 2 $_{0}$χλ λ − ρ 2 $_{0}$γ$^{2}$χ , (B8) χ$_{u}$$_{L}$$_{ρ}$ ( q ) = β 〈 u$_{L}$ ( q ) δρ ( − q ) 〉 = iρ 2 $_{0}$χγ q ( λ − ρ 2 $_{0}$γ$^{2}$χ ) . (B9) a$_{jk}$z 3 + b$_{jk}$z 2 + d$_{jk}$q $^{2}$z + q $^{2}$e$_{jk}$ ( z 2 − c 2 $_{L}$q 2 + izD$_{L}$q $^{2}$)( z 2 − c 2 $_{2}$q 2 + izD$_{2}$q $^{2}$) = ˜ A$_{jk}$z + ˜ B$_{jk}$ z 2 − c 2 $_{L}$q 2 + izD$_{L}$q 2 + ˜ C$_{jk}$z + ˜ D$_{jk}$ z 2 − c 2 $_{2}$q 2 + izD$_{2}$q $_{2}$, (C1) ˜ A$_{jk}$ = a$_{jk}$ [ c 4 L − c 2 $_{2}$c 2 L + q $^{2}$D$_{L}$ ( D$_{L}$c 2 − D$_{2}$c 2 $_{L}$) ] ( c 2 L − c 2 $_{2}$) 2 + q $^{2}$( D$_{L}$ − D$_{2}$ )( D$_{L}$c 2 − D$_{2}$c 2 $_{L}$) + ib$_{jk}$ ( c 2 $_{2}$D$_{L}$ − D$_{2}$c 2 $_{L}$) + d$_{jk}$ ( c 2 L − c 2 $_{2}$) + ie$_{jk}$ ( D$_{L}$ − D$_{2}$ ) ( c 2 L c 2 $_{2}$) 2 + q $^{2}$( D$_{L}$ − D$_{2}$ )( D$_{L}$c 2 − D$_{2}$c 2 $_{L}$) , (C2) dE$_{SS}$ = T ds − λ$_{ik}$dR$_{ik}$ − ( v$_{n}$$_{i}$ − v$_{s}$$_{i}$ ) v$_{n}$$_{j}$dρ$_{s}$$_{ij}$ + [ µ + 1 2 (2 v$_{n}$ 2 i − 2 v$_{n}$$_{i}$v$_{s}$$_{i}$ + v$_{s}$ 2 $_{i}$) ] dρ + ρ$_{s}$$_{ij}$v$_{s}$$_{j}$dv$_{s}$$_{i}$ + ( ρδ$_{ij}$ − ρ$_{s}$$_{ij}$ ) v$_{n}$$_{i}$dv$_{n}$$_{j}$. (B4) dϵ = T ds + µdρ − λ$_{ik}$dR$_{ik}$ +( v$_{n}$$_{i}$ − v$_{s}$$_{i}$ ) d [ ( ρδ$_{ij}$ − ρ$_{s}$$_{ij}$ )( v$_{n}$$_{j}$ − v$_{s}$$_{j}$ ) ] . (B3) E$_{SS}$ = 1 2 ρv 2 s + ( ρδ$_{ij}$ − ρ$_{s}$$_{ij}$ )( v$_{n}$$_{j}$ − v$_{s}$$_{j}$ ) v$_{s}$$_{i}$ + ϵ, (B2) E$_{SS}$ = 1 2 ρ$_{s}$$_{ij}$v$_{s}$$_{i}$v$_{s}$$_{j}$ + 1 2 ( ρδ$_{ij}$ − ρ$_{s}$$_{ij}$ ) v$_{n}$$_{i}$v$_{n}$$_{j}$ + U$_{SS}$ ( ρ, ρ$_{s}$$_{ij}$, s, R$_{ij}$ ) . (B1) where Appendix C: Calculation of the density-density correlation function Appendix B: Thermodynamic Relations and the Static Correlation Functions of Supersolids 11