content_image/385744.jpg content_image/385754.jpg content_image/385753.jpg content_image/385752.jpg content_image/385745.jpg content_image/385755.jpg content_image/385746.jpg content_image/385747.jpg content_image/385749.jpg content_image/385750.jpg content_image/385748.jpg content_image/385751.jpg
Taking ψ = √$_{ne}$ i ˜ φ with the number density n , the Gross- Pitaevskii Lagrangian density becomes 

<img src='content_image/385744.jpg'>

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 

## Appendix B: Thermodynamic Relations and the Static Correlation Functions of Supersolids 

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 

<img src='content_image/385754.jpg'>

where ϵ has a thermodynamic relation 

<img src='content_image/385753.jpg'>

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 

<img src='content_image/385752.jpg'>

This thermodynamic relation agrees with Eq. (2.18) of Saslow 4 and Eq. (2.1) of Liu 5 after identifying 

<img src='content_image/385745.jpg'>

<img src='content_image/385755.jpg'>

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). 

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$_{j}$ : 

<img src='content_image/385746.jpg'>

Using Eqs. (42) - (44) for an isotropic supersolid, the free energy (in Fourier space) can be written as 

<img src='content_image/385747.jpg'>

where 

Then the static correlation functions can be easily read off from Eq. (B6): 

<img src='content_image/385749.jpg'>

## Appendix C: Calculation of the density-density correlation function 

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, 

<img src='content_image/385750.jpg'>

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$_{2}$ ): 

<img src='content_image/385748.jpg'>

<img src='content_image/385751.jpg'>

