content_image/420670.jpg content_image/420666.jpg content_image/420663.jpg content_image/420669.jpg content_image/420665.jpg content_image/420668.jpg content_image/420667.jpg content_image/420664.jpg
In the following we demonstrate that the equations of motion derived above are equivalent to the non- dissipative supersolid hydrodynamics developed by An- dreev and Lifshitz, 1 Saslow, 4 and Liu. 5 First, the taking the gradient of Eq. (11) produces the Josephson equation 

where we have used the thermodynamic relation for ∂U$_{ss}$/∂ρ given in Eq. (3). Second, we derive the momen- tum conservation equation; the following identity simpli- fies the derivation: 

<img src='content_image/420670.jpg'>

where D/Dt ≡ ∂$_{t}$ + v$_{i}$∂$_{i}$ . Take D$_{n}$/Dt of Eq. (14), and eliminate the Lagrange multipliers by using Eqs. (7), (8), (13)-(15). The result is 

<img src='content_image/420666.jpg'>

Third, combine Eq. (18) with the thermodynamic rela- tion, Eq. (3), the continuity equation, Eq. (5), and the Josephson equation, Eq. (16). After some algebra, we obtain the momentum conservation law 

where j$_{i}$ is the mass current given in Eq. (6), and Π$_{ij}$ is the (non-dissipative) stress tensor 

<img src='content_image/420663.jpg'>

where p$_{i}$ ≡ ( ρδ$_{ij}$ − ρ$_{s}$$_{ij}$ )( v$_{n}$$_{j}$ − v$_{s}$$_{j}$ ) and ϵ satisfies a ther- modynamic relation given by Eq. (B3). Note that the Josephson equation, Eq. (16), and the momentum con- servation equation, Eq. (19), are Eqs. (9) and (12) of Andreev and Lifshitz 1 [Andreev and Lifshitz neglected nonlinear strain terms, effectively replacing R$_{ik}$ by δ$_{ik}$ in the last term of Eq. (20) above]. Moreover, the mo- mentum conservation equation is equivalent to Eq. (4.16) of Saslow 4 when v$_{s}$ is taken as a Galilean velocity, and Eq. (3.40) of Liu 5 in the case where the superthermal current vanishes. 

The Lagrangian density used to derive the hydrody- namics of supersolids, Eq. (9), can be recast into a more familiar and compact form by using the equations of mo- tion, as illustrated for an ideal fluid in Appendix A. To see this, we integrate the terms involving the Lagrange multipliers by parts (neglecting boundary terms), and 

<img src='content_image/420669.jpg'>

<img src='content_image/420665.jpg'>

use Eqs. (10) and (13) to eliminate α and β$_{i}$ . We then obtain 

<img src='content_image/420668.jpg'>

where f ≡ U$_{SS}$ − T s satisfies the thermodynamic relation 

<img src='content_image/420667.jpg'>

When cast in this form, we see that the coupling be- tween the superfluid and the normal fluid [the fourth term in Eq. (21)] is − ( ρδ$_{ij}$ − ρ$_{s}$$_{ij}$ ) v$_{n}$$_{i}$v$_{s}$$_{j}$ ; this is a “current- current” interaction, where the coupling constant is the normal fluid density. This coupling coefficient is univer- sal –it is determined by conservation laws and Galilean invariance. 

As mentioned earlier, several other authors have re- cently proposed Lagrangian descriptions for supersolids. Son 35 used symmetry-based arguments to derive an ef- fective Lagrangian for a supersolid. To connect to Son’s results, we first invert Lin’s constraint, Eq. (8), to obtain 

where R − 1 ji ≡ ∂x$_{i}$/∂R$_{j}$ and R$_{ij}$ R − 1 jk = δ$_{ik}$ . Substituting this into Eq. (21), we obtain a Lagrangian similar in form to Eq. (23) of Son’s paper (however, our energy density f depends upon ρ , ρ$_{s}$ , and T in addition to R$_{ij}$ ). A differ- ent approach was used by Josserand et al. , 36 who applied a homogenization procedure to a nonlocal version of the Gross-Pitaevskii equation to systematically derive a long- wavelength Lagrangian for a supersolid. On the whole, our Eq. (21) agrees with their Eq. (4), once we identify their ρ ( n ) with our normal fluid density ρδ$_{ij}$ − ρ$_{s}$$_{ij}$ . Fi- nally, Ye 37 proposed a phenomenological supersolid La- grangian; however, the Lagrangian in his Eq. (1) is not manifestly Galilean invariant, so that his coupling con- stants a$_{αβ}$ are arbitrary. His approach also misses certain nonlinear terms that are important when treating topo- logical defects. 

## III. QUADRATIC LAGRANGIAN DENSITY AND THE LINEARIZED HYDRODYNAMICS OF SUPERSOLIDS 

In this section we discuss the propagation of collec- tive modes in supersolids by examining the response to small fluctuations away from equilibrium. The number of collective hydrodynamic modes of a system can be inferred by enumerating the system’s conservation laws and broken symmetries. 38 Since we are more interested in the effect of density or defect fluctuations than ther- mal fluctuations, for simplicity we ignore thermal fluctu- 

<img src='content_image/420664.jpg'>

