content_image/393217.jpg content_image/393213.jpg content_image/393215.jpg content_image/393220.jpg content_image/393221.jpg content_image/393214.jpg content_image/393216.jpg content_image/393219.jpg content_image/393218.jpg
where we have used Eq. (29) and the identity 

Equation (32) shows that a density fluctuation in an isothermal supersolid is caused either by a lattice defor- mation or by a defect density fluctuation δρ$_{Δ}$ , just as in a normal solid. 43–45 Following Zippelius et al. (ZHN), 44 we can identify ( ∂ρ/∂w$_{ij}$ )$_{ρ}$$_{Δ}$ = − ρ$_{0}$δ$_{ij}$ and ( ∂ρ/∂ρ$_{Δ}$ )$_{w}$$_{ij}$ = 1. We finally obtain 

which illustrates the roles of the lattice deformation w$_{ii}$ = ∇ · u and the defect density fluctuation in determining the total density fluctuation. We note in passing that in a higher order expansion of the Lagrangian density the terms proportional to the superfluid density fluctuation must also be retained in Eq. (34). This would resemble the “three-fluid” scenario proposed by Saslow 46 in which the lattice density and velocity are introduced for the third fluid component. 

We can now use Eq. (34) to rewrite the quadratic La- grangian density in terms of the defect density, with the result 

<img src='content_image/393217.jpg'>

where we introduced θ = φ + µ$_{0}$t . Next, we derive the linearized equations of motion from the Lagrangian den- sity. First, note that the variation with respect to δρ$_{Δ}$ reproduces Eq. (31) because ∂$_{t}$θ is − δµ . Second, taking the variation with respect to θ produces the linearized equation of continuity, expressed in terms of δρ$_{Δ}$ : 

where the defect current density is given by 

We see that the defect current arises from counterflow between the superfluid velocity ∇ θ and the normal fluid velocity ∂$_{t}$ u , and vanishes when ρ$_{s}$$_{0}$$_{ij}$ = 0, in the normal state. In other words, ∂$_{t}$δρ$_{Δ}$ = 0 in the normal state, in agreement with the non-dissipative description of nor- mal solids 44 in which defect currents are only produced through diffusion (i.e., the defect current is dissipative in 

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the normal solid). The last equation of motion derived from the variation of u$_{i}$ is 

<img src='content_image/393214.jpg'>

When the time derivative of Eq. (36) is combined with Eq. (31), we obtain 

<img src='content_image/393216.jpg'>

Our linearized equations of motion, Eqs. (38) and (39), are equivalent to Eq. (19) of Andreev and Lifshitz. 1 

In the particular case in which the lattice sites are fixed (so that u = 0) we recover from Eq. (39) the fourth sound modes obtained by Andreev and Lifshitz, 1 which have the dispersion relation 

where we have used ( ∂/∂ρ$_{Δ}$ )$_{w}$$_{ij}$ = ( ∂/∂ρ )$_{w}$$_{ij}$. On the other hand, when there are no defect fluctuations ( δρ$_{Δ}$ = 0), Eqs. (38) and (39) are combined into 

A mode analysis of this equation would produce six sound modes of an anisotropic normal solid. We see that with- out defects there are no additional sound modes, as ex- pected. 

## IV. DENSITY-DENSITY CORRELATION FUNCTION OF A MODEL SUPERSOLID 

In this section we will calculate the density-density correlation function of a model supersolid, a measurable quantity in a light scattering experiment. However, be- fore delving into the calculations for a supersolid let’s first review what’s revealed by scattering light from a struc- tureless, normal fluid [for example, see Ref. 47]. The mode counting for the fluid is simple–there are five col- lective modes, due to conservation of mass, energy, and three components of momentum (in three dimensions). The five collective modes are a pair of transverse mo- mentum diffusion modes and three longitudinal modes: 

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