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 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 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: 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. 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 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 Our linearized equations of motion, Eqs. (38) and (39), are equivalent to Eq. (19) of Andreev and Lifshitz. 1 When the time derivative of Eq. (36) is combined with Eq. (31), we obtain the normal solid). The last equation of motion derived from the variation of u$_{i}$ is 5 ∂x ∂y ∣ 0 = ∂x ∂y ∣ z + ∂x ∂z ∣ y ∂z ∂y ∣ 0 . (33) ρ$_{n}$$_{0}$$_{ij}$ ∂ 2 t u$_{j}$ − ( ∂µ ∂w$_{ji}$ ∣ ρ − ρ$_{n}$$_{0}$$_{ij}$ ∂µ ∂ρ$_{Δ}$ ∣ w$_{ij}$ ) ∂$_{j}$ δρ$_{Δ}$ − ( ∂λ$_{ji}$ ∂w$_{lk}$ ∣ ρ − ρ$_{n}$$_{0}$$_{ij}$ ∂µ ∂w$_{lk}$ ∣ ρ Δ − ρ$_{0}$ ∂µ ∂w$_{ij}$ ∣ ρ δ$_{lk}$ ) ∂$_{j}$ w$_{lk}$ = 0 . (38) δρ = − ρ$_{0}$w$_{ii}$ + δρ$_{Δ}$, (34) ∂ 2 t δρ$_{Δ}$ − ρ$_{s}$$_{0}$$_{ij}$ ∂µ ∂ρ$_{Δ}$ ∣ w$_{ij}$ ∂$_{i}$∂$_{j}$ δρ$_{Δ}$ − ρ$_{s}$$_{0}$$_{ij}$ ∂$_{i}$∂ 2 t u$_{j}$ − ρ$_{s}$$_{0}$$_{ij}$ ∂µ ∂w$_{lk}$ ∣ ρ Δ ∂$_{i}$∂$_{j}$ w$_{lk}$ = 0 . (39) L quad SS = ρ$_{0}$w$_{ii}$∂$_{t}$θ − ρ$_{n}$$_{0}$$_{ij}$ ∂$_{t}$u$_{i}$∂$_{j}$ θ − 1 2 ρ 2 0 ∂µ ∂ρ ∣ w$_{ij}$ w 2 ii − δρ$_{Δ}$∂$_{t}$θ − 1 2 ρ$_{s}$$_{0}$$_{ij}$ ∂$_{i}$θ∂$_{j}$ θ − ∂µ ∂w$_{ij}$ ∣ ρ δρ$_{Δ}$w$_{ij}$ + ρ$_{0}$ ∂µ ∂ρ$_{Δ}$ ∣ w$_{ij}$ w$_{ii}$δρ$_{Δ}$ − 1 2 ∂µ ∂ρ$_{Δ}$ ∣ w$_{ij}$ δρ$_{Δ}$ 2 − 1 2 ∂λ$_{ji}$ ∂w$_{lk}$ ∣ ρ w$_{ij}$ w$_{lk}$ + ρ$_{0}$ ∂µ ∂w$_{ij}$ ∣ ρ w$_{ij}$ w$_{kk}$ + 1 2 ρ$_{n}$$_{0}$$_{ij}$ ∂$_{t}$u$_{i}$∂$_{t}$u$_{j}$ , (35) ω 2 = ρ$_{s}$$_{0}$$_{ij}$ ∂µ ∂ρ ∣ w$_{ij}$ q$_{i}$q$_{j}$ , (40) ρ$_{0}$∂ 2 t u$_{i}$ = ∂λ$_{ji}$ ∂w$_{lk}$ ∣ ρ ∂$_{j}$ w$_{lk}$ − ρ$_{0}$ ∂µ ∂w$_{lk}$ ∣ ρ Δ ∂$_{i}$w$_{lk}$ − ρ$_{0}$ ∂µ ∂w$_{ij}$ ∣ ρ ∂$_{j}$ w$_{kk}$ . (41) ∂$_{t}$δρ$_{Δ}$ + ∂$_{i}$j Δ i = 0 , (36) j Δ i = ρ$_{s}$$_{0}$$_{ij}$ ( ∂$_{j}$ θ − ∂$_{t}$u$_{j}$ ) . (37) IV. DENSITY-DENSITY CORRELATION FUNCTION OF A MODEL SUPERSOLID