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# Hydrodynamic theory of supersolids: Variational principle and effective Lagrangian 

C.-D. Yoo 1 and Alan T. Dorsey 1 

$^{1}$Department of Physics, University of Florida, P.O. Box 118440, Gainesville, FL 32611-8440 

(Dated: November 4, 2018) 

We develop an effective low-energy, long-wavelength theory of a bulk supersolid—a putative phase of matter with simultaneous crystallinity and Bose condensation. Using conservation laws and gen- eral symmetry arguments we derive an effective action that correctly describes the coupling between the Bose condensation and the elasticity of the solid. We use our effective action to calculate the correlation and response functions for the supersolid, and we show that the onset of supersolidity produces peaks in the response function, corresponding to propagating second sound modes in the solid. Throughout our work we make connections to existing work on effective theories of superflu- ids and normal solids, and we underscore the importance of conservation laws and symmetries in determining the number and character of the collective modes. 

In 1969 Andreev and Lifshitz proposed a novel phase of matter in quantum Bose crystals, wherein a Bose conden- sate of point defects would coexist with the crystallinity of the solid. 1 This is perhaps the most conceptually clear picture of what we now call a “supersolid,” although sug- gestions of the coexistence (or non-coexistence) of Bose condensation and crystallinity can be traced to the ear- lier work of Penrose and Onsager 2 and Chester. 3 An- dreev and Lifshitz provided an elegant (albeit incom- plete) formulation of the hydrodynamics of supersolids, and predicted propagating modes analogous to second (or fourth) sound in liquid $^{4}$He. Their hydrodynamic formu- lation was further extended by Saslow 4 and by Liu. 5 Ex- perimental searches for signatures of the supersolid phase proved fruitless 6 until recently, when Kim and Chan ob- served rotational inertia anomalies in solid $^{4}$He that they interpreted as evidence for supersolidity. 7–9 Their work fueled extensive searches for further evidence of this elu- sive phase of matter, 10–17 and there are now a number of extensive reviews of the experimental and theoretical progress in this area—see Refs. 18, 19, 20, and 21. 

The present work is a detailed—and we believe novel— study of the hydrodynamics of bulk supersolids, of the type originally proposed by Andreev and Lifshitz. Our work uses conservation laws and general symmetry prin- ciples to derive an effective action for a supersolid. We rely extensively on a variational principle used to obtain the dynamic equations in various continuum systems: normal fluids, 22–24 superfluids, 24–29 normal solids, 22,30 liquid crystals, 31 trapped superfluid gases, 32 and rela- tivistic fluids. 33 This effective action is a powerful tool for calculating and elucidating the collective modes of the supersolid phase, and one of our important new results is a calculation of the correlation and response functions in the supersolid phase. Moving beyond linearized hydrody- namics, the effective action can also be use to study the dynamics and interaction of topological defects—vortices and dislocations—in the supersolid, the topics of subse- quent publications. 34 We should also state what this work 

PACS numbers: 67.80.bd, 67.25.dg 

## I. INTRODUCTION 

is not —it is not an explanation of the recent experiments on possible supersolidity in $^{4}$He, as the prevailing wisdom suggests that structural disorder plays a key role in most of the experiments, and our simplified model assumes an ordered solid. 

This work is organized as follows. In Sec. II we de- rive the supersolid hydrodynamics and the effective La- grangian density using the variational principle. We show that the equation of motion are equivalent (up to a term nonlinear in the elastic strain) to the hydrodynamic equa- tions of motion derived by Andreev and Lifshitz $^{1}$. We also discuss the connection to the work of Saslow, 4 Liu, 5 and Son. 35 In Sec. III we use a quadratic version of the Lagrangian to investigate the linearized hydrodynamics of a supersolid. Finally, in Sec. IV the collective modes and the density-density correlation function of a model supersolid are calculated in detail. The Appendices pro- vide additional detail, as an aid to the reader. 

## II. VARIATIONAL PRINCIPLE AND AN EFFECTIVE LAGRANGIAN OF SUPERSOLIDS 

We start with a Lagrangian density for a supersolid in the Eulerian description (in which all quantities are depicted at fixed position x and time t ), 

where ρ$_{s}$$_{ij}$ is the superfluid density tensor, ρ is the total density, v$_{s}$ is the velocity of the super-components, v$_{n}$ is the velocity of the normal-components, s is the entropy density, and 

is the deformation tensor, with R the coordinate affixed to material elements ( ∂$_{i}$ ≡ ∂/∂x$_{i}$ and ∂$_{t}$ ≡ ∂/∂$_{t}$ in what follows). The first two terms in the Lagrangian density are the kinetic energy densities of the super-component 

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