1. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Phenomenology of Supersolids Alan Dorsey & Chi-Deuk Yoo Department of Physics University of Florida Paul Goldbart Department of Physics University of Illinois at Urbana-Champaign John Toner Department of Physics University of Oregon A. T. Dorsey, P. M. Goldbart, and J. Toner, “Squeezing superfluid from a stone: Coupling superfluidity and elasticity in a supersolid,” Phys. Rev. Lett. 96, 055301 (2006). C.-D. Yoo and A. T. Dorsey, work in progress.

2. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Outline  Phenomenology-what can we learn without a microscopic model? Landau theory of the normal solid to supersolid transition: coupling superfluidity to elasticity. Assumptions:  Normal to supersolid transition is continuous (2 nd order).  Supersolid order parameter is a complex scalar (just like the superfluid phase).  What is the effect of the elasticity on the transition? Hydrodynamics of a supersolid:  Employ conservation laws and symmetries to deduce the long-lived hydrodynamic modes.  Mode counting: additional collective mode in the supersolid phase.  Use linearized hydrodynamics to calculate S(q,).

3. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Landau theory for a superfluid  Symmetry of order parameter  Broken U(1) symmetry for T<T c.  Coarse-grained free energy:  Average over configurations:  Fluctuations shift T 0 ! T C, produce singularities as a function of the reduced temperature t=|(T-T c )/T c |.  Universal exponents and amplitude ratios.

4. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Specific heat near the transition Lipa et al., Phys. Rev. B (2003).  The singular part of the specific heat is a correlation function:  For the transition, = -0.0127. Barmatz & Rudnick, Phys. Rev. (1968)

5. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Sound speed  What if we allow for local density fluctuations  in the fluid, with a bare bulk modulus B 0 ? The coarse-grained free energy is now  The “renormalized” bulk modulus B is then  The sound speed acquires the specific heat singularity (Pippard-Buckingham-Fairbank):

6. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Coupling superfluidity & elasticity  Structured (rigid) superfluid: need anisotropic gradient terms:  Elastic energy: Hooke’s law. 5 independent elastic constants for an hcp lattice:  Compressible lattice: couple strain to the order parameter, obtain a strain dependent T c.  Minimal model for the normal to supersolid transition:

7. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Related systems  Analog: XY ferromagnet on a compressible lattice. Exchange coupling will depend upon the local dilation of the lattice. Studied extensively: Fisher (1968), Larkin & Pikin (1969), De Moura, Lubensky, Imry & Aharony (1976), Bergman & Halperin (1976), … Under some conditions the elastic coupling can produce a first order transition.  Other systems: Charge density waves: Aronowitz, Goldbart, & Mozurkewich (1990). Spin density waves: M. Walker (1990s). A15 superconductors: L.R. Testardi (1970s).

8. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Universality of the transition  De Moura, Lubensky, Imry & Aharony (1976): elastic coupling doesn’t effect the universality class of the transition if the specific heat exponent of the rigid system is negative,which it is for the 3D XY model. The critical behavior for the supersolid transition is in the 3D XY universality class.  But coupling does matter for the elastic constants:  Could be detected in a sound speed experiment as a dip in the sound speed.  Anomaly appears in the “longitudinal” sound in a single crystal. Should appear in both longitudinal and transverse sound in polycrystalline samples.

9. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Specific heat J.A. Lipa et al., Phys. Rev. B 68, 174518 (2003). High resolution specific heat measurements of the lambda transition in zero gravity. Specific heat near the putative supersolid transition in solid 4 He. Lin, Clark, and Chan, PSU preprint (2007)

10. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Inhomogeneous strains  Inhomogeneous strains result in a local T c. The local variations in T c will broaden the transition.  Could “smear away” any anomalies in the specific heat.  Strains could be due to geometry, dislocations, grain boundaries, etc.  Question: could defects induce supersolidity?

11. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Supersolidity from dislocations?  Dislocations can promote superfluidity (John Toner). Recall model:  Quenched dislocations produce large, long-ranged strains. For an edge dislocation (isotropic elasticity)  For a screw dislocation,  Even if t 0 >0 (QMC), can have t<0 near the dislocation! Edge dislocation Screw dislocation

12. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Condensation on edge dislocation  Euler-Lagrange equation:  To find T c solve linearized problem; looks like Schrodinger equation:  For the edge dislocation,  Need to find the spectrum of a d=2 dipole potential.  Expand the free energy

13. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Details: Quantum dipole problem  Instabililty first occurs for the ground state:  Variational estimate:  Edge dislocations always increase the transition temperature!  What about screw dislocations? Either nonlinear strains coupling to || 2 or linear strain coupling to gradients of  [E. M. Chudnovsky, PRB 64, 212503 (2001)].  J. Toner: properties of a network of such superfluid dislocations (unpublished).

14. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Interesting references V.M. Nabutovskii and V.Ya. Shapiro, Sov. Phys. JETP 48, 480 (1979).

15. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Hydrodynamics I: simple fluid  Conservation laws and broken symmetries lead to long-lived “hydrodynamic” modes (lifetime diverges at long wavelengths).  Simple fluid: Conserved quantities are , g i, e. No broken symmetries. 5 conserved densities ) 5 hydrodynamic modes.  2 transverse momentum diffusion modes.  1 longitudinal thermal diffusion mode .  2 longitudinal sound modes.

16. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Light scattering from a simple fluid  Intensity of scattered light:  Longitudinal modes couple to density fluctuations. Sound produces the Brillouin peaks. Thermal diffusion produces the Rayleigh peak (coupling of thermal fluctuations to the density through thermal expansion). Rayleigh peak (thermal diffusion) P. A. Fleury and J. P. Boon, Phys. Rev. 186, 244 (1969) Brillouin peak (adiabatic sound)

17. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Hydrodynamics II: superfluid  Conserved densities , g i, e.  Broken U(1) gauge symmetry  Another equation of motion:  6 hydrodynamic modes: 2 transverse momentum diffusion modes. 2 longitudinal (first) sound modes. 2 longitudinal second sound modes.  Central Rayleigh peak splits into two new Brillouin peaks.

18. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Light scattering in a superfluid Winterling, Holmes & Greytak PRL 1973Tarvin, Vidal & Greytak 1977

19. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Solid “hydrodynamics”  Conserved quantities: , g i, e.  Broken translation symmetry: u i, i=1,2,3  Mode counting: 5 conserved densities and 3 broken symmetry variables ) 8 hydrodynamic modes. For an isotropic solid (two Lame constants and ): 2 pairs of transverse sound modes (4), 1 pair of longitudinal sound modes (2), 1 thermal diffusion mode (1).  What’s missing? Martin, Parodi, and Pershan (1972): diffusion of vacancies and interstitials.

20. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Vacancies and interstitials  Local density changes arise from either lattice fluctuations (with a displacement field u) or vacancies and interstitials.  In classical solids the density of vacancies is small at low temperatures.  Does 4 He have zero point vacancies?

21. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Supersolid hydrodynamics  Conserved quantities: , g i, e  Broken symmetries: u i, gauge symmetry.  Mode counting: 5 conserved densities and 4 broken symmetry variables ) 9 hydrodynamic modes. 2 pairs of transverse sound modes (4). 1 pair of longitudinal sound modes (2). 1 pair of longitudinal “fourth sound” modes (2). 1 longitudinal thermal diffusion mode.  Use Andreev & Lifshitz hydrodynamics to derive the structure function (isothermal, isotropic solid). New Brillouin peaks below T c.

22. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Structure function for supersolid Second sound First sound

23. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Supersolid Lagrangian  Lagrangian Reversible dynamics for the phase and lattice displacement fields Lagrangian coordinates R i, Eulerian coordinates x i, deformation tensor Respect symmetries (conservation laws): rotational symmetry, Galilean invariance, gauge symmetry.  Reproduces Andreev-Lifshitz hydrodynamics. Agrees with recent work by Son (2005) [disagrees with Josserand (2007), Ye (2007)].  Good starting point for studying vortex dynamics in supersolids (Yoo and Dorsey, unpublished). Question: do vortices in supersolids behave differently than in superfluids?

24. Phenomenology of Supersolids Supersolid State of Matter, 25 July 2007 Summary  Landau theory of the normal solid to supersolid transition. Coupling to the elastic degrees of freedom doesn’t change the critical behavior. Predicted anomalies in the elastic constants that should be observable in sound speed measurements. Noted the importance of inhomogeneous strains in rounding the transition.  Structure function of a model supersolid using linearized hydrodynamics. A new collective mode emerges in the supersolid phase, which might be observable in light scattering.  In progress: Lagrangian formulation of the hydrodynamics. Vortex and dislocation dynamics in a supersolid.