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					          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,” e-print available at cond-mat/0508271.
 C.-D. Yoo and A. T. Dorsey, “Dynamic structure function of a
 model supersolid,” work in progress.

Phenomenology of Supersolids               UF CMT Journal Club, September 2005
 Read all about it!
• Nature, January 2004: Glimpse
of a new type of matter
• Chemical and Engineering News,
January 2004: Helium supersolid
has superfluid properties
• Physics World, February 2004:
Supersolid is seen in the lab
• New York Times, September
2004: Only in Quantum Physics:
Spinning While Standing Still
• Physics Today, November 2004:
Solid Helium-4 in the bulk doesn’t
go with the flow
• Scientific American, January
2005: Glimpse of a supersolid




 Phenomenology of Supersolids        UF CMT Journal Club, September 2005
Outline
  A brief history of supersolids.
       Theoretical background.
       Experimental searches.
       Recent torsional oscillator experiments on solid 4He
        by Kim and Chan.
  Phenomenology-what can we learn without a
   microscopic model?
       Landau theory of the normal solid to supersolid
        transition (ATD, Goldbart, and Toner).
       Hydrodynamics of a model supersolid and the
        dynamic structure function S(q,w) (C.-D. Yoo and
        ATD).




Phenomenology of Supersolids           UF CMT Journal Club, September 2005
Phase diagram of 4He
                                               a=3.53 A, c=5.93 A (at
                                              2 K, 26 atm).
                                               c/a=1.68 [ideal HCP has
 Solid 4He is soft: shear                    c/a=p(8/3)=1.632].
                                               1 atm=1.013£105 Pa
  modulus of 20 MPa (Al
  is 26 GPa, butter is 5
  MPa).
 Solid 4He is
  light:density of 0.2
  g/cm3 (like cork or
  balsa wood).                                      Lambda transition

 qDebye =25 K.


                               http://ltl.tkk.fi/research/theory/helium.html




Phenomenology of Supersolids       UF CMT Journal Club, September 2005
 A history of supersolids: theory
 What is a supersolid?
        Coexisting crystalline order (spontaneous) and off diagonal long range
         order (superfluidity). Likely candidates are Bose solids (solid 4He).
 Can this happen?
        O. Penrose and L. Onsager (1956): no.
        G. Chester (1970): well, maybe.
        A. F. Andreev and I. M. Lifshitz (1969): specific example of Bose
         condensation of defects in solid 4He.
        A. J. Leggett (1970): suggested signature would be “nonclassical
         rotational inertia” (NCRI).
        Saslow, Liu: further development of hydrodynamics.
 Recent results
        D. M. Ceperley and B. Bernu (2004): path integral Monte Carlo-no
         supersolid for a perfect lattice.
        N. Prokof’ov and B. Svistunov (2005): ground state vacancies a
         necessary condition for supersolidity.
        E. Burkovski et al., J. G. Dash and J. S. Wettlaufer: think about
         superfluid interfaces.

 Phenomenology of Supersolids                     UF CMT Journal Club, September 2005
History II: theory




Phenomenology of Supersolids   UF CMT Journal Club, September 2005
  History III: experiments
Review: M. W. Meisel, Physica B 178, 121 (1992).




 Phenomenology of Supersolids                  UF CMT Journal Club, September 2005
Recent experiments: Kim & Chan
 (New York Times, 21 September 2004)




      3 different systems:
        4He in vycor glass: Nature 427, 225 (2004).
        4He in porous gold: J. Low Temp. Phys. 138, 859 (2005).
        Bulk 4He: Science 305, 1941 (2004).
      Recent report of effect in solid H2 (LT24 and ULT).

Phenomenology of Supersolids              UF CMT Journal Club, September 2005
Details
   Resonant period of oscillation is

   Changes in the period can be due
    to either I or G.
   Pressures ranged from 26 to 66
    bars.
   Decoupling observed below 230
    mK.
   Amplitude of about 1 nm,
    maximum velocities of about 10 m
    m/s.
   Frequency: 1 kHz (period of 106
    ns). Period shifts of order 40 ns.
   Cell size: 10 mm OD, 0.63 mm
    width, 5 mm height.
   Barrier inserted: no effect.
   400 ppm of 3He quenches effect.



Phenomenology of Supersolids             UF CMT Journal Club, September 2005
Results




Phenomenology of Supersolids   UF CMT Journal Club, September 2005
Landau theory of superfluids
  Most carefully studied
   2nd order transition.
  Symmetry of order
   parameter
  Broken U(1) symmetry
   for a<0.




Phenomenology of Supersolids   UF CMT Journal Club, September 2005
Normal solid phase
 Invariant under translations and
  rotations so that only symmetric
  strains matter:

 Hooke’s law (elastic energy):

 HCP lattice, with 5 independent
  elastic constants. Isotropic in
  hexagonal planes.
 Can’t separate sound into
  transverse and longitudinal modes in
  general.


Phenomenology of Supersolids    UF CMT Journal Club, September 2005
Coupling superfluidity & elasticity
 Structured (rigid) superfluid: need anisotropic
  gradient terms:
 Compressible lattice: couple strain to the order
  parameter, obtain a strain dependent Tc.




 Analog: XY ferromagnet on compressible
  lattice. Exchange coupling will depend upon the
  local dilation of the lattice.




Phenomenology of Supersolids   UF CMT Journal Club, September 2005
Universality of the transition
 De Moura, Lubensky, Imry & Aharony (1976):
  coupling doesn’t effect the universality class of
  the transition if the specific heat exponent a<0,
  with
 For the l transition, a= -0.0127.
 So the critical behavior for the supersolid
  transition is in the 3D XY universality class.
 But coupling does matter for the elastic
  constants! anomalies.




 Could be detected in a sound speed experiment
  (“longitudinal” sound in a single crystal).
Phenomenology of Supersolids   UF CMT Journal Club, September 2005
 Specific heat

 High resolution specific heat    Specific heat near the putative
 measurements of the lambda       supersolid transition in solid 4He.
 transition in zero gravity.




J.A. Lipa et al.,                 A.C. Clark & M.H.W. Chan,
Phys. Rev. B 68, 174518 (2003).   J. Low Temp. Phys. 138, 853 (2005).

Phenomenology of Supersolids             UF CMT Journal Club, September 2005
Inhomogeneous strains
 Inhomogeneous strains result
  in a local Tc. The local
  variations in Tc will broaden the
  transition.
 Could “smear away” any
  anomalies in the specific heat.
 Strains could be due to
  geometry, dislocations, grain
  boundaries, etc.
 Need to prepare strain-free
  samples.
 N.B.: similar effects observed
  in superconductors (A15
  compounds).

Phenomenology of Supersolids    UF CMT Journal Club, September 2005
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 r, gi, e.




         No broken symmetries.
         5 conserved densities) 5 hydrodynamic modes.
           2 transverse momentum diffusion modes
                        .
           1 longitudinal thermal diffusion mode
                        .
           2 longitudinal sound modes            .



Phenomenology of Supersolids                UF CMT Journal Club, September 2005
Light scattering from a simple fluid
                                        Rayleigh peak (thermal diffusion)




                                                                 Brillouin peak (adiabatic sound)




              P. A. Fleury and J. P. Boon, Phys. Rev. 186, 244 (1969)


     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).


Phenomenology of Supersolids                                   UF CMT Journal Club, September 2005
Hydrodynamics II: superfluid
 Conserved densities r, gi, 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.


Phenomenology of Supersolids       UF CMT Journal Club, September 2005
Light scattering in a superfluid




 Winterling, Holmes & Greytak PRL 1973   Tarvin, Vidal & Greytak 1977

Phenomenology of Supersolids             UF CMT Journal Club, September 2005
Solid “hydrodynamics”
 Conserved quantities: r, gi, e .
 Broken translation symmetry: ui, i=1,2,3
 Mode counting: 5 conserved densities and 3
  broken symmetry variables) 8
  hydrodynamic modes. For an isotropic solid
  (two Lame constants l and m):
   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.

Phenomenology of Supersolids   UF CMT Journal Club, September 2005
Vacancies and interstitials

 Local density changes
  arise from either lattice
  fluctuations or vacancies
  and interstitials.

 Neglect of vacancies will
  leads to a missing mode.
 In classical solids
  vacancies diffuse slowly.
  Density of vacancies is
  small at low temperatures.
 Does 4He have zero point
  vacancies?

Phenomenology of Supersolids   UF CMT Journal Club, September 2005
Supersolid hydrodynamics
 Conserved quantities: r, gi, e
 Broken symmetries: ui, 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 “second sound” modes (2).
       Vacancy waves (?).
      1 longitudinal thermal diffusion mode.
 Structure function will exhibit new Brillouin
  peaks below Tc. Need to determine the spectral
  weight (Chi-Deuk) to see if they are observable
  in light scattering experiments from solid 4He.

Phenomenology of Supersolids          UF CMT Journal Club, September 2005
Summary
 Interpretation of Kim & Chan’s torsional
  oscillator results remain controversial. Need
  other thermodynamic measurements to
  corroborate the results.
 Constructed a 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.
 Derived a hydrodynamic model for the
  supersolid. A new collective mode emerges in
  the supersolid phase, which might be
  observable in light scattering.
Phenomenology of Supersolids         UF CMT Journal Club, September 2005
Bibliography




Phenomenology of Supersolids   UF CMT Journal Club, September 2005

				
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