Helium-4: The Once and Future Supersolid

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Helium-4The Once and Future Supersolid

• Michael Ma
• University of Cincinnati

Hong Kong Forum, 2006
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Supersolid Solid with Superfluid Properties
Introduction Solids - Quantum or Otherwise
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Supersolid Solid with Superfluid Properties
Introduction Solids - Quantum or
Otherwise Living in the Past This is the
Moment Days of Future Passed
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Classical Solid

• Static density

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• ??gaussian like
• harmonic approximation valid
• ltu2gt1/2 ltlt a
• Lindemanns Rule Melts when ltu2gt1/2 0.14a
• Particles are localized.

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Quantum Solid
He4 Shallow potential well light mass
large zero point motion
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Quantum Solid
conventional solid
He4
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• Lindemanns Rule does not hold
• ltu2gt 0.3 a, pressure dependent
• Short-ranged correlations important
• Deviation of density from gaussian
• strong anharmonicity
• Solid caused by steep repulsive core
• Particle exchange

Nosanow
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• Two-particle exchange not favored due to
  repulsive core
• Three and Four particle ring exchange
• Jex mK, Debye T 25 K

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• Lindemanns Rule does not hold
• ltu2gt 0.3 a, pressure dependent
• Short-ranged correlations important
• Deviation of density from gaussian
• strong anharmonicity
• Solid caused by steep repulsive core
• Particle exchange

Nosanow
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• Intriguing possibility
• but atoms mobile
• mobile atoms (bosons) can Bose condense
• exhibit superfluidity

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Bose-Einstein condensation - Non-interacting
bosons at low T, n0/N O(1)

• Bose condensation / Off-diagonal long range order
• Generalization to interacting bosons by Penrose
  and Onsager
• Further generalization by Yang as ODLRO
• Largest eigenvalue of the density matrix O(N)
• - Applicable for non-translational invariant
  system also

• Superfluidity
• zero resistance flow
• irrotational flow
• ODLRO sufficient condition for superfluidity

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PAST
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A quantum solid may Bose condense and be a
supersolid!

• Microscopic ring exhange may lead to macroscopic
  exchange
• Andreev and Lifshitz - quantum fluctuations
• may favor finite density of vacancies even at
  T0.
• Vacancies are mobile and can Bose condense.
• Chester - Jastrow wavefunctions generally have
  ODLRO,
• including ones describing solid order.
  Speculate due to vacancy
• condensation.
• Leggett - Supersolid exhibits non-classical
  rotational inertia.
• Provided expression for upper bound.

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Andreev-Lifshitz

• Vacancy motion is diffusive at high T due to
  scattering off phonons
• Wave-like at low T --gt tight binding band
• Delocalization energy may overcome local
  activation E
• Vacancies spontaneously generated
• Bose condense at low T

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Chester

• Jastrow wavefuntion generically
  has ODLRO (Reatto)
• Write and
  consider
• as partition function of a classical system at
  temp Teff
• Transition from liquid to solid with increasing
  density
• solid will have ODLRO
• postulate due to BC of vacancies.

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Irrotational Flow Meissner Effect
Lab frame
Rotating frame
?
H H - ?L
H
v p/m - A A ? x r
v p/m

• Meissner effect gt v lt ?r
• gtmoment of inertia I lt I0
• Non-classical Rotational Inertia (NCRI)
• ?I/I0 ?s/?
• ??I can be measured very accurately from resonant
  frequency

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For 30years, expt search overwhelmingly negative
Meisel. Physica
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Expt gt vacancies activated
X-ray data Simmons
data fit to c(T) exp -(f/kT)
Ev 10 K
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Present
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Kim and Chan, Science 2004
Detection of NCRI of solid He4 in torsional
oscillator
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• Effect goes away if He4 replaced by He3
• Effect significantly reduced if annulus blocked
• NCRI also observed by
• Shirahama group at Keio U
• Kubota group at U of Tokyo
• Rittner and Reppy (Cornell)

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• Effect goes away if He4 replaced by He3
• Effect significantly reduced if annulus blocked
• NCRI also observed by
• Shirahama group at Keio U
• Kubota group at U of Tokyo
• Rittner and Reppy (Cornell)
• NCRI disappears upon annealing
• Cubic cell

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Still No Evidence for Infinite Conductivity

• Day and Beamish
• No pressure driven flow
• vc lt 10-14 m/s

• Sasaki et al
• No observed flow
• without grain boundaries

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Kim and Chan
Critical velocity single quantum of circulation
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He3 dependence
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He3 dependence
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Bulk EquilBm Supersolid?

• Pro
• Phase coherence
• NCRI does not anneal to 0
• No difference between bulk and vycor
• ?s increases with Xtal quality
• specific heat anomaly

• Con
• no evidence of zero resistance
• NCRI may anneal to 0
• ?s temp dependence
• He3 impurities effects
• geometry dependence
• tiny entropy, 10-6 kB/He4

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Commensurate vs. Incommensurate Supersolid
filling
Incomm. SS
MI
Commensurate SS
x
1
KE
Incomm SS
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Commensurate SS

• Pro
• Galli and Reatto
• (Variational SW)

• Con
• Ceperley and Bernu (Ring Exchange)
• Boninsegni et al (Worm Algorithm)
• Prokofev and Svistunov (Proof)

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Incommensurate SS

• If incommensurate gt SS (Galli and Reatto)

• Anderson-Brinkman-Huse
• T7 correction to CV
• gt ?n T4
• Commensurate solid metastable

But T7 can be due to anharmonic effect
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local distortion of lattice and density vacancy
hopping given by (heavy) polaron mass attraction
between vacancies (Troyer)
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Dai Xi, FCZ, MM HuaiBin Zhuang

• With finite vacancy density, distortion can be
  static and uniform
• vacancies have light mass
• Bose condensation energy can overcome
  activation energy
• First order transition
• At T0, nv 0 in normal solid
• finite in supersolid
• Normal-Supersolid transition accompanied by
• Commensurate-incommensurate transition
• Change in local density profile

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Change in Local Density Profile
?(r)
?(r)
Supersolid
Normal Solid
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Qualitative Agreement with Penn State Expts
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Pressure Dependence of T0 Superfluid Density
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Finite T Superfluid Density
Finite T

• data suggests transiton
• smeared by disorder
• specific heat shows no
• critical behavior

• Two possibilities for pure system
• second order transition not in X-Y universality
  class
• first order transition

Transition is first order in our model
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He3 Impurities

• Expt, with increasing He3 concentration
• Tc increases
• low T ?s decreases
• NCRI not observeable beyond 0.1 He3
  concentration

• Qualitative agreement
• Impurties weaken solid ordering and favors
  defects
• gt Tc increases
• Impurities localize vacancies
• gt reduce ?s and eventually destroys Bose
  condensation
• (dirty bosons)

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Future
Is it or isnt it? Smoking gun?
If helium is not SS, is there a deeper reason
than energetics?
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Thank You!