Download Solid Helium-4: A Supersolid?

Helium-4:
The Once and Future
Supersolid
Michael Ma
University of Cincinnati
Hong Kong Forum, 2006
Supersolid = Solid with Superfluid Properties
Introduction: Solids - Quantum or Otherwise
Supersolid = Solid with Superfluid Properties
Introduction: Solids - Quantum or Otherwise
Living in the Past
This is the Moment
Days of Future Passed
Classical Solid
• Static density
(r)  0 cos(G r)
  (ri  Ri )
 gaussian like
 harmonic approximation valid
 <u2>1/2 << a

 Lindemann’s Rule: Melts when <u2>1/2 ~ 0.14a
r / a
 Particles are localized.
P(r) ~ e
Quantum Solid
He4
Shallow potential well
light mass
large zero point motion
Quantum Solid
conventional
solid
He4
 Lindemann’s Rule does not hold
<u2> ~ 0.3 a, pressure dependent
 Short-ranged correlations important
Nosanow
   f (rij ) (ri  Ri )
i j
 Deviation of density from gaussian

 strong anharmonicity
 Solid caused by steep repulsive core
 Particle exchange
 Two-particle exchange not favored due to repulsive core
 Three and Four particle ring exchange
 Jex ~ mK, Debye T ~ 25 K
 Lindemann’s Rule does not hold
<u2> ~ 0.3 a, pressure dependent
 Short-ranged correlations important
Nosanow
   f (rij ) (ri  Ri )
i j
 Deviation of density from gaussian

 strong anharmonicity
 Solid caused by steep repulsive core
 Particle exchange
Intriguing possibility:
• (r)  0 cos(G r)
• but atoms mobile
• mobile atoms (bosons) can Bose condense
• exhibit superfluidity
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
  vs  0
- irrotational flow
- ODLRO sufficient condition for superfluidity

PAST
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 T=0.
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.
Andreev-Lifshitz
 Vacancy motion is diffusive at high T due to scattering
off phonons
 Wave-like at low T --> tight binding band
 Delocalization energy may overcome local activation
E
 Vacancies spontaneously generated
 Bose condense at low T
Chester
 Jastrow wavefuntion
ODLRO (Reatto)
   f (rij )
generically has
i j


 Write f (rij )  exp(V
ij /Teff ) and consider
as partition function of a classical system at temp Teff
2

 
Transition from liquid to solid with increasing
density
 solid will have ODLRO
 postulate due to BC of vacancies.
Irrotational Flow ~ Meissner Effect
Lab frame
QuickTime™ and a
TIFF (LZW) decompress or
are needed to s ee this pic ture.
H
v = p/m
Rotating frame

QuickTime™ and a
TIFF (LZW) decompress or
are needed to s ee this pic ture.
H’= H - L
v’ = p/m - A
A=xr
 “Meissner effect” => v < r
 =>moment of inertia I < I0
 Non-classical Rotational Inertia (NCRI)
 I/I0 ~ s/
I can be measured very accurately from resonant frequency
For 30+years, expt search overwhelmingly negative
Meisel.
Physica
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Expt => vacancies activated
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
data fit to c(T) ~ exp -(f/kT)
Ev ~ 10 K
X-ray data
Simmons
Present
Kim and Chan, Science 2004
Detection of NCRI of
solid He4 in torsional oscillator
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
G
f res 
I

QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
 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)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
 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
Still No Evidence for Infinite
Conductivity
 Day and Beamish
No pressure driven flow
vc < 10-14 m/s
 Sasaki et al
No observed flow
without grain boundaries
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Kim and Chan
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Critical velocity ~ single quantum of circulation
He3 dependence
QuickTime™ and a
TIFF (LZW ) decompressor
are needed to see this picture.
He3 dependence
QuickTime™ and a
TIFF (LZW ) decompressor
are needed to see this picture.
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
Commensurate vs. Incommensurate Supersolid
filling
Incomm. SS
1
MI
x
Incomm SS
Commensurate SS
“KE”
Commensurate SS
Pro

Galli and Reatto
(Variational SW)
Con
 Ceperley and Bernu (Ring
Exchange)
 Boninsegni et al (Worm
Algorithm)
 Prokof’ev and Svistunov
(“Proof”)
Incommensurate SS
 If incommensurate => SS (Galli and Reatto)
 Anderson-Brinkman-Huse
T7 correction to CV
=> n ~ T4
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Commensurate solid metastable
But T7 can be due to anharmonic effect
local distortion of lattice and density
vacancy hopping given by (heavy) polaron mass
attraction between vacancies (Troyer)
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 T=0, nv = 0 in normal solid
finite in supersolid
 Normal-Supersolid transition accompanied by
Commensurate-incommensurate transition
Change in local density profile
Change in Local Density Profile
(r)
(r)
Normal Solid
Supersolid
Qualitative Agreement with Penn State Expts
Pressure Dependence of T=0 Superfluid
Density
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Finite T Superfluid Density
Finite T
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
• 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
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
=> Tc increases
- Impurities localize vacancies
=> reduce s and eventually destroys Bose condensation
(dirty bosons)
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
Future
Is it or isn’t it?
Smoking gun?
If helium is not SS, is there a deeper reason than energetics?
Thank You!