helium-4: the once and future supersolid michael ma university of cincinnati hong kong forum, 2006

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

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ρ(r) = ρ 0 cos(G ⋅r)

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Ψ= φ(ri∏ − Ri)

gaussian like harmonic approximation valid <u2>1/2 << a Lindemann’s Rule: Melts when <u2>1/2 ~ 0.14a Particles are localized.

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P(r) ~ e−r / a

Quantum Solid

He4Shallow potential welllight mass large zero point motion

Quantum Solid

conventionalsolid

He4

Lindemann’s Rule does not hold<u2> ~ 0.3 a, pressure dependent

Short-ranged correlations important

Deviation of density from gaussian

strong anharmonicity

Solid caused by steep repulsive core

Particle exchange

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Ψ= f (riji< j

∏ ) φ(ri∏ − Ri)Nosanow

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

Deviation of density from gaussian

strong anharmonicity

Solid caused by steep repulsive core

Particle exchange

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Ψ= f (riji< j

∏ ) φ(ri∏ − Ri)Nosanow

Intriguing possibility:

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

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ρ(r) = ρ 0 cos(G ⋅r)

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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∇×vs = 0

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 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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ψ = f (riji< j

∏ )

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ψ 2

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f (rij ) = exp(−Vij /Teff )

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H H’= H - L

v = p/m v’ = p/m - AA = x r

Irrotational Flow ~ Meissner Effect

Lab frame Rotating frame

“Meissner effect” => v < r =>moment of inertia I < I0

Non-classical Rotational Inertia (NCRI) I/I0 ~ ρs/ρ I can be measured very accurately from resonant frequency


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For 30+years, expt search overwhelmingly negative

Meisel.Physica



X-ray dataSimmons

data fit to c(T) ~ exp -(f/kT)

Expt => vacancies activated

Ev ~ 10 K

Present

Kim and Chan, Science 2004

Detection of NCRI of solid He4 in torsional oscillator





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fres ∝G

I



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)



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





Kim and Chan

Critical velocity ~ single quantum of circulation



He3 dependence

Bulk EquilBm Supersolid?

Pro Phase coherence

NCRI does not anneal to 0 No difference between bulk

and vycor ρs increases with Xtal

quality specific heat anomaly

Con no evidence of zero

resistance NCRI may anneal to 0 ρs temp dependence

He3 impurities effects geometry dependence tiny entropy, ~10-6 kB/He4

filling

“KE”1 x

MI Commensurate SS

Incomm SS

Incomm. SS

Commensurate vs. Incommensurate Supersolid

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

Commensurate solid metastable

But T7 can be due to anharmonic effect

local distortion of lattice and densityvacancy hopping given by (heavy) polaron massattraction between vacancies (Troyer)

With finite vacancy density, distortion can be static and uniform vacancies have light mass Bose condensation energy can overcome activation energy

First order transitionAt T=0, nv = 0 in normal solid finite in supersolid

Normal-Supersolid transition accompanied byCommensurate-incommensurate transitionChange in local density profile

Dai Xi, FCZ, MM; HuaiBin Zhuang

Change in Local Density Profile

ρ(r) ρ(r)

Normal Solid Supersolid

Qualitative Agreement with Penn State Expts

Pressure Dependence of T=0 Superfluid Density



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

He3 ImpuritiesExpt, with increasing He3 concentration:-Tc increases- low T ρs decreases- NCRI not observeable beyond 0.1% He3 concentration

Qualitative agreement:- Impurties weaken solid ordering and favors defects => Tc increases- Impurities localize vacancies => reduce ρs and eventually destroys Bose condensation (dirty bosons)

Future

Is it or isn’t it?Smoking gun?

If helium is not SS, is there a deeper reason than energetics?

Thank You!

helium-4: the once and future supersolid michael ma university of cincinnati hong kong forum, 2006

Documents

physica slide

present slide

low t slide

anharmonic effect slide

he3 dependence slide

vacancies troyer slide

b he4 slide

reppy cornell slide