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

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

Quantum Solid He4 Shallow potential well light mass large zero point motion

Quantum Solid conventional solid He4

 Lindemann’s Rule does not hold ~ 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

 Two-particle exchange not favored due to repulsive core  Three and Four particle ring exchange  J ex ~ mK, Debye T ~ 25 K

 Lindemann’s Rule does not hold ~ 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

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

Bose-Einstein condensation - Non-interacting bosons at low T, n 0 /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

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

H H’= H -  L v = p/m v’ = p/m - A A =  x r  Irrotational Flow ~ Meissner Effect Lab frameRotating frame  “Meissner effect” => v <  r  =>moment of inertia I < I 0  Non-classical Rotational Inertia (NCRI)   I/I 0 ~  s /    I can be measured very accurately from resonant frequency

For 30+years, expt search overwhelmingly negative Meisel. Physica

X-ray data Simmons data fit to c(T) ~ exp -(f/kT) Expt => vacancies activated E v ~ 10 K

Present

Kim and Chan, Science 2004 Detection of NCRI of solid He4 in torsional oscillator

 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 v c < 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   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 k B /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 T 7 correction to C V =>  n ~ T 4 Commensurate solid metastable But T 7 can be due to anharmonic effect

local distortion of lattice and density vacancy hopping given by (heavy) polaron mass attraction 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 transition At T=0, n v = 0 in normal solid finite in supersolid  Normal-Supersolid transition accompanied by Commensurate-incommensurate transition Change in local density profile Dai Xi, FCZ, MM; HuaiBin Zhuang

Change in Local Density Profile  (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 Impurities Expt, with increasing He3 concentration: -T c increases - low T  s decreases - NCRI not observeable beyond 0.1% He3 concentration Qualitative agreement: - Impurties weaken solid ordering and favors defects => T c increases - Impurities localize vacancies => reduce  s 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!