1. Could a quantum solid flow like a superfluid ? S. Sasaki, R. Ishiguro, F. Caupin, H.J. Maris* and S. Balibar Laboratoire de Physique Statistique (ENS-Paris) * Brown University, Providence (RI, USA) Oxford, 25 jan 2007 A reference: Science 313, 1098 (25 aug. 2006)

2. Evangelista Torricelli (1608-1647) Galileos friend invented the first barometer liquid Hg 1 atm = 760 mmHg vacuum two communicating vessels (inside and outside the tube) hydrostatic equilibrium the weight of the liquid column is compensated by the atmospheric pressure

3. under vacuum: same level when Torricelli pumped through E: liquid-gas equilibrium in A and B same temperature same vapor pressure same levels because a liquid allows the mass flow which is necessary to achieve hydrostatic equilibrium we did the same experiment with solid 4 He in eq. with liquid 4 He E. Torricelli, Florence 1644

4. Motivation : is solid 4 He « supersolid »? E. Kim and M. Chan (Penn. State U. 2004): a torsional oscillator (~1 kHz) a change in the period of oscillation below 200 mK below 200 mK 1 % of the solid mass decouples from the oscillating walls ? Be-Cu Torsion Rod Torsion Bob containing helium Drive Detection

5. 1% superfluid density in solid 4He ? NCRI (non classical rotational inertia) ~1% at 51 bar no effect in 3 He the effect is strongly reduced with a barrier in the rotating annulus

6. early theoretical ideas Penrose and Onsager 1956: BEC is impossible in a solid (but they used non-symetrized wave fonctions) Andreev and Lifshitz 1969: delocalized defects (vacancies) could exist at T=0 ( the crystal would be « incommensurate ») BEC => superplasticity at low velocity or long times Reatto, Chester and Leggett 1969-70: NCRI is possible if atoms are delocalized (if there are free vacancies ?) Imry and Schwartz (1975): no supersolidity in a true crystal without free vacancies (a lattice gas is different)... E0E0E0E0

7. recent theoretical ideas Prokofev and Svistunov 2005: no BEC in crystals without free vacancies (commensurate crystal, vacancy-interstitial pairs); BEC in a 4 He glass (Boninsegni et al. PRL 2006) Galli and Reatto 2006: superfluidity in simulations with trial functions (« SWF ») which reproduce the properties of solid 4He Clark and Ceperley (2006) : superfluidity depends on the trial functions not found in quantum Monte Carlo simulations; the crystal is commensurate, no vacancies at T =0 Anderson Brinkman and Huse 2005: a new analysis of the T variation of the lattice spacing (old experiments by Simmons) and the specific heat C v (T) = AT 3 + BT 7  a low density of zero-point vacancies (< 10 -3 ?); T BEC ~ a few mK ;  s ? PG de Gennes (CR-Physique 2006): quantum dislocations are mobile at low T...

8. puzzling experimental results Kim and Chan: the critical velocity is 10  m/s, independent of P is 10  m/s, independent of P The critical temperature is also independent of P also independent of P the superfluid fraction increases before decreasing as a fct of P although atoms should be less mobile and vacancies should disappear as P increases

9. annealing the crystals, adding 3 He Rittner and Reppy (Cornell, 2006): annealing destroys supersolid behavior Kim and Chan (Penn State, 2006): annealing enhances supersolid behavior ! Shirahama et al. (Tokyo, 2006): no effect of annealing but the supersolid density  s = 0.1%, not 1%... Kim and Chan (Penn State, 2006): 3 He impurities increase Tc but decrease  s but ultrapure 4He shows very small  s thermodynamic quantities : very small change in the specific heat (Kim and Chan) no singularity in the melting curve (Todoshchenko et al. Helsinki 2006)

10. two previous experiments on superflow... Day, Herman and Beamish (PRL 2005): no flow in Vycor glass the lattice is probably pinned at low T, mass flow requires motion of the lattice But probably not in the new expt through capillaries (PRL 2006) crystalliquid Bonfait, Godfrin and Castaing (J. Physique 1989) growth inside a thin capacitor at T < 20 mK blocked by a facet at the entrance ?

11. ENS 2006: experimental setup Fill a test tube (1 cm  ) at 1.3 K lower T down to 50 mK melt the outside follow the level inside any change in the level inside requires a mass flow through the solid (  C = 1.1  L ) melting velocity V = 3 mm/h if critical velocity 10  m/s and superfluid density  s /  C = 10 -2 V liquid solid

12. Ishiguro’s tube

13. the ENS fridge with optical access large optical access through sets of windows through sets of windows down to 30 mK

14. filling the tube with solid 4He makes defects liquid solid the inside crystallizes only if a substantial stress is applied. For example if the outside is warmed up to 1.4K for a few seconds while the inside is at 1.3K P m ( 1.4 K) - P m ( 1.3 K) = 0.3 bar fast growth under inhomogeneous stress creates defects liquidliquid

15. cusps and grain boundaries crystal 1 crystal 2 grain boundary liquid mechanical equilibrium of surface tensions at the liquid-solid interface: each cusp signals the existence of an emerging grain boundary (GB) most cusps move away in a few hours (melting-crystallization + pinning) some GBs stay pinned

16. no flow in good quality crystals for 10 crystals with no or very few cusps the tube we could see no flow no mass leak along the glass wall if supersolidity were due to a 1% superfluid density in the bulk with a critical velocity v c = 10  m/s the interface should relax at V = [  s /(  C -  L )]v c = 1  m/s that is 3.6 mm in 1 hour Instead, we see no flow within 50  m in 4 hours, meaning 300 times less => supersolidity is not due to the superfluidity of a 1% equilibrium density of vacancies

17. mass flow in crystals with enough grain boudaries for 3 crystals with some cusps inside the tube we observed a mass flow If the cusps disappear, the mass flow stops (see crystal #1) Mass flows along grain boudaries Solids with grain boudaries may be supersolid (polycrystals) but not single crystals

18. crystal 1 relaxed 1 mm down and stopped

19. crystal 1

20. crystal 2 had many defects Many grain boundaries more in the lower part faster flow down to equilibrium at h = 0

21. crystal 2 relaxed down to eq. (h = 0) time x 250 5 s = 20 min

22. crystal 2: relaxation at 50 mK relaxation is not exponential but linear with two successive regimes, constant velocity : 6  m/s for 0 < t < 500 s 11  m/s for 500 < t < 1000 s more defects in the lower part of crystal 2 typical of superfluid flow at a critical velocity

23. crystal 1 : a single grain boundary The relaxation at V = 0.6  m/s stops when the cusp disappears (the grain boundary moves away, unpinning from the wall somewhere)

24. grain boundaries at P m are comparable to liquid films with atomic thickness If we assume the existence of a single grain boundary with thickness e, width w, the critical velocity inside is v c GB = (  D 2 /4ew  s )(  C -  L )V = 1.5 (a/e)(D/w)(  C /  s ) m/s comparable to 2 m/s measured by Telschow et al. (1974) on free adsorbed films of liquid He agreement with the prediction by Burovski, Prokof’ev and Svistunov (PRL 2005) Svistunov (PRL 2005) in a general model. simulations of GBs in solid helium 4 are in progress in their group (U. Mass. Amherst) and at Urbana (Ceperley and Clark)

25. Numerical simulation of grain boundaries Nature 21 octobre 2006

26. crystal 4 at 1.13 K a highly distorted crystal ; final relaxation at 0.9  m/s grain boundaries are superfluid up to 1.13 K at least consistent with e ~ 2a and  s ~  C at P = P m

27. have we seen the same effect as Kim and Chan ? the effect of annealing: Rittner and Reppy (2006) vs Kim and Chan (2004) large scatter of data evidence for the importance of quenched disorder not an intrinsic property of He crystals most natural defect: grain boundaries increase of  s (P) : more grain boundaries ? decrease of at large P: superfluidity disappears at high density

28. T c and v c are different at P = P m, equilibrium with the liquid: Partial wetting of grain boundaries by the liquid phase (long range van der Waals forces) (long range van der Waals forces) The thickness is microscopic (a few times a) Out of equilibrium at high P: prewetting near P m, e(P) should decrease, T c et v c as well below one layer Pe ( or T c ) ( or v c ) PmPmPmPm

29. 1% superfluid density is large In torsional oscillator experiments, crystallization at constant V from the normal liquid At variable T and P => polycrystals => polycrystals grain boundaries every 100 à 200 a, about 50nm ?? 1% vacancies would be very large too

30. crystals grown from the normal liquid at 1.9 K dendritic growth strong light scattering by a high density of defects

31. work in progress The research is now focusing on the effect of disorder, especially grain boundaires (GB): calculate the thickness e and superfluid transition temperature T c of GBs measure the T c of GBs with variable misorientation measure v c in fixed GBs, find a model for it measure GBs at P > P m : thinner ? lower T c ? lower v c ? measure the adsorption of 3He on GBs characterize the density of GBs in crystals grown at cst V : X rays, light scattering study the pinning of GBs on different walls torsional oscillator experiments in good quality crystals grown at cst T and P supersolidity under rotation reproduce the measurement of the vacaqncy density vs T change the frequency of torsional oscillator measurements...

33. 1% superfluid density is large In torsional oscillator experiments, all crystals have been grown at constant V from the normal liquid phase variable T and P => polycrystals => polycrystals grain boundaries every 100 to 200 a ~ 50 nm ? a very high density

34. facets block the growth no growth if the crystal level is raised again outside except if a large  P is applied: facets are easily pinned to wall defects facets disappear during melting ( a geometrical effect) => no pinning