Helium-4: The Once and Future Supersolid

1. Helium-4:The Once and Future Supersolid Michael Ma University of Cincinnati Hong Kong Forum, 2006

2. Supersolid = Solid with Superfluid Properties Introduction: Solids - Quantum or Otherwise

3. Supersolid = Solid with Superfluid Properties Introduction: Solids - Quantum or Otherwise Living in the Past This is the Moment Days of Future Passed

4. Classical Solid • Static density

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

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

8. Quantum Solid conventional solid He4

9. 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 Nosanow

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

12. 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 Nosanow

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

14. 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

15. PAST

16. 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.

17. 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

18. 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.

19. Irrotational Flow ~ Meissner Effect Lab frame Rotating frame  H’= H - L H v’ = p/m - A A =  x r v = p/m • “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

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

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

22. Present

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

25. 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)

26. 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

27. 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

29. Kim and Chan Critical velocity ~ single quantum of circulation

30. He3 dependence

31. He3 dependence

32. 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 Bulk EquilBm Supersolid?

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

34. Pro Galli and Reatto (Variational SW) Con Ceperley and Bernu (Ring Exchange) Boninsegni et al (Worm Algorithm) Prokof’ev and Svistunov (“Proof”) Commensurate SS

35. 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

38. local distortion of lattice and density vacancy hopping given by (heavy) polaron mass attraction between vacancies (Troyer)

39. 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

40. Change in Local Density Profile (r) (r) Supersolid Normal Solid

41. Qualitative Agreement with Penn State Expts

42. Pressure Dependence of T=0 Superfluid Density

43. 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

44. 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)

46. Future Is it or isn’t it? Smoking gun? If helium is not SS, is there a deeper reason than energetics?

48. Thank You!