Superfluidity , BEC and d imensions of liquid 4 He in n anopores

1. Superfluidity, BEC and dimensions of liquid 4He in nanopores Henry R. Glyde Department of Physics & Astronomy University of Delaware APS March Meeting 17 March, 2016

2. Collaborator\Leader: Path Integral Monte Carlo Leandra Vranjes-Markic: University of Split Fulbright Scholar, University of Delaware, USA 2013-4

3. B. HELIUM IN POROUS MEDIA AEROGEL* VYCOR (Corning) 30% porous • Å pore Diameter GELSIL (Geltech, 4F) 50% porous 25 Å pores 44 Å pores 34 Å pores NANOPORES: MCM-4130% porous 47 Å pores FSM-16 28 Å pores and smaller down to 15 Å Focus on small diameter (d) nanopores, 15 < d < 32 Å

4. Experiments: Low Dimensional He in Nanopores

5. Theory: Low Dimensional He in Nanopores

6. Goals Path integral Monte Carlo (PIMC) calculations: • Can we predict the superfluid fraction, ρS\ρ,and one body density matrix (OBDM) (BEC) of helium in nanopores? e.g. low suppression of TC to TC ~ 1 K below bulk Tλ= 2.17 K? TC < T BEC ~ Tλ • Is a static, zero frequency, PIMC theory sufficient in nanotubes? • What is the effective dimensionality of the helium in nanopores, of the superfluid fraction ρS\ρ and the OBDM? …… 1D, 2D or 3D?

7. Phase Diagram Bulk helium

8. SUPERFLUID: Bulk Liquid SF Fraction s(T) Critical Temperature Tλ = 2.17 K From Boninsegni et al. PRB (2006)

9. Bose-Einstein Condensation: Bulk Liquid Expt: Glyde et al. PRB (2000)

10. Helium in Porous Media

11. Phase Diagram in FSM-16: 28 A pore diameter - Taniguchi et al, Phys. RevB82, 104509 (2010)

12. Phase Diagram in MCM-41: 47 Å pore diameter Diallo et al. PRL (2014)

13. BEC: Liquid 4He in MCM-41 Diallo et al. PRL (2014)

14. Focus on Helium in MCM-41Consists of pores 47 A diameter, a Powder

15. Model of Liquid 4He in a Nanopore Liquid pore diameter dL = 2R Nanopore diameter d = dL + 10 Å

16. Hamiltonian and Potentials Pore Potential seen by a He atom : v (ri– rj) = He – He pair potential, R. A. Aziz et al. (1979)

17. Confining potential arising from inert layers and nanopore walls: R = dL/2 (d = dL +10 Å)

18. Radial density profile of liquid 4He as a function of liquid pore radius R

19. Superfluid fraction vs T as a function of liquid pore radius R R = 6 to 11 Å Pore Diameter: d = 2R + 10 Å

20. Finite size scaling of ρS/ρ vs T and pore length L for R = 3 Å

21. Finite size scaling of ρS/ρ vs T and pore length L R = 3 Å d = 2R + 10 Å

22. R = 3 ÅOne Body Density Matrix n (z) (z along pore)

23. Finite size scaling of ρS/ρ vs T and pore length L for R = 4 Å

24. Finite size scaling of ρS/ρ vs T and pore length L R = 4 Å d = 2R + 10 Å

25. Superfluid fraction vs T as a function of liquid pore radius R R = 6 to 11 Å

26. Finite size scaling of ρS/ρ vs T and pore length L as expected for a 2D fluid

27. Finite size scaling of ρS/ρ vs T and pore length L for R = 11 Å: 2D and 3D scaling compared

28. Radial density profile of liquid 4He as a function of liquid pore radius R

29. Superfluid fraction vs T (K) Increasing density top to bottom, R = 7.3 A ρ = 0. 016 Å -3 ρ = 0.0214 Ǻ -3 ρ= 0.0241 Ǻ-3

30. Bose-Einstein Condensation: R = 11 ÅOne Body Density Matrix n (z) (z along pore)

31. R = 6 ÅOne Body Density Matrix n (z) (z along pore)

32. One Body Density Matrix (OBDM): n(z) Scaling of n(z) at Kosterlitz –Thouless TC: Algebraic decay of n(z): n(z) ~ z – η(T) At KT transition TC, η(TC) = ¼ Find TC from n(z) by finding T at which n(z) ~ z – 1/4 For pore diameters 22 < d < 32 Å, both ρS/ρ and OBDM predict 2D behaviour and a cross over to 3D at d ~ 32 • At dL ~ 22 Å (d~ 32 Å) there is a cross over from 2D to 3D like scaling. Anticipate 3D like behaviour at larger d.

33. Superfluid fraction ρS\ρ vs T (K) with Point Disorder, R = 7.3 A

34. Conclusions: Experiment • PIMC predicts ρS\ρ = 0 for nanopores d ≤ 16 Å, as observed. Liquid is 1D in nanopore . ρS\ρ and OBDM scale as expected for a 1D Luttinger Liquid. • For nanopores 18 < d < 32 Å where many measurements made, PIMC predicts standard, static superflow. TC (1.4 K) is close to observed value (e.g. TC= 0.9 K). Liquid fills pores in 2D cylindrical layers. ρS\ρ and OBDM scale as a 2D liquid. • Cross-over to 3D scaling at larger nanopore diameter, d > 32 Å. OBDM and condensate fraction consistent with measurements. • OBDM predicts TC (using KT theory) that is consistent with TC predicted from ρS\ρ. • TBEC obtained from OBDM > TC for superflow as observed.

35. Conclusions: Dimensions • Predicted cross-over from 1D (no superflow) to 2D (superflow) at d = 16 Å agrees with experiment. • PIMC predicts standard, static superflow in nanopore range 18 < d < 32 Å as observed. Frequency theories not needed. Liquid is 2D like. • A low TC for ρS\ρ in nanopores can be predicted especially if disorder added as observed. TC decreases with density as predicted. . 3D behavior predicted for d > 35 Å, n0 similar to bulk predicted, as observed in d = 47 Å MCM-41.

39. Superfluid fraction ρS\ρ vs T (K) with “Neck” Disorder

40. Superfluid Density in Gelsil (Geltech) – 25 A diameter -Yamamoto et al.

41. Normal Liquid He in bulk and in MCM-41 Pressure dependence Bossy et al. (unpublished )

43. N. N. Bogoliubov

44. Landau

45. BEC, P-R modes, Superfluidity Bose Einstein Condensation (neutrons) 1968- Collective Phonon-Roton modes (neutrons) 1958- Superfluidity (torsional oscillators) ` 1938- He in porous media integral part of historical superflow measurements.

46. Collective (Phonon-roton) Modes, Structure Collaborators: (ILL) JACQUES BOSSY Institut Néel, CNRS- UJF, Grenoble, France Helmut Schober Institut Laue-Langevin Grenoble, France Jacques Ollivier Institut Laue-Langevin Grenoble, France Norbert Mulders University of Delaware

47. PHONON-ROTON MODE: Dispersion Curve ← Δ Donnelly et al., J. Low Temp. Phys. (1981)  Glyde et al., Euro Phys. Lett. (1998)

48. SUPERFLUIDITY 1908 – 4He first liquified in Leiden by Kamerlingh Onnes 1925 – Specific heat anomaly observed at Tλ= 2.17 K by Keesom. Denoted the λ transiton to He II. 1938 – Superfluidity observed in He II by Kaptiza and by Allen and Misener. 1938 – Superfluidity interpreted as manifestation of BEC by London vS = grad φ (r)

49. BOSE-EINSTEIN CONDENSATION 1924 Bose gas : Φk = exp[ik.r] , Nk k = 0 state is condensate state for uniform fluids. Condensate fraction, n0 = N0/N = 100 % T = 0 K Condensate wave function: ψ(r) = √n0 e iφ(r)

50. Kamerlingh Onnes