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

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 % 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. - Taniguchi et al, Phys. Rev B82, 104509 (2010)
Phase Diagram in FSM-16: 28 A pore diameter
- Taniguchi et al, Phys. Rev B82, (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-41 Consists 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
d = 2R + 10 Å

22. R = 3 Å One 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
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
Superfluid fraction vs T (K) Increasing density top to bottom, R = 7.3 A
ρ = Å -3
ρ = Ǻ -3
ρ = Ǻ-3

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

31. R = 6 Å One 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
Jacques Ollivier Institut Laue-Langevin
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

51. London

52. Landau Theory of Superfluidity
Superfluidity follows from the nature of the excitations:
- that there are phonon-roton excitations only and no other low energy excitations to which superfluid can decay.
- have a critical velocity and an energy gap (roton gap ).

53. 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
MCM % porous
47 Å pores
FSM-16
28 Å pores and smaller down to 15 Å
Focus on small diameter (d) media 15 < d < 32 Å

54. Phonon- Roton Mode in MCM-41: SVP

55. Phonon- Roton Mode in MCM-41: SVP

56. Roton in Bulk Liquid 4He
Talbot et al., PRB, 38, (1988)

57. Maxon in bulk liquid 4He
Talbot et al., PRB, 38, (1988)

58. Liquid 4He in Nanopores Conclusions:
Sharply defined phonon-roton mode at higher wave vector, Q > 1 Å -1, exist where there is BEC, exist because there is BEC.
Brings unity to Landau (phonon-roton) and BEC phase coherence based theories of superfluidity since modes and BEC phase coherence are common properties of a Bose condensed liquid.

59. Liquid 4He in Nanopores Conclusions:
Below Tc in the superfluid phase, have extended BEC.
Superfluid – non superfluid liquid transition is associated with an extended to localized BEC cross over.
Above Tc have localized BEC (separated islands of BEC). Tc < T < TBEC have BEC, no superflow.
Tc and TBEC are separated: Tc < TBEC

60. Finite size scaling of ρS/ρ vs T and pore length L

61. Finite size scaling of ρS/ρ vs T and pore length L

62. Goals Path integral Monte Carlo calculations:
Can we predict the effective dimensionality of the helium, the superfluid fraction ρS\ρ and the one body density matrix (OBDM) of helium in nanopores?
1D, 2D or 3D?
Can we predict a low TC in nanopores?
Is there static, zero frequency superflow in nanotubes?
Is a frequency dependent theory needed?

63. Phase Diagram of Bulk Helium

64. SUPERFLUID: Bulk Liquid SF Fraction s(T)
Critical Temperature Tλ = 2.17 K

65. Excitations, BEC, and Superfluidity in Nanopores
Organization of Talk
Bulk liquid 4He –Review (short)
Superfluid density, ρS
BEC condensate fraction, n0
Phonon-roton excitations.
2. Liquid 4He in Porous media
Review ρS, BEC, Modes
Can separate BEC from ρS in nanopores
Well defined Phonon-roton modes beyond sound region exist only where there is BEC.
3. Present PIMC calculations: ρS, BEC
Determine effective dimensions from scaling of ρS and OBDM
and liquid density in pores.

66. Goals Path integral Monte Carlo calculations:
Can we predict the effective dimensionality of the helium, the superfluid fraction ρS\ρ and the one body density matrix (OBDM) of helium in nanopores?
1D, 2D or 3D?
Can we predict a low TC in nanopores?
Is there static, zero frequency superflow in nanotubes?
Is a frequency dependent theory needed?

67. Superfluid fraction vs T (K) for liquid pore radius R = 3 Å and R = 4 Å

68. Conclusions: Dimensions
At very small liquid pore diameter, dL , dL ≤ 2σ where σ ~2.5 Å, He hard core diameter, liquid He fills pore in a 1D line (d = dL + 10 Å). Have a 1D system. PIMC ρS\ρ scales as a 1D LL liquid. No Superflow, no BEC.
At dL > 8 Å (dL > 2σ) i.e. at nanopore diameters d = dL + 10 Å ~ 18 Å, He fills pores in 2D cylindrical layers. Have a 2D layered system. Cross- over to 2D scaling of ρS\ρ and OBDM at d ~ 18 Å,. Standard, static superflow predicted. Low TC predicted.
At nanopores 18 < d < 32 Å where many measurements made, conventional 2D static ρS\ρ is predicted by PIMC.
Cross-over to 3D scaling at larger nanopore diameter,
d > 32 Å. OBDM and condensate fraction consistent with measurements.

69. Superfluid fraction vs T (K) as a function of liquid pore radius R

70. BEC, Excitations and Superfluidity
Bulk Liquid 4He
1. Bose-Einstein Condensation,
2. Well-defined phonon-roton modes, at Q > 0.8 Å-1
3. Superfluidity
All co-exist in same p and T range.
They have same “critical” temperature,
Tλ = 2.17 K SVP
Tλ = 1.76 K bar

71. - Yamamoto et al, Phys. Rev. Lett. 93, 075302 (2004)
Phase Diagram in gelsil: 25 A pore diameter
- Yamamoto et al, Phys. Rev. Lett. 93, (2004)