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

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

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-41 30% 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

Experiments: Low Dimensional He in Nanopores

Theory: Low Dimensional He in Nanopores

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?

Phase Diagram Bulk helium

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

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

Helium in Porous Media

- Taniguchi et al, Phys. Rev B82, 104509 (2010) Phase Diagram in FSM-16: 28 A pore diameter - Taniguchi et al, Phys. Rev B82, 104509 (2010)

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

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

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

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

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

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

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

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

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

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

R = 3 Å One Body Density Matrix n (z) (z along pore)

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

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

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

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

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

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

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 = 0. 016 Å -3 ρ = 0.0214 Ǻ -3 ρ = 0.0241 Ǻ-3

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

R = 6 Å One Body Density Matrix n (z) (z along pore)

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.

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

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.

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.

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

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

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

N. N. Bogoliubov

Landau

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.

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

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

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)

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)

Kamerlingh Onnes

London

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 ).

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-41 30% 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

Phonon- Roton Mode in MCM-41: SVP

Phonon- Roton Mode in MCM-41: SVP

Roton in Bulk Liquid 4He Talbot et al., PRB, 38, 11229 (1988)

Maxon in bulk liquid 4He Talbot et al., PRB, 38, 11229 (1988)

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.

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

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

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

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?

Phase Diagram of Bulk Helium

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

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.

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?

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

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.

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

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 25 bar

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