Phononroton excitations and quantum phase transitions in liquid

Phonon-roton excitations and quantum phase transitions in liquid 4 He in nanoporus media Henry R. Glyde Department of Physics & Astronomy University of Delaware Recent Progress in Many Body Theories Barcelona, 16 -20 July, 2007

Excitations, BEC, and Superfluidity Collaborators: Jonathan Pearce University of Delaware, ILL National Physical Laboratory Teddington, UK Jacques Bossy Centre de Recherche sur Les Très Basses Temperature CNRS, Grenoble, Francesco Albergamo -ESRF, Grenoble, France Bjorn Fåk - Commissariat à l’Energie Atomique, Grenoble, France Norbert Mulders -University of Delaware Richard T. Azuah - NIST Center for Neutron Research, Gaithersburg, Maryland, USA

Excitations, BEC, and Superfluidity Collaborators (Con’t): Oliver Plantevin -Université de Paris Sud Helmut Schober - Institut Laue Langevin, Grenoble, France

Excitations, BEC, and Superfluidity Goals: Explore the interdependence of Bose. Einstein Condensation (BEC), phononroton excitations, and superfluidity. Reveal origin of superfluidity in disorder and confinement. -BEC or well defined excitations. Neutron scattering studies of excitations of liquid 4 He in confinement and disorder. Compare with measurements of superfluid density.

Excitations, BEC, and Superfluidity Landau Theory: Superfluidity follows from existence of well defined phonon-roton modes. The P-R mode is the only mode in superfluid 4 He. Bose-Einstein Condensation: Superfluidity follows from BEC. An extended condensate has a well defined magnitude and phase, <ψ> = √n 0 eιφ ; vs ~ grad φ Bose-Einstein Condensation (BEC): Well defined phonon-roton modes follow from BEC. Single particle and P-R modes have the same energy when there is BEC. No low energy single particle modes.

Bosons in Disorder Liquid 4 He in aerogel, Vycor, gelsil (Geltech) Bose gases in traps with disordered potentials Josephson Junction Arrays Granular Metal Films Cooper Pairs in High Tc Superconductors Flux Lines in High Tc Superconductors Specific Present Goals: Impact of finite size (confinement) and disorder on excitations and Bose-Einstein condensation. Localization of Bose-Einstein Condensation by disorder Search for a Quantum Phase Transition Explore liquid helium at higher pressure Helium at negative pressure and on nanotubes (1 D)

Excitations, BEC, and Superfluidity Organization of Talk 1. Bulk liquid 4 He --review Superfluid density, ρS BEC condensate fraction, n 0 Phonon-roton excitations. 2. Porous media – p ~ 0, T dependence Review ρS , TC Present phonon-roton data. Evidence for localized BEC at temperatures above TC 3. Porous media –high pressures, low T Phonon-roton modes disappear at 37 bars and T ~ 0 K, evidence for a superfluid-normal transition at T ~ 0 K, a quantum phase transiton? Or just solidification.

BULK HELIUM: Phase Diagram

SUPERFLUIDITY 1908 – 4 He 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 v. S = grad φ (r)

Kamerlingh Onnes

London

Superfluid Density s(T) Bulk Liquid 4 He Superfluid Density ρS (T) = 0 at T = Tλ

Phase Diagram of Bulk Helium

BOSE-EINSTEIN CONDENSATION Atoms in Traps

Bose-Einstein Condensation: Atoms in Traps

Bose-Einstein Condensation Glyde, Azuah, and Stirling Phys. Rev. B 62, 14337 (2000)

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

Condensate fraction bulk 4 He L. Vranjes and J. Boronat et al. PRL (2005)

Condensate fraction bulk 4 He Moroni and Boninsegni JLTP (2004) 50 bars

Bose-Einstein Condensation Solid Helium p = 41 bars Diallo et al. PRL 98, 205301 (2007)

PHONONS AND ROTONS Donnelly et al. , J. Low Temp. Phys. (1981) Glyde et al. , Euro Phys. Lett. (1998)

Roton Energy versus Pressure Roton energy at Q ~ 2. 1 Å-1 as a function of pressure. Vranjes et al. PRL (2005)

Liquid 4 He at Negative Pressure

Liquid 4 He at Negative Pressure Dispersion curve at SVP and - 5 bar

Liquid 4 He at Negative Pressure MCM-41 Adsorption isotherm Pores are full with 4 He at negative pressure at fillings C to H. C = -5. 5 bar.

Maxon Energy versus Pressure Maxon energy at Q = 1. 1 Å-1 as a function of pressure.

Phonon-roton mode of 4 He under pressure, 24. 7 bars

Phonon-roton mode of 4 He under pressure, 31. 2 bars

Temperature dependence of mode intensity: Maxon, bulk liquid 4 He Talbot et al. , PRB, 38, 11229 (1988)

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

Beyond the Roton in Bulk 4 He Data: Pearce et al. J Phys Conds Matter (2001)

Phonons and Rotons (sharply defined modes) arise From Bose. Einstein Condensation Bogoliubov (1947) showed: Bose gas with BEC -- quasiparticles have energy: - phonon (sound) form Quasiparticle mode coincides with sound mode. Only one excitation when have BEC.

Phonons and Rotons Arise From Bose-Einstein Condensation Gavoret and Nozières (1964) showed: Dense liquid with BEC – only one excitation: density and quasiparticle modes have the same energy, as in Bose gas. -- no other excitations at low energy (could have vortices). Ma and Woo (1967), Griffin and Cheung (1973), and others showed: Only a single mode at all Q with BEC -the phonon-roton mode.

Excitations in a Bose Fluid ρ+

Excitations, BEC, and Superfluidity Bulk Liquid 4 He BEC, well-defined phonon-roton modes at Q > 0. 8 Å-1 and superfluidity coincide. e. g. , all have some “critical” temperature, Tλ = 2. 17 K SVP Tλ = 1. 76 K 25 bar

Phase Diagram of Bulk Helium

Superfluidity Landau Theory 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 ). Via P-R excitations, superflow arises from BEC and Phase Coherence, Ø (r) Superfluidity follows directly from BEC, phase conherence.

Landau

POROUS MEDIA AEROGEL Open 95% porous 87% porous A 87% porous B -- grown with deuterated materials or flushed with D 2 VYCOR 30% porous 70 Å pore Diameter -- grown with B 11 isotope GELSIL (GELTECH) 50% porous 44 Å pore Diameter 34 Å pore Diameter 25 Å pore Diameter MCM-41 47 Å pores 30% porous

Superfluid Properties in Confinement/Disorder Confinement reduces Tc below . Confinement modifies (T dependence). Confinement reduces (magnitude). Porous media is a “laboratory” to investigate the relation between superfluidity, excitations, and BEC. Measure corresponding excitations and condensate fraction, no(T). (new, 1995)

Tc in Porous Media

Superfluid Density in Porous Media Chan et al. (1988) Miyamoto and Takeno (1996) Geltech (25 Å pores)

Superfluid Density in gelsil (Geltech) – 25 A diameter - Yamamoto et al. Phys. Rev. Lett. 93, 075302 (2004)

Schematic Phase Diagram of Helium Confined to Nanoscales e. g. 2 - 4 nm

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

Bose-Einstein Condensation Liquid 4 He in Vycor Tc (Superfluidity) = 2. 05 K Azuah et al. , JLTP (2003)

Bose-Einstein Condensation Vycor Azuah et al. , JLTP (2003)

Phonons, Rotons, and Layer Modes in Vycor and Aerogel

Temperature Dependence of Roton Energy Fåk et al. , PRL, 85 (2000)

Excitations of Liquid 4 He in Confinement Conclusions: • Liquid helium in porous media supports well defined phonon-roton excitations – up to wave vectors Q ≈ 2. 8 Å. • Energies and widths (within precision) are the same as in bulk 4 He at all T. • Liquid also supports “layer modes” at roton wave vectors. • At partial fillings, can also see ripplons on 4 He liquid surfaces. (Lauter et al. Appl. Phys. A 74, S 1547 (2002))

Intensity in P-R Mode vs. T Glyde et al. , PRL, 84 (2000)

Mode Intensity in Vycor: T = 1. 95 K

Mode Intensity in Vycor T = 2. 05 K

Mode Intensity in Vycor: T = 2. 15 K

Mode Intensity in Vycor: T = 2. 25 K

Fraction, fs(T), of Total Intensity in Phonon-Roton Mode Vycor Tc = 2. 05 K Albergamo et al. Phys. Rev. B 69, 014514 (2004)

Mode Intensity in 44 A Gelsil: versus T. Tc = 1. 92 K Albergamo et al. PRB (2007)

Fraction, fs(T), of total scattering intensity in Phonon-Roton Mode - gelsil 44 A pore diameter

Liquid 4 He in 25 A gelsil (Geltech) Tc (Superfluidity) ~ 1. 3 K

Localization of Bose-Einstein Condensation in disorder Conclusions: • Observe phonon-roton modes up to T = Tλ = 2. 17 K in porous media, i. e. above Tc for superfluidity • Well defined phonon-roton modes exist because there is a condensate. Thus have BEC above Tc in porous media. Vycor Tc = 2. 05 K gelsil (44 Å) Tc = 1. 92 K gelsil (25 Å) Tc = 1. 3 K • At temperatures Tc < Tλ - BEC is localized by disorder - No extended phase coherence across the sample - No superflow

Liquid 4 He in Disorder and Boson Localization Conclusions: • Extended BEC at temperature below Tc in superfluid phase. • Superfluid - Normal liquid transition associated with an extended to localized BEC cross over at SVP.

Schematic Phase Diagram of BEC in Nanoporous media

PRESSURE DEPENDENCE Phonon-Roton modes, Low T Liquid 4 He up 57 bars in gelsil • gelsil 44 Å mean pore diameter, – Pearce et al. PRL (2004) • gelsil 34 Å mean pore diameter - Pearce et al. Preprint (2006) • gelsil 25 Å mean pore diameter - being analysed (2006) - Compare with Yamamoto et al. PRL (2004) , superfluid density in 25 Å gelsil.

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

Phonon-roton mode of 4 He under pressure, 31. 2 bars

Pressure dependence: 44 Å gelsil phonon (Q = 0. 7 Ǻ-1) roton (Q=2. 1Å-1)

Pressure dependence of S(Q, ω) at the roton (Q=2. 1Å-1) 34 A gelsil

Pressure dependence of S(Q, ω) at the roton (Q=2. 1Å-1) 25 A gelsil

Roton energy and intensity in roton peak vs pressure gelsil 34 Å Pearce et al. (2006)

Phase diagram of modes of liquid 4 He in 34 Å pore diameter gelsil

4 He remains liquid in 34 A gelsil up to what pressure? Δp = p. L – p. S = 2α / Rc p. S = 25. 3 bars Rc = 14 Å (a) α = 0. 17 erg/cm 2 -- constant p. L = 50 bars (b) α = -increases with pressure (Maris and Caupin, JLTP 131, 145 (2003)) p. L = 70 bars Vycor, p. L = 45 bars Rc = 35 Å

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

Schematic Phase Diagram QPT in Nanoporous media

Net Scattering intensity gelsil 34 Å Pearce et al. PRL ( rejected 2006 -7) Compare with L. Vranjes, J. Boronat et al. PRL, 95, 145302 (2005)

Net Scattering intensity, gelsil 34 Å and bulk liquid simulation compared. ← 60 bars Bulk liquid Pearce et al. (in progress)

Scattering intensity, gelsil 70 Å and p = 70 bars Wallacher et al. JLTP 138, 1013 (2005)

Schematic Phase Diagram QPT in Nanoporous media

Liquid 4 He in Disorder and Boson Localization Conclusions: • Extended BEC at temperature below Tc in superfluid phase at SVP. • Superfluid - Normal liquid transition associated with an extended to localized BEC cross over at SVP. • Quantum Phase Transition at p ~ 35 bars Only localized BEC at p > 35 bars.

Liquid 4 He in Disorder and Boson Localization Conclusions (QPT): • At T ~ 0 K and higher pressure, ( p > 25 bars) BEC condensate fraction is small. (n 0 ~ 1 % at p = 70 bars, bulk 4 He). Speculation: At T ~ 0 K and pressures p > pc - BEC is localized by disorder - No extended phase coherence across the sample - No superflow Quantum Phase Transition at 35 bars. Phonon – roton modes disappear, p ~ 38 bars - Have liquid up to 38 bars and liquidsolid co-existence above 38 bars, probably up to 45 -50 bars.

Excitations of superfluid 4 He at pressures up to 40 bars

Phase diagran and excitations of superfluid 4 He in 44 Å gelsil Pearce et al. , PRL (2004)

Bose-Einstein Condensation

PHONONS AND ROTONS Donnelly et al. , J. Low Temp. Phys. (1981) Glyde et al. , Euro Phys. Lett. (1998)

Superfluid Properties in Confinement/Disorder Confinement reduces Tc below . Confinement modifies (T dependence). Confinement reduces (magnitude). Porous media is a “laboratory” to investigate the relation between superfluidity, excitations, and BEC. Measure corresponding excitations and condensate fraction, no(T). (new, 1995)

Excitations of liquid 4 He in 34 Å pore diameter gelsil Pearce et al. , (2006) (in progress)

BEC, Excitations, and Superfluidity

BEC in 2 D Boninsegni et al. PRL 96, 070601 (2006)

Condensate fraction bulk 4 He L. Vranjes and J. Boronat et al. PRL (2005)

Condensate fraction bulk 4 He Moroni and Boninsegni JLTP (2004) 50 bars

Sum rule for condensate component of S(Q, ω) HRG, PRL (1995)

Topic of Talk: • Well defined p-r excitations (Q > 0. 8 Å) exist because there is Bose-Einstein condensation (BEC). • Measure superfluid density ρs (T) and determine the normal to superfluid transition temperature Tc in Vycor (same sample). Find: Tc = 2. 05 K (Vycor) < Tλ = 2. 17 K (Bulk) - disorder suppresses Tc below Tλ • Find well defined phonon–roton excitations in Vycor at temperatures T > Tc, up to T = Tλ = 2. 17 K • Thus BEC in Vycor above Tc , at temperatures Tc < Tλ. - localized BEC.

Momentum distribution solid 4 He

Layer Mode in Porous Media

Layer Mode in Vycor and Aerogel

Liquid 4 He in Disorder and Boson Localization Conclusions: • Observe phonon-roton modes up to T = Tλ = 2. 17 K in porous media, i. e. above Tc for superfluidity • Well defined phonon-roton modes exist because there is a condensate. Thus have BEC above Tc in porous media. Vycor Tc = 2. 05 K Geltech (44 Å) Tc = 1. 92 K Geltech (25 Å) Tc = 1. 0 K • At temperatures Tc < Tλ - BEC is localized by disorder - No extended phase coherence across the sample - No superflow

Quantum Liquids in Confinement Lopatin and Vinokur (2002): Same model as Huang & Meng -- disorder arising from random impurities Reduction of critical temperature for BEC by disorder Reduction of critical temperature for superfluidity by disorder.

Quantum Liquids in Confinement Giorgini et al. (1994): Same model as Huang & Meng -- disorder arising from random impurities Sound velocity Half width of phonons

Quantum Liquids in Confinement Huang and Meng (1992): Dilute Bose gas in disorder (T = OK). Disorder potential arises from hard sphere impurities placed at random. Condensate fraction Superfluid density where Astrakharchik et al (2002) -- Monte Carlo extension to Bose fluid.

Beyond the Roton in Bulk Liquid 4 He

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

Bose-Einstein Condensation Liquid 4 He in Vycor Tc (Superfluidity) = 1. 95 -2. 05 K Azuah et al. , JLTP (2003)

Phonon in Bulk Liquid 4 He Q= 0. 4 Å-1 Stirling and Glyde, PRB, 41, 4224 (1990)

Excitations, BEC, and Superfluidity Liquid 4 He in confinement, disorder BEC and well-defined phonon-roton modes are separated from superfluidity. Below Tc – have superfluidity, BEC and well-defined phonon-roton modes. BEC is extended. Have extended phase coherence. Above Tc - have phonon-roton modes and BEC but no superflow. BEC is localized by disorder. No extended phase coherence. Localized BEC at Tc < Tλ. Localized BEC at p > pc New Here Measurements of phonon-roton excitations and BEC in disorder

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

Physics & Astronomy Superfluid and Normal 4 He J(Q, s) = r 1(s) R(Q, s) J(Q, s) - Fourier transform of J(Q, y) Shows difference arising from the condensate

Excitations, BEC, and Superfluidity Neutron scattering studies of excitations of liquid 4 He in confinement and disorder. • phonons and rotons in helium at nanoscale size, in disorder, near surfaces. • identify new excitations. • temperature and pressure dependence. Explore the interdependence of Bose. Einstein Condensation (BEC), phononroton excitations, and superfluidity. Reveal origin of superfluidity, BEC or well defined excitations.

Phonon-roton mode of liquid 4 He in 34 Å pore diameter gelsil Pearce et al. (2006)

Pressure dependence of S(Q, ω) at the roton (Q=2. 1Å-1)

Excitations, BEC, and Superfluidity Conclusions -- porous media At SVP and lower p, have localized BEC in “normal” liquid phase, i. e. for temperatures Tc < Tλ. Have order in the normal phase up to Tλ At SVP, superfluid-normal transition in porous media is associated with an extended to localized BEC cross over. At pressures, p > 35 bars, liquid 4 He no longer supports well- defined P-R modes. No roton for p > 35 bars. Loss of P-R modes coincides with a superfluid –normal Quantum Phase Transition at pc ~ 35 bars Localized BEC at Tc < Tλ. No phonon - roton mode at p > pc

Excitations, BEC, and Superfluidity Future program: * Observe BEC in solid helium. * Observe P-R modes in 25 Å gelsil (same sample as used by Yamamoto et al). Compare directly with ρS (p, T), pc , Tc. * Observe OBDM in 2 D. (peak in n(k) in 2 D). • 1 D 4 He on nanotubes, observe vibrational density of states.

Carl Weyman and Eric Cornell