phonon-roton excitations and quantum phase transitions in liquid 4 he in nanoporus media

Phonon-roton excitations Phonon-roton excitations and quantum phase and quantum phase

transitions in liquid transitions in liquid 44He in He in nanoporus mediananoporus media

Henry R. GlydeDepartment of Physics & Astronomy

University of Delaware

Recent Progress in Many Body Theories

Barcelona, 16-20 July, 2007

Excitations, BEC, and SuperfluidityExcitations, BEC, and Superfluidity

Collaborators:

Jonathan Pearce University of Delaware, ILLNational Physical Laboratory

Teddington, UK

Jacques Bossy Centre de Recherche sur LesTrès Basses TemperatureCNRS, Grenoble, France

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

Helmut Schober Institut Laue-Langevin

Grenoble, France


Collaborators (Con’t):

Oliver Plantevin - Université de Paris Sud

Helmut Schober - Institut Laue Langevin, Grenoble, France


Goals:

Explore the interdependence of Bose-Einstein Condensation (BEC), phonon-roton excitations, and superfluidity.

Reveal origin of superfluidity in disorder and confinement. -BEC or well defined excitations.

Neutron scattering studies of excitations of liquid 4He in confinement and disorder. Compare with measurements of superfluid density.


Landau Theory: Superfluidity follows from

existence of well defined phonon-roton modes. The P-R mode is the only mode in superfluid 4He.

Bose-Einstein Condensation: Superfluidity follows from BEC. An extended condensate has a well defined magnitude and phase, <ψ> = √n0eιφ ;

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 DisorderBosons in Disorder

Liquid 4He 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: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 (1D)


Organization of Talk

1. Bulk liquid 4He --review Superfluid density, ρS

BEC condensate fraction, n0 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 TPhonon-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 DiagramBULK HELIUM: Phase Diagram

SUPERFLUIDITYSUPERFLUIDITY

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)

Kamerlingh OnnesKamerlingh Onnes

LondonLondon

Superfluid Density Superfluid Density ss(T)(T)

Superfluid Density ρS (T) = 0 at T = Tλ

Bulk Liquid 4He

Phase Diagram of Bulk HeliumPhase Diagram of Bulk Helium

BOSE-EINSTEIN CONDENSATIONBOSE-EINSTEIN CONDENSATION

Atoms in TrapsAtoms in Traps

Bose-Einstein Condensation: Bose-Einstein Condensation: Atoms in TrapsAtoms in Traps

Bose-Einstein CondensationBose-Einstein Condensation

Glyde, Azuah, and StirlingPhys. Rev. B62, 14337 (2000)


Expt: Glyde et al. PRB (2000)

Condensate fraction bulk Condensate fraction bulk 44HeHe

L. Vranjes and J. Boronat et al. L. Vranjes and J. Boronat et al. PRL (2005)PRL (2005)


Moroni and Boninsegni JLTP (2004)Moroni and Boninsegni JLTP (2004)

50 bars

Bose-Einstein CondensationBose-Einstein CondensationSolid Helium p = 41 barsSolid Helium p = 41 bars

Diallo et al. PRL 98, 205301 (2007)

PHONONS AND ROTONSPHONONS AND ROTONS

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

Roton Energy versus PressureRoton Energy versus Pressure

Roton energy at Q ~ 2.1 Å-1 as a function of pressure.

Vranjes et al. PRL (2005)

Liquid Liquid 44He at Negative PressureHe at Negative Pressure

Liquid Liquid 44He at Negative Pressure He at Negative Pressure

Dispersion curve at SVP and - 5 bar

Liquid Liquid 44He at Negative Pressure He at Negative Pressure MCM-41 MCM-41

Adsorption isotherm

Pores are full with 4He at negative pressure at fillings C to H. C = -5.5 bar.

Maxon Energy versus PressureMaxon Energy versus Pressure

Maxon energy at Q = 1.1 Å-1 as a function of pressure.

Phonon-roton mode of Phonon-roton mode of 44He He under pressure, 24.7 barsunder pressure, 24.7 bars

Temperature dependence of mode Temperature dependence of mode intensity: Maxon, bulk liquid intensity: Maxon, bulk liquid 44He He

Talbot et al., PRB, 38, 11229 (1988)

Roton in Bulk Liquid Roton in Bulk Liquid 44HeHe

Talbot et al., PRB, 38, 11229 (1988)

Beyond the Roton in Bulk Beyond the Roton in Bulk 44HeHe

Data: Pearce et al. J Phys Conds Matter (2001)

Phonons and Rotons (sharply Phonons and Rotons (sharply defined modes) arise From Bose-defined modes) arise From Bose-Einstein CondensationEinstein Condensation

Bogoliubov (1947) showed: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.

cQQ

Phonons and Rotons Arise From Phonons and Rotons Arise From Bose-Einstein CondensationBose-Einstein Condensation

Gavoret and NoziGavoret and Nozièreères (1964) showed:s (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).

cQQ

Ma and Woo (1967), Griffin and Ma and Woo (1967), Griffin and Cheung (1973), and others showed:Cheung (1973), and others showed:

Only a single mode at all Q with BEC -- the phonon-roton mode.

Excitations in a Bose FluidExcitations in a Bose Fluid

ρ+


Bulk Liquid Bulk Liquid 44HeHe

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 HeliumPhase Diagram of Bulk Helium

SuperfluiditySuperfluidity

Landau TheoryLandau 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.

BEC and Phase Coherence, BEC and Phase Coherence, Ø (r)Ø (r)

Superfluidity follows directly from BEC, phase conherence .)(r

s

LandauLandau

POROUS MEDIAPOROUS MEDIA

AEROGEL 95% porousOpen 87% porous A

87% porous B-- grown with

deuterated materials or flushed

with D2

VYCOR 30% porous70 Å pore Diameter -- grown with B11 isotope

GELSIL (GELTECH) 50% porous44 Å pore Diameter34 Å pore Diameter25 Å pore Diameter

MCM-41 30% porous

47 Å pores

Superfluid Properties in Superfluid Properties in Confinement/DisorderConfinement/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)

2.17KTλ

)(Ts

)(Ts

TTcc in Porous Media in Porous Media

Geltech (25 Å pores)

Superfluid Density in Porous MediaSuperfluid Density in Porous Media

Chan et al. (1988)

Miyamoto and Takeno (1996)

- - Yamamoto et al. Yamamoto et al. Phys. Rev. Lett. 93, 075302 (2004) Phys. Rev. Lett. 93, 075302 (2004)

Superfluid Density in gelsil Superfluid Density in gelsil (Geltech) – 25 A diameter(Geltech) – 25 A diameter

Schematic Phase Diagram of Schematic Phase Diagram of Helium Confined to NanoscalesHelium Confined to Nanoscales

e.g. 2 - 4 nme.g. 2 - 4 nm

- - Yamamoto et al,Yamamoto et al, Phys. Rev. Lett. 93, 075302 (2004) Phys. Rev. Lett. 93, 075302 (2004)

Phase Diagram of gelsil: Phase Diagram of gelsil: 25 25 ÅÅ pore diameter pore diameter

Bose-Einstein CondensationBose-Einstein CondensationLiquid Liquid 44He in VycorHe in Vycor

Azuah et al., JLTP (2003)

Tc (Superfluidity) = 2.05 K

Bose-Einstein Condensation Bose-Einstein Condensation VycorVycor


Phonons, Rotons, and Layer Modes Phonons, Rotons, and Layer Modes in Vycor and Aerogelin Vycor and Aerogel

Temperature DependenceTemperature Dependenceof Roton Energyof Roton Energy

Fåk et al., PRL, 85 (2000)

• 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 4He at all T.

• Liquid also supports “layer modes” at roton wave vectors.

• At partial fillings, can also see ripplons on 4He liquid surfaces. (Lauter et al. Appl. Phys. A 74, S1547 (2002))

Conclusions:Conclusions:

Excitations of Liquid Excitations of Liquid 44He in He in ConfinementConfinement

Intensity in P-R Mode vs. Intensity in P-R Mode vs. TT

Glyde et al., PRL, 84 (2000)

Mode Intensity in Vycor:Mode Intensity in Vycor:T = 1.95 KT = 1.95 K

Mode Intensity in VycorMode Intensity in Vycor T = 2.05 K T = 2.05 K

Fraction, Fraction, ffss(T)(T), of Total Intensity , of Total Intensity

in Phonon-Roton Modein Phonon-Roton Mode

Vycor TVycor Tcc = 2.05 K = 2.05 K

Albergamo et al. Phys. Rev. B69, 014514 (2004)

Mode Intensity in 44A Gelsil:Mode Intensity in 44A Gelsil:versus T. Tversus T. Tcc = 1.92 K = 1.92 K

Albergamo et al. PRB (2007)

Fraction, Fraction, ffss(T)(T), of total scattering , of total scattering

intensity in Phonon-Roton Modeintensity in Phonon-Roton Mode- gelsil 44 A pore diameter - gelsil 44 A pore diameter

Liquid Liquid 44He in 25 A gelsil (Geltech)He in 25 A gelsil (Geltech)

Tc (Superfluidity) ~ 1.3 K


Localization of Bose-Einstein Localization of Bose-Einstein Condensation in disorderCondensation in disorder

• 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 < T < Tλ - BEC is localized by disorder- No extended phase coherence across the sample- No superflow


Liquid 4He in Disorder and Boson Liquid 4He in Disorder and Boson LocalizationLocalization

• 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 DiagramSchematic Phase Diagram of BEC in Nanoporous media of BEC in Nanoporous media

PRESSURE DEPENDENCE PRESSURE DEPENDENCE Phonon-Roton modes, Low TPhonon-Roton modes, Low T

• 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.

Liquid 4He up 57 bars in gelsil


Quantum Phase Transition in Quantum Phase Transition in 25 A pore diameter gelsil ?25 A pore diameter gelsil ?

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


25 A gelsil

Roton energy and intensity Roton energy and intensity in roton peak vs pressurein roton peak vs pressure

gelsil 34 gelsil 34 ÅÅ

Pearce et al. (2006)

Phase diagram of modes of Phase diagram of modes of liquid liquid 44He in 34 He in 34 ÅÅ pore diameter pore diameter gelsilgelsil

44He remains liquid in 34 A gelsil He remains liquid in 34 A gelsil up to what pressure?up to what pressure?

Δp = pL – pS = 2α / Rc

pS = 25.3 bars Rc = 14 Å

(a) α = 0.17 erg/cm2 -- constant

pL = 50 bars

(b) α = -increases with pressure (Maris and Caupin, JLTP 131, 145 (2003))

pL = 70 bars

Vycor, pL = 45 bars Rc = 35 Å

Schematic Phase DiagramSchematic Phase Diagram QPT in Nanoporous media QPT in Nanoporous media

Net Scattering intensity Net Scattering intensity gelsil 34 gelsil 34 ÅÅ

Pearce et al. PRL ( rejected 2006-7) Compare with L. Vranjes, J. Boronat et al. PRL,95, 145302 (2005)

Net Scattering intensity, Net Scattering intensity, gelsil 34 gelsil 34 Å Å

and bulk liquid simulation compared.and bulk liquid simulation compared.

Pearce et al. (in progress)

← 60 bars

Bulk liquid

Scattering intensity, Scattering intensity, gelsil 70 gelsil 70 ÅÅ

and p = 70 barsand p = 70 bars

Wallacher et al. JLTP 138, 1013 (2005)

Schematic Phase DiagramSchematic Phase Diagram QPT in Nanoporous media QPT in Nanoporous media



• 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 Quantum Phase Transition at p ~ 35 bars Only localized BEC at p > 35 bars.Only localized BEC at p > 35 bars.

Conclusions (QPT):Conclusions (QPT):


• At T ~ 0 K and higher pressure, ( p > 25 At T ~ 0 K and higher pressure, ( p > 25 bars) BEC condensate fraction is small.bars) BEC condensate fraction is small.

(n(n00 ~ 1 % at p = 70 bars, bulk ~ 1 % at p = 70 bars, bulk 44He)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 liquid-solid co-existence above 38 bars, probably up to 45-50 bars.

Excitations of superfluid Excitations of superfluid 44He at He at pressures up to 40 barspressures up to 40 bars

Phase diagran and excitations of Phase diagran and excitations of superfluid superfluid 44He in 44 He in 44 ÅÅ gelsil gelsil

Pearce et al., PRL (2004)

PHONONS AND ROTONSPHONONS AND ROTONS

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

Superfluid Properties in Superfluid Properties in Confinement/DisorderConfinement/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)

2.17KTλ

)(Ts

)(Ts

Excitations of liquid Excitations of liquid 44He in 34 He in 34 ÅÅ pore diameter gelsilpore diameter gelsil

Pearce et al.,(2006) (in progress)

BEC, Excitations, and SuperfluidityBEC, Excitations, and Superfluidity

BEC in 2DBEC in 2D

Boninsegni et al. PRL 96, 070601 (2006)


L. Vranjes and J. Boronat et al. L. Vranjes and J. Boronat et al. PRL (2005)PRL (2005)


Moroni and Boninsegni JLTP (2004)Moroni and Boninsegni JLTP (2004)

50 bars

Sum rule for condensate component Sum rule for condensate component of S(Q, of S(Q,ωω))

HRG, PRL (1995)

Topic of Talk:Topic of Talk:

cT

• 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 < Tλ = 2.17 K

(Vycor) (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 < T < Tλ . - localized BEC.

Momentum distribution solid Momentum distribution solid 44HeHe

Layer Mode in Porous MediaLayer Mode in Porous Media

Layer Mode in Vycor and AerogelLayer Mode in Vycor and Aerogel



• 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 < Tc < Tλ - BEC is localized by disorder- No extended phase coherence across the sample- No superflow

Quantum Liquids in ConfinementQuantum Liquids in Confinement

Lopatin and Vinokur (2002):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.

]6

)/(1[(BEC) 02

320

Rn

mkTTT cocc

])(27

321[ 2

00 RTTcc


Giorgini Giorgini et al.et al. (1994): (1994):

Same model as Huang & Meng -- disorder arising from random impurities

Sound velocity

Half width of phonons

]3

51[2

02

N

Ncc R

0222

4

)/(24)( R

cm

QQ


Huang and Meng (1992):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.

0)( xu

)()()( yxRyuxu o

)(xu

N

Nna

nN

N Ro 2/13

3)(

81

N

N Rs

3

41

02/1

02/3

22

)(8

)/(Ran

n

m

n

n

N

N RR

Beyond the Roton in Bulk Liquid Beyond the Roton in Bulk Liquid 44HeHe


Phase Diagram of gelsil: Phase Diagram of gelsil: 25 A pore diameter25 A pore diameter

Bose-Einstein CondensationBose-Einstein CondensationLiquid Liquid 44He in VycorHe in Vycor


Tc (Superfluidity) = 1.95-2.05 K

Phonon in Bulk Liquid Phonon in Bulk Liquid 44HeHe

Q= 0.4 Q= 0.4 ÅÅ-1-1

Stirling and Glyde, PRB, 41, 4224 (1990)


Liquid Liquid 44He in confinement, disorderHe 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 < T < Tλ .Localized BEC at p > pc

New HereNew Here

Measurements of phonon-roton excitations and BEC in disorder

Superfluid and Normal 4He

J(Q,s) = 1(s) R(Q,s)

J(Q,s) - Fourier transform of J(Q,y)

Shows difference arising from the condensate

Physics &

Astronomy


Neutron scattering studies of excitations of liquid 4He 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), phonon-roton excitations, and superfluidity.

Reveal origin of superfluidity, BEC or well defined excitations.

Phonon-roton mode of liquid Phonon-roton mode of liquid 44He He in 34 in 34 Å pore diameter gelsilÅ pore diameter gelsil

Pearce et al. (2006)


Conclusions -- porous mediaConclusions -- porous media

At SVP and lower p, have localized BEC in “normal” liquid phase, i.e. for temperatures Tc < T < 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 4He 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 < T < Tλ .No phonon - roton mode at p > pc


Future program: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 2D. (peak in n(k) in 2D).

• 1D 4He on nanotubes, observe vibrational density of states.

Carl Weyman and Eric CornellCarl Weyman and Eric Cornell

phonon-roton excitations and quantum phase transitions in liquid 4 he in nanoporus media

Documents

localized bec

manifestation of bec

defined excitations

n0 phononroton excitations

france excitations

defined phononroton

s bec condensate fraction

studies of excitations