Dynamics of superﬂuid 4He: Single– and multiparticle ... · Correlated wave functions: Bragbook Dynamic Many-Body Theory 2 The Helium Liquids Confronting Theory and Experiment

Dynamics of superfluid 4He:Single– and multiparticle excitations

Dec. 7, 2016

Theory: C. E. Campbell, E. Krotscheck, T. LichteneggerExperiments: Ketty Beauvois, Bjorn Fak, Henri Godfrin, Hans Lauter,Jacues Olivier, Ahmad Sultan...

Sub-eV, Dec. 7-9, 2016

Outline

1 Generalities - Setting the sceneWhy helium physics ?Many-Body TheoryCorrelated wave functions: BragbookDynamic Many-Body Theory

2 The Helium LiquidsConfronting Theory and ExperimentDynamic Many–Body Theory

3 The physical mechanismsWhat is a roton ?Experimental challenge: 4He in 2DConsequence of roton energyMode-mode couplings

4 Summary5 Acknowledgements

What is interesting about helium physics ?Quantum Theory of Corresponding States:

How “quantum” is a (quantum) liquid ?

Let VJL(r) = 4ǫ[

(σ

r

)12−(σ

r

)6]

x =rσ

VLJ(r) = ǫ v(x)

V(r

)

(s

ome

units

)

r (some units)

ε

σ

Molecules/AtomsHard spheresCoulomb

Then1ǫ

H(x1 . . . , xN) = −Λ2

2

∑

i

∇2x i+∑

i<j

v(|x i − x j |)

Quantum Parameter: Λ =

(

~2

mǫσ2

)

12

Λ ≈ 3 for He, Λ ≈ 1 − 2 for H2, HD, D2, Λ < 0.1 for rare gases.

Generalities - Setting the scene Why helium physics ?

Observables: What neutron scatterers measureUnderstanding the dynamics of the helium liquids

Double differential cross section: What experimentalists measure

∂2σ

∂Ω ∂~ω= b2

(

kf

ki

)

S(k, ~ω)




∂2σ

∂Ω ∂~ω= b2

(

kf

ki

)

S(k, ~ω)

Dynamic structure function: The definitions

S(k, ~ω) =1N

∑

n

∣

∣

⟨

Ψn∣

∣ρk∣

∣Ψ0⟩∣

∣

2δ(~ω − εn)

H∣

∣Ψ0⟩

= E0∣

∣Ψ0⟩

H∣

∣Ψn⟩

= [E0 + εn]∣

∣Ψn⟩




∂2σ

∂Ω ∂~ω= b2

(

kf

ki

)

S(k, ~ω)

Dynamic structure function: The definitions

S(k, ~ω) =1N

∑

n

∣

∣

⟨

Ψn∣

∣ρk∣

∣Ψ0⟩∣

∣

2δ(~ω − εn)

H∣

∣Ψ0⟩

= E0∣

∣Ψ0⟩

H∣

∣Ψn⟩

= [E0 + εn]∣

∣Ψn⟩

Density-density response function: What theorists calculate

S(k, ~ω) = −1πℑmχ(k, ~ω)

δρ1(r, t) =

∫

d3r ′dt ′ χ(r, r′; t − t ′)δVext(r′, t ′)


Early experimentsR. A. Cowley and A. D. B. Woods, Can. J. Phys. 49, 177 (1971).

Spectrum by Cowley andWoods 1971




The characteristic features:

Phonon branch





Phonon branch

The famous “roton minimum”,Energy ∆ and wave number k∆





Phonon branch

The famous “roton minimum”,Energy ∆ and wave number k∆The “maxon”





Phonon branch


The “Pitaevskii Plateau”, Energy≈ 2∆, wave number up to 2k∆





Phonon branch



Questions:

What are the physical mechanisms behind these features ?





Phonon branch



Questions:

What are the physical mechanisms behind these features ?

Is there anything else to be seen ?


The tools.. of the theorist and

The theorist’s tools:

H(t)∣

∣ψ(t)⟩

= i~∂

∂ t

∣

∣ψ(t)⟩


The tools.. of the theorist and


H(t)∣

∣ψ(t)⟩

= i~∂

∂ t

∣

∣ψ(t)⟩

+


The tools.. of the theorist and the experimentalist


H(t)∣

∣ψ(t)⟩

= i~∂

∂ t

∣

∣ψ(t)⟩

+

The experimentalist’s tools: (IN5)




H(t)∣

∣ψ(t)⟩

= i~∂

∂ t

∣

∣ψ(t)⟩

+


Getting the sameanswer ?




H(t)∣

∣ψ(t)⟩

= i~∂

∂ t

∣

∣ψ(t)⟩

+


Getting the sameanswer ?

Is there anything elseto be seen ?


Microscopic Many-Body TheoryHamiltonian, wave functions, observables

Postulate. . .1 An empirical, non-relativistic microscopic Hamiltonian

H = −∑

i

~2

2m∇2

i +∑

i

Vext(i) +∑

i<j

V (i , j)

Generalities - Setting the scene Many-Body Theory



H = −∑

i

~2

2m∇2

i +∑

i

Vext(i) +∑

i<j

V (i , j)

2 Particle number, mass




H = −∑

i

~2

2m∇2

i +∑

i

Vext(i) +∑

i<j

V (i , j)

2 Particle number, mass3 Interactions




H = −∑

i

~2

2m∇2

i +∑

i

Vext(i) +∑

i<j

V (i , j)


4 Statistics (Fermi, Bose)




H = −∑

i

~2

2m∇2

i +∑

i

Vext(i) +∑

i<j

V (i , j)


4 Statistics (Fermi, Bose)5 Temperature




H = −∑

i

~2

2m∇2

i +∑

i

Vext(i) +∑

i<j

V (i , j)



Turn on your computer and..

Calculate from no other information. . .




H = −∑

i

~2

2m∇2

i +∑

i

Vext(i) +∑

i<j

V (i , j)





1 Energetics




H = −∑

i

~2

2m∇2

i +∑

i

Vext(i) +∑

i<j

V (i , j)





1 Energetics2 Structure




H = −∑

i

~2

2m∇2

i +∑

i

Vext(i) +∑

i<j

V (i , j)





1 Energetics2 Structure3 Excitations

4 Thermodynamics




H = −∑

i

~2

2m∇2

i +∑

i

Vext(i) +∑

i<j

V (i , j)






4 Thermodynamics5 Finite-size properties




H = −∑

i

~2

2m∇2

i +∑

i

Vext(i) +∑

i<j

V (i , j)






4 Thermodynamics5 Finite-size properties6 Phase transitions (?). . .


Correlated wave functions: Bragbook

A “simple quick and dirty” method:

Ψ0(1, . . . ,N) = exp12

[

∑

i

u1(r i) +∑

i<j

u2(r i , r j) + . . .

]

Φ0(1, . . . ,N)

≡ F (1, . . . ,N)Φ0(1, . . . ,N)

Φ0(1, . . . ,N) “Model wave function” (Slater determinant)

Equation of state for 4He and 3He:

−7.4

−7.2

−7.0

−6.8

−6.6

−6.4

−6.2

−6.0

0.016 0.018 0.020 0.022 0.024 0.026 0.028

E/N

(K

)

r (Å−3)

HNC−ELDMC

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

0.010 0.012 0.014 0.016 0.018 0.020

E/N

[K

]

ρ (Å-3)

FHNC-EL/CFHNC-EL/5FHNC-EL/5 +TFHNC-EL/5 +T +CBFFHNC-EL/5 +T +CBF +σ

VMC-JVMC-JTVMC-JTBDMC-RN

Generalities - Setting the scene Correlated wave functions: Bragbook

Correlated wave functions: Bragbook

A “simple quick and dirty” method:

Ψ0(1, . . . ,N) = exp12

[

∑

i

u1(r i) +∑

i<j

u2(r i , r j) + . . .

]

Φ0(1, . . . ,N)

≡ F (1, . . . ,N)Φ0(1, . . . ,N)

Φ0(1, . . . ,N) “Model wave function” (Slater determinant)

Structure functions of 4He and 3He

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

S(k

)

k (Å−1)

PIMCHNC−ELHNC−EL’Svensson et al.Robkoff and Hallock

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 1 2 3 4 5

S(k

)

k (Å-1)

FHNC-EL DMC

Achter and Meyer

Hallock, T = 0.36 K

Hallock, T = 0.41 K

Generalities - Setting the scene Correlated wave functions: Bragbook

Dynamic Many-Body Theory (DMBT)(Multi-)particle fluctuations for bosons

Build on the success story for the ground state:Make the correlations time dependent !

|Φ(t)〉 = e−iE0t/~ 1N (t)

Fe12 δU |Φ0〉 ,

∣

∣Φ0⟩

: model ground state, δU(t): excitation operator, N (t):normalization.Bosons:

δU(t) =∑

i

δu(1)(r i ; t) +∑

i<j

δu(2)(r i , r j ; t) + . . .

Generalities - Setting the scene Dynamic Many-Body Theory

Dynamic Many-Body Theory (DMBT)(Multi-)particle fluctuations for bosons and fermions


|Φ(t)〉 = e−iE0t/~ 1N (t)

Fe12 δU |Φ0〉 ,

∣

∣Φ0⟩

: model ground state, δU(t): excitation operator, N (t):normalization.Fermions:

δU(t) =∑

p,h

δu(1)p,h(t)a

†pah +

∑

p,h,p′,h′

δu(2)p,h,p′,h′(t)a

†pa†

p′ahah′




|Φ(t)〉 = e−iE0t/~ 1N (t)

Fe12 δU |Φ0〉 ,

∣

∣Φ0⟩


δU(t) =∑

p,h

δu(1)p,h(t)a

†pah +

∑

p,h,p′,h′


†pa†

p′ahah′

δu(2) describes fluctuations of the short-ranged structure




|Φ(t)〉 = e−iE0t/~ 1N (t)

Fe12 δU |Φ0〉 ,

∣

∣Φ0⟩


δU(t) =∑

p,h

δu(1)p,h(t)a

†pah +

∑

p,h,p′,h′


†pa†

p′ahah′

δu(2) describes fluctuations of the short-ranged structure

The physical content of δu(2) is beyond mean field theory !


Dynamic Many-Body Theory (DMBT)What these amplitudes do for bosons

δU(t) =∑

i

δu(1)(r i ; t)


Dynamic Many-Body Theory (DMBT)What these amplitudes do for bosons and fermions

δU(t) =∑

p,h

δu(1)p,h(t)a

†pah


Dynamic Many-Body Theory (DMBT)What these amplitudes do for bosons

δU(t) =∑

i

δu(1)(r i ; t)

+∑

i<j

δu(2)(r i , r j ; t) + . . .


Dynamic Many-Body Theory (DMBT)What these amplitudes do for bosons and fermions

δU(t) =∑

p,h

δu(1)p,h(t)a

†pah

+∑

p,h,p′,h′


†pa†

p′ahah′


Bosons: 4He in 3DConfronting Theory and Experiment: Experiments by Godfrin group in Grenoble

Experiments

Feynman (RPA)-Theory

S(k,ω)

ρ = 0.022 (Å−3)

k (Å−1)0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

2.0

E (

meV

)

One-body fluctuations:Feynman(RPA) spectrum

δU(t) =∑

i

ei(k·r i−ωt)

~ω(k) = ~2k2/2mS(k)

~ω(k) = ck as k → 0

The Helium Liquids Confronting Theory and Experiment


Experiments

Single-Pair-Fluctuations

0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

1.5

2.0

S(k,ω)

ρ = 0.022 (Å−3)

k (Å−1)

E (

meV

)

Two-body fluctuations:Feynman-Cohen “backflow”

δU(t) =∑

i

ei(k ·r i−ωt)

r i = r i +∑

j 6=i

η(rij)r ij



Experiments

Multi-Pair-Fluctuations

0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

1.5

2.0

S(k,ω)

ρ = 0.022 (Å−3)

k (Å−1)

E (

meV

)

Many-body fluctuations:

δU(t) =∑

i

δu(1)(r i ; t)

+∑

i<j

δu(2)(r i , r j ; t) + . . .

Stationarity principle

δ

∫

dt⟨

Φ(t)∣

∣

∣H + δH(t)− i~

∂

∂t

∣

∣

∣Φ(t)

⟩

= 0



Experiments

Multi-Pair-Fluctuations

0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

1.5

2.0

S(k,ω)

ρ = 0.022 (Å−3)

k (Å−1)

E (

meV

)

Many-body fluctuations:

δU(t) =∑

i

δu(1)(r i ; t)

+∑

i<j

δu(2)(r i , r j ; t) + . . .

Stationarity principle

δ

∫

dt⟨

Φ(t)∣

∣

∣H + δH(t)− i~

∂

∂t

∣

∣

∣Φ(t)

⟩

= 0

Brillouin-Wigner perturbationtheory in the basis ei

∑i k·r i

∣

∣Ψ0⟩

.


Dynamic Many–Body TheoryA few equations

Dynamic Structure Function

S(k, ω) = −1πℑm

∫

d3reik·r)χ(r, r′;ω).

Density–density response function

χ(k , ω) =S(k)

ω − Σ(k , ω)+

S(k)−ω − Σ(k ,−ω)

,

Self-energy

Σ(k , ω) = ε0(k)+12

∫

d3pd3q(2π)3ρ

δ(k − p − q) |V3(k; p, q)|2

ω − Σ(p, ω − ε0(q))− Σ(q, ω − ε0(p)).

3-phonon vertex V3(k; p, q)

The Helium Liquids Dynamic Many–Body Theory

Dynamic Many–Body TheoryA few diagrams

(a) (b) (c) (d)

(e) (f) (g) (h)

The Helium Liquids Dynamic Many–Body Theory

The physical mechanismsWhat is a roton ?

Is it

The physical mechanisms What is a roton ?


Is it

Or



Is it

Or

If so, could there be a second Bragg peak ?(None found in 3D 4He)


4He in 2DTheoretical predictions – An experimental challenge

Below saturation (ρ = 0.044 A−2)strong anomalous dispersion;

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

k [Å−1]

0

2

4

6

8

10

12

− hω [K

]

− hω [m

eV]

ρ=0.044 Å−2

Feynman

CBF

EOM

Arrigoni et al.

0.0

0.2

0.4

0.6

0.8

1.0

The physical mechanisms Experimental challenge: 4He in 2D



Roton paramaters within errorbars of Monte Carlo calculations;

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

k [Å−1]

0

2

4

6

8

10

12

− hω [K

]

− hω [m

eV]

ρ=0.054 Å−2

Feynman

CBF

EOM

Arrigoni et al.

0.0

0.2

0.4

0.6

0.8

1.0





Around saturation, not much new;

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

k [Å−1]

0

2

4

6

8

10

12

− hω [K

]

− hω [m

eV]

ρ=0.054 Å−2

Feynman

CBF

EOM

Arrigoni et al.

0.0

0.2

0.4

0.6

0.8

1.0






Close to the liquid-solid phasetransition, a weak, secondary“roton” !

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

k [Å−1]

0

2

4

6

8

10

12

− hω [K

]

− hω [m

eV]

ρ=0.064 Å−2

Feynman

CBF

EOM

Arrigoni et al.

0.0

0.2

0.4

0.6

0.8

1.0







Monte Carlo calculationsdynamically inconsistent

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

k [Å−1]

0

2

4

6

8

10

12

− hω [K

]

− hω [m

eV]

ρ=0.064 Å−2

Feynman

CBF

EOM

Arrigoni et al.

0.0

0.2

0.4

0.6

0.8

1.0







Monte Carlo calculationsdynamically inconsistent

Still a challenge for neutronscatterer !

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

k [Å−1]

0

2

4

6

8

10

12

− hω [K

]

− hω [m

eV]

ρ=0.064 Å−2

Feynman

CBF

EOM

Arrigoni et al.

0.0

0.2

0.4

0.6

0.8

1.0


Consequence of roton energyWhat does this tell us ?

Sum rules∫

dωℑmχ(k , ω) = S(k)

∫

dωωℑmχ(k , ω) =~

2k2

2m

S(k,ω)

ρ = 0.022 (Å−3)

k (Å−1)0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

2.0

E (

meV

)

The physical mechanisms Consequence of roton energy


Sum rules∫


∫


2k2

2m

If we assume only onephonon, we get Feynman (offby a factor of 2)

S(k,ω)

ρ = 0.022 (Å−3)

k (Å−1)0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

2.0

E (

meV

)



Sum rules∫


∫


2k2

2m

If we assume only onephonon, we get Feynman (offby a factor of 2)

Need a multi(quasi-)particlecontinuum to get theenergetics right !

S(k,ω)

ρ = 0.022 (Å−3)

k (Å−1)0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

2.0

E (

meV

)


The physical mechanisms:Mode-mode couplings

Experiments

0

5

10

15

20

25

0 1 2 3 4 5

hω

(K)

q (Å−1)

∆

q∆

Theory

0 1 2 3 4 5q (Å−1)

0

5

10

15

20

25

hω (

K)

The ”Pitaevskii plateau”

A perturbation with momentum q andenergy ω can decay into two rotonsq(1)∆ and q(2)

∆ with |q(1)∆ | = |q(2)

∆ | = q∆

under momentum and energy andconservation ω = 2∆.

The roton momenta may bealigned

|q| ≤ 2qR

The physical mechanisms Mode-mode couplings


Experiments

Theory

0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

1.5

2.0

S(k,ω)

ρ = 0.022 (Å−3)

k (Å−1)

E (

meV

)

The ”Pitaevskii plateau”

A perturbation with momentum q andenergy ω can decay into two rotonsq(1)∆ and q(2)

∆ with |q(1)∆ | = |q(2)

∆ | = q∆

under momentum and energy andconservation ω = 2∆.

The roton momenta may bealigned

|q| ≤ 2qR

or anti-aligned

|q| ≥ 0



Experiments

Theory

0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

1.5

2.0

S(k,ω)

ρ = 0.022 (Å−3)

k (Å−1)

E (

meV

)

The ”ghost phonon”

Phonon dispersion relation

ω(q) = cq(1 + γq2)“normal dispersion”: γ < 0⇒ phonons are stable

“anomalous dispersion”: γ > 0⇒ phonons can decay

⇒ 4He at zero pressure is borderlinebetween normal and anomalous,γ ≈ 0.1⇒ Perturbations with momentum (q,ω) can decay into two phonons with(q/2, ω/2) as long as the dispersionrelation is almost linear up to q/2.



Experiments

Theory

0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

1.5

2.0

S(k,ω)

ρ = 0.022 (Å−3)

k (Å−1)

E (

meV

)

Maxon-roton coupling

A similar but less sharply definedprocess


Summary

What we know today

Quantitative agreement between experiments in 3D;

Summary

Summary

What we know today


Prediction of a secondary roton–like mode in 2D 4He;

Summary

Summary

What we know today



Prediction of maxon damping at high pressure 4He

Summary

Summary

What we know today




Structures are more pronounced in 2D;

Summary

Summary

What we know today




Structures are more pronounced in 2D;

1 → 2 and 2 → 1 processes are not the end of the story but do notlead to sharp features.

Summary

Thanks to collaborators in this project

C. E. Campbell Univ. MinnesotaF. M. Gasparini University at BuffaloH. Godfrin (and his team) CNRS GrenobleT. Lichtenegger University at Buffalo

Thanks for your attentionand thanks to our funding agency:

Acknowledgements

Documents

Dynamics of superﬂuid 4He: Single– and multiparticle ... · Correlated wave functions: Bragbook Dynamic Many-Body Theory 2 The Helium Liquids Confronting Theory and Experiment