Dynamics of Superfluid Transitions in Helium Three

-- from Baked Alaska to Kibble-Zurek mechanism

Physics Department Colloquium U. of Kyoto (July 20, 2011)

Hiroshi Fukuyama

Dept. of Phys., The Univ. of Tokyo

Phase diagram of superfluid 3He

1 2 3 T (mK)

0

1

2

3

4

P (M

Pa)

superfluid A-phase

superfluid B-phase

Normal

Solid

TCP

u2d2

超流動3Heの相図 Ａ相 （ABM状態）

スピン空間と軌道空間の 対称性が独立に破れている

l 運動量空間 の秩序変数

d スピン空間 の秩序変数

SO(3) × SO(3) × U(1) → U(1) × U(1) × Z2

B相 （BW状態） n

SO(3) × SO(3) × U(1) → SO(3)

スピン空間と軌道空間の 対称性が相対的に破れている

↑↑ , ↓↓

↑↑ , ↑↓ + ↓↑( ), ↓↓

クーパー対の対称性が p波スピン三重項の超流動状態

大きく過冷却するが 必ずB相は生まれる

一次相転移

Large supercooling of 3He-A phaseIt was recognized even in the first observation of superfluid phases.

D.D. Osheroff, R.C. Richardson and D.M. Lee, Phys. Rev. Lett. 28, 885 (1972)

TBcooling / TB

warming ≈ 0.7

Easily detectable in NMR: • Large magnetization difference between A and B phases. • frequency shit in A-phase

Supercooling in water

H. Kanno, R.J. Speedy and C.A. Angell, Science 189, 880 (1975)

Tmcooling / Tm = 234 K / 273 K = 0.86

AA-B相転移 (一次相転移) のミステリー

半径RのB相核が生成したときの自由エネルギー変化 (ΔG) : ΔG = 4πR2σAB − 4 3( )πR3 GA −GB( )

σAB: AB界面エネルギー (GA - GB) : バルクA相とB相の

自由エネルギー密度差

均一核生成理論の範囲では、宇宙の年齢ほど待ってもB相は生成されないはず！ ••• (GA – GB) が異常に小さい

均一核生成理論

Δ = 16 3( )πσAB3 GA −GB( )2

106 kBT !

エネルギー障壁： Rc = 2σAB GA −GB( )

2 μm ! (T = 0.7 Tc, P = 3.4 MPa) (Rc ≈ 4 nm for H2O)

臨界核半径：

生成確率 ∝ ω0exp(-Δ/kBT) ≈ 0 (ω0 ≤ 1012 s-1)•••熱活性

　　　　 ∝ ω0exp(-SαEL/ ) ≈ 10-20000 ≈ 0 (αEL：Euler-Lagrange方程式の解) •••量子トンネリング D. Bailin and A. Love, J. Phys. A 13, L271 (1980)

D. Bailin and A. Love, J. Phys. A 13, L271 (1980)

Free energy difference between A- and B-phases

Tc TAB T

F

0

ΔF

A-phase

B-phase

Normal

F0B

ΔF << F0

Spin-fluctuation (Paramagnon) feedback stabilizes A-phase at high-T and high-P, otherwise B-phase is stable in a whole T-range.

P.W. Anderson and W.F. Brinkman, Phys. Rev. Lett. 30, 1108 (1973)

ΔF = −αdμi∗ dμi + β1dμi

∗ dμi∗ dνjdνj + β2dμi

∗ dμidνj∗ dνj + β3dμi

∗ dνi∗dμjdνj + β4dμi

∗ dνidνj∗ dμj + β5dμi

∗ dνidνjdμj∗

4th order terms in expansion of GL free energy with dμi and spin-fluctuation parameter (I);

ΔFABM =α 2

2β1 2 −1.05I( )

So, |ΔFABM| < |ΔFBW| for I > 0.46.

ΔFBW =α 2

2β1 5 3 − 0.33I( ),

(FABM – FBW) ≈ (1/2)(χABM - χBW)BAB2

<< kBTc

tiny free-energy difference

Interfacial energy between A and B phases

σAB = 0.7ξF0B

D.D. Osheroff and M.C. Cross, PRL 38, 905 (1977)

  Early measurements at melting pressure (P = 3.4 MPa)

T-sweep

T (μ

K)

GA – GB =2σABr0

r0 : orifice radius

B-sweep

r0 A

B ξ = ξ0(1-T/Tc)-1/2：コヒーレンス長

F0B : B相凝縮エネルギー密度

σAB = 0.38ξF0B

  More recent measurements at low pressure (P = 0)

M. Bartkowiak et al., PRL 93, 045301 (2004)

ξD = 10-40 ξ(T)

Order-parameter textures at the A-B interface

l-vector texture

A-phase

B-phase

  l-vectors in both phases are parallel each other at the interface.

  Long tailed only in the B-phase side (Δx ≈ ξD).

  Interface energy vanishes in the weak coupling limit.

: E.V. Thuneberg, PRB 44, 9685 (1991) GL calculations in B = 0

in B ≠ 0: N. Schopohl, PRL 58, 1664 (1987)

l-texture in a cylindrical container at B ≠ 0

B

non-zero order parameters

B = 0 P = TCP

planar-phase

A-phase B-phase

Heterogeneous nucleation at surfaces? w

all

l-vector

A-phase

A-phase

B-phase

  No impurities in bulk superfluid 3He.

  Energetically, A-phase is more stable near walls than B-phase.

→ less wall effects

 Near surface irregularities, the energy barrier for B-phase may be suppressed.

Heterogeneous nucleation seems to almost impossible.

However,

A.J. Leggett and S.K. Yip, Helium Three, edited by W.P. Halperin and L.P. Pitaevskii (North-Holland, Amsterdam, 1990), p.523.

Is this always true?

Detailed calculations show that those effects are not large enough to explain the observations.

Degree of supercooling of A-phase

 Nucleation does not depend on • cooling rate • rotation of sample • particle size of heat exchanger • presence of solid 3He

 Nucleation occurs only during the sample is cooling.

  Almost all data fall on a single "catastrophe line".

 The lowest AB transition ever reported was 0.63TB (1.6 mK) at melting pressure

"catastrophe line"

Baked Alaska model

1. Cosmic-ray muons passing through the sample chambers produce secondary electrons.

A.J. Leggett, PRL 53, 1096 (1984) A. J. Leggett and S. K. Yip, Helium 3, W. P. Halperin and L. P. Pitaevskii

(eds.), Elsevier Science Publishers B.V. (1990), p. 523. A.J. Leggett, JLTP 87, 571 (1992)

2

3

3. Quasiparticles travel ballistically away from the hot spot with the Fermi velocity.

ionizing radiation hypothesis

2. The secondary electron dumps most of its kinetic energy within a short distance just before stopping (hot spot).

This energy will overcome the barrier for B-phase nucleation.

4 4. Travelling quasiparticles form a hot shell protecting the inside

colder region. The inner region cools quickly below Tc, and occasionally B-phase appears there.

Baked Alaska model

5. If the size of hot cell exceeds Rc before dissipating, the inner B-phase bubble can expands freely subsequently.

Earlier experiments of Los Alamos group G.W. Swift and D.S. Buchanan, Jap. J. Appl. Phys. 26, Supplement 26-3, 1828 (1987)

D.S. Buchanan, G.W. Swift and J.C. Wheatley, Phys. Rev. Lett. 57, 341 (1986)

 No meaningful correlation between passage of cosmic-ray for the two detectors and B-phase nucleation (537 events).

But, the detectors cover only 30% of the total solid angle.

inconsistent with BA model

Mechanism of B-phase nucleation is still unresolved.

Why？ (14C in epoxy?)

 There were two preferable locations for nucleation in the sample cell.

reasonable (A phase is more stable in higher fields)

 Application of magnetic field (10-20 mT) reduced the A-B transition temperature.

Stanford experiment-1 on B-phase nucleation

  Use fused silica tube (without 14C) of small diameter with very smooth inner surfaces (roughness ≤ 10 nm < ξ << Rc).

  Nuclepore dust filters (pore ≈ 0.1 μm) and magnetic valves (NeFeB) at the top and bottom ends.

  Having longer and shorter tubes with different lengths by a factor of two, ... expecting deeper supercooling for the loner tube.

P. Schiffer et al., PRL 69, 120 (1992); Rev. Mod. Phys. 67, 491 (1995)

avoid "Boojum" texture by high magnetic field

NeFeB

m

agnet

Degree of supercooling of A-phase

Supercooled down to significantly lower temperatures.

◎ B = 3-10 mT M.T. O'Keefe, Ph.D. thesis (Stanford Univ.,1997) ▽ B = 0 H. Fukuyama et al., PRB 36, 8921 (1987) ○ B = 0 G.W. Swift and D.S. Buchanan, Jpn. J. Appl.

Phys. 26, Suppl.3, 1828 (1987) ◇ B =10 mT Swift and D.S. Buchanan (1987) □ B = 20 mT Swift and D.S. Buchanan (1987) △ B = 28 mT P.J. Hakonen et al., PRL 54, 245 (1985) × B = 57 mT P.J. Hakonen et al. (1985) + B = 5 mT R.L. Kleinberg et al., JLTP 17, 521 (1974)

◆ ■ B = 28 mT P. Schiffer et al., PRL 69, 120 (1992) N t( ) = N 0( )exp − t τ( )

T = 1.2 mK, P = 3.4 MPa

stochastic process! → determination of the lifetime

▽ ◎

"catastrophe line" : earlier exp.

Stanford exp-1 (1992)

Dramatic reduction of lifetime of supercooled A-phase by γγ-ray or neutron irradiation

Mean liquid temperature is unchanged.

γ

The short tube always survives longer than the long tube.

Local heating by neutron capture

3He + n → 3H- + p + 0.764 MeV

superfluid 3He

Stanford exp. strongly supports BA scenario

τ = 2.11×10−4 exp 5.25 Rc R0( )3 2{ }Rc T( ) =

R0 1−T Tc( )1 2

1−T TAB − Heff Hc( )2{ }Rc = 2σAB ΔG

Heff = χA χB( )1 2HHc = 0.63 T, R0 = 0.45 μm

: constant magnetization process

  All the T-dependences can be fitted well by the same functional form:

  A single mechanism should be dominant.

  Field dependence can also be explained by the same formula.

calculated for B = 100 mT

calculated for B = 14 mT

deviation

Stanford experiment-II

  6 fused silica tubes (2 mm o.d./1 mm i.d.) with controlled surface morphology

  removal of 3H from 3He sample

  Studies of secondary nucleation as well as primary one

9 cm

M.T. O'Keefe, B. Barker and D.D. Osheroff, Czech J. Phys. Suppl. S1 46, 163 (1996) M.T. O'Keefe, Ph.D. thesis (Stanford University, 1997)

#1. Empty, 0.47 Tesla, 9 cm long

#2. Empty, low field, 5 cm long

#3. Empty, 0.43 T, 9 cm long

#4. SiC grit (nominal size: 8 μm), 9 cm long

#5. Pitted silicon wafer (105 micropits of 1-5 μm diam. and 10 μm deep): loosely fixed, 9 cm long

#6. Blanked silicon wafer, 9 cm long

Stanford experiment-II

Data can be well fitted with the Baked Alaska model without adjustable parameters.

Empty tubes

with γ-ray

background

Pitted wafer tube

background

with γ-ray

105 micro-pits (1-5 μm dia., 10 μm deep) giving "Lobster-pot"

Memory effect in secondary nucleation in Stanford experiment II

Each curve denotes a temperature, where nucleation takes place with designated probability, calculated by Monte Carlo method based on the Baked Alaska model.

Pitted wafer tube

memory effect

Empty tubes

no memory effect

other mechanism than BA

Lancaster experiments on B-phase nucleation

  Irradiation (γ-ray, neutron) and heat pulses cannot trigger the B-phase primary nucleation.   Observed small but metastable under-demagnetization ≈ 30 mT.

M. Bartkowiak et al., PRL 85, 4321 (2000)

mm

B ≈ 0.34T P = 0 T ≈ 0.15 mK

• Could be the same reason why the life time of supercooled A-phase is very short at low pressures in Stanford exp.

• A sintered Ag dust found at the bottom of the tube may play important roles. • The sapphire tube was not assembled in a clean room.

  Only mechanical shocks can trigger the B-phase primary nucleation.

sapphire tube

• absence of normal component (or counter flow)

Primary and secondary nucleation

1 2 3 T (mK)

0

1

2

3

4

P (M

Pa)

A-phase

B-phaseNormal

Solid

TCP

U

primary nucleation

secondary nucleation

In most of experiments where higher-T nucleation was observed, the secondary nucleation took place at temperatures much closer to TAB than the primary nucleation events.

Memory effect

No memory effect only in Stanford experiments

The extrapolation of measured life time of supercooled A-phase cannot account for nucleation observed at much higher temperatures in the previous experiments.

There must be other mechanisms, which are far more efficient to nucleate B-phase than the Baked Alaska mechanism, in other experiments with much more surface irregularities. To observe Baked Alaska event, one must remove those irregularities carefully.

Yes Yes

Classification of existing experiments

Nucleation is stimulated by particle irradiation.

Nucleation occurs even in thermal equilibrium (reasonably deep supercooling).

Absence of memory effect

Lightly rough surfaces with B-phase pockets.

Stanford Exp-1

(smooth tubes) Schiffer et al.,1992

Lancaster experiment (sapphire tube

with a dust) Bartkowiak et al., 2000

Many other experiments

(dirty walls)

Supercooled A-phase is stable against other external disturbances.

Stanford Exp-2

(smooth tubes) O'Keefe et al.,1997

Stanford Exp-2

(rough tubes) O'Keefe et al.,1997

Leggett's ionizing radiation hypothesis

Absence of particle irradiation effect

Nucleation is stimulated by mechanical disturbances.

Yes Yes

Yes Yes

Yes

Yes

Yes

Yes

Extremely rough surfaces with B-phase pockets.

Yes Yes

Yes Yes

No No

No

No

Yes

memory effect

Yes

AA-B相転移でもK-Z機構は有効か？ BA機構以外の可能性

Yu.M. Bunkov and O.D. Timofeevskaya, PRL 80, 4927 (1998)

無数のB相核がパーコレーションしたときA-B転移が起こる。

F

Ａ相核、Ｂ相核が生まれる確率は、超流動3Heの18次元の秩序変数空間での自由エネルギー構造が決める。

急急速2次転移に伴う位相欠陥生成 (Kibble-Zurek機構)

・超流動3Heに粒子線(n, γ, β)を照射した後の、N相→B相急速2次相転移に伴う量子渦糸の観測

高温相 秩序相

秩序変数の位相欠陥：量子渦 ・超流動3Heと4Heで結果が矛盾 ・他の量子渦検出法を試す ・中性子核反応によるenergy depositの正しい見積もりとその時定数

再実験の必要性

ビビッグバン後の宇宙の急冷に伴う真空の相転移

位相欠陥 (宇宙ひも)の生成　⇨　宇宙の大規模構造として残存？

WMAP観測結果

Grenoble-Lancasterグループの実験 C. Bäuerle et al., Nature 382, 332(1996)

Grenoble-Lancasterグループの実験 V.M.H. Ruutu et al., Nature 382, 334(1996)

Phase diagram of superfluid 3He

1 2 3 T (mK)

0

1

2

3

4 P

(MP

a)

A-phase

B-phase

Normal

Solid

TCP

0

Stanford (1992,1997)

Gre

nobl

e-La

ncas

ter

(199

6)

0.5 Lancaster (2000)

Needs comprehensive survey of the whole phase diagram.

R0/l(Tc) << 1

BA model is relevant at

KZ model is relevant at

R0/l(Tc) >> 1

Stanford (1992,1997)

新新しい実験計画のポイント

• 最低温度 (T 0.13 mK) に固定して、磁場掃引(B 0.45T)でA-B転移を観測する。

　　　　　　← 高い制御性と時間の節約

• ST実験と同じ溶融石英管を使い、BAモデルをより広い圧力範囲(P = 0 - 3.4 MPa)で定量的に検証する。

• 通常物質 (合金等) の管も同時に設置し、catastrophe line の磁場依存性を観測する。

　← catastrophe lineの機構解明 (織目構造? Lobster pots?)

ベベークド・アラスカ (Baked Alaska) モデル

A.J. Leggett, PRL 53, 1096 (1984) Anthony James Leggett