RECENT TOPICS IN THE THEORY OF

SUPERFLUID 3He

Lecture at ISSP Tokyo 13.5.2003

Erkki Thuneberg

Department of physical sciences, University of Oulu

Janne Viljas and Risto Hanninen

Low temperature laboratory, Helsinki university of technology

Content

Introduction to superfluid 3He

I. Josephson effect

II. Vortex sheet in 3He-A

III. 3He in aerogel

These lecture notes will be placed at

http://boojum.hut.fi/research/theory/

Phase diagram of superfluid 3He

3He is the less common isotope of helium. It is a fermion.

0.001 0.01 100

2

1010.10.0001

3

4

1

0

super-fluid

phases

normalFermiliquid

solid

A

BPre

ssur

e (M

Pa)

Temperature (K)

gas

classicalliquid

Length scales

chargesmasses, ...

1 m

1 mm

1 µm

1 nm

1 pm

atom

nucleus

density,viscosity, ...

atomic scale

hydrodynamic

scale

nuclear scale

Length scales in superfluid 3He

1 m

1 mm

1 µm

1 nm

1 pm

atom

nucleus

density,superfluid density,stiffness parameters,viscosity parameters

density, Fermi velocity, interaction parameters ,Tc, ...

spin = 1/2charge = 2 emass = 5.01 10-27 kggyromagnetic ratio = 2.04 108 1/Ts

quantum theory

Ginzburg-Landau theory

quasiclassical theory

hydrodynamic theory

α', βi, K, γ

Cooper pairs in superfluid 3He

(- + )

( + )

i( + )

Spin wave functions (S=1)

Orbital wave functions (L=1)

x y

z

S =0:x

S =0:y

S =0:z

A xx A xy A xz

A yx A yy A yz

A zx A zy A zz

order parameter

Part I. JOSEPHSON EFFECT

superfluidL

weak link

superfluidR

∆φ = φR− φL

F ≈ −F0 cos(∆φ)

J ≈ J0 sin(∆φ)currentphase

differenceπ π π π

π π π π ∆φ

energy

superfluid orsuperconductor

ΨL = AL exp(iφL) ΨR = AR exp(iφR)

Experiments:

- superconductors: 1960’s

- 3He: Avenel and Varoquaux 1988

New experiments in 3He-B- S. Backhaus, S. Pereverzev, R. Simmonds, A. Loshak, J. Davis, R. Packard, A.Marchenkov (1998, 1999)- O. Avenel, Yu. Mukharsky, E. Varoquaux (1999)

∆φ

current

π π π π

π π π π ∆φ

energyπ state

Superfluid B phase

(- + )

( + )

i( + )

Spin wave functions (S=1)

Orbital wave functions (L=1)

x y

z

S =0:x

S =0:y

S =0:z

A xx A xy A xz

A yx A yy A = A0eiφyz

A zx A zy A zz

R xx R xy R xz

R yx R yy R yz

R zx R zy R zz

order parameter

amplitude(real)

phase factor(complex)

rotation matrix(real)

angle θ = 104 axis n

Pinhole

size ξ0 ≈ 10 nm

ξ0

J (∆φ,RLµj , RRµj)current

path of a quasiparticle

= 2m3AvFN(0)πT∑εn

∫d2k

4π(k · z )g(k,Rhole, εn)

Pinhole current-phase relation

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

T/Tc= 0.1 T/Tc= 0.1 0.20.3

0.40.2

0.3

0.4

0.5

0.5

0.7

0.7

0.8

0.8

0.9

0.90.6

0.6

phase phaseπ π 2π0

0

0.1

-0.1

0.2

0.3

0.4

0.5

π state

curr

ent

nn

π state occurs only at very low temperatures

⇒ not consistent with experiments

Aperture array

n

0 π 2π∆φ

Current

0 π 2π

∆φ

Energy

n

array of pinholes

spin-orbit rotationaxis

“Anisotextural” π state

Results

- asymmetry of the junction- one fitting parameter η

n n

-100

-50

0

50

100

Cur

rent

[ng/

s]

0 π 2πφ

0 π 2πφ

0.3Tc

0.9Tc

Experiment:S. Backhaus, S. Pereverzev, R. Simmonds, A. Loshak, J. Davis, R. Packard, A. Marchenkov (1998, 1999)

η

Theoretically strong dependence on magnetic field and surroundings- to be tested experimentally

Large aperture

x

cR

D

W

y

z

Ginzburg-Landau theory- valid only for T near- partial differential equation:

- at surfaces

Order parameter Boundary condition

when

=

A =

Axx Axy AxzAyx Ayy AyzAzx Azy Azz

∂j∂jAµi+(γ − 1)∂i∂jAµj = [−=

A+ β1

=

A∗ Tr(

=

A=

AT )

+β2

=

ATr(=

A=

AT∗) + β3

=

A=

AT

=

A∗ + β4

=

A=

AT∗=A+ β5

=

A∗=AT

=

A]µi

z → ±∞

=

A = exp(± i2∆φ)

1 0 0

0 1 00 0 1

Tc

=

A = 0

0 2ππ0 2ππ-0.5

0

0.5

curr

ent

0 2ππφ http://boojum.hut.fi/research/theory/jos.html

Part II. ROTATING SUPERFLUID

An uncharged superfluid cannot rotate homogeneously

Rotation takes place via vortex lines

Vortex cores in 3He-B

Numerical minimization of Ginzburg-Landau functional in 2 D

∂j∂jAµi+ (γ − 1)∂i∂jAµj = [−A + β1A∗Tr(AAT ) + β2ATr(AAT∗)

+β3AATA∗+ β4AAT∗A + β5A∗ATA]µi. (1)

⇒ broken symmetry in vortex cores

x

yy

0 1 2 30

1

2

3

4

normalfluid

temperature (mK)

pres

sure

(M

Pa) solid

A

B-superfluid

Virtual half-quantum vortices

x

y

z

half-quantum (π) vortex

half-quantum (π) vortex

The A phase

The order parameter Aµj = ∆dµ(mj + inj)

( +i )( + )

orbital wave function spin wave function

d

m n

A phase factor eiχ corresponds to rotation of m and n around l:

eiχ(m + in) = (cosχ+ i sinχ)(m + in)

= (m cosχ− n sinχ) + i(m sinχ+ n cosχ).

Superfluid velocity

vs =

2m∇χ =

2m

∑j

mj∇nj. (2)

Vortices in the A phase

Consider the structure

field:

Here l sweeps once trough all orientations (once a unit sphere).

⇒ m and n circle twice around l when one goes around this object.

⇒ This is a two-quantum vortex. It is called continuous, because ∆ (the

amplitude of the order parameter) vanishes nowhere.

Hydrostatic theory of 3He-A

Assume the order parameter (m, n, l, d) changes slowly in space. Then wecan make gradient expansion of the free energy

F =∫d3r

[− 1

2λD(d · l)2 + 12λH(d ·H)2

+12ρ⊥v2 + 1

2(ρ‖ − ρ⊥)(l · v)2 + Cv · ∇ × l− C0(l · v)(l · ∇ × l)

+12Ks(∇ · l)2 + 1

2Kt|l · ∇ × l|2 + 12Kb|l× (∇× l)|2

+12K5|(l · ∇)d|2 + 1

2K6[(l×∇)idj)]2

]. (3)

Vortex phase diagram in 3He-A

hard coresoft core

Continuous unlocked vortex (CUV)Vortex sheet (VS)

Singular vortex (SV)rotationvelocity(rad/s)

magneticfield (mT)

d≈constant

d≈constant

field:

field:

field:

d≈constant

0.50.250

2

0

4

6

8

= d field:

Locked vortex 1 (LV1) Locked vortex 3 (LV3)

xy

z

= d field:

Vortex sheet

Vortex sheets are possible in3He-A

Sheets were first suggested toexist in 4He, but they were foundto be unstable.

Why stable in 3He-A?

Dipole-dipole interaction (3)

fD = −12λD(d · l)2

⇒

or ⇒d dd d

domain wall’soliton’

Vortex sheet = soliton wall towhich the vortices are bound.

field:

d≈constant

Shape of the vortexsheetThe equilibrium configuration of thesheet is determined by the minimum of

F =

∫d3r1

2ρs(vn − vs)

2 + σA.

Here A is the area of the sheet and σ itssurface tension.

The equilibrium distance b between twosheets is

b =

(3σ

ρsΩ2

)1/3

. (4)

This gives 0.36 mm at Ω = 1 rad/s.

The area of the sheet

A ∝ 1

b∝ Ω2/3, (5)

as compared to Nvortex line ∝ Ω.

velocity

vn = Ω × r

vs

radiusb

Connection lines with the side wallallow the vortex sheet to grow andshrink when angular velocity Ω changes.

NMR spectra of vortices

potentialenergy

forspin waves

vortex sheet

bound stateΨ

Larmorfrequency

100 20

Frequencyshift (kHz)

NM

R a

bsor

ptio

n

Soliton

Singularvortex

Vortexsheet

Lockedvortex

http://boojum.hut.fi/research/theory/sheet.html

Part III. IMPURE SUPERFLUID 3He

3He is a naturally pure substance

Impurity can be introduced by porous aerogel

- strands of SiO2

- typically 98% empty

- small angle x-ray scattering ⇒ homogeneous on scale above ≈ 100 nm

Experimental phase diagram

0 1 2 30

1

2

3

4

Normal fluid

Superfluid

Temperature (mK)

Pre

ssur

e (M

Pa)

Solid

Porto et al (1995), Sprague et al (1995), Matsumoto et al (1997)

Models

Main effect of aerogel is to

scatter quasiparticles

Simplest model: homogeneous

scattering model

⇒ only qualitative agreement

with experiments

Better models:

Random voids - not simple enough to calculate

Periodic voids - not simple enough to calculate

trajectory of a quasiparticle

Isotropic inhomogeneous scattering - spherical unit-cell approximation - (quasi)periodic boundary condition

Isotropic inhomogeneous scattering model (IISM)

Isotropic on large scale

⇒ Hydrodynamics the same as in bulk but renormalized parameter values

Better fit with experiment than in HSM

⇒ realistic parameter values

Slab model

0 1 20.0

0.5

1.0

ξ0 /L

HSM

R = , steep

R = 2, steep

R = , gentle

TcTc0

http://boojum.hut.fi/research/theory/aerogel.html

Conclusion

Superfluid 3He forms a very rich system because of the several differentlength scales.

1 m

1 mm

1 µm

1 nm

1 pm

atom

nucleus

quantum theory

Ginzburg-Landau theory

quasiclassical theory

hydrodynamic theoryJosephson effectvorticessuperfluid in aerogel