Vortices at the A-B phase boundary in superfluid 3 He

R. H¨

anninen

Low Temperature Laboratory

Helsinki University of Technology

Finland

Theory:

E.V. Thuneberg

G.E. Volovik

Experiments:

R. Blaauwgeers

V.B. Eltsov

A.P. Finne

M. Krusius

Outline:

1. Introduction

2. Experimental setup

3. Hydrostatic theory in

He

4. A-B phase boundary

5. Results

6. Summary

Introduction:

•

two main phases: A and B phase

•

several textures, especially in the A phase

•

new experiments with the phase boundary

•

what is the effect of the A-B phase boundary?

•

we have calculated the effect on the A phase vortices

Hydrostatic theory in

3

He:

•

p-wave spin triplet

=

⇒

order parameter

A

is a 3

×

3 matrix

A phase order parameter:

A

= ∆

d

ˆ

( ˆ

m

+ iˆ

n

)

,

where ˆ

m

⊥

n

ˆ

•

vector ˆ

d

defines the axis along which the spin of the Cooper pair vanishes.

•

vector ˆ

l

= ˆ

m

×

n

ˆ

gives the orbital angular momentum axis

•

superfluid velocity:

v

=

h

¯

2

m

X

ˆ

m

∇

n

ˆ

Hydrostatic free energy:

F

=

R

d

r

(

f

+

f

+

f

)

•

dipole term:

f

=

−

1

2

λ

(ˆ

d

·

ˆ

_l

₎

,

•

magnetic anisotropy term:

f

=

1

2

λ

(ˆ

d

·

H

)

2

_,

•

kinetic terms + gradient energy density (

v

= 0):

2

f

=

ρ

v

+ (

ρ

−

ρ

)(ˆ

l

·

v

)

+ 2

C

v

·

∇

×

ˆ

l

−

2

C

(ˆ

l

·

v

)(ˆ

l

·

∇

×

ˆ

l

)

+

K

(

∇

·

ˆ

l

)

+

K

(ˆ

l

·

∇

×

ˆ

l

)

+

K

|

ˆ

l

×

(

∇

×

ˆ

l

)

|

+

K

|

(ˆ

l

·

∇

)ˆ

d

|

+

K

X

[(ˆ

l

×

∇

)

d

ˆ

]

.

Characteristic scales:

•

dipole length:

ξ

= (¯

h/

2

m

)

q

ρ

/λ

≈

10

µ

m

•

dipole field:

H

=

p

λ

/λ

≈

2 mT

•

dipole velocity:

v

=

q

λ

/ρ

≈

1 mm/s

B phase order parameter:

A

= ∆

R

(ˆ

n

, θ

)

e

,

•

R

rotation matrix around ˆ

n

with angle

θ

•

superfluid velocity:

v

=

h

¯

2

m

∇

φ

Hydrostatic free energy:

•

kinetic term (

v

= 0):

f

=

1

2

ρ

v

,

•

dipole term:

f

=

λ

(

R

R

+

R

R

)

=

⇒

θ

= 104

•

gradient term:

f

=

λ

∂

R

∂

R

+

λ

∂

R

∂

R

=

⇒

R

= const

Vortices in the B phase:

P

µi

|

A

|

2

•

singular (in the scale of

ξ

)

•

larger critical velocity (

v

v

)

A-B phase boundary:

•

requirements at the A-B boundary (normal ˆ

s

= ˆ

z

k

Ω

):

ˆ

d

=

_R

·

ˆ

s

( ˆ

m

+ iˆ

n

)

·

ˆ

s

=

e

ˆ

_l

_·

_ˆ

_s

_{= 0}

co

ordi

nate

sy

stem

where

v

=

0

Results:

•

calculations done for

v

= 0 deep in the A phase

•

assumed GL-region (

T

≈

T

)

•

minimization using conjugate gradient method

•

two different textures obtained

–

texture depends on the rotation velocity (density of the vortices at the boundary)

•

both textures have half-quantum vortex cores at the phase boundary

•

d

ˆ

≈

x

ˆ

everywhere

Low density texture (low rotation velocity):

0

2

4

6

8

0

1

2

3

4

B phase (z < 0)

A phase (z > 0)

y/

ξ

_d

z/

ξ

_d

Low density texture (low rotation velocity):

8

3

7

4

3

2

1

0

6

5

4

l

x

3

2

l

y

1

0

1

0

−1

l

m

n

y/

ξ

d

x/

ξ

d

z/

ξ

d

∇ ×

v

s

for the low density texture:

0

2

4

6

8

0

1

2

3

y/

ξ

d

z/

ξ

d

Superfluid current for the low density texture:

0

2

4

6

8

−1

0

1

2

y/

ξ

d

z/

ξ

d

A phase

B phase

High density texture (high rotation velocity):

0

1

2

3

4

5

0

1

2

B phase (z < 0)

A phase (z > 0)

y/

ξ

_d

z/

ξ

_d

High density texture (high rotation velocity):

5

2

2

1

0

4

l

x

3

2

l

y

1

0

1

0

−1

l

n

m

y/

ξ

d

z/

ξ

d

x/

ξ

d

∇ ×

v

s

for the high density texture:

0

1

2

3

4

5

0

1

2

y/

ξ

d

z/

ξ

d

Superfluid current for the high density texture:

0

1

2

3

4

5

−0.5

0

0.5

1

1.5

ξ

z/

ξ

d

A phase

B phase

Summary:

•

calculated the vortex structure at the A-B boundary

•

two different textures obtained (low density & high density)

–

half-quantum vortex cores

To be done:

•

generalize to other pressures

•

possible other textures??

•

calculate the NMR spectrum