Spin and Orbital

7/29/2019 Spin and Orbital

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DOI: 10.1007/s10909-005-8224-2

Journal of Low Temperature Physics, Vol. 141, Nos. 3/4, November 2005 ( 2005)

Coherently Precessing Spin and Orbital Statesin Superfluid 3HeB

S. N. Fisher1 and N. Suramlishvili1,2

1Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK

E-mail: [email protected]

Andronikashvili Institute of Physics, Tbilisi, 0177, Georgia

(Received June 29, 2005; revised August 4, 2005)

The Leggett equations for the spin dynamics of superfluid 3He give a gooddescription of the whole range of NMR phenomena observed at relativelyhigh temperatures. However these equations assume that the orbital angularmomentum of the condensate may only change on timescales much longerthan the spin precession period. At the lowest achievable temperatures, theorbital viscosity of the B-phase of superfluid 3He becomes vanishingly small,giving rise to the possibility of rapid orbital motion. We have reformulatedLeggetts equations for the B-phase to allow for fast orbital dynamics in theabsence of dissipation. The resulting non-linear equations of motion couplespin and orbital degrees of freedom resulting in qualitatively new phenomena.In particular, they allow for phase-locked precession of the spin and orbitalangular momentum around an applied magnetic field. The coupled spin-orbitdynamics may eventually explain the exotic ultra long-lived NMR signalsfound at the lowest temperatures in 3HeB.

KEY WORDS: Superfluid3

He; precessing spin-orbit system; Hamiltonian;Free energy.

1. INTRODUCTION

Superfluid 3He is formed by Cooper pairs in a triplet state with spin

and orbital angular momentum quantum numbers equal to unity. Con-

sequently, the superfluid exhibits not only the usual superfluid properties

arising from broken gauge symmetry but also displays macroscopic quan-

tum rotation phenomena arising from the broken spin and orbital rotation

symmetries. The superfluid phases are characterized by multidimensional

order parameters which describe rich equilibrium and dynamic properties.

One of these is coherent spin precession. The stability of this precession

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112 S. N. Fisher and N. Suramlishvili

is essentially governed by the spin stiffness of the order parameter and

by spin-orbit coupling. Spin supercurrents provide the feedback which acts

on the spin-orbit coupling stabilizing coherent spin precession even under

inhomogeneous external conditions. In this sense, superfluid 3He may be

considered as a spin superfluid.

Superfluid 3He, however, also has orbital angular momentum. The

orbital motion, being a mutual precession of neutral 3He atoms, has no

magnetic moment and therefore there is no analogue of the external mag-

netic field to set the system into precession. Nevertheless, the orbital and

spin moments are coupled via the spin-orbit interaction. Therefore, in

principle, orbital precession can be generated by spin precession. An orbi-

tal moment only appears in the B-phase on the application of an external

magnetic field which also generates axial anisotropy in the normal fluid

component along the orbital momentum axis. Consequently, orbital pre-

cession is accompanied by a redistribution of the normal fluid resulting in

dissipation which may be described in terms of an orbital viscosity. 1

The first homogeneously precessing domain (HPD) in 3He-B was

discovered experimentally2 in 1984 and was quantitatively explained3 by

Leggetts equations for spin dynamics.4 It was first thought that the HPD

should be the only stable mode of coherent precession at low tempera-

tures. However, experiments show that the HPD decays rapidly in the low

temperature limit.5 At the lowest temperatures an ultra long-lived mode

of precession is observed6 called the persistent precessing domain (PPD),

which can have a free decay lasting more than half an hour.7 The PPD

(formerly known as the persistent induction signal, PIS) has many unusual

properties,8 which are hard to understand within the framework of the

usual Leggett equations.

An important, but often overlooked, assumption in Leggetts origi-

nal formulation4 is that the orbital degrees of freedom are frozen. This

assumption is quite reasonable at high temperatures since orbital viscosity

rapidly damps any such motion.1 However, in the B-phase at the lowest

temperatures, the normal fluid fraction falls exponentially and orbital

viscosity becomes vanishingly small.1 In this case, the orbital angular

momentum becomes free to precess under the driving torque of the

dipoledipole interaction. Below, we generalize Leggetts Hamiltonian and

equations of spin dynamics to include the effects of orbital motion. This

leads us to confirm the existence of orbital precession which has been

suggested by experimental results on the PPD, at low temperatures.8 In

general, under inhomogeneous conditions, there will be associated orbital

supercurrents. So at low enough temperatures, in the absence of significant

dissipation, the B-phase can be considered as a combined mass, spin and

orbital superfluid. It is quite unique amongst condensed matter systems.


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Coherently Precessing Spin and Orbital States in Superfluid 3HeB 113

The plan of the paper is as follows. In Sec. 2, we obtain the

Hamiltonian and equations describing the coupled spin and orbital dynam-

ics under homogeneous conditions. We include here a discussion of the

physical meaning of the orbital momentum as used through out this paper.

In Sec. 3, we obtain a general expression for the free energy, under homo-

geneous conditions, and formulate the requirements for the general solu-

tions of the spin-orbit dynamics. In Sec. 4, we discuss how the, already

known, high temperature solutions arise from the general equations when

the orbital degrees of freedom are frozen. In Sec. 5, we discuss some of

the new solutions which arise at low temperatures when the damping of

orbital motion becomes negligible. In Sec. 6, we summarize our findings

and discuss briefly how these might relate to recent experiments.

2. GENERAL EQUATIONS OF MOTION

In the absence of an applied magnetic field, the B-phase has equal

populations of Cooper pairs with the different allowed spin and orbital

angular momentum projections. The nett spin and orbital angular momen-

tum densities are, therefore, both zero. A finite field polarizes the Cooper

pairs, producing a nett spin density S and, via spin orbit symmetry, a nettorbital momentum density L. The general dynamic states are classified byspecifying the spin and orbital momentum of the Cooper pairs relative to

the direction of the static magnetic field H. The B-phase is characterizedby the order parameter9

(k) = i0( d(k) )y , d(k) = Ri ki . (2.1)

Here 0 is the gap parameter, i are Pauli spin matrices, R is an orthogo-

nal matrix describing relative 3D rotations of spin and orbital spaces andk is the unit vector parallel to the quasiparticle momentum p. There are

many ways of parameterizing the rotation matrix. To describe the generalprecessing states, it is natural to introduce separate rotations R(S) and R(L)

in spin and orbital spaces10,11

Ri = R(S) (R

(L)i )

1 = R(S) R(L)i = d

u

i , (2.2)

where d = R

(S) and u

i = R

(L)i relate to spin and orbital degrees of free-

dom, respectively. The two rotations can then be parameterized by Euler

angles as discussed below.

The spin-orbit dynamics of the B-phase are described by the motions

of S and L together with their respective parts of the order parameter,d

and u

i . We assume, consistent with typical experimental conditions,

that the characteristic frequencies are small compared to the gap frequency


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/h and that the length scale for spatial variations is large compared to

the coherence length =hvF/. Under such conditions the magnitude ofthe order parameter is fixed so the dynamics correspond to rotations in

spin and orbital spaces. These rotations may be described by the angu-

lar velocities S and L of spin and orbital motions respectively, that isd = S d

and u = L u.

It is possible to find general equations describing spin and orbital

dynamics by considering conservation laws (continuity equations for the

spin and orbital momentum densities) and the symmetry properties of the

B-phase,12 in a similar manner to that done previously for spin dynamics

alone.13 Below, we obtain a closed set of equations for the dynamics by

investigating how the Hamiltonian varies with the spin and orbital rota-tions. We use a model Hamiltonian which has been used extensively to

study both spin and orbital collective modes in the superfluid phases1315

H = H0 + HL + HD, where

H0 =1

2m

d3r

(r) (r)

+1

2 d

3r d3r g(r r ) (r)

(

r ) ( r )(r),

HL = 1

2

d3r

(r) L (r), (2.3)

HD =2

2

d3r d3r

( 3ee )

|r r |3

(r)() (r)

(

r )( ) ( r ).

H0 describes the kinetic energy and the BCS-like pairing interaction,

where (r) is the field operator of3He atoms with mass m. HL is the

Larmor energy in the externally applied magnetic field H = L/, where

is the nuclear magnetic moment and L is known as the Larmor fre-quency. HD is the dipole interaction energy between the nuclear spins (the

spin orbit interaction energy), where e = (r r )/|r r |. We use units inwhich h = 1. The spin and the orbital angular momentum densities aredefined by

S= 12 (r) (r), (2.4)

L = 12 (r)l (r), (2.5)

where l = i r / r is the orbital angular momentum operator.In the absence of the dipole interaction, the Hamiltonian (2.3) is

invariant against separate rotations of spin and orbital spaces. Thus the


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original symmetry of the Hamiltonian is broken in the superfluid state.

This symmetry is formally restored by the existence of collective modes

(Goldstone bosons). Therefore it is natural to expect both spin and orbi-

tal collective modes in the B-phase at low temperatures. The dipole energy

will introduce a finite gap in the energy spectra of the corresponding

bosons.

We need to consider how the Hamiltonian transforms under the local

rotations R(S) and R(L) of spin and orbital spaces, respectively,

(r) R(S) R(L)(r). (2.6)

The rotations produce additional terms in the Hamiltonian, H H+

H1 + H2 which, to first-order, are given by Refs. 1315.

H1 = 1

2

d3r

(r)[ S + lL] (r),

H2 = i

4m

d3r

(r)

A

S + A

Ll

(r) + H.C.

,

(2.7)

where

S = 12

R(S) t

R(S) , lL = 12

R(L) t

R(L),

(AS)i = 12

R(S)i R(S) , lk(AL)ik = 12

R(L)i R(L).(2.8)

The first term, H1, gives the change due to time variations of the

rotations which subsequently generate spin and orbital polarizations ( Sand L). The second term, H2, produced by spatial variations, generatespin and orbital super-currents. Under typical experimental conditions the

angular velocities are small compared to the gap frequency, |S|, |L| /h. Also, the length scale for spatial variations is large compared to the

coherence length, =hvF/. Therefore, the additional terms H1 and

H2 are small compared to the model Hamiltonian (2.3). In the follow-ing, we will only consider homogeneous conditions. The gradient terms

and associated super-currents will be discussed elsewhere.12

The free energy associated with the homogeneous motion of the spin

and the orbital parts of the order parameter is obtained from the pertur-

bation H1. In Appendix A, we show that this can be written as

F= H1 =

d3r

1

2 SS S +

1

2 LijLi Lj +

SLj S Lj

, (2.9)

where S, Lij and

SLj are the static spin, orbital and spin-orbit corre-

lation functions. As discussed in Appendix B, the spin correlation func-

tion is simply related to the static spin susceptibility B of the B-phase by


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S = B where, ignoring Fermi liquid corrections,

B =14

N (0) 2 + Y ( T )3

. (2.10)

Here N (0) is the density of states at the Fermi surface and Y ( T ) is the

Yoshida function which falls from unity at the transition temperature to

zero at T = 0. In Appendix B, we show that the orbital and spin-orbit cor-relation functions are given by

Lij = 14 N (0) 23 (1 Y (T ))ij,

SLj = 14

N (0) 23

(1 Y (T ))d ui .

(2.11)

The change in the free energy (2.9) determines the kinetic energy of the

spin and orbital motions. In the presence of the dipole interaction, local

rotations of spin and orbital spaces introduce an additional potential

energy, HD =

d3rFD, where the dipole interaction energy is

FD =2

152B

trR

1

2

2.

Here B is known as the longitudinal resonance frequency of the B-phase.

To obtain the effective Hamiltonian describing the coupled spin-orbit

dynamics, we have to relate the velocities of spin and orbital rotations

to the spin and orbital angular momentum densities. This is derived in

Appendix C. In the low temperature limit (i.e., assuming a negligible nor-

mal fluid component) and using units where B = = 1, we find that

S = L + S12 {

S+ d(u L)},

L = 12 {

L + u( d S)}.(2.12)

The total free energy associated with spin-orbit dynamics in the

B-phase can now be written as Ftotal = d3rHeff, where the effective

Hamiltonian is given by

Heff =14

S2 14L2 L S

12

( Sd)( Lu) + FD . (2.13)


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The corresponding equations of motion become

S = S L +FD

d d, (2.14)

L =FD

u u, (2.15)

d = d

L S+12 {

S+ d(u L)}

(2.16)

u = 12 u

L + u( d S)

. (2.17)

These four equations describe, self-consistently, the motion of S, L, d,and u under homogeneous conditions for the general case in whichorbital motion is allowed. Under inhomogeneous conditions, both spin

and orbital supercurrents must be included. This will be considered else-

where.12 The first two Eqs. (2.15) and (2.16), represent Newtons laws for

the spin and the orbital momenta. The last two Eqs. (2.17) and (2.18),

express explicitly the angular velocities, S and L, of d and u in termsof S and L. The equations obtained by Bunkov and Golo,16 who have alsoinvestigated how the Leggett equations are modified when orbital motion

is allowed, differ from those given here since they did not account for the

effects of orbital dynamics on the order parameter.

The terms in curly brackets in Eqs. (2.13), (2.17), and (2.18) may be

considered as representing the total angular momentum J= L + R S in spinand orbital spaces appropriately. At high temperatures, orbital motion is

heavily damped1 so L = 0 and the total angular momentum of the con-densate is zero. The equations of motion in this case reduce to Leggetts

equations [i.e. Eqs. (2.15) and (2.17) after setting the curly bracket to

zero]. However, from Eqs. (2.13) we see that, in general, J = 2L andthus rotation of the orbital parts of the order parameter induces a non-

zero total angular momentum (as discussed in Appendix C). In this case

all four Eqs. (2.15)(2.18), are required to determine the coupled spin-orbit

dynamics.

We end this section with a comment on the orbital angular momen-

tum L as defined by Eq. (2.5) and evaluated in Appendix C. This is theintrinsic (internal) orbital angular momentum of the Cooper pairs and

does not contain contributions from macroscopic supercurrents (which

will, in general, arise under inhomogeneous conditions). It is generated

by the magnetic field and by spin-orbit rotations of the pair wave func-

tion. The orbital rotations imply the internal rotations of Cooper pairs

and do not include their center-of-mass motion. The dynamics of the

orbital momentum is defined by the dipole torque acting in orbital space


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as given by Eq. (2.16). This was previously introduced phenomenologically

by Bunkov and Golo.16 The orbital momentum defined in this way cor-

responds directly to the average value of the operator K introduced by

Leggett and Takagi in their theory of orbital dynamics of the A-phase.17

In the absence of spin dynamics, Yip,18 starting from the same definition,

obtained an identical result for the orbital momentum, as did Combescot

and Dombre19 using a rather different approach.

3. GENERAL DESCRIPTION OF STATIONARY SOLUTIONS AND

THE FREE ENERGY OF THE PRECESSING SPIN-ORBIT SYSTEM

Rotations of the order parameter in spin and orbital spaces may be

described by the Euler angles (S, S, S), and (L, L, L), respectively.Following the usual definition of the Euler angles we can set

RS = R(S, S, S) = Rz(S)Ry (S)Rz(S),

RL = R(L, L, L) = Rz(L)Ry (L)Rz(L),(3.1)

where Rz(S) is a matrix describing a rotation by the angle S about the

z axis etc. It is convenient to introduce rotating coordinate systems in spin

and orbital spaces (S, S, S) and (L, L, L) rigidly coupled to the vec-

tors d

and u

, respectively. This is analogous to the coordinates oftenused in describing spin dynamics alone.20 The Euler angles are canoni-

cally conjugate to the projections (Sz, S, S ) and (Lz, L, L ), respectively,

where Sz is the projection of S on to the z (field) axis, S is the projectionon to the axis S= R

(S) z and S is the projection on to the axis S= S z.Analogous definitions apply for L. The angles are illustrated in Fig 1a. Wenote that L = R

(L)z is the orbital anisotropy axis in the B-phase.10 This

H

L

L

L

L

L L

(a) (b)

Fig. 1. (a) Shows the Euler angles and axes used to describe the orbital rotations. Analogouscoordinates are used to describe spin rotations. (b) Shows the spin and orbital momentum

precessing around the field axis at a common frequency with a finite total angular momen-

tum corresponding to the particular solution described in Sec. 5.1.


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is the symmetry axis for the changes in the quasiparticle excitation ener-

gies induced by the magnetic field and its motion is damped at higher tem-

peratures by orbital viscosity.1

The effective Hamiltonian (2.14), when expressed in terms of these

angles and associated projections, is a function of the angular variables

S, L, S, and L only through the subtractions = S L and =S L. To describe stationary solutions, it is more convenient to expressthe Hamiltonian in terms of the variables S, , S, and . After perform-

ing a canonical transformation with the generating function fder = SPz +(S L)Qz + s P + (S L)Q, we move to new variables S, andS, and to the corresponding momenta Pz = Sz + Lz, Qz = Lz, andP

= S

+ L

, Q

= L

. Now we have the following pairs of canonically

conjugate variables: (S, Pz), (S, P),(S, S ),(, Lz), (, L),and (L, L ). The effective Hamiltonian becomes

H =1

4sin2 S[(P L)

2 + (Pz Lz)2 2(P L)(Pz Lz) cos S]

+1

4S2 L(Pz Lz)

1

4sin2 L[L2 + L

2Z 2LzL cos L]

1

4L2

cos

2 [(Pz Lz) (P L) cos S](Lz L cos L) + S L

sin S sin L

sin

2

L

sin S[(Pz Lz) (P L) cos S]

S

sin L(Lz L cos L)

1

2(P L)L + FD, (3.2)

where

FD =2

152B

cos S cos L

1

2 +1

2(1 + cos S)(1 + cos L) cos( + )

+1

2(1 cos S)(1 cos L) cos( ) + sin S sin L(cos + cos )

2.

(3.3)

The equations of motion generated by the Hamiltonian (3.2) describe the

dynamics of the angular variables given by;

S =HPz

, S =HP

, S =HS

,

= HLz , = H

L, L =

HL

.(3.4)


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The canonically conjugate projections of the moments are given by;

Pz = HS , P =

HS , S =

HS ,

Lz =H

, L =H

, L = HL

.(3.5)

These equations are given explicitly in Appendix D.

The variables S and S do not appear in Hamiltonian (3.2) explicitly.

They play the role of cyclic variables and, consequently, the corresponding

moments Pz and P are conserved, becoming integrals of motion. We wish

to find stationary solutions which satisfy the conditions:

Pz = P = S = Lz = L = L = S = L = 0. (3.6)

Let us define four angular frequencies

S = p, = , S = s , = . (3.7)

Of these, only the spin precession frequency p can be directly measuredin an NMR experiment. For stationary solutions, the other frequencies

must correspond to either absolute or local minima of the free energy. The

general solutions for the orbital momentum projections which satisfy the

conditions (3.6) are given by:

Lz = (p ) + ( 2s ) cos L

+ (cos S cos L + sin S sin L cos ),

L = (p ) + ( 2s ) + cos S,

L = sin S sin . (3.8)

Here = p L is the frequency shift which arises when there is a finitedipole torque. From the definitions of Pz and P, the corresponding solu-

tions for the projections of the spin are given by:

Sz = (p )(cos S cos L + sin S sin L cos ) + cos S ,

S = (p ) cos L + cos S,

S = (p ) sin L sin

(3.9)


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Also, from the conditions (3.6), the solutions must satisfy:

FD

= 0, (3.10)

FD

= (p ) sin S sin L sin , (3.11)

FD

S= (p )(sin S cos L cos S sin L cos )

+ sin S, (3.12)

FD

L= (p )(sin L cos S cos L sin S cos )

+ (p )( 2s ) sin L. (3.13)

We note that the general solutions for the moments (3.8) and (3.9), are

strongly dependent on the four frequencies p, , , and s . This is in

contrast to the high temperature regime where the orbital momentum is

strongly damped and spin dynamics only depend on the two frequencies

p and s . Consequently, there are a much greater variety of solutions at

low temperatures.

The free energy corresponding to the stationary conditions (3.6) is:

F = H+ pPz s P Lz + L, (3.14)

where H is the Hamiltonian (3.2). Substituting the general solutions (3.8)

and (3.9) gives:

F = 12 (p )2 12

2 + ( s )

2 + (p )( 2s ) cos L

+ (p )(cos S cos L + sin S sin L cos )

+ cos S + FD 12

2. (3.15)

Under typical experimental conditions, p

, so the first four terms usu-ally dominate the free energy. These correspond to the kinetic energies of

the rotations of the spin and orbital momenta and their respective parts of

the order parameter. The other, smaller, terms arise from the dipoledipole

interaction and from the so-called spectroscopic energy. The spectroscopic

energy (which may be written more simply as Sz) describes the action of

the magnetic field on the spin momentum in the precessing frame. 10,21,22

The stationary precessing states develop in either the absolute or in

the local minima of the free energy. For the case where orbital motion

is suppressed (i.e. at high temperatures) many of these states have been

extensively studied both experimentally2,3 and theoretically.11,21,22,24,25.

Below, in Sec. 4, we show how these states arise within the framework of

the general theory given above.


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At very low temperatures we have to allow for the possibility of orbi-

tal motion. This leads to qualitatively new behavior and an entirely differ-

ent set of coupled spin-orbit stationary states. In Sec. 5, we discuss some

of the properties of these new states and how they might arise from the

well known conventional states at higher temperatures.

4. STATIONARY SOLUTIONS WITH STATIC ORBITAL DEGREES

OF FREEDOM

Here we consider the solutions when the orbital degrees of freedom

are assumed to be static. This is the approximation which is usually made

and is valid only at sufficiently high temperatures where the orbital angu-

lar momentum is clamped by orbital viscosity.1 As discussed above, in this

case the equations of motion (2.15)(2.18) reduce to the usual Leggett

equations4 and the total angular momentum J = L + R S is zero. Theapproximation is equivalent to setting = p and = s correspondingto stationary L while S precesses around the magnetic field H with fre-quency p. In this case, the free energy (3.15) reduces to:

F = 12 2s + s cos S + FD . (4.1)

Here we have omitted the term containing 2 as we assume p andso this term plays no role in the subsequent solutions. The orbital and

spin momentum projections become:

Lz = s cos L + (cos S cos L + sin S sin L cos ),L = s + cos S,L = sin S sin

(4.2)

and

Sz = s cos S ,S = s cos S,S = 0.

(4.3)

The dipole interaction allows three classes of solutions with static orbital

degrees of freedom. Each corresponds to a different resonance between the

dynamics of the spin momentum and the spin parts of the order param-

eter. The resonances occur at the frequencies s = 0, s = p/2, and s =p. Each has a different combination of and which acts as the slow

variable (slow on the time scale of the spin precession). Each resonance

also has a different magnitude for the spin and orbital momentum.


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4.1. Zero Magnetization States

When is the slow variable, s = 0 gives a local minimum of the

free energy. This corresponds to the so-called zero magnetization precess-ing states.24,25 From Eqs. (4.2) and (4.3), we see that the magnitudes of

the spin and orbital momenta are rather small, | L| = | S| .

4.2. Half Magnetization States

When the slow variable is = + 2, then s = p/2 gives alocal free energy minimum. This is the so-called half magnetization state

which was theoretically proposed 11 and experimentally observed 23 quite

recently. In this case the dipole energy FD averaged over the fast variablesbecomes11

FD,S=1/2 =1

102B

1 + 2cos2 S cos

2 L + sin2 S sin

2 L

+2

3sin S sin L(1 + cos S)(1 + cos L) cos

. (4.4)

From Eqs. (4.2) and (4.3) we find that the magnitudes of the momenta

are roughly half of their equilibrium values, | L| = | S| p/2 (assuming,as is usually the case, that the field is relatively large so the precession

frequency is much larger than the frequency shift).

4.3. Equilibrium Magnetization States

When the slow combination of and is = + then we havesolutions with s = p. This gives us the usual situation observed in mostNMR experiments where the magnitudes of the spin and the orbital angu-

lar momenta are close to their equilibrium values, | S| = | L| p. The aver-age dipole energy becomes

FD =2

152B

cos S cos L +

1

2(1 + cos S)(1 + cos L) cos

1

2

2

+1

8(1 cos S)

2(1 cos L)2 + sin2 S sin

2 L(1 + cos )

. (4.5)

This expression was used by Volovok.22 to find a series of metastable

solutions, corresponding to local minima of the free energy, in addition

to the well-known (HPD like) solution2,3 corresponding to the absolute

minimum. The equilibrium magnetization states are the most energetically


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favorable. The free energy is

F =

2p

2 + p cos S + FD . (4.6)

The stationary values of S and L are found by minimizing the dipole

energy (4.5) with respect these variables. The solutions which correspond

to absolute minima of the dipole energy are 22

cos L = 1, 14 cos S1, (4.7)

cos S = 1, 14 cos L1. (4.8)

The HPD corresponds to the first solution (4.7) where the orbital momen-

tum is along the magnetic field direction around which the deflected spin

precesses. When the spin deflection exceeds the magic Leggett angle,

cos1(1/4), the dipole torque increases the precession frequency. In thecase of HPD, this compensates for any field gradient allowing coherent

precession at the global frequency p. The second solution (4.8) is the sta-

tionary domain where the spin is along the magnetic field direction and

the orbital momentum may be deflected.

5. STATIONARY STATES WITH DYNAMIC ORBITAL DEGREESOF FREEDOM

We now consider the more general problem in which the orbital

degrees of freedom are also free to precess. This situation will occur at

sufficiently low temperatures where orbital viscosity is not large enough to

clamp the orbital degrees of freedom.1 In this case = p and = s .With more degrees of freedom, the solutions to this problem are far more

complex and varied. Below, we will concentrate on some particular types

of solutions which believe will be more relevant to experiments.

Minimizing the dipole energy with respect to [Eq. (3.10)] gives

cos =(1 + cos S cos L) cos + sin S sin L

1 + cos S cos L + sin S sin L cos . (5.1)

The corresponding dipole energy minimum is zero provided that cos

(1/4), and is otherwise given by

FD =8

152B

1

4+ cos

2, (5.2)

where

cos = cos S cos L + sin S sin L cos . (5.3)


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Here is the rotation angle around the axis ni = i l d u

l /2sin which

moves the triad of orbital vectors (ux , uy , uz) to the triad of spin vectors( dx , dy , dz). This is analogous to notations often used at higher temper-atures where the rotation matrix can be considered to represent a relative

rotation of spin and orbital coordinates by an angle about an axis n.In general, the frequencies , , and s can have arbitrary values.

To search for stationary solutions which are likely to exist in real experi-

ments, we will try to find solutions which can be continuously generated

from the conventional high temperature solution discussed in Sec. 4.3.

There, the stationary solutions were shown to correspond to resonances

between the dynamics of the spin and the spin parts of the order param-

eter. The type of resonance was characterized by the relationship between

the frequencies S = p and S = s . The free energy was at its abso-lute minimum when the two frequencies were equal. We expect that when

the orbital degrees of freedom become active, similar resonances between

the dynamics of orbital angular momentum (characterized by L) and the

orbital part of the order parameter (characterized by L) will develop.

Such a set of resonances must correspond to absolute or local minima of

the free energy (3.15).

Below we consider two types of resonances. In Sec. 5.1, we consider

the case where the spin and orbital angular momenta precess coherently

at a frequency p while their respective parts of the order parameter also

precess coherently but at a different frequency s . In Sec. 5.2, we consider

various solutions corresponding to a resonance in which the spin and orbi-

tal momenta precess at the same frequencies as their respective parts of

the order parameter, but the frequencies of the spin and orbital motions

may differ.

5.1. Coherent Spin-orbit Precession

First, we consider the case = = 0. This corresponds to the orbi-tal angular momentum precessing around the magnetic field coherently

with the spin at a frequency p while the orbital parts of the order param-

eter precess coherently with the spin parts at a frequency s . The free

energy in this case may be written as

F = 12 2p +

2s 2ps cos L + p cos + FD, (5.4)

where is defined by Eq. (5.3). Minimization of this with respect the angu-

lar variables shows that when the dipole energy is finite, i.e., when cos

(1/4), the orbital anisotropy axis is parallel to the field (cos L = 1) and


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the spin axis is deflected to an angle S which determines the frequency shift

= 1615

2B

p

14

+ cos S

.

This is the same expression for the frequency shift as that which occurs

at high temperatures described by the usual Leggett equations. However,

now the orbital momentum has, in addition of the longitudinal component

Lz = p 2s + cos S, a small transverse component L = sin Sexcited by the dipole torque. This precesses coherently with the spin at

a frequency p. This solution is illustrated in Figure 1b. The precession

of the orbital parts of the order parameter induce a finite total angularmomentum J as described by Eq. (2.13). In this case J is parallel to thefield. As s decreases from p, | J| increases from zero and Lz becomesmore positive, passing through zero when S p/2. As s approacheszero, the total angular momentum is maximized, | J| 2p and the orbi-tal momentum becomes almost parallel to the field with Lz p. Whenthe spin has a smaller deflection, it precesses at exactly the Larmor fre-

quency, the dipole torque disappears and the orbital momentum becomes

static and vertical with magnitude |p 2s |. The minimum value of the

free energy (5.4), is achieved when s = p. So in this case the resonanceresembles that which occurs for the high temperature equilibrium spin

precessing states discussed in Sec. 4.3. The free energy in this case also

reduces to that of the high temperature expression (4.6). Indeed, physi-

cally this state is identical to that considered in Section 4.3 with cos L = 1and since the orbital anisotropy axis is static, there is no extra dissipation

from orbital viscosity. The solutions however arise in very different man-

ners and have very different stabilities towards perturbations in L.

Before considering other types of solutions, we comment on the pos-

sibility of observing this state experimentally. As we have already stressed,a proper treatment of real experimental conditions will require inhomoge-

neous solutions which are outside the scope of this article. However, one

might anticipate that near the axis of a long cylinder which is parallel to

an applied field (the exotic PPD signals are found experimentally to be rel-

atively stable here), the initial texture will be almost vertical and hence

the solutions discussed above might be excited. One would expect that the

free energy should not deviate very far from its absolute minimum (sshould be close to p). However, in practice s will be determined by the

conservation of P and the initial conditions which will depend not onlythe magnetic field profile, but also the initial texture in the experimental

cell.


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5.2. Stationary Solutions of Spin and Orbital Precession at Different

Frequencies

In the low temperature limit the orbital degrees of freedom are freeto precess anywhere except near the walls of an experimental cell. Close

to the walls, surface energy clamps the orbital anisotropy axis along the

normal to the wall so orbital motion is suppressed. Here, stationary solu-

tions are given by the non-precessing domain discussed in Sec. 4.3 with

cos L = 0, cos S= 1, s = p, and = = p. Experimentally, the longlived PPD signals at low temperatures can not be excited close to the walls

of a cell.26 Over some bending length from the wall the direction of the

orbital anisotropy axis becomes preferentially parallel to the field direc-

tion27 and free to precess at low temperatures.1 Far from the cell walls,cos L = 1 and the solution discussed above with s = p and = = 0may develop. Between these two extremes, in the region where 0 < cos L )

sin L

=(p or b)

(p + orb)orbsin

S. (5.15)

This relationship is shown schematically in Fig. 2. At lower orbital fre-

quencies, sin L = 1. As discussed above, owing to the relatively small mag-nitude of the dipole torque, as the orbital precession frequency increases,

the deflection of the orbital angular momentum and the anisotropy axis

must decrease.

orb/p

sinL

orbsinS

0 10.5

0.5

1

Fig. 2. The relationship between the deflection of the anisotropy axis L and the precession

frequency of the orbital degrees of freedom for the solution described in Sec. 5.2.4.


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6. SUMMARY

The Leggett equations, which give a good description of NMR

phenomena at higher temperatures, represent the spin dynamics of thesuperfluid phases of liquid 3He as a coupled motion of the spin (nuclear

magnetization) and the spin part of the order parameter. We have extended

these equations to include the low temperature regime where orbital

degrees of freedom may also be involved in the motion (orbital viscos-

ity which damps the motion at higher temperatures, becomes vanishingly

small at very low temperatures). The resulting equations couple spin and

orbital momenta along with their respective parts of the order parameter.

The dynamics of the spin can be driven by the externally applied magnetic

field. This then drives orbital dynamics via the dipole torque arising fromthe spin-orbit interaction. We have found several types of stationary solu-

tions for the coupled spin-orbit dynamics under homogeneous conditions.

The characteristic frequency of orbital motion depends on the deviation of

the orbital anisotropy axis from the direction of the magnetic field. In

spite of the smallness of the dipole torque, orbital motion can be excited

at close to the spin precession frequency provided that the deviation of

l from the field direction is correspondingly small, and consequently the

transverse component of the orbital momentum L is also small. Orbital

motion which is coherent with the spin motion is also possible when orbi-tal anisotropy axis is exactly parallel to the external magnetic field. In this

case, the orbital momentum L retains a small residual transverse compo-nent excited by the dipole torque.

Experimentally, orbital motion can be inferred by its influence on the

spin dynamics which is observed directly in NMR measurements. In prin-

ciple, the orbital dynamics ought to be observable directly by, for instance,

its effect on the quasiparticle excitations. In practice, however, this would

be difficult to observe since the field has a relatively small effect on the

quasiparticle energies.Finally, we would like to speculate on how the stationary solutions,

which we have discussed above, might be related to existing NMR exper-

iments at very low temperatures, namely the exotic PPD signals. Early

observations of these signals6 found them to be highly irreproducible.

However, more recently, very reproducible and extremely long lived sig-

nals7 have been excited in a cylindrical chamber with its axis along the

field direction. These signals are found to be only excited in a field min-

imum28 and decay much more rapidly when placed close to a cell wall.26

This finding alone suggests that orbital motion is very likely to be involvedin the long lived signals since the orbital momentum is clamped at the cell

walls. One of the outstanding properties of these signals is that they can


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be excited and driven by an excitation frequency which can differ substan-

tially from the signal frequency8 (the excitation frequency is always higher

than the signal frequency and the frequency difference can vary contin-

uously from a few H z to in excess of a few kHz). More recently, it has

been found that the PPD signals can even be excited by applying white

noise to the excitation coil.29 At first sight it is very difficult to under-

stand how energy can be injected into the superfluid at one frequency and

excite motion at a different, seemingly unrelated, frequency. It was spec-

ulated8 that orbital motion might be involved since having another cou-

pled energy bath might allow for a different output frequency. The above

solutions show that this can indeed occur. The spin and orbital motions

can have different frequencies. The frequency of the spin precession is

determined by the applied field (the Larmor frequency) plus the frequency

shift arising from the dipole interaction. However, we have shown that the

frequency of the orbital motion can vary continuously depending on the

orientation of . The two systems possess substantial energies and are cou-

pled by the dipole interaction. This allows energy to be absorbed by the

spin system at one frequency, close to the cell walls for instance where the

(non-stationary) spin precession frequency is relatively high, transferred to

orbital precession at a different (lower) frequency and then transferred via

super-currents to other regions (away from the cell walls) where stationary,

long-lived coupled spin-orbit precession may be excited.

A better comparison with existing experiments at low temperatures

will require solutions under inhomogeneous conditions. The freeing of

the orbital degrees of freedom will, almost certainly, reveal many new

phenomena in the low temperature regime where the B-phase effectively

becomes a combined mass, spin and orbital superfluid.

APPENDIX A. DERIVATION OF THE FREE ENERGY

ASSOCIATED WITH THE SPIN AND ORBITAL ROTATIONS

The free energy associated with the shift in the Hamiltonian H1 due

to homogeneous rotations is calculated as a perturbation:

F = T

T10

dH1( )con

+T

2

T10

d1

T10

d2T{H1(1)H1(2)}con, (A.1)

where H1( ) is the Hamiltonian H1 expressed in the interaction repre-

sentation, , 1, and 2 are imaginary times, T is the Wicks time-order-

ing operator and T is the temperature. The suffix con implies that only


23/31


connected diagrams are used. The first term in Eq. (A.1) vanishes identi-

cally, while the second term can be written as

F =T

2

T1

0

d1

T1

0

d2

d3r1

d3r2

1

4{T( (1) (2))[S S ]

+T(li (1)lj(2))[Li Lj] + T((1)lj(2))[S Lj]

+T(li (1) (2))[Li S]}. (A.2)

Here (1) and (2) refer to the four coordinates (r1, 1) and (r2, 2), respectively.In the low temperature limit, the thermal quasiparticle density becomes

vanishingly small and this expression simplifies to

F = 12

d3r1

d3r2{

S (0)[S S ] +

Lij(0)[Li Lj]

+ SLj (0)[S Lj] + SLi (0)[Li S ]}, (A.3)

where

S(0) =1

4

T10

dT((r1,i) (r2, 0)), (A.4)

Lij(0) =

1

4T1

0dT(li (r1,i)lj(r2, 0)), (A.5)

SLj (0) =1

4

T10

dT((r1,i)lj(r2, 0)). (A.6)

The correlation functions are approximated by local correlation functions:

S(0) = S(r1 r2),

Lij(0) = Lij(r1 r2), (A.7)

SLj (0) = SLj (r1 r2).

So the additional free energy becomes

F =

d3r

1

2 SS S +

1

2 LijLi Lj +

SLj S Lj

. (A.8)

APPENDIX B. CALCULATION OF THE CORRELATION

FUNCTIONS

The calculation of S has already been discussed by Maki14 and by

Balian and Werthamer.30 We will not repeat a similar calculation here, but

simply quote their result:


24/31


S = SB ,

SB =

1

4N (0)

2

3+

1

3Y ( T )

. (B.1)

Here, Y ( T ) is the Yoshida function defined as

Y ( T ) = 1 2 T

n=0

2o

(2n + 2o)

3/2, (B.2)

where n are the Matsubara frequencies.

To calculate the other correlation functions, it is convenient to use the

Greens function describing the B-phase:14

G1o = in 3 22(Rk),

Go = 1

2n+2+2o

(in + )I ,

, (in )I

.

(B.3)

Here = (1/2m)p2 , i and i are Pauli matrices operating on particle-

hole and on spin spaces respectively, is the order parameter given byEq. (2.1) and I is the two dimensional unit matrix. The Pauli spin matri-

ces and the orbital angular momentum operator L are defined in the

four-dimensional representation as:

=

, 00,

, L =

l, 0

0, l

, (B.4)

where l is the orbital angular momentum operator defined as l = i p

p .In this representation,

Lij =1

4

T10

dT(li (i),lj(0))

= T

n

d3p

(2 )3T r{Li Go( p, n)LjGo( p, n)}. (B.5)

Here T r is the trace in both particle-hole and spin spaces. The integral is

carried out by replacing d3p by N (0)(d/4)d. After substituting Eqs.

(B.3) and (B.4) in Eq. (B.5) and taking the trace in particle-hole space we


25/31


obtain,

Lij =

N (0)

4 T

n

d

d4

2o

(2n + 2 + 2o)

2

T r( + ( ))li dljd

=N (0)

42 T

n=0

2o

(2n + 2o)

3/2

d

4

i k

k

i

Rpkp

i k

k

j

Rq kq

=

N (0)

4 (1 Y(T)) d

4

k

k

ikp

k

k

jkp. (B.6)

Here T r is the trace in spin space. We finally obtain the orbital correla-

tion function as

Lij = 1

4N (0)

2

3(1 Y (T ))ij. (B.7)

The spin-orbit correlation function is found in the same manner:

SLj =1

4

T1

0

dT((i),lj(0))

= T

n

d3p

(2 )3T r{Go( p, n)LjGo( p, n)}

=N (0)

4T

n

d

d

4

2o

(2n + 2 + 2o)

2

T r( + ( ))dljd

= N (0)4

2 T

n=0

2o

(2n + 2o)

3/2

i Rq

d

4Rpkp

i k

k

j

Rq kq

= N (0)

4

1

3(1 Y (T )) jpq RpRq . (B.8)

Using the Eq. (2.2) and taking into account that ( d d ) = d and

(u u )j = uj , we finally obtain,

SLj = SLj =

14

N (0) 23

(1 Y (T ))d uj . (B.9)


26/31


In the low temperature limit the normal fluid component vanishes

and so the correlation functions have equal magnitudes S = | L| =| SL | =

B.

APPENDIX C. CALCULATION OF THE SPIN AND ORBITAL

MOMENTUM DENSITIES

To calculate the spin and the orbital momentum densities we use

the Greens function for the B-phase in the presence of a magnetic field

together with spin and the orbital rotations. The effects of the magnetic

field and the rotations may be considered as perturbations defined by

V ( L, S, L) = 12

[(S + L) + LL]. (C.1)

Here and L are given by Eq. (B.4). The complete Greens function

becomes

G( p, n) = Go( p, n) + G( p, n), (C.2)

where Go( p, n) is given by Eq.(B.3) and the perturbed component is

G( p, n) = Go( p, n)V ( L, S, L)Go( p, n)

=1

2(2n + 2 + 2o)

2

G11, G12G12, G22

, (C.3)

where

G11 = (in + )2 ( L + S)

( L + S) + li Li ,

G12 = ((in + ) (in ) )( L + S) + (in + )li Li ,

G21 = ((in + ) (in )

)( L + S) + (in )li Li ,

G22 = (in )2 ( L + S) +

( L + S) + li Li .

(C.4)

The spin density is given by

S =1

2T

n

d3p

(2 )3T r{Go( p, n)}

+1

2T

n

d3p

(2 )3T r{G( p, n)}. (C.5)


27/31


The first term in this equation is zero. The second term, after taking the

trace in particle-hole space, becomes

S= 14

T

n

d3p

(2 )31

(2n + 2 + 2o)

2T r(G11 G22

). (C.6)

The spin is finally obtained as

S=1

4N (0)

2

3+

1

3Y ( T )

( L + S)

2

3(1 Y(T)) d(uL)

. (C.7)

In the same manner,

L = 12T

n

d

3

p(2 )3

T r{Go( p, n)L}

+1

2T

n

d3p

(2 )3T r{G( p, n)L}.

(C.8)

The first term is again zero and the second term becomes

L =1

4T

n

d3p

(2 )31

(2n + 2 + 2o)

2T r(G11l + G22 l). (C.9)

Finally, the orbital angular momentum is obtained as

L = 14 N (0)23

(1 Y(T))[L + u( d( L + S))]. (C.10)

In the low temperature limit, the normal fluid fraction vanishes so we can

set the Yoshida function to zero giving

S= B [ L + S d(uL)],

L = B [L + u( d( L + S))].

(C.11)

Here B = N (0)/6 is the static spin susceptibility of 3HeB at zero temper-ature. In the following, we use units setting B = 1.

The spin and orbital momenta may be considered to have two com-

ponents. We can consider the intrinsic component to be that which is

induced by the magnetic field and by spin rotations:

S0 = L + S,L0 = u

( d( L + S)).(C.12)

These intrinsic components satisfy the well known relation

J0 = S0 + R L0 = 0, (C.13)

where the orthogonal matrix R is defined by Eq. (2.2).


28/31


The other components of the momenta are induced by orbital rota-

tions and are given by

Si = d(uL),

Li = L.(C.14)

These components are responsible for the appearance of nonzero total

angular momentum. From Eq. (C.7) we see, that at low temperatures the

total angular momentum becomes J = S+ R L = 2L.

APPENDIX D. EXPLICIT FORM OF THE EQUATIONS

OF MOTION

The equations of motion generated by the Hamiltonian (3.2) have the

following explicit form;

S =H

Pz= L +

1

2sin2 S[(Pz Lz) (P L) cos S]

1

2sin S sin L(Lz L cos L) cos

1

2sin2 SL sin , (D.1)

S =H

P=

1

2sin2 s[(P L) (Pz Lz) cos S]

+cos S

2sin S sin L(Lz L) cos +

cos S

2sin SL sin

L

2, (D.2)

S =

H

S =

S

2

L

2 cos +

1

2sin L (Lz L) sin , (D.3)

= H

Lz= L +

1

2sin2 s[(Pz Lz) (P L) cos S]

+1

2sin2 L(Lz L cos L)

1

2

L

sin S+

S

sin L

sin

+1

2sin S sin L{[(Pz Lz) (P L) cos S]

(Lz L cos L)} cos , (D.4)


29/31


= H

L=

1

2sin2 s[(P L) (Pz Lz) cos S]

+ 12sin S sin L

{cos S(Lz L cos L)

cos L[(pz Lz) (P L) cos S]} cos

+1

2sin2 L(L Lz cos L)

+1

2

cos S

sin SL +

cos L

sin LS

sin +

P

2 L, (D.5)

L

=H

L=

L

2

S

2cos

1

2sin S[(Pz Lz) (P L) cos S]sin , (D.6)

Pz = H

S, (D.7)

P = H

S, (D.8)

S = HS

= 1

2sin3 S[(Pz Lz) (P L) cos S]

[(P L) (Pz Lz) cos S]

+1

2sin L sin2 S

(Lz L cos L)[(P L) (Pz Lz) cos S]cos

+L

2sin2 S[(P L) (Pz Lz) cos S]sin

FD

S, (D.9)

Lz =H

=F

D

, (D.10)

L =H

=

1

2

1

sin S sin L[(Pz Lz) (P L) cos S]

(Lz L cos L) + S L

sin

1

2

L

sin S[(Pz Lz) (P L) cos S]

Ssin L

(Lz L cos L) cos

+ FD

, (D.11)


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L = H

L=

1

2sin3 L(Lz L cos L)(L Lz cos L)

S2sin2 L

(L Lz cos L) sin + 12sin S sin

2 L(L Lz cos L)

[(Pz Lz) (P L) cos S]cos FD

L. (D.12)

The Hamiltonian (3.2) does not contain the angular variables S and Sexplicitly so Eqs. (D.7) and (D.8) automatically satisfy Pz = 0 and P = 0.

ACKNOWLEDGMENT

We would like to thank A. J. Leggett and G. R. Pickett for useful dis-

cussions and the U.K. EPSRC for funding.

REFERENCES

1. S. N. Fisher and N. Suramlishvili, J. Low Temp. Phys. 138, 771 (2005).2. A. S. Borovik-Romanov, Yu. M. Bunkov, V. V. Dmitriev, Yu. M. Mukharsky, JETP Lett.

40, 1033 (1984).3. I. A. Fomin JETP Lett. 40, 1037 (1984).4. A. J. Leggett, Ann. Phys. 85, 11 (1974).5. Yu. M. Bunkov, S. N. Fisher, A. M. Guenault, C. J. Kennedy, and G. R. Pickett,

Phys. Rev. Lett. 68, 600 (1992).6. Yu. M. Bunkov, S. N. Fisher, A. M. Guenault, and G. R. Pickett, Phys. Rev. Lett. 69,

3092 (1992).7. S. N. Fisher, A. M. Guenault, A. J. Hale, G. R. Pickett, P. A. Reeves, and G. Tvalashvili

J. Low Temp. Phys. 121, 303 (2000).8. D. J. Cousins, S. N. Fisher, A. I. Gregory, G. R. Pickett, and N. S. Shaw, Phys. Rev. Lett.

82, 4484 (1999).9. D. Vollhardt and P. Wolfle, The Superfluid Phases of Helium 3, Taylor & Francis, London

(1990).10. T. Sh. Misirpashaev and G. E. Volovik, Sov. Phys. JETP 75, 650, (1992).11. G. Kharadze and G. Vachnadze, JETP Lett. 56, 458 (1992).12. S. N. Fisher and N. Suramlishvili, To be published.13. R. Combescot, Phys. Rev. A. 10, 1700 (1974); R. Combescot Phys. Rev. B 13, 126 (1975).14. K. Maki, Phys. Rev. B 11, 4264 (1975).15. K. Maki and H. Ebisawa, J. Low Temp. Phys. 22, 285 (1976).16. Y. M. Bunkov and V. L. Golo, J. Low Temp. Phys. 137, 625 (2004).17. A. J. Leggett and S. Takagi, Ann. Phys. 110, 353 (1978).18. S. Yip, J. Phys. C: Solid State Phys. 19, 1491 (1986).19. R. Combesot and T. Dombre, Phys. Lett. 76A, 293 (1980); T. Dombre and R. Combe-

scot Phys. Rev. B 32, 1751 (1985).20. I. A. Fomin Sov. Phys. JETP 57, 1227 (1983).21. G. Kharadze, N. Suramlishvili, and G. Vachnadze, J. Low Temp. Phys. 110, 851 (1998).22. G. E. Volovok, J. Phys: Condens. Matt. 5, 1759 (1993).23. V. V. Dmitriev, J. V. Kosarev, M. Krusius, D. V. Ponarin, V. M. H. Ruutu, and G. E.

Volovik, Phys. Rev. Lett. 78, 86 (1997).24. E. Sonin, Sov. Phys. JETP 67, 1791 (1988).25. G. Kharadze and N. Suramlishvili, J. Low Temp. Phys. 126, 539 (2002).


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26. D. I. Bradley, D. O. Clubb, S. N. Fisher, A. M. Gunault, C. J. Matthews, G. R. Pickett,and P. Skyba. J. Low Temp. Phys. 134, 351 (2004).

27. Yu. Bunkov, O. D. Timofeevskaya, and G. E. Volovik, Phys. Rev. Lett. 73, 1817 (1994).

28. S. N. Fisher, A. M. Guenault, C. J. Matthews, P. Skyba, G. R. Pickett, and K. L. Zaki,J. Low Temp. Phys. 138, 777 (2005).

29. S. N. Fisher, A. M. Gunault, G. R. Pickett, and P. Skyba, Physica B 329, 80 (2003).30. R. Balian and N. R. Werthamer, Phys. Rev. 131, 1553 (1963).

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Spin and Orbital