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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
111
0022-2291/05/1100-0111/0 2005 Springer Science+Business Media, Inc.
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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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114 S. N. Fisher and N. Suramlishvili
/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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Coherently Precessing Spin and Orbital States in Superfluid 3HeB 115
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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116 S. N. Fisher and N. Suramlishvili
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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Coherently Precessing Spin and Orbital States in Superfluid 3HeB 117
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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118 S. N. Fisher and N. Suramlishvili
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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Coherently Precessing Spin and Orbital States in Superfluid 3HeB 119
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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120 S. N. Fisher and N. Suramlishvili
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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Coherently Precessing Spin and Orbital States in Superfluid 3HeB 121
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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122 S. N. Fisher and N. Suramlishvili
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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Coherently Precessing Spin and Orbital States in Superfluid 3HeB 123
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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124 S. N. Fisher and N. Suramlishvili
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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Coherently Precessing Spin and Orbital States in Superfluid 3HeB 125
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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126 S. N. Fisher and N. Suramlishvili
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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Coherently Precessing Spin and Orbital States in Superfluid 3HeB 127
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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Coherently Precessing Spin and Orbital States in Superfluid 3HeB 131
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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132 S. N. Fisher and N. Suramlishvili
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
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Coherently Precessing Spin and Orbital States in Superfluid 3HeB 133
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:
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134 S. N. Fisher and N. Suramlishvili
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
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Coherently Precessing Spin and Orbital States in Superfluid 3HeB 135
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)
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136 S. N. Fisher and N. Suramlishvili
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)
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Coherently Precessing Spin and Orbital States in Superfluid 3HeB 137
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).
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138 S. N. Fisher and N. Suramlishvili
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)
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Coherently Precessing Spin and Orbital States in Superfluid 3HeB 139
= 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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140 S. N. Fisher and N. Suramlishvili
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.
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