ISSN 0021-3640, JETP Letters, 2008, Vol. 88, No. 3, pp. 205209. Pleiades Publishing, Ltd., 2008.

205

The phenomenon of the spin superfluidity wasintensively studied experimentally and theoretically inthe 1970s and 1980s (see reviews [13] and referencestherein). Nowadays, there is a revival of interest in thephenomenon of the spin superfluidity in

3

HeB [48].Efforts to observe a similar phenomenon in

3

HeAwere also reported [9]. Meanwhile, there still remainssome unresolved controversy, which was inheritedfrom previous studies of this problem: what is the struc-ture of the spin-precession vortex,

1

which participatesin the phase-slip process, and what determines the sta-bility of the spin-precession supercurrent in

3

HeB.

The concept of the spin vortex as a phase-slip tooldetermining the stability of the superfluid spin currentwas introduced in 1978 [10]. For the superfluid

3

HeB,the spin-precession vortex was considered in 1987 [11](see also [12]). It was obtained that the core radius

r

c

ofthe vortex is on the order of the dipole length:

r

c

~

d

=

c

/

, where

is the longitudinal-NMR frequency and

c

is the velocity of transversal spin waves. Using theLandau criterion, it was shown that the critical-phasegradient is also determined by the inverse dipole length.The barrier for the vortex growth in the phase-slip pro-cess vanishes at phase gradients on the order of theinverse core radius. Hence, the threshold for the vortexinstability agrees with the critical gradient in the Lan-

The text was submitted by the author in English.

1

As well as in [11, 12], it is preferable to use the term the

spin

pre-cession

rather than the

spin

vortex, since, strictly speaking in

3

HeB, one deals not with currents of the spin, but with currentsof the precession moment (see below).

dau criterion. This is usual in the superfluidity theory[1].

One year later, Fomin [13] suggested that the vortexcore must be determined by another scale

F

=

c

/ , where

P

and

L

are the precessionand the Larmor frequencies. This was supported byMisirpashaev and Volovik [14] on the basis of a topo-logical analysis. Since 1/ plays a role in the chemicalpotential for the precession moment and is directly con-nected with

d

, the question of whether the core radiusis determined by

d

or

F

is similar to the question ofwhether the core radius in the Bose liquid is determinedby the liquid density or by the chemical potential: bothanswers are correct since the quantities are connectedby the thermodynamic relations. A real important dif-ference was that, according to Fomin, if the precessionangle

approaches the critical value

c

= 1.82 rad (or104

) the core radius becomes

r

c

~

F

~

d

/(

c

), i.e.,by the large factor 1/(

c

), differs from the estimation

r

c

~

d

carried out in [11]. Thus, the latter is valid onlyfar from the critical angle, where

c

~ 1.Since no barrier impedes the vortex expansion

across a channel if the gradient is on the order of 1/

r

c

,

the large core

r

c

~

d

/(

c

) at

c

leads to astrange (from the point of view of the conventionalsuperfluidity theory) conclusion: the instability withrespect to the vortex expansion occurs at the phase gra-dients ~1/

r

c

, essentially less than the Landau-criticalgradient ~1/

d

obtained in [11] for any

>

c

. Thepresent letter suggests a resolution of this paradox. Itdemonstrates that, at precession angles close to 104

P L( )L

F2

Spin-Precession Vortex and Spin-Precession Supercurrent Stability in

3

HeB

E. B. Sonin

Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904, Israele-mail: sonin@cc.huji.ac.il

Received May 12, 2008; in final form, June 26, 2008

The stability of the spin-precession currents in superfluid

3

HeB is analyzed for the precession angle very closeto 104

. In this limit, a spin-precession vortex has a very large core, and the barrier that blocks motion of theselarge-core vortices across the current streamlines (phase slip) disappears at precession-phase gradients muchsmaller than critical gradients estimated from the Landau criterion. Nevertheless, spin-precession currentsremain stable up to the Landau-critical gradients, since, in this case, there is a barrier, which blocks the phaseslip at a very early stage of the vortex-core nucleation. The second-order phase transition between the parity-symmetric and parity-asymmetric spin-precession vortex cores at the precession angle of 126.5

is also pre-dicted.PACS numbers: 67.30.Hj, 67.30.He

DOI:

10.1134/S0021364008150137

206

JETP LETTERS

Vol. 88

No. 3

2008

SONIN

and at phase gradients less than the Landau-critical gra-dient but larger than the inverse core radius, no barrierimpedes the phase slips at the stage of the vortexmotion across streamlines, but there is a barrier, whichblocks the phase slips on the very early stage of nucle-ation of the vortex core. Thus, for these gradients, thestability of the current states is determined not by thevortices but by the vortex-core nuclei.

This analysis also addresses the possible symme-tries of the vortex core. It was expected that the paritysymmetry (its definition is given below) is always bro-ken [14]. The present letter presents a numerical calcu-lation demonstrating the second-order transitionbetween a parity-symmetric and a parity-asymmetricvortex at the precession angle 126.5

. In the past, thefirst-order transition in cores of

3

HeB mass vorticeswas detected in NMR experiments upon rotating

3

HeB [15]. It was theoretically explained in [16, 17] interms of the transition between the axisymmetric andnonaxisymmetric cores. Later, this theory was con-firmed experimentally by the direct observation of thenonaxisymmetric core in one of the two vortices [18].

The spin dynamics of the superfluid phases of

3

He isdescribed by the theory of Leggett and Takagi [19]. Fol-lowing Fomin [2], the Euler angles

,

, and

are intro-duced in the spin space of the

3

HeB order parameter.The angle

is the precession angle, and

is the preces-sion phase. The angle

=

+

characterizes the result-ant rotation of the order parameter in the laboratoryframe, and in the limit

0 (no precession)becomes the angle of rotation about the

z

axis. Themoments canonically conjugate to the angles

,

, and

are, respectively,

P

=

M

z

M

,

M

, and

M

, where

M

z

is the

z

component of the magnetization

M

in the labo-ratory frame,

M

is the projection of

M

on the

axis ofthe rotating coordinate frame, and

M is the projectionof M on the axis perpendicular to the z and axes.

For phenomena observed experimentally, only onedegree of freedom is essential, which is connected withthe conjugate pair precession phase -precessionmoment P. The Hamilton equations for the precessionmode are

(1)

Since the degree of freedom connected with the conju-gate pair M is not active, the angle is determinedfrom the minimization of the energy: F/ = 0. Thefree energy F = FZ + F + V includes the Zeemanenergy FZ = M H = MHu = /2, where H = His an external constant magnetic field, the gradientenergy (the spin current is assumed to be normal to themagnetic field H ),

(2)

t-------

FP------,

Pt------

F-------.= =

L2

z

z

Fc

2

2--------- A u( )

2

2----------c||

2

c2-----

22-----------

u2

2 1 u2( )----------------------+ + ,=

where

(3)

and the dipole energy V = v(u, )/2 , where

(4)

Here, is the magnetic susceptibility, is the gyromag-netic ratio, L = H is the Larmor frequency, u = cos,and the ratio c||/c of the velocities of the longitudinaland transversal spin waves will be chosen to be [13]. In the state of the stationary precession, the pre-cession angular velocity is constant: /t = P. Thisstate corresponds to the extremum of the Gibbs thermo-dynamic potential, which is obtained from the freeenergy with the Legendre transformation G = F +PP/. Thus, the precession frequency P plays the roleof the chemical potential conjugate to the precessionmoment density P. The distribution of the parametersu = cos and is determined from the two EulerLagrange equations G/u = 0 and G/ = 0.

For uniform precession, the minimization withrespect to u yields the relation

(5)

and the minimization with respect to (only the dipoleenergy depends on ) gives the equation

(6)

The solution of this equation yields

(7)

for < 104 (u > 1/4) and

(8)

for > 104 (u > 1/4).The spin-precession vortex state is nonuniform, and

the gradient energy becomes essential. For an axiallysymmetric vortex with 2 circulation of the precessionphase ( = 1/r), the EulerLagrange equations are

(9)

A u( ) c||2

c2----- 1 u( )

2 1 u2,+=

c2 d2

v u ,( ) 215------ 1 cos+( )u 12---cos+

2.=

4/3

P L( )L 2

----------------------------------

Vu-------+

1F2-----

1d2-----

vu-------+ 0,=

1 cos+( )u cos 12---+ 1 u+( ) sin 0.=

cos 1/2 u1 u+----------------, v u ,( ) 0= =

cos 1, v u ,( ) v 0 u( ) 815------14--- u+

2= = =

1F2-----

4 u3r2

-----------

u

1 u2( )2---------------------

dudr------

2 11 u2--------------

1r---

dudr------

d2udr2--------+

JETP LETTERS Vol. 88 No. 3 2008

SPIN-PRECESSION VORTEX AND SPIN-PRECESSION SUPERCURRENT STABILITY 207

(10)

There are two types of vortices corresponding totwo types of symmetry. The solution with = 0 is par-ity symmetric, while in the structure with 0 sym-metry with respect to parity transformation, is broken. At the periphery of the vortex core, where 1 and u + 1/4 1, the solution of Eqs. (9) and (10)is

(11)

where the constant C is zero for the symmetric vortexand is on the order of unity for the asymmetric vortex.The ratio between the Fomin and the dipole length isdetermined by the value of u at infinity: d =4F . Equation (11) demonstrates that, inthe limit d F (|u + 1/4| 1), the first two terms inthe expression for u can be neglected everywhereexcept for at very large distances r on the order or largerthan Fln(F/d). It is the approximation of Fomin, whoused the relation in Eq. (7) between and u = cosobtained for the uniform state for u > 1/4. This allowsus to reduce two coupled Eqs. (9) and (10) to one equa-tion after the exclusion of the terms 1/ . The result-ing equation does not contain the dipole length explic-itly, but the length F which determines the core size,certainly depends on it.

In order to find whether the symmetric = 0 vortexcan exist, the stability of the solution of Eq. (9) is stud-ied for = 0. This requires an analysis of a linear equa-tion for ,

(12)

which is analogous to the 2D Schrdinger equation fora particle in a potential well. Here, u is determined fromEq. (9) at = 0. The = 0 vortex becomes unstable ifEq. (12) has a solution with a negative energy < 0. Itis known from quantum mechanics that the 2D poten-tial well always has a bound (localized) state [20].However, since for u < 1/4 (the necessary conditionfor a stable spin-precession current) the potentialenergy 2(u + 1/4)(1 + u)/5 in Eq. (12) is positive at

+4

15d2----------- 1 cos+( )u cos 12---+ 1 cos+( ) 0,=

1r---

ddr-------

d2dr2----------+

+1

5d2-------- 1 cos+( )u cos 12---+ 1 u+( ) sin 0.=

u14---+

15d216F2-----------

85d264r2-----------

32

16----------,+ +

C Fr

-----e3r/4 2F

,=

u

1/4+ /15

d2

1r---

ddr-------

d2dr2----------+ 2 u 1/4+( ) 1 u+( )5d2------------------------------------------ ,=

d2

infinity, the existence of a localized state leads to thecondition + 2(u

+ 1/4)(1 + u

)/5 < 0, which does

not rules out that > 0, i.e., the solution = 0 can bestable (except for the limits u

1/4 and u

1). The phase transition between two types of vorticesis determined by the condition = 2(u

+ 1/4)(1 +

u)/5 . Studying Eq. (12) numerically, it is found that

reaches its critical value at u = 0.594, which corre-

sponds to the precession angle = arccos(u) = 126.5.

Using the solution for in the critical point for the esti-mation of the energy contribution of the terms on theorder of 4, it is found that this contribution is posi-tive.2 Therefore, the phase transition between the sym-metric and the asymmetric vortices is on the secondorder. The numerical solutions of Eq. (9) (with = 0)and Eq. (12) are shown in the figure.

The spin-precession vortex in the absence of thespin current at infinity was analyzed above. If an uni-form spin current is present, the motion of the vorticesacross current streamlines leads to precession-phaseslips suppressing the current. If the precession-phasegradient connected with the current exceeds the inversecore radius, there is no barrier impeding this process.The analysis of the Landau criterion [11] showed thatthe spin-precession current is stable up to the criticalgradient on the order of the inverse dipole length 1/d.In the limit u 1/4, the latter can essentially exceedthe inverse core radius /d. However, howcan currents with gradients between 1/d and

/d be stabilized? The answer to this questionis that the barrier blocking phase slip is present at thevery early stage of nucleation of the vortex rings in 3Dsystems (or vortex pairs in 2D systems). Vortex nucle-ation starts from a slight localized depression of the2 Estimating the terms 4, one must take into account corrections

on the order of 2 to u.

d2

d2

u 1/4+

u 1/4+

Vortex structure at the transition from parity-symmetric toparity-asymmetric vortices. The plot of shown by thedashed line is normalized to the value (0) = 1.

208

JETP LETTERS Vol. 88 No. 3 2008

SONIN

superfluid density (determined by A(u) in our case). Inorder to analyze this protonucleus of the vortex corein the spin-current state, one should consider a newGibbs thermodynamic potential = G j/2:

(13)

where j is a Lagrange multiplier. The nucleus, which isrelated to the peak of a barrier, corresponds to a saddlepoint in the functional space. Thus, its structure shouldbe found from the solution of the EulerLagrange equa-tions for the introduced Gibbs potential. The first stepis to vary the Gibbs potential with respect to . Let usrestrict ourselves to a 1D problem, when the distribu-tion in the nucleus depends only on one coordinate x.Then, the distribution of is given by = j/A(u),where the reduced spin-precession current j = A(u

)0

is determined by the gradient 0 far from the nucleuscenter. Expanding with respect to the small deviationg = u u

from the equilibrium value of u at infinity,

one obtains

(14)

where the relation d3v(u)/du3 = 0 is taken into account.The terms linear in g must vanish at the stationary cur-rent state. The term quadratic in g determines the stabil-ity of the current state: it vanishes at the Landau-criticalcurrent

(15)

which was derived in [11]. Considering the case of acurrent close to the critical value and using the Taylorexpansion of A1(u) around u = 1/4, one obtains

(16)

G c2

Gc

2

2--------- A u( )

2

2----------u2

2 1 u2( )----------------------

u

F2-----+ +=

+v u( )

d2------------ j ,

2

c2---------G

j2

2A u( )---------------u( )2

2 1 u2( )----------------------

u

F2-----

v u( )d2

------------+ + +=

g( )2

2 1 u

2( )

-----------------------

+ g ddu------j2

2A u

( )------------------v u

( )d2

---------------+1F2-----+

+g2

2-----d2

du2--------

j22A u

( )------------------

v u

( )d2

---------------+g3

6-----j22----

d3A u

( ) 1du3

------------------------,+

jc 1d2-----

d2v u

( )u

2d--------------------

d2 A u

( ) 1[ ]du2

-----------------------------

1

,=

G16c

2

15 2---------------

g( )22-------------- a

g2

2----- bg3

6-----+ ,=

where

(17)The EulerLagrange equation for this Gibbs potential,g = ag bg2/2 = 0, determines the distribution g:

(18)

where g0 = 3a/b = 1.24( j2)/ is the value of g inthe nucleus center and rp = 2/ = 4.1d/ is thenucleus size. The energy of the nucleus,

(19)

determines the barrier for the process of the vortex-corenucleation. Here, S is the cross-section area of the chan-nel. Since, in the limit j jc the nucleus size rp isdivergent, our 1D description is always valid closeenough to the critical point, where rp . When rpbecomes smaller than the transverse size of the channel,one should consider the 3D or 2D (in the case of a thinlayer) nucleus. The first stage of this problem is tofind the distribution of from the continuity equa-tion [g] = 0. Its solution demonstrates that, outsidethe nucleus, the distribution of is the same as thataround the vortex ring (3D case) or the vortex dipole(2D case). In particular, in the 2D case,

(20)

In contrast to the 1D case, the relation between andg is not local, so that the following variation of theGibbs potential with respect to g leads to an integrodif-ferential equation. However, on the basis of our solu-tion to the 1D problem and using the scaling arguments,one may conclude that the nucleus size can be roughlyestimated from the expression in Eq. (19) for the 1Dcase replacing S by or by rpd for the 3D and 2D case,respectively (d is the thickness of the 2D layer).

In the analysis, it is assumed that the value of theprecession angle

= arccosu

far from the centers of

a vortex or a nucleus was fixed and exceeded 104 (u 1/4), the dipole energy van-

ishes, and without the dipole energy, no stable currentis possible. Meanwhile, Fomin [21] suggested that thespin current can be stable even if u > 1/4 and the Lan-dau criterion is violated. He argued that the emission ofthe spin waves, which comes into play after exceedingthe Landau-critical gradient, is not essential in the

a 0.239 jc2 j2( ), b 0.577 jc2.= =

g g0 1x

rp----tanh2 ,=

jc2 jc2

a jc2 j2

16c

2 S15 2

------------------ ag2 bg3

3-------- gd0

3a/b

64c225 2---------------a5/2

b2--------S= =

= 0.214Sc

2

2---------

jc2 j2( )5/2

jc4------------------------,

S

0 g r1( )r12 r10

r2----------

2r r 0( )r

4---------------------------- .d0

=

rp2

JETP LETTERS Vol. 88 No. 3 2008

SPIN-PRECESSION VORTEX AND SPIN-PRECESSION SUPERCURRENT STABILITY 209

experimental conditions (see also a similar conclusionafter Eq. (2.39) in the review by Bunkov [3]). This argu-ment is conceptually inconsistent. If the experimental-ists observed dissipationless spin transport simplybecause dissipation was weak, it would be ballisticrather than superfluid transport. The essence of the phe-nomenon of superfluidity is not the absence of sourcesof dissipation, but the ineffectiveness of these sourcesfollowing from energetic and topological consider-ations. The Landau criterion is an absolutely necessarycondition for superfluidity. Fortunately for the superflu-idity scenario in 3HeB, Fomins estimation of the roleof dissipation by the spin-wave emission triggered bythe violation of the Landau criterion is not conclusive.He found that this dissipation is weak compared to thedissipation by spin diffusion. However, this is an argu-ment in favor of the importance rather than the unim-portance of the Landau criterion. Indeed, spin diffu-sion, regardless of how high the diffusion coefficientcould be, is ineffective in the subcritical regime, inwhich the gradient of chemical potential P is absent.On the other hand, in the supercritical regime, thechemical potential is no longer constant and this trig-gers the strong spin-diffusion mechanism of the dissi-pation.

It is worthwhile to recall that the Landau criterion isa necessary but not sufficient condition for transportwithout dissipation. At the Landau-critical gradient(current), the current state ceases to be metastable,because the barrier leading to the metastability van-ishes. However, it is well known that, in reality, the cur-rent dissipation via phase slips is possible even in thepresence of barriers (due to thermal fluctuations orquantum tunneling). Therefore, the present work onlyaddresses ideal critical currents (the upper bound forthem) leaving actual critical currents beyond thescope of analysis.

In summary, this letter analyzes the stability of thespin-precession currents in superfluid 3HeB when theprecession angle is very close to c = 104 and thespin-precession vortex has a core radius of ~d/much larger than the dipole length d. Although the bar-rier for motion of these large-core vortices across thecurrent streamlines disappears at rather small preces-sion-phase gradients ~ /d, the spin-precessioncurrents remain stable up to much large gradients ~1/d,which were estimated from the Landau criterion [11].The stability of the currents, in this case, is provided bythe barriers at a very early stage of the vortex-corenucleation. It was also demonstrated that, at the preces-

c

c

sion angle 126.5, there is a second-order phase transi-tion between the parity-symmetric and parity-asym-metric spin-precession vortex cores.

I am grateful to I.A. Fomin and G.E. Volovik forcritical comments, which influenced the conclusions ofthis work.

REFERENCES1. E. B. Sonin, Usp. Fiz. Nauk 137, 267 (1982) [Sov. Phys.

Usp. 25, 409 (1982)].2. I. A. Fomin, Physica B 169, 153 (1991).3. Yu. M. Bunkov, Progress of Low Temperature Physics,

Ed. by W. P. Halperin (Elsevier, Amsterdam, 1995),Vol. 14, p. 68.

4. G. E. Volovik, cond-mat/0701180.5. Yu. M. Bunkov and G. E. Volovik, Phys. Rev. Lett. 98,

265302 (2007).6. Yu. M. Bunkov, J. Magn. Magn. Mater. 310, 1476

(2007).7. Yu. M. Bunkov and G. E. Volovik, J. Low Temp. Phys.

150, 609 (2008).8. Yu. M. Bunkov and G. E. Volovik, to be published in

Physica C 468, 135 (2008).9. T. Sato, T. Kunimatsu, K. Izumina, et al., arXiv:

0804.2994.10. E. B. Sonin, Zh. ksp. Teor. Fiz. 74, 2097 (1978) [Sov.

Phys. JETP 47, 1091 (1978)].11. E. B. Sonin, Pisma Zh. ksp. Teor. Fiz. 45, 586 (1987)

[JETP Lett. 45, 747 (1987)].12. E. B. Sonin, Zh. ksp. Teor. Fiz. 94, 100 (1988) [Sov.

Phys. JETP 67, 1791 (1988)].13. I. A. Fomin, Zh. ksp. Teor. Fiz. 94, 112 (1988) [Sov.

Phys. JETP 67, 1148 (1988)].14. T. Sh. Misirpashaev and G. E. Volovik, Zh. ksp. Teor.

Fiz. 102, 1197 (1992) [Sov. Phys. JETP 75, 650 (1992)].15. O. T. Ikkala, G. E. Volovik, P. J. Khakonen, et al., Pisma

Zh. ksp. Teor. Fiz. 35, 338 (1982) [JETP Lett. 35, 416(1982)].

16. E. V. Thuneberg, Phys. Rev. Lett. 56, 359 (1986).17. M. M. Salomaa and G. E. Volovik, Phys. Rev. Lett. 56,

363 (1986).18. Y. Kondo, J. S. Korhonen, M. Krusius, et al., Phys. Rev.

Lett. 67, 81 (1991).19. A. J. Leggett and S. Takagi, Ann. Phys. (N.Y.) 106, 79

(1977).20. L. D. Landau and E. M. Lifshitz, Course of Theoretical

Physics, Vol. 3: Quantum Mechanics: Non-RelativisticTheory (Nauka, Moscow, 1989, 4th ed.; Pergamon,Oxford, 1977, 3rd ed.), Sec. 45.

21. I. A. Fomin, Pisma Zh. ksp. Teor. Fiz. 45, 106 (1987)[JETP Lett. 45, 135 (1987)].

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 150 /GrayImageDepth -1 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputCondition () /PDFXRegistryName (http://www.color.org?) /PDFXTrapped /False

/Description >>> setdistillerparams> setpagedevice