Dispersion-Induced Splitting of the Collective Mode Spectrum in Axial and Planar Phases of Superfluid 3 He

J Low Temp Phys (2009) 155: 200–217DOI 10.1007/s10909-009-9878-y

Dispersion-Induced Splitting of the Collective ModeSpectrum in Axial and Planar Phases of Superfluid 3He

Peter Brusov · Pavel Brusov

Received: 22 December 2008 / Accepted: 25 February 2009 / Published online: 20 March 2009© Springer Science+Business Media, LLC 2009

Abstract The whole collective mode spectrum in axial and planar phases of super-fluid 3He with dispersion corrections is calculated for the first time. In axial A-phasethe degeneracy of clapping modes depends on the direction of the collective modemomentum k with respect to the vector l (mutual orbital moment of Cooper pairs),namely: the mode degeneracy remains the same as in case of zero momentum k fork‖l only. For any other directions there is a threefold splitting of these modes, whichreaches maximum for k⊥l.

In planar 2D-phase, which exists in the magnetic field (at H > HC) we find thatfor clapping modes the degeneracy depends on the direction of the collective modemomentum k with respect to the external magnetic field H , namely: the mode degen-eracy remains the same as in case of zero momentum k for k‖H only. For any otherdirections different from this one (for example, for k⊥H ) there is twofold splittingof these modes.

The obtained results imply that new interesting features can be observed in ultra-sound experiments in axial and planar phases: the change of the number of peaks inultrasound absorption into clapping mode. One peak, observed for these modes byLing et al. (J. Low Temp. Phys. 78:187, 1990), will split into two peaks in a planarphase and into three peaks in an axial phase under the change of ultrasound directionwith respect to the external magnetic field H in a planar phase and with respect tothe vector l in an axial phase.

P. Brusov (�)Finance Academy under the Government of the Russian Federation, 49-55, Leningradsky Ave.,Moscow 125993, Russiae-mail: [email protected]

P. BrusovPhysical Research Institute, South Federal University, 194 Stachki Ave., Rostov-on-Don 344090,Russia

J Low Temp Phys (2009) 155: 200–217 201

In planar phase, some Goldstone modes in the magnetic field become massive(quasi-Goldstone) and have a similar twofold splitting under the change of ultrasounddirection with respect to the external magnetic field H .

The obtained results as well will be useful under interpretation of the ultrasoundexperiments in axial and planar phases of superfluid 3He.

Keywords Superfluid 3He · Axial phase · Planar phase · Collective-modespectrum · Dispersion corrections

PACS 67.30.H-

1 Introduction

A renewal of the interest to investigation of superfluid 3He is caused by the intensivestudy during last two decades of superfluid 3He in porous media, like aerogel, Vycorglasses, etc. [1]. In such complex systems superfluid phase analogs to axial A-phasehave been observed. Another cause of such an interest is related to the study of su-perfluid 3He in confined geometry, where a very old problem of the boundary statein the isotropic B-phase has been solved recently [2]. One more cause of the renewalof the interest to superfluid 3He relates to recent study of the possibility of subdomi-nant f-wave pairing in superfluid 3He (in addition to the conventional p-wave pairing)[25, 26].

A study of the collective excitations in superfluid 3He plays a very important role.A lot of very interesting features of the collective mode spectrum still remain exper-imentally unexplored [3, 4]. The collective excitation spectrum in different phasesof superfluid 3He at zero momentum of collective excitation has been calculated bymany authors. But in sound experiments, where the collective modes are excited, onehas the collective excitations with nonzero momentum k. So, the knowledge of thecollective excitation spectrum at zero momentum k is not sufficient to compare thetheoretical results with the sound experiment data and we must take into account thedispersion corrections.

In the present paper we calculate the whole collective mode spectrum in axial andplanar phases of superfluid 3He with dispersion corrections. We show that in axialA-phase the degeneracy of clapping modes depends on the direction of the collectivemode momentum k with respect to the vector l (mutual orbital moment of Cooperpairs): the mode degeneracy remains the same as in the case of zero momentum k

for k‖l only. For any other direction, there is a threefold splitting of these modes,which reaches maximum for k⊥l.

In a planar 2D-phase, which exists in the magnetic field (at H > HC ), we find thatfor clapping- and quasi-Goldstone modes the degeneracy depends on the direction ofthe collective mode momentum k with respect to the external magnetic field H : themode degeneracy remains the same as in the case of zero momentum k for k‖Honly. For any other direction different from this one (for example, for k⊥H ), there isa twofold splitting of these modes.

202 J Low Temp Phys (2009) 155: 200–217

2 Axial Phase

In the bulk superfluid 3He the A-phase gives us an anisotropic superfluid quantumliquid and thus is perhaps the most interesting object in superfluid 3He. The mainfeatures of the A phase of 3He are related to the existence of two nodes in the gapof a single-particle spectrum on the Fermi-surface. This leads to the existence ofchiral fermions, gauge fields, analogs of W and Z bosons, zero-charge phenomenon,the damping of collective excitations (CE) at zero momentum, and to many otherconsequences for the system [5].

The collective excitation spectrum in 3He–A at zero momentum of collective ex-citation has been calculated by many authors, in particular by Wolfle [6] using kineticequation method and by Brusov and Popov [6] using path integration technique. Thelatter authors have taken the damping of collective excitations into account and thatled to some differences in results of Refs. [7] and [6]. The precise experiments onmeasurement of the clapping-mode frequency [8] are in excellent agreement with theBrusov and Popov theory [9].

The case of nonzero momentum k has been considered by Wolfle [6], Combescot[10], Brusov and Popov [11]. Wolfle has considered for simplicity the case of k‖l,where 3 × 3 matrices are diagonal, and obtained the dispersion laws for Goldstonemodes: sound mode E = cF k/

√3, orbital waves E = cF k‖, and diffusive mode, as-

sociated with the breakdown of the rotation symmetry in orbital spaces and quadraticdispersion corrections to the spectrum of nonphonon modes, for example, to normalflapping (nfl-) mode

E2nfl = E2

nfl(k = 0) + (cF k)2.

Combescot [7] pointed out that for T = 0 the small particle–hole asymmetric termschange the dispersion law of orbital mode from linear to quadratic in the low fre-quency regime E � TC(TC/TF ).

Brusov and Popov [11] have investigated the stability of Goldstone modes with re-spect to decay into several Bose excitations. They have shown that stability of Gold-stone modes depends on the angle θ between the excitation momentum k and l: themode is stable when its momentum k lies inside some cones and unstable when out-side them.

2.1 The Model of Superfluid 3He

In the method of functional integration, the initial Fermi-system (3He) is describedby anticommuting functions χs(x, τ ), χ̄s(x, τ ) defined in the volume V = L3, whichare antiperiodic in time τ with a period β = T −1. Here s is the spin index. Thesefunctions can be expanded into a Fourier series

χs(x) = (βV )−1/2∑

p

as(p) exp(i(ωτ + k · x)

), (1)

J Low Temp Phys (2009) 155: 200–217 203

where p = (k,ω); ω = (2n + 1)πT are Fermi-frequencies and x = (x, τ ). Let usconsider the functional of action for an interacting Fermi-system

S =∫ β

0dτ d3x

∑

s

χ̄s(x, τ )∂τχs(x, τ ) −∫ β

0H ′(τ ) dτ, (2)

which corresponds to the Hamiltonian

H ′(τ ) =∫

d3x∑

s

(2m)−1∇χ̄s(x, τ )∇χs(x, τ ) − (λ + sμ0H)χ̄s(x, τ )χs(x, τ )

+ 1

2

∫d3x d3y U(x − y)

∑

ss′χ̄s(x, τ )χ̄s′(y, τ )χs′(y, τ )χs(x, τ ). (3)

In order to obtain the effective functional of action, we shall use the method of di-vision of Fermi-fields into “fast” and “slow” fields with subsequent successive inte-gration over these fields. Fast fields χ1s and χ̄1s are determined by components ofexpansion (1) either with frequencies |ω| > ω0, or with momenta |k − kF | > k0.The remaining components χ0s = χs − χ1s of the Fourier expansion define slowfields χ0s . Integrating over fast fields, we insert into the integral over the Fermi-fields,a Gaussian integral over the complex functions cia(x, τ ) and c̄ia(x, τ ) with the vectorindex i and the isotopic index a (i, a = 1,2,3):

∫Dc̄ia(x, τ )Dcia(x, τ ) exp

(g−1

0

∑

p,i,a

c̄ia(p)cia(p)

). (4)

We then shift the Bose-fields by a quadratic form of the Fermi-fields so as to annihi-late the quartic form in the Fermi-fields. After integrating over slow fields we arriveat the “effective” or “hydrodynamic action” functional

Seff = g−10

∑

p,i,a

c+ia(p)cia(p) + 1

2ln det

M̂(cia, c+ia)

M̂(c(0)ia , c

(0)+ia )

. (5)

It defines the point of the phase transition of the initial Fermi-system as a point of theBose-condensation of the fields c and c̄, and Seff determines also the density of thecondensate at T < TC and the spectrum of the collective excitations.

The fourth-order matrix M̂(p1,p2) has the following elements Mab(p1,p2):

M11 = Z−1[iω − ξ + μ(Hσ )

]δp1p2,

M22 = Z−1[−iω + ξ + μ(Hσ )

]δp1p2 ,

M12 = (βV )−1/2(n1i − n2i )cia(p1 + p2)σa,

M21 = −(βV )−1/2(n1i − n2i )c+ia(p1 + p2)σa,

(6)

where ξ = cF (k − kF ), ni = ki/kF , β = T −1, H is the magnetic field and μ is themagnetic moment of the quasi-particle, V -volume, σa(a = 1,2,3) are 2 × 2 Pauli-matrices, and ω = (2n + 1)πT are the Fermi frequencies.

204 J Low Temp Phys (2009) 155: 200–217

In the first approximation the quadratic form of action functional describes thecollective excitations in A-phase in case of nonzero momenta as well. We use it tocalculate the dispersion corrections.

2.2 The Equations for the Collective Mode Spectrum in an Arbitrary MagneticField and at Arbitrary Collective Mode Momenta

We will now begin the investigation of the spectrum of collective excitations. Tomake this in the region TC − T ∝ TC one should expand the functional ln det inpowers of the deviations cia(p) from the condensate value c

(0)ia (p), which is different

for different phases and in A-phase has a form

c(0)ia (p) = (βV )1/2cδp0(δi1 + iδi2)δa1. (7)

So, we apply the shift

cia(p) → c(0)ia (p) + cia(p)

and separate the quadratic form

∑

p

Aijab(p)c+ia(p)cjb(p) + 1

2

∑

p

Bijab(p)[cia(p)cjb(−p) + c+

ia(p)c+jb(−p)

](8)

from Seff. After calculation of this quadratic form we obtain the following result [2](here uia = Re cia, via = Im cia):

[g−1

0 + 2Z2

βV

∑

p1+p2=p

D3 cos2 θ

](u2

33 + v233

)

+[g−1

0 + 2Z2

βV

∑

p1+p2=p

(D1 + D2) cos2 θ

]((u31 + v32)

2 + (u32 + v31)2)

+[g−1

0 + 2Z2

βV

∑

p1+p2=p

(D1 − D2) cos2 θ

]((u31 − v32)

2 + (u32 − v31)2)

+[g−1

0 + Z2

βV

∑

p1+p2=p

D3 sin2 θ

]((v31 + u32)

2 + (u13 − v23)2)

+[g−1

0 + Z2

βV

∑

p1+p2=p

(−Δ2 sin2 θ∂3 + D3) sin2 θ

](u13 + v23)

2

+[g−1

0 + Z2

βV

∑

p1+p2=p

(Δ2 sin2 θ∂3 + D3) sin2 θ

](v13 − u23)

2

+[g−1

0 + Z2

βV

∑

p1+p2=p

(D1 + D2) sin2 θ

]((u12 + v11 + u21 + v21)

2

J Low Temp Phys (2009) 155: 200–217 205

+ (u11 + v12 − u22 − v21)2)

+[g−1

0 + Z2

βV

∑

p1+p2=p

(D1 − D2) sin2 θ

]((u12 − v11 − u21 + v22)

2

+ (u11 − v12 + u22 − v21)2)

+[g−1

0 + Z2

βV

∑

p1+p2=p

sin2 θ(D1 + D2 − Δ2 sin2 θ(∂1 + ∂2)

)]

× (u12 + v11 − u21 − v21)2

+[g−1

0 + Z2

βV

∑

p1+p2=p

sin2 θ(D1 − D2 − Δ2 sin2 θ(∂1 − ∂2)

)]

× (u11 − v11 + u21 − u22)2

+[g−1

0 + Z2

βV

∑

p1+p2=p

sin2 θ(D1 + D2 + Δ2 sin2 θ(∂1 + ∂2)

)]

× (u11 + v12 + u22 + v21)2

+[g−1

0 + Z2

βV

∑

p1+p2=p

sin2 θ(D1 − D2 + Δ2 sin2 θ(∂1 − ∂2)

)]

× (u11 − v12 − u22 + v21)2. (9)

Here

a = Z−1(iω − ξ), b = Z−1μH, q1 = Z−1Δ(n1 + in2)α+,

q2 = iZ−1Δ(n1 + in2)α−,

d1,2 = Z−2(ω2 + (ξ ± μH)2 + Δ2 sin2 θ(α+ ± α−)2),

∂1,2 = (α+ + α−)2

d1(1)d1(2)± (α+ − α−)2

d2(1)d2(2),

D1,2 = (a+(1) + b)(a+(2) + b)

d1(1)d1(2)± (a+(1) − b)(a+(2) − b)

d2(1)d2(2),

∂3 = (α2+ + α2−

)( 1

d1(2)d2(1)+ 1

d1(1)d2(2)

),

D3 = (a+(1) + b)(a+(2) − b)

d1(1)d2(2)± (a+(1) − b)(a+(2) + b)

d2(1)d1(2).

The equation detQ = 0, where Q is the matrix of quadratic form (9), gives us 18equations which completely determine the 18 collective modes in 3He–A in an arbi-trary magnetic field and at arbitrary collective excitation momenta.

206 J Low Temp Phys (2009) 155: 200–217

2.3 The Dispersion Corrections to Collective Mode Spectrum in 3He–A

Considering the limit T → 0, we replace the summation in equation detQ = 0, bythe integration near the Fermi-sphere, in accordance with the rule

(βV )−1∑

p1

→ k2F (2π)−4c−1

F

∫dω1 dξ1 dΩ1, (10)

where∫

dΩ1 is an integral with respect to the angle variables. We then calculatedirectly the integrals with respect to ω1 and ξ1. Putting H = 0 and considering smallmomenta of collective modes, we obtain the whole collective mode spectrum in A-phase of superfluid 3He with dispersion corrections.

It consists of the following:Nine Goldstone modesThree sound modes

E = cF k/√

3, v13 − u23, u11 − v21, u22 − v12,

and six orbital waves

E = cF k‖, u33, v33, u32, v32, u31, v31. (11)

Six clapping modes

u11 + v21, u22 + v12, u23 + v13,

E2cl = Δ2

0(1.17 − i0.13)2 + c2F

⌊k2‖(0.159 − i0.090) + k2⊥(0.284 + i0.061)

⌋.

v11 + u21, v22 − u12,

E2cl = Δ2

0(1.17 − i0.13)2 + c2F

⌊k2‖(0.159 − i0.090) + k2⊥(0.557 + i0.029)

⌋.

u13 − v23

E2cl = Δ2

0(1.17 − i0.13)2 + c2F

⌊k2‖(0.159 − i0.090) + k2⊥(0.694 + i0.013)

⌋.

(12)

Three pairbreaking modes

u13 + v23, u12 − v11, u21 + v22,

E2pb = Δ2

0(1.96 − i0.31)2 + c2F

⌊k2‖(0.096 + i0.0004) + k2⊥(0.898 + i0.509)

⌋.

(13)

Here k2⊥ = k21 + k2

2 , k2‖ = k23 = (k, l)2.

We have obtained the dispersion corrections for whole collective mode spectrumfor arbitrary direction of collective excitation momentum. As we expected, the dis-persion corrections (except the ones for Goldstone modes) turn out to be complex aswell as the frequencies (energies) of collective excitations. This is related to the factthat the excitation with nonzero energy and small momentum k can decay into initialfermions (see Fig. 1).

J Low Temp Phys (2009) 155: 200–217 207

Fig. 1 The decay of the phonon(collective excitation) withmomentum k into initialfermions with momenta k1and k2, the directions of whichare close to axis of anisotropy l(H ) in anisotropicA-phase (2D)

It is interesting to note that for the case of k‖l (which has been considered byWolfle [6]) the clapping mode remains fully degenerated while for other directions ofcollective excitation momentum, a threefold splitting of clapping mode takes place.Note, that the value of splitting increases with k2⊥ and reaches maximum at k =k⊥ or at k⊥l. The maximal ratio of the splitting of the clapping mode is equal toΔα1c

2F k2/Δα2c

2F k2 = 0.117 : 0.059 ≈ 2 : 1.

3 Planar Phase

The superfluid phases of 3He, in addition to the isotropic B-phase, the anisotropicA-phase and the A1-phase, also include a 2D-phase [12, 13], known as planar andhaving the order parameter

c(0)ia (p) = c(βV )1/2δp0(δi1δa1 + δi2δa2), (14)

or in a matrix form:

c(0)ia (p) = c(βV )1/2δp0

⎛

⎝1 0 00 1 00 0 0

⎞

⎠ . (15)

Contrarily to the A-, B- and A1-phases, 2D-phase has not yet been observed, butits existence under certain conditions was deduced by many researchers. In partic-ular, Popov et al. [12] predicted a phase transition from the B- to the 2D-phase atH = HC and proved the stability of the 2D-phase to small perturbations for H > HC .Fujita et al. [14] by considering the B-phase in semibounded space have shown thata 2D-phase is realized on the boundary: in this situation it is energetically more fa-vored than the A-phase. One of the possible explanations of the double splitting of thesquashing mode in the B-phase, observed experimentally [15], was an assumed exis-tence on the sell boundary of a 2D-phase, one of the collective modes of which leadsto the appearance of a second peak in ultrasound absorption (our recent study [2],however, has shown that not 2D-phase, but a deformed B-phase is realized in thevicinity of the boundary; see Ref. [2] for details).

208 J Low Temp Phys (2009) 155: 200–217

These examples suffice to understand the importance of investigation of the pla-nar 2D-phase and particularly the spectrum of its collective excitations. Below wecalculate this spectrum by path integration technique.

3.1 Stability of 2D-Phase

All properties of superfluid 3He are determined by the functional Sh of the hydrody-namical action, given by

Sh = g−1∑

p,i,a

c+ia(p)cia(p) + 1

2ln det

M̂(cia, c+ia)

M̂(c(0)ia , c

(0)+ia )

. (16)

Here cia(p) is the Fourier transform of the Bose-field cia(x, τ ) describing the Cooperpairs of the quasi-fermions on the Fermi surface, the operator M̂ is given by

M̂ =(

Z−1(iω − ξ + μHσ3)δp1p2 (βV )−1/2(n1i − n2i )σacia(p1 + p2)δp1+p2,0

−(βV )−1/2(n1i − n2i )σac+ia(p1 + p2)δp1+p2,0 Z−1(−iω + ξ + μHσ3)δp1p2

),

(17)

where ξ = cF (k − kF ), ni = ki/kF , H is the magnetic field and μ is the magneticmoment of the quasi-particle, σa(a = 1,2,3) are 2×2 Pauli-matrices, and ω = (2n+1)πT are the Fermi frequencies. Expanding the functional (17) in the Ginsburg–Landau region TC − T ∝ TC into powers of the fields c and c+, we obtain

Sh = −20k2F (ΔT )2βV

21ξ(3)cF

�, (18)

where

� = −Tr(AA+) + ν Tr

(AA+P

) + (TrAA+)2 + Tr

(AA+AA+) + Tr

(AA+A∗AT

)

− Tr(AAT A∗A+) − 1

2Tr

(AAT

)Tr

(A+A∗). (19)

Here

ν = 7ξ(3)μ2H 2/4π2TCΔT,

P is the projector on the third axis along which the field is directed. Minimizing �,we obtain the matrix A that determines the condensate density. The equation δ� = 0or

−A+νAP +2(trA+A

)A+2AA+A+2A∗AT A−2AAT A∗ −A∗ trAAT = 0 (20)

has several nontrivial solutions corresponding to the superfluid phases. One of themhas an order parameter

1

2

(1 0 00 1 00 0 0

). (21)

J Low Temp Phys (2009) 155: 200–217 209

This is in fact the planar 2D-phase. Calculations of the second variation δ2� yield

δ2F2 = (ν − 1/2)u233 + (ν + 1/2)v2

33 + ν(u2

13 + u223

) + (ν + 2)(v2

13 + v223

)

+ (1/2)[3u2

11 + 3u222 + 2u11u22 + (u12 + u21)

2]

+ (1/2)[3v2

12 + 3v221 − 2v12v21 + (v11 − v22)

2].

Here uia = Re cia , via = Im cia .For ν < 1/2, the second variation δ2� is of an alternating sign, while for ν > 1/2 it

is non-negative. This means that 2D-phase is stable in the magnetic field H > HC =[πμ2TCΔT/7ξ(3)]1/2. As indicated above, at H = HC a phase transition takes placefrom the B- to the 2D-phase.

3.2 The Equations for Collective Mode Spectrum in 3He–2D

To calculate the collective mode spectrum in 3He–2D at T → 0, the functional Sh

must be expanded in terms of fluctuations of the fields cia(p). Making in Sh a shiftcia(p) → cia(p) + c

(0)ia (p) and providing the calculations which are similar to the

ones for the axial phase, we obtain from the equation detQ = 0 the following equa-tions for the collective mode spectrum [13]:

pb:∫ 1

0

(1 − x2)I (c)(1 + 4c) dx = 0, u11 + u22, v12 − v21; (22)

cl:∫ 1

0

(1 − x2)I (c)(1 + 2c) dx = 0,

v11 − v22, u12 + u21, u11 − u22, v12 + v21; (23)

Gd:∫ 1

0

(1 − x2)I (c) dx = 0, u12 − u21, v11 + v22; (24)

Gd:∫ 1

0

[(1 + 2c)I (c) − 1

]x2 dx = 0, u31, u32, v31, v32; (25)

∫ 1

0x2[(1 + 4c+)I (c+) + (1 + 4c−)I (c−) − 2

]dx = 0, u33; (26)

∫ 1

0x2[I (c+) + I (c−) − 2

]dx = 0, v33; (27)

qGd:∫ 1

0

(1 − x2)[(1 + 4c+)I (c+) + (1 + 4c−)I (c−)

]dx = 0,

u13, u23; (28)

qpb:∫ 1

0

(1 − x2)[I (c+) + I (c−)

]dx = 0, v13, v23. (29)

210 J Low Temp Phys (2009) 155: 200–217

Here

I (c) = 1

(1 + 4c)1/2ln

(1 + 4c)1/2 + 1

(1 + 4c)1/2 − 1,

c± = Δ20(1 − x2)

ω2 + [cF (n,k) ± 2μH ]2, c = Δ2

0(1 − x2)

ω2 + c2F (n,k)2

,

uia = Re cia, via = Im cia.

(30)

Let us examine (22)–(29) at zero momenta (k = 0) of the collective excitations. Inthis case (22)–(24) coincide with those obtained by Brusov and Popov [16] for theA-phase without the magnetic field, while (27)–(30) go over into aforementionedBrusov–Popov equations for an A-phase without the magnetic field, following thesubstitution ω2 + 4μ2H 2 → ω2. These equations can thus be solved by using theresults of Ref. [13]. Finding also the roots of (25) and (26), we obtain the results ofthe collective mode spectrum at k = 0, which are listed in Table 1.

Thus, the spectrum of a planar 2D-phase in the magnetic field contains modes sim-ilar to those in the A-phase without the magnetic field, as well as a number of newmodes. The former consist of six Goldstone modes, four clapping modes, and twopairbreaking modes. Two quasi-Goldstone modes and two quasi-pairbreaking modesare obtained from the Goldstone- and pairbreaking modes, respectively, by substitut-ing E2 → E2 − 4μ2H 2. The gap in the quasi-Goldstone-mode spectrum is ∝2μH .Finally, we obtained two new modes having no analogs in the A-phase. They corre-spond to the variables u33 and v33, are not degenerate, and the difference betweentheir frequencies is small. Interestingly, whereas for the clapping- and pairbreakingmodes there exists in the A-phase a linear Zeeman effect (threefold splitting in themagnetic field), the frequencies of these modes in the 2D-phase are independent ofthe magnetic field, while the energies of the quasi-pairbreaking modes and of thetwo “new” modes are quadratic in the field. Note also that the energies of all thenonphonon modes, except the two “new” ones, have imaginary parts due, just as inA-phase, to the vanishing of a Fermi-spectrum gap in a special direction (that of themagnetic field). The frequencies of all the nonphonon modes of the spectrum turn

Table 1 The collective mode spectrum in planar 2D-phase at zero momentum of collective excitation(k = 0)

No. Type Frequency Variables

6 Goldstone E = 0 u31, u32, v31, v32, u12 − u21,

v11 + v22

4 clapping E = (1.17 − i · 0.13)Δ0 v11 − v22, u12 + u21, u11 − u22,

v12 + v21

2 pairbreaking E = (1.96 − i · 0.31)Δ0 u11 + u22, v12 − v21

2 quasi-Goldstone E = 2μH u13, u23

2 quasi-pairbreaking E2 = (1.96 − i · 0.31)2Δ20 + 4μ2H v13, v23

1 E2 = (0.518)2Δ20 + 4μ2H 2 u33

1 E2 = (0.495)2Δ20 + 4μ2H 2 v33

J Low Temp Phys (2009) 155: 200–217 211

out to be complex, in view of the possible decay of the collective excitations into theinitial fermions (owing to the vanishing of the Fermi-spectrum gap along the fielddirection). Just as in the A- and B-phases, collective modes can be excited in the2D-phase in ultrasound and NMR experiments.

Note that notwithstanding some similarity between the spectra of the A- and 2D-phases, they also have substantial differences, that can possibly help identify the 2D-phase. Just as in the latter, there exist some nonphonon modes absent in the A-phase(and also in the B-phase), and the behavior of the spectrum (and even of the analogmodes) in the 2D-phase and in the A-phase is quite different: in the A-phase we havea linear splitting of the pairbreaking- and clapping modes, while in the 2D-phase onepart of the spectrum is independent of the field, whereas the other part has a quadraticfield dependence.

Collective mode spectrum in the 2D-phase was studied also by Hirashimaet al. [17] who, however, considered a 2D-phase without the magnetic field. Sincethe 2D-phase is stable only for H > HC , the meaning of their calculations is notclear. Obviously, they could not obtain six collective modes with frequencies depen-dent on the magnetic field. Comparing nonetheless results of Brusov et al. [13] withthose of Ref. [17], we note the following:

1. The main conclusions of both studies, that the 2D-phase spectrum coincides inpart with the A-phase spectrum, but modes, typical of the 2D-phase, are presentand are close to each other.

2. The correspondence between that fraction of the modes which is the same in bothphases as in the A-phase spectrum investigation by the kinetic-equation [18] andpath-integration methods. A frequency ωcl = 1.23Δ0(T ) was thus obtained inRef. [18] for the clapping mode, as against ωcl = 1.17Δ0(T ) in Brusov et al. [16]paper, in much better agreement with the experiments in the A-phase (see Ref. [19]and the citations therein). The reason is that we have taken into account the col-lective mode damping due to decay of Cooper pairs in view of the vanishing ofthe Fermi-spectrum gap (see Ref. [20] for details).

3. In Ref. [13] one mode was obtained, typical only of the 2D-phase and having anenergy somewhat lower than that of the super-flapping mode at all temperatures.This new mode is due to spin waves with a coupling coefficient O(k2). It is neitherresonant nor diffuse. Note that Brusov et al. [13, 16] have obtained not the super-flapping mode, but additional Goldstone modes whose appearance is due to thepresence of latent symmetry. As noted above, Brusov et al. [13] have obtainedin the magnetic field two modes, that are indicative only of the 2D-phase. Theirfrequencies are close to each other and depend on the field.

3.3 The Equations for Collective Mode Spectrum in 3He–2D with DispersionCorrections

In previous section we determined the complete collective excitation spectrum in3He–2D at zero momentum of collective excitations. But knowledge of the collec-tive excitation spectrum at zero momentum of excitations is not enough. First of allin sound experiments, which are used to study the collective excitation spectrum,the collective modes are created with nonzero momentum k, and for more detailed

212 J Low Temp Phys (2009) 155: 200–217

comparison of the theoretical results with sound experiment data we must take thedispersion corrections into account. On the other hand, taking the dispersion cor-rections into account can lead to the lift of the degeneracy of the collective modessimilarly to the case of 3He–B , where the dispersion-induced splitting of collectivemodes takes place [21] and has been observed experimentally [22]. Below we calcu-late the dispersion corrections for collective mode spectrum in planar 2D-phase.

Since the quadratic form (1.2) from Ref. [13] describes the collective excitationsin 2D-phase in the case of nonzero momenta as well, we can use it to calculate thedispersion corrections. For this we need to take traces and then to make replacement

(βV )−1∑

p1

→ k2F (2π)−4c−1

F

∫dω1 dξ1 dΩ1

= k2F (2π)−4c−1

F

∫dω1 dξ1 sin θ dθ dϕ.

After integrating in (1.2) from Ref. [13] with respect to ω1, ξ1, we must expand thecoefficients of quadratic form in powers of k up to k2. Then equating detQ to zero,where Q is the matrix of the coefficients of the quadratic form, we obtain the follow-ing equations for the collective mode spectrum.

Goldstone modes

∫ 1

0

(1 − x2)

[I +

((B − 4A)

(1 − x2)k2⊥

2+ (B − 2A)x2k2‖

)c2F

ω2

]dx = 0,

v11 + v22, u12 − u21;∫ 1

0x2

[(1 + 2c)I − 1 +

((B − A)(1 − x2)

k2⊥2

+ Bx2k2‖)

c2F

ω2

]dx = 0, v32, u31;

∫ 1

0x2

[(1 + 2c)I − 1 +

((B + A)(1 − x2)

k2⊥2

+ Bx2k2‖)

c2F

ω2

]dx = 0, u32, v31.

Clapping modes

∫ 1

0

(1 − x2)

[(1 + 2c)I +

((B − 2A)

(1 − x2)k2⊥

2+ Bx2k2‖

)c2F

ω2

]dx = 0,

u11 − u22, v12 + v21;∫ 1

0

(1 − x2)

[(1 + 2c)I +

((B + 2A)

(1 − x2)k2⊥

2+ Bx2k2‖

)c2F

ω2

]dx = 0,

v11 − v22, u12 + u21.

Pairbreaking modes

∫ 1

0

(1 − x2)

[(1 + 4c)I +

((B + 4A)

(1 − x2)k2⊥

2+ (B + 2A)x2k2‖

)c2F

ω2

]dx = 0,

u11 + u22, v12 − v21.

J Low Temp Phys (2009) 155: 200–217 213

Quasi-Goldstone modes

∫ 1

0

(1 − x2)

[IH + F2

((1 − x2)3k2⊥

4+ x2k2‖

)c2F

ω2

]dx = 0, u13;

∫ 1

0

(1 − x2)

[IH + F2

((1 − x2)k2⊥

4+ x2k2‖

)c2F

ω2

]dx = 0, u23.

(31)

Quasi-pairbreaking modes

∫ 1

0

(1 − x2)

[(1 + 4cH )IH + F1

((1 − x2)3k2⊥

4+ x2k2‖

)c2F

ω2

]dx = 0, v13;

∫ 1

0

(1 − x2)

[(1 + 4cH )IH + F1

((1 − x2)k2⊥

4+ x2k2‖

)c2F

ω2

]dx = 0, v23.

Mode E2 = (0.518)2Δ2 + 4μ2H 2

∫ 1

0x2

[IH − 1 + F2

((1 − x2)

k2⊥2

+ x2k2‖)

c2F

ω2

]dx = 0, u33.

Mode E2 = (0.495)2Δ2 + 4μ2H 2

∫ 1

0x2

[(1 + 4cH )IH − 1 + F1

((1 − x2)k2⊥

2+ x2k2‖

)c2F

ω2

]dx = 0, v33.

Here

k2⊥ = k21 + k2

2, k2‖ = k23, ω2

H = ω2 + (2μH)2,

c = Δ0(1 − x2)/ω2, cH = Δ0

(1 − x2)/ω2

H ,

I = I (c) = 1

(1 + 4c)1/2ln

(1 + 4c)1/2 + 1

(1 + 4c)1/2 − 1, IH = I (cH ),

uia = Re cia, via = Im cia,

A = c(1 − I (1 + 2c)

)/(1 + 4c), B = c

(1 + 2c − 4c2I

)/(1 + 4c),

F1 = (2μH)2

ω2H

(−2 − 20cH − 48c2H + IH 8cH

(1 + 7cH + 7c2

H

))/(1 + 4cH )2

+ B + 2A,

F2 = (2μH)2

ω2H

(−2 − 4cH − IH 8cH (1 + cH ))/(1 + 4cH )2 + B − 2A.

For each equation we have pointed out the corresponding variables.

214 J Low Temp Phys (2009) 155: 200–217

3.4 The Collective Mode Spectrum in 3He–2D with Dispersion Corrections

Solving (31) we obtain the 18 collective modes in planar 2D-phase with dispersioncorrections.

1. Goldstone modesThe spectrum of Goldstone modes consists of two sound modes with E(k) = cF k√

3and

of four orbital waves with E(k) = cF k‖. For latter ones there is a twofold splittingunder taking into account the next terms of the expansion [20]

E1(k) = cF k√3

(1 − sin2 θ cos2 θ

2(cos2 θ − 13 ) ln

4Δ20

f1(θ,k)

),

E2(k) = cF k‖(

1 − 11 cos2 θ − 3

24 cos2 θ ln4Δ2

0f2(θ,k)

),

E3(k) = cF k‖(

1 − 51 cos4 θ − 40 cos2 θ + 5

72 cos2 θ(cos2 θ − 13 ) ln

4Δ20

f3(θ,k)

),

(32)

where

f1(θ, k) = c2F k2

(cos2 θ − 1

3

), f2(θ, k) = 1

12c2F k2(11 cos2 θ − 3

),

f3(θ, k) = 1

36c2F k2 51 cos4 θ − 40 cos2 θ + 5

cos2 θ(cos2 θ − 13 )

.

(33)

The obtained equations show that the stability of the spectrum in the 2D-phase de-pends on the angle θ between the excitation momentum and the preferred direction.The first (acoustic) mode is stable inside the cones cos2 θ > 1/3, and the second in-side the cones cos2 θ > 3/11. The third mode is stable in the regions

cos2 θ >20 + √

145

51,

1

3> cos2 θ >

20 − √145

51.

Outside the stability regions, the energy of the excitation becomes complex becauseof the imaginary parts of the logarithms in (32). Physically this is related to the pos-sibility of the decay of the excitation into constituted fermions whose momenta areclose to the preferred direction (see Fig. 1).

Note that in the considered approximation (neglecting the coupling between theGoldstone- and nonphonon modes) the Goldstone modes of the spectrum in the A-and 2D-phases coincide and differ only in the degeneracy multiplicity (3 in theA-phase and 2 in the 2D-phase).

J Low Temp Phys (2009) 155: 200–217 215

2. Clapping modes

Ecl = Δ0

{(1.17 − i0.13) + c2

F

Δ20

[k2‖(0.14 − i0.06) + k2⊥(0.26 + i0.05)

]},

u11 − u22, v12 + v21,

Ecl = Δ0

{(1.17 − i0.13) + c2

F

Δ20

[k2‖(0.14 − i0.06) + k2⊥(0.23 + i0.05)

]},

v11 − v22, u12 + u21.

(34)

3. Pairbreaking modes

Epb = Δ0

{(1.96 − i0.31) + c2

F

Δ20

[k2‖(0.15 + i0.02) + k2⊥(0.93 + i0.90)

]},

u11 + u22, v12 − v21. (35)

4. Quasi-Goldstone modes

Eqgd = 2μH + c2F

Δ0

[1

3k2‖ + 1

2k2⊥

], u13,

Eqgd = 2μH + c2F

Δ0

[1

3k2‖ + 1

6k2⊥

], u23.

(36)

5. Quasi-pairbreaking modes

Eqpb = Δ0

[(1.96 − i0.31)2 + (2μH)2

Δ20

]−1/2[(1.96 − i0.31)2 + (2μH)2

Δ20

+ c2F

Δ20

[k2‖(0.093 + i0.0004) + k2⊥(0.1009 − i0.0001)

]

+ (2μH)2c2F

Δ40

[k2‖(0.2573 − i0.4796) + k2⊥(0.1884 − i0.6175)

]], v13,

(37)

Eqpb = Δ0

[(1.96 − i0.31)2 + (2μH)2

Δ20

]−1/2[(1.96 − i0.31)2 + (2μH)2

Δ20

+ c2F

Δ20

[k2||(0.093 + i0.0004) + k2⊥(0.1009 − i0.0001)

]

+ (2μH)2c2F

Δ40

[k2||(0.2573 − i0.4796) + k2⊥(1.1884 − i0.6175)

]], v23.

(38)

216 J Low Temp Phys (2009) 155: 200–217

6. Mode E2 = (0.518)2Δ20 + 4μ2H 2

E = Δ0

[(0.518)2 + (2μH)2

Δ20

]−1/2[(0.518)2 + (2μH)2

Δ20

+ c2F

Δ20

[0.468k2‖ + 0.016k2⊥

] + (2μH)2c2F

Δ40

[α‖k2‖ + α⊥k2⊥

]], u33. (39)

7. Mode E2 = (0.495)2Δ20 + 4μ2H 2

E = Δ0

[(0.495)2 + (2μH)2

Δ20

]−1/2[(0.495)2 + (2μH)2

Δ20

+ c2F

Δ20

[0.467k2‖ + 0.016k2⊥

] + (2μH)2c2F

Δ40

[β‖k2‖ + β⊥k2⊥

]], v33. (40)

Note that terms with the coefficients α‖, α⊥ and β‖, β⊥ are of the higher order ofsmallness, because we suppose that both momentum of the collective mode and mag-netic field are small.

We have obtained the whole collective mode spectrum in 3He–2D with dispersioncorrections. It is interesting to note that for clapping modes the degeneracy dependson the direction of the collective mode momentum k with respect to the externalmagnetic field H , namely: the mode degeneracy remains the same as in the caseof zero momentum k for k‖H only. For any other direction different from this one(for example, k⊥H ), there is a twofold splitting of these modes. For details, seeRefs. [23, 24].

4 Conclusions

We have for the first time calculated the whole collective mode spectrum in axial andplanar phases of superfluid 3He with dispersion corrections.

The obtained results imply that new interesting features can be observed in ultra-sound experiments in axial and planar phases: the change of the number of peaksin ultrasound absorption into clapping mode. One peak, observed for these modesby Ling et al. [8] in axial phase, will split into two peaks in planar phase and intothree peaks in axial phase under the change of ultrasound direction with respect tothe external magnetic field H in planar phase and with respect to the vector l in axialphase.

In planar phase, some Goldstone modes in the magnetic field become massive(quasi-Goldstone) and have a similar twofold splitting under the change of ultrasounddirection with respect to the external magnetic field H .

The obtained results as well will be useful under interpretation of the ultrasoundexperiments in axial and planar phases of superfluid 3He, because knowledge of thedispersion corrections allows to make a more careful comparison between experi-mental data and theoretical results.

J Low Temp Phys (2009) 155: 200–217 217

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