Linear NMR in the polar phase of $^3$He in aerogel

0.39 MB·

by  V. V. Zavjalov

Linear NMR in the polar phase of 3He in aerogel
V. V. Zavjalov1),
LowTemperatureLaboratory,DepartmentofAppliedPhysics,AaltoUniversity,POBox15100, FI-00076AALTO,Finland
3He is an example of the system with non-trivial Cooper paring. A few different superfluid phases are
known in this system. Recently the new one, the polar phase, have been observed in 3He confined in ne-
matically ordered aerogel [1]. As in other phases, a numberof various topological defects can exist the polar
phase. One of them, a half-quantum vortex can couple rotation of the sample and its spin dynamics. The
6
1 two-dimentional defects, d-solitons, appear if the magnetic field is tilted from the direction of the aerogel
0 strands. These solitons connect half-quantum vortices and can be observed by means of nuclear magnetic
2 resonance (NMR) [2]. Here we present theoretical and numerical studies of linear NMR in the polar phase
n both in theuniform order-parameter textureand in thepresence of half-quantumvortices and d-solitons.
a
J
Order-parameter field and energies γ2
6 F = (S γH)+ S2+δ (d S)2 , (7)
M
1 We are studying the polar phase of 3He in nemati- − · 2χ⊥ (cid:2) · (cid:3)
1
] cally ordered aerogel. The order parameter in this sys- FSO =2∆2gD (d l)2 , (8)
rtem is h · − 3i
e 1 ∆2
th Aaj = √3∆ eiϕdalj, (1) F∇ = 2 Kjk [(∇jϕ)(∇kϕ)+(∇jda)(∇kda)], (9)
o
.where ϕ is the phase, and d and l are unit vectors in where symmetric matrix K =K δ +(K +K )l l
t jk 1 jk 2 3 j k
aspin and orbital spaces respectively. The orbital unit is introduced. Motion of the phase ϕ (sound) is not
m
vector l is directed along the aerogel strands and can coupled with the motion of d (spin waves). Terms with
-not move. thephasegradientsgiveonlyaconstantcontributionto
d
n There are three components of the Hamiltonian the energy and can be skipped.
owhich are important for spin dynamics: magnetic en-
c
ergy, energy of spin-orbit interaction and gradient en-
[ Equilibrium texture
ergy:

1 =F +F +F , (2) Let’s first study the static picture. In the equilib-
M SO ∇
v H
rium ∂ /∂S =0. This means
0 γ2 H a
19 FM =−(S·γH)+ 2 χ−ab1SaSb, (3) S0+δ (d0 S0) d0 = χ⊥ H, (10)
4 2 · γ
F =3g A∗ A +A∗ A A∗ A , (4)
0 SO Dh jj kk jk kj − 3 jk jki whereS0andd0areequilibriumvaluesofSandd. Mul-
01. F∇ = 32hK1(∇jA∗ak)(∇jAak) (5) tiplyingthisbyd0 wecanfind(d0·S0)=χk/γ (d0·H).
then substituting it back to (10) we find the value for
6
+K ( A∗ )( A )+K ( A∗ )( A ) ,
1 2 ∇j ak ∇k aj 3 ∇j aj ∇k ak i the spin in the equilibrium:
:
v
XiwbihlietryeχSiissasnpiisnotarnodpiHc, tishethaeximsaogfnaentiicsofitreoldp.ySisusdceapntid- γSa0 =hχ⊥δab−(χ⊥−χk)d0ad0biHb =χabHb (11)
rminimumofthemagneticenergycorrespondstoS d.
For calculation of the equilibrium distribution (tex-
a ⊥
This can be written as ture) of the d vector we will use a coordinate system
1 where H ˆz and l is in ˆz yˆ plane (See Fig.1). This
χ−1 = (δ +δ d d ), δ =(χ χ )/χ >0. (6) k −
ab χ ab a b ⊥− k k can be written as
⊥
Substituting the order parameter (1) into energies H=ˆzH, l=yˆsinµ+ˆzcosµ, (12)
and using the fact that l is uniform we have d0 =(xˆcosα+yˆsinα)sinβ+ˆzcosβ.
Hereµisanglebetweenlandmagneticfield,itissetby
the experimental setup because direction of l is deter-
1)e-mail: vladislav.zavyalov@aalto.fi minedbyaerogel;β isanglebetweend0 andthefield; α
1
2 V. V. Zavjalov
is azimuthal angle of the d in the plane, perpendicular 2ξ¯2 α=sin2α, (19)
Djk ∇j∇k
to the magnetic field, it is counted from the line, per- ξ
where ξ¯2 = Djk .
pendicular to both H and l which corresponds to the Djk sin2µ
minimum of energy.
One canseethatin the caseofH l(or µ=0)there is
k
z nolengthscaleinthisproblem. dcanfreelymoveinthe
plane perpendicular to the field and only the gradient
µ
termisimportant. Tiltingthe magneticfieldfromthel
β H directionmakestheξ¯ finite. AtH lthelengthscale
D
⊥
reaches its minimum value, ξ .
D
l
Textural defects
x d0
α y Equation (19) shows that in a tilted magnetic field
there are two possible uniform textures with α = 0
Fig. 1. Angles, used in the texturecalculations
and α = π. Vector d is oriented perpendicularly to
both H and l and can point in two possible directions.
Between this two states there is a d-soliton. One can
The energies (7)-(9) (without constant terms) are:
also imagine a spin vortex in which vector d rotates
by 2π around the vortex line. Two d-solitons should
1
FM = 2(χ⊥−χk)H2 cos2β, (13) end at this vortex. Looking at the order parameter
formula (1) one can see that there can be also a half-
F =2g ∆2 (sinαsinβsinµ+cosβcosµ)2, (14)
SO D
quantum vortex, in which both vector d and phase φ
∆2
F = K [sin2β( α)( α)+( β)( β)](15) rotate by π around the vortex line. This is possible
∇ jk j k j k
2 ∇ ∇ ∇ ∇
becauseA (d,φ)=A ( d,φ+π). Inthetiltedmag-
αj αj
−
There are two scales introduced by these energies. netic field one d-soliton should end at the vortex. On
Ratio of magnetic and gradient energies gives the mag- Fig. 2 two types of vortices are shown.
netic length ξ and ratio of spin-orbitand gradienten-
H
ergies gives the dipolar length ξ . Since the gradient Half-quantum vortex Spin vortex
D
energy is anisotropic, we have different values in direc-
tions perpendicular and parallel to the l vector: d-soliton
π
ξH2jk = H2(Kχj⊥k∆2χk), ξD2jk = 4KgjDk (16) =0...
−
φ
In the high-field limit ξ ξ . Magnetic energy is drotates byπ, drotatesby2π,
D H
≫ φchanges byπ φisuniform
in the minimum everywhere excluding small regions of
the ξ size (for example cores of spin vortices). The Fig.2. Thehalf-quantumvortexandthespinvortexin
H
thepolarphaseof3He. Vectorlisperpendiculartothe
small volume of this regions makes them invisible in
picture plane. Angle α = 0 is changing by π between
NMR experiments. In the rest of the volume β = π/2,
upperandlowerpartsofthepicture. Thiscanbedone
only variations of α are important and the energy is:
via either a d-soliton or a π jump in the phase (which
1 is shown by color gradient).
= K ∆2 ( α)( α)+2g ∆2 sin2αsin2µ (17)
jk j k D
H 2 ∇ ∇
The equilibrium state corresponds to the mini-
The form of the infinite d-soliton can be found ana-
mum: δ /δα = 0. Since the energy depends on the
lytically. Inthisone-dimentionalproblemequation(19)
H
gradient we have to use variational derivative
has a form of static sine-Gordon equation:
δ ∂ ∂
1
H = H j H . (18) ξ¯2 α′′(x)= sin2α(x), (20)
δα ∂α −∇ ∂ jα D 2
∇
Usingthisforenergy(17)wehaveasimpleequationfor where x is a coordinateperpendicular to the wall. Here
the distribution of α: the value of ξ¯ depends on the wall orientation: if x
D
Linear NMR in the polar phase of 3He in aerogel 3
coordinate goes perpendicular or parallelto l, it should Using these equations one can show that d(d S) = 0
be ξ¯ or ξ¯ respectively. and thus the value (d S) is an integral ofdmt oti·on.
D⊥ Dk
·
The analytical solution can be obtained by multi- Derivatives of the Hamiltonian are:
plying the equation by a′ and integrating with proper
δ γ2
boundary conditions. Then for a single soliton with H = γH + [S +δ (d S)d ], (26)
a a a
δS − χ ·
sinα( )=0 and α′( )=0 we have a ⊥
±∞ ±∞ δ δ γ2
H = (d S)S (27)
ξ¯D2 (α′)2 =sin2α, (21) δda χ⊥ · a
+4g ∆2 (d l)l K ∆2 ( d ).
D a jk j k a
· − ∇ ∇
and then for the domain wall at x=0:
Substituting (26), (27), and (23) into equations (24-25)
α(x)=2arctan exp(x/ξ¯D) (22) one has:
(cid:0) (cid:1)
In the 2D case with isotropic ξ (which takes place S˙ =[S γH] (28)
D
×
when the texture is uniform along l-direction) the sine- +4g ∆2 (d l)[l d] K ∆2 [ d d],
D jk j k
Gordon equation has analytic solutions for a number · × − ∇ ∇ ×
of configurations with spin vortices and solitons [3, 4]. d˙ =γ d H γ S . (29)
This includes, in particular, the kink on soliton, which (cid:20) ×(cid:18) − χ⊥ (cid:19)(cid:21)
represents the 2π spin vortex with two π-solitons be-
Note that the anisotropy of susceptibility do not affect
ing on the opposite sides of it (see right part of Fig. 2).
dynamics.
The linearchainofthe alternating2π and 2π vortices
−
(kinks on straight soliton) has also analytic solution.
The configurationwith two solitons crossing eachother Linearized dynamics
may alsorepresentthe spinvortex,if eachsolitonhas a
Consider small oscillations near the equilibrium:
kink andthe positions of twokinks coincide. This is 4π
spin vortex, from which four π-solitons emerge. Such S=S0+δS(t), d=d0+δd(t) (30)
analytic solutions do not take into account the pinning
of vortices which exists in the real system. Linearize equations, differentiate the first one and
exclude δd. The result can be written as:
Spin dynamics δS¨ =[δS˙ γH] +Λ δS , (31)
a a ab b
×
TostudyspindynamicswewriteHamiltonequation Λ =Ω2 (d0 l)2δ [l d0] [l d0] (d0 l)d0l
using Poisson brackets. Motion of any value A in this ab P · ab− × a × b− · a b
+c2 (cid:2)(δ d0d0) 2d0( d0) (cid:3)
approachisgivenbyA˙ = ,A . Choiceofcoordinates jk ab− a b ∇j∇k− b ∇j a ∇k
isquitearbitraryasfaras{wHekn}owPoissonbracketsfor (cid:2)+ d0a(∇j∇kd0b)−d0b(∇j∇kd0a)
them. Brackets can be found from microscopic consid- (cid:3)
where we introduced parameters
erations,fromcommutationrulesinquantummechanic,
or from symmetry [5]. For spin S and a vector d in the ∆2γ2 ∆2γ2
Ω2 =4g , c2 =K =Ω2ξ2 , (32)
spin space the Poisson brackets are P D χ jk jk χ P Djk
⊥ ⊥
Sa,Sb = eabcSc, da,db =0, (23) and use the fact that cjk =ckj.
{ } − { }
Consider H zˆ and look for a harmonic solution
k
d ,S = S ,d = e d , δS=sexp(iωt). Then the equation can be written as
a b a b abc c
{ } { } −
and equations of motion: ω2s =Λ s +iω ω s , (33)
x xb b L y
−
δ δ ω2sy =Λyb sb iωLω sx,
S˙ = ,S = H S ,S + H d ,S (24) − −
a {H a} δSb{ b a} δdb{ b a} −ω2sz =Λzb sb
δ δ
= H S+ H d, In high field (comparing with dipolar and gradient
δS × δd ×
δ δ effects) motion of the spin is close to a Larmor preces-
d˙a ={H,da}= δSHb{Sb,da}+ δdHb{db,da} (25) sion with frequency ω ≈ ωL = γH and Λ ≪ ωL2. One
can separate equations by putting s from the second
δ y
= H d. equation to the first one and vise versa and neglecting
δS ×
4 V. V. Zavjalov
small terms. We get the same equations for s and s . 4
x y
π
This can be written as a single equation for a complex
coordinate s =(s +is )/√2: x)2 a
+ x y (
α
0
(ω2 ω2)s =i(Λ Λ )s +(Λ +Λ )s (34)
L− + xy− yx + xx yy +
-6 -4 -2 0 2 4 x/ξ¯D⊥6
In high field d0 is perpendicular to the field and we 0
can use angles (12) with β =π/2. Then
n
)-1
x b
Λxx+Λyy =Ω2P (1+sin2α)sin2µ−1 +c2jk ∇j∇k U(-2
1 (cid:2) (cid:3)
Λ Λ = Ω2 sin2α sin2µ (35)
xy− yx −2 P -6 -4 -2 0 2 4 x/ξ¯D⊥6
+2c2jk (∇j∇kα)+(∇jα)∇k . 1 c
(cid:2) (cid:3)
Substituting this into (34) and using (19) we have x)
(0
+
¯s
(ω2 ω2)s =Ω2 cos2µ sin2αsin2µ s (36)
− L + P n − o + -1
c2 +i ( α)+2( α) s . -6 -4 -2 0 2 4 x/ξ¯D⊥6
− jk n∇j∇k (cid:2) ∇j∇k ∇j ∇k(cid:3)o +
d
One can rewrite the equation in the form:
(ω2 ω2)s =Ω2 cos2µ sin2αsin2µ s (37)
− L + P n − o +
2
−c2jk n−(cid:18)∇i +∇α(cid:19)jk+(∇α)2jko s+. Fspigin. 3w.avAeninex1aDmspolleitoonf tshtreuccatulcruelsa.teBdlatcekxtcuurreveasncdortrhee-
spondtoaninfinitesoliton,blueandpurpleonescorre-
where we use notation (A)2 = A A (Discussion
jk j k spondtoperiodicstructureswithsameandalternating
of Aharonov-Bohm effect, hermitian operators and
soliton orientations. The distance between solitons is
real/complex solutions.) D = 4ξD. (a) Texture α(x). (b) Potential for a real-
It is useful to make a substitution s¯+ =s+exp(iα). value wave s¯+. Energy levels for all three textures are
Then the equation contains no imaginary terms: the same, λ = −1. (c) The real-value wave s¯ . Note,
+
that phaseof theactualmagnetization s rotates byπ
+
(ω2 ω2)s¯ =Ω2 cos2µ sin2αsin2µ s¯ (38) across each soliton in the direction determined by the
− L + P n − o + soliton orientation. One can check that the total mag-
−c2jk n∇j∇k+(∇jα)(∇kα)o s¯+, nDeitsitzraibtiuotnion|Rosf+thdex|amforplibtoutdhewanadvepshiassenoonf-zthereo.act(uda)l
The inversetransformationis neededif oneneed to cal- magnetization s+ =s¯+exp(−iα).
culate the actual distribution of magnetization.
whereasin(22)the valueofξ depends onthedomain
D
NMR in the uniform texture and in the
wall orientation. The frequency is
d-soliton
ω = ω2 +Ω2 cos2µ. (41)
ToobtainfrequencyoftheuniformNMRintheuni- s L P
q
form texture we put α=0 in (38). Then the frequency
OnNMRexperimentstwopeaksareobserved,onefrom
is
theuniformtextureandanotherfromthestatelocalized
ωu = ωL2 +Ω2P cos2µ. (39) in solitons. The difference between peaks is
q
This formula can be used to measure Ω . Ω2
P δω P sin2µ (42)
To find the spin wave, localized in the infinite d- ≈ 2ω
solitonweuse(38)andthe solitonequation(22)forthe Animportantparameterofthewaveistheintegralratio
distribution of α. This gives us
2
s
s¯+ =cosh−1(x/ξ¯D), (40) IM = (cid:12)(cid:12)RVV|s++|(cid:12)(cid:12)2 . (43)
R
Linear NMR in the polar phase of 3He in aerogel 5
a b c
π
Fig. 4. An example of the calculated texture and the spin wave in the soliton between two half-quantum vortices. (a)
The calculation grid made of 4696 cells covers one-forth of the whole area (8×8)ξD with two vortices separated by
D=7ξD. Densityofthegridischosenaccordingwithgradientsofthetexture,itishighernearvortices. (b)Calculated
valueofα. Onecanseeasmooth rotationbyπ inthesoliton betweenvorticesandπ jumpontheothersideofvortices
where phase also changes by π. (c) The calculated real-value wave s¯ .
+
ItconnectsthetotaltransversemagnetizationM ,mea- The solution for this problem is shown on Fig. 3.
⊥
sured in NMR experiments and energy E stored in the Parameter λ for both periodic structures has the same
wave: M2 = 2χ IME. For the infinite soliton with value 1 as for the infinite soliton.
⊥ ⊥ −
length L (L ξ ) the ratio is IM =2L. Let’s also study an effect of a finite-length soli-
D
≫
ton. Consider a two-dimensional problem with two
half-quantum vortices parallel to the l vector. Dis-
Numerical study of soliton structures
tance between vortices is D. The same equations (44)
It is interesting to study how various effects can and(45)aresolvednumericallyin2Dspaceusingdeal.II
change the frequency of the wave in the soliton. We library [7]. The code is available in [8]. An example of
willdoitnumericallyinoneandtwo-dimentionalcases, the calculation is presented on Fig. 4.
withsolitonsperpendiculartothe lvector. Usingcoor-
dinates in units of ξ¯2 one can write the equation (19) D D
D⊥ a b c d e
for the texture as
D
1 D
2α= sin2α, (44)
∇ 2
3
and equation (38) for the real-value waves as: /L b λ-0.4 e
M
2 s¯ +U(x) s¯ =λ s¯ , (45) I
+ + + 2
−∇ -0.6
d
where potential U(x)= ( α)2 sin2α and e
− ∇ − -0.8
1 c
ω2 ω2 Ω2 cos2µ ω2 ω2 a d a
λ= − L− P = − u. (46) -1.0
Ω2 sin2µ −ω2 ω2 b,c
P s − u 0
1 10 1 10
In the case of an infinite soliton ω =ωs and λ= 1. D/ξ¯D⊥ D/ξ¯D⊥
−
Using the equation (44) we can numerically calcu- Fig. 5. Calculated values of λ and IM/L for various
late distribution of α. Then, using equation (45) we soliton structures.
can calculate eigenvalues λ.
Firstconsidera1Dperiodicstructureofparallelsoli-
tons, locatedat some distance D from eachother. Soli- Near a half-quantum vortex, at a distance much
tonshaveanorientation(directionof α),andtwosim- smaller then ξ¯ , the textural angle α ϕ/2+const.,
D⊥
∇ ≈
pleststructureswhichwestudyaresequencesofsolitons where ϕ is the azimuthal coordinate. One can see that
with same and alternating orientations. the potential in (45) is U(x) ( α)2 1/4r2 (where r
≈ ∇ ≈
6 V. V. Zavjalov
is distance from the vortex core). The real-value wave
s¯ cannotfallintothisholebecauseofAharonov-Bohm
+
effect: it should be zero along some radial direction to
allow a smooth s distribution. The symmetry rea- 1. Polar phase of superfluid 3He in anisotropic aerogel.
+
sons tell, that in the case of two vortices with a soliton V.V.Dmitriev,A.A.Senin,A.A.Soldatov,A.N.Yudin,
Phys. Rev. Lett., 115, 165304 (2015), arXiv:1507.04275
the wave is zero on the line connecting vortices outside
them. The correspondingsolutionof the waveequation
2. Observationofhalf-quantumvorticesinsuperfluid3He.
is s¯ cos(φ/2+const.), this kind of discontinuity is
+ ≈ S.Autti,V.V.Dmitriev,V.B.Eltsov,J.M¨akinen,G.E.
clearly seen on the calculated wave near vortices.
Volovik,A.N. Yudin,V.V. Zavjalov, arXiv:1508.02197
OnFig.5calculatedvaluesofλandIM/L(whereL
is the soliton length) are plotted as a function of some 3. On vortex configurations in two-dimensional sine-
structure dimension D/ξ¯2 . There are five structures Gordon systems with applications to phase transitions
D⊥
which are shown on the upper part of the figure: a sin- of the Kosterlitz-Thouless type and to Josephson junc-
tions. O. Hudak,Phys.Lett. 89A, 245–248 (1982).
gle soliton with a finite length D; A periodic structures
of infinite solitons with the period D and same or al-
4. Relation Between Certain Quasi-Vortex Solutions and
ternating soliton orientations; the combination of both
Solitons of the Sine-Gordon Equation and Other Non-
effects, periodic structures of finite solitons with equal
linear Equations. A. Nakamura, J. Phys. Soc. Jpn. 52,
length and period (this corresponds to a square lattice 1918–1920 (1983).
of vortices). For large D all curves come to the values
for a single infinite soliton: λ = 1, IM = 2L. The 5. Poisson brackets in condensed matter physics. I.E.
−
numerical studies show that noticeable deviation of λ Dzyaloshinskii and G.E. Volovick, Annals of Physics,
from the asymptotic value appears only at high vortex 125, 67-97 (1980)
dencities, when the inter-vortex distance D is less then
6. Satellite magnetic resonances of a bound pair of half-
a few ξ¯ .
D quantum vortices in rotating superfluid 3He-A. Chia-
Ren Hu, Kazumi Maki, Phys. Rev. B, 36, 6871–6880
(1987)
Acknowledgements 7. https://www.dealii.org
I thank G.E. Volovik for useful discussions. This 8. https://github.com/slazav/dealii progs
work has been supported in part by the Academy of
Finland (project no. 284594).