научная статья по теме ANISOTROPIC 2D LARKIN-IMRY-MA STATE IN POLAR DISTORTED ABM PHASE OF 3HE IN “NEMATICALLY ORDERED” AEROGEL Физика

Текст научной статьи на тему «ANISOTROPIC 2D LARKIN-IMRY-MA STATE IN POLAR DISTORTED ABM PHASE OF 3HE IN “NEMATICALLY ORDERED” AEROGEL»

Pis'ma v ZhETF, vol. 100, iss. 10, pp. 747-753

© 2014 November 25

Anisotropic 2D Larkin-Imry-Ma state in polar distorted ABM phase of

3He in "nematically ordered" aerogel

It. Sh. Askhadullin+, V. V. Dmitriev^, P. N. Martynov+, A. A. Osipov+, A. A. Senin, A. N. Yudin

Kapitza Institute for Physical Problems of the RAS, 119334 Moscow, Russia + Leypunsky Institute for Physics and Power Engineering, 249033 Obninsk, Russia Submitted 21 October 2014

We present results of experiments in superfluid phases of 3He confined in aerogel which strands are nearly parallel to one another. High temperature superfluid phases of 3He in this aerogel (ESP1 and ESP2) are biaxial chiral phases and have polar distorted ABM order parameter which orbital part forms 2D Larkin-Imry-Ma state. We demonstrate that this state can be anisotropic if the aerogel is squeezed in direction transverse to the strands. Values of this anisotropy in ESP1 and ESP2 phases are different, what leads to different NMR properties.

DOI: 10.7868/S0370274X14220135

1. Introduction. A so-called "nematically ordered" (N-) aerogel differs from standard silica aerogels by a high value of a global anisotropy. This aerogel consists of A1203-H20 strands which are nearly parallel to one another [1], i.e. it may be considered as aerogel with infinite stretching anisotropy. Investigations of superfluid 3He confined in N-aerogel are especially interesting because according to a theory [2] such a strong anisotropy may make a superfluid polar phase more favorable than Anderson-Brinkman-Morel (ABM) phase which corresponds to A phase of bulk 3He and to A-like phase of 3He in isotropic or weakly anisotropic silica aerogels [35]. A superfluid phase diagram of 3He in N-aerogel is different from the case of 3He in silica aerogel with similar porosity [6]. The superfluid transition temperature (Tca) is slightly (by 3-6%) suppressed in comparison with the transition temperature (Tc) of bulk 3He. Depending on prehistory, pressure and temperature, three superfluid phases are observed: two Equal Spin Pairing phases (ESP1 or ESP2) and Low Temperature phase (LTP). The ESP1 phase appears on cooling from the normal state. On further cooling the first order transition into the LTP takes place. Due to inhomogeneities of the aerogel, this transition occurs in a wide temperature range 0.05 Tc). On warming from the LTP the back transition into the ESP phase is observed. At high pressures (P > 10 bar) the NMR frequency shift in this phase, called ESP2 phase, is greater than in the ESP1 phase at the same conditions.

The LTP has a polar distorted Balian-Werthamer (BW) order parameter [7]. As for ESP phases, their NMR properties point out that they both have ABM order parameter with a strong polar distortion [6]. This distortion is larger at low pressures and at higher temperatures. It was also found that the order parameter orbital vector 1 of the distorted ABM phase in N-aerogel is in a spatially inhomogeneous Larkin-Imry-Ma (LIM) state similar to that predicted in [8] and observed in Alike phase of 3He in silica aerogel [4, 9]. In N-aerogel we get the two-dimensional LIM state because the aligned strands orient 1 normal to their axis.

In this paper we present results of nuclear magnetic resonance (NMR) studies of liquid 3He confined in N-aerogel which was slightly squeezed in direction transverse to the strands. In particular, these experiments allow us to explain the difference between properties of ESP1 and ESP2 phases.

2. Theory. Transverse NMR frequency shift can be found from the following equation [10]:

Acj = —

9

dU

D

xH d cos ¡3

(1)

where g is the gyromagnetic ratio, x ~~ the spin susceptibility, H - the external magnetic field, /3 - the tipping angle of the magnetization, and UD - the density of the dipole energy, averaged over a fast spin precession. For the LIM state the dipole energy should also be averaged over the space (see e.g. [4, 11]). The order parameter of the ABM phase with polar distortion is:

-^e-mail: dmitriev@kapitza.ras.ru

Ajk = Aqe^dj (arrik + ibrik

(2)

IfiicbMa b >K3TO tom 100 Htm. 9-10 2014

747

where An is the gap parameter, d is the unit spin vector, m and n are mutually orthogonal unit vectors in the orbital space, and a2 + b2 = 1. For the ABM phase a = b, for polar distorted ABM phase a2 > 52, and for polar phase a = 1, b = 0. Similarly to pure ABM phase, the distorted ABM phase is a chiral phase and we can introduce the orbital vector 1 = m x n which orientation defines two Weyl points in the momentum space: the energy gap of this phase equals 0 along 1 and equals \/2oAo and \/25Ao along m and n. Note that the polar phase is not chiral and its gap has line of zeroes in the plane normal to m. The dipole energy density for the order parameter (2) is:

6

Ud = -9d [a2 (dm)2 + 62(dn)2]

(3)

where go = 9d{T) is the dipole constant. In weak coupling limit gd can be expressed in terms of the Leggett frequency of the pure ABM phase Qa [12]:

9D

3 - 4o262

9AD

£ A

3 -4a262 V6 g

O2

2 A

(4)

where g^ is the dipole constant of the ABM phase. Strong coupling corrections to (4) do not exceed ±5 % [13], therefore we do not consider them below.

Following [11, 14], we use two coordinate frames: an orbital frame (£, C) bound to the aerogel sample and a spin frame (x,y,z). We choose H = Hz and fix (axis along aerogel strands. Then strands of N-aerogel orient m || ( and 1 _L ( [2]. In the isotropic 2D LIM state vectors 1 and n are randomly distributed in £ — ?} plane and (l^j = (/2) = = = 1/2, where

angle brackets mean the space averaging. We introduce the angle A = A(r) which defines the orientation of specific 1 and the corresponding n: = —nv = cos A and lv = n^ = sin A. For uniaxially anisotropic in plane 2D LIM state we fix the £-axis along the direction corresponding to the maximum value of Consequently

1 > (l^j > 1/2 > (/2) and we assume that the distribution of = A) is symmetric.

An orientation of H with respect to the aerogel (Fig. 1) is described by angles of rotation of the orbital frame: ¿t (rotation around £) and <p (rotation around (). Then we get:

mx = 0, my = — sin yit, m~ = cos ¿t, nx = sin(<£> + A), ny = — cos/iCos(<£> + A), (5) nz = — sin yit cos^ + A).

Motions of d in the spin frame are described by Euler angles (a, /3,7), where a corresponds to the phase

B

Fig. 1. Orientation of H with respect to N-aerogel axes

of spin precession and /3 is the tipping angle. After an averaging over the fast spin precession we obtain:

^ = i(cos2$)(l+cos/3)2 + i(l - cos/3)2, 4 = i(sin2$)(l+cos/3)2 + i(l - cos/3)2,

d~L — 2 r,1"*J

ij sin" /3, dxdz = dydz = 0,

(6)

dxdy = -g (sin2$) (1 +cos/3)2,

where $ = a + 7 is a slow variable. Then the dipole energy (3) averaged over the space is:

Tl 6

Ud = -9D

5

,2 c J1

(d;

v v

(Em2)

+ K) + 4 (4) + ^ ("=> + . (7)

where (w2) = sin2 ip (cos2 A) + cos2 ip (sin2 A), (??.2) = cos2 ytt(cos2 ip (cos2 A) + sin2 p (sin2 A)), (??2) = sin2 yit(cos2 ip (cos2 A) + sin2 ip (sin2 A)), and (nxny) = (2 (sin2 A) — 1) cosyit sin cos p. The angle $ may be spatially homogeneous (the spin nematic state, SN) or random (the spin glass state, SG) [4]. The SN state is more favorable and corresponds to the homogeneous spatial distribution of d, but the SG state may be created e.g. in pulse NMR experiments after an application of large tipping pulses. In the isotropic SG state (sin2 $) = (cos2 $) = 1/2 and (sin 2$) = 0 while in the SN state $ is determined by minimization of (7).

The result of the minimization is shown in Fig. 1 where the shaded area corresponds to orientations of H with sin2 $ = 1 while for other orientations the minimum of (7) corresponds to sin2 $ = 0. The border of the shaded area satisfies to the following condition:

1? [((/|) COS2 ip + (/2) Sill2 ip) COS2 (J, -

- (Z|) sin2 <p - (l2) cos2 ip] + a2 sin2 ¿t = 0. (8)

In particular, if ip = 90° then sin2 $ = 1 for fj, < yitc and sin2 $ = 0 for ytt > yitc, where

52(l-2</2))

Sill yltc

1 - 62 - 62 </2> '

(9)

The critical angle ¿tc corresponds to an orientational transition: in the equilibrium SN state d _L fj for ¿t < ¿tc, while d || i) for fj, > fj,c.

The NMR frequency shift from the Larmor value can be obtained from (1) and (7):

1

Aw = -K 4

2 2 a my

-5>2>+5>2>)x x [l - 2sin2$(l + COS/3)] + + [4 - 5a2m2 - 62(7 (n2x) + 5 (n2))] cos/?}, (10)

where

K

2 Q24 3 -Aa?b2~

and uj = gH. Let consider 4 cases: <p = 0, 0 < ¿t < 90° (the case A); ¿t = 90°, 0 < p < 90° (B); <p = 90°, 0 < (j, < Atc (Ci); ip = 90°, Atc < (j, < 90° (Co). In Fig. 1 these orientations of H correspond to arcs marked A, B, Ci, and Co. Then for the case of continuous wave (CW) NMR (cos/3 « 1) we get:

A: A w = K(D sin2 ¿ i + E cos2 ¿t),

B : Alv = KD(l — 2sin2 ip),

C'i : Acj = KEcos2fj,,

C2: Auj = K(E cos2 fi-D),

(11)

where D = 62(1 -2 (l2)) > 0 and E = 1 -b2-b2 (l2) > > 0. The dependence of Aui on ¿t for ip = 90° is shown in Fig. 2. This dependence is fully determined by 2 values of the frequency shift: Aw^ = —KD (H || £) and Aw( = KE (H || C) so that sin2 f.ic = -Awe/Aw(.

In the isotropic 2D LIM state (l2) = 1/2 (i.e. D = 0) and for yit = 90° (the case B) Aui = 0 in agreement with [6]. If the 2D LIM state is anisotropic and (l2) < 1/2, then for yit = 90° the shift equals 0 for ¡p = 45°. For other values of <p the shift is 0 only for pure polar phase (6 = 0). In pure ABM phase or in the ABM phase with

Acó»-

S <

r Aoo = L 2 kûç(l -2sin p.) y 2 Aro = Acûç(1 - sin n) + Ao>!

t . 2 Sin |IC 1 2 COS 1xc 1

Aciv

0 0.25 0.50 0.75 1.0

. 2 Sin

Fig. 2. CW NMR frequency shift versus /li for cp = 90" as follows from (11)

polar distortion the shift is positive (if p < 45°) or negative (if ip > 45°).

3. Experimental setup. The experimental chamber used in the present work is similar to the chamber described in [6]. The chamber has two cells with N-aerogel samples. The samples (named below as 1 and 2) have a form of a cuboid with characteristic sizes of 4 mm. Initially the sampl

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