Pis’ma v ZhETF, vol. 110, iss. 11, pp. 748 - 749

Superfluid³He in squeezed nematic aerogel

V. V. Dmitriev⁺¹⁾, M. S. Kutuzov^∗, A. A. Soldatov⁺, A. N. Yudin^+×

⁺P. L. Kapitza Institute for Physical Problems of Russian Academy of Sciences, 119334 Moscow, Russia

^∗Metallurg Engineering Ltd., 11415 Tallinn, Estonia

^×National Research University Higher School of Economics, 101000 Moscow, Russia

Submitted 3 November 2019

Resubmitted 4 November 2019

Accepted 5 November 2019

DOI: 10.1134/S0370274X19230061

Introduction. Nematic aerogels consist of nearly

measurements of spin diffusion in normal³He we have

parallel strands. In liquid 3He in these aerogels, the

determined effective mean free paths of³He quasiparti-

strands lead to an anisotropy of³He quasiparticle scat-

cles (at T = 0) along (λ_∥) and normal (λ_⊥) to z: λ_∥ =

tering. It makes favorable new superfluid phases: polar,

= 900 nm, λ_⊥ = 235 nm in mullite-F, and λ_∥ = 550 nm,

polar-distorted A (PdA) and polar-distored B (PdB) [1].

λ_⊥ = 130 nm in mullite-S.

Polar and PdA are Equal Spin Pairing (ESP) phases and

Theory. The strands fix m ∥ z and in the PdA

have the order parameter:

phase destroy the long-range order. As a result the 2D

Larkin-Imry-Ma (2D LIM) state is formed where ℓ is

A_νk = Δ₀e^iϕd_ν(am_k + ibn_k),

(1)

random in the plane normal to z [4, 5]. In isotropic 2D

where Δ₀ is the gap parameter, e^iϕ is the phase factor,

LIM state, projections of ℓ, averaged over space, are:

d is the unit spin vector, m and n are mutually or-

ℓ^2x

= ℓ^2y

= 1/2, ℓ^2z

= 0. The squeezing along ŷ

thogonal unit orbital vectors, and a² + b² = 1. The PdA

orients ℓ, on average, along x ( ℓ^2x

> 1/2, ℓ^2y

< 1/2)

phase (a² > b²) is an intermediate state between the po-

and changes NMR properties in the PdA phase [5] but

lar phase (a = 1, b = 0) and the A phase (a = b). PdA

does not in non-chiral polar phase.

and A phases are chiral with two nodes in the gap along

We identify the ESP phases by measuring cw NMR

ℓ = m×n. The polar phase has only one orbital vector

frequency shift (Δω) from the Larmor value (ω_L). This

m, and its gap is zero in the plane normal to m. Previ-

shift in the isotropic 2D LIM state is given by [2, 5]:

ous experiments with³He in nematic aerogel were done

using Obninsk aerogel or nafen (with AlOOH or Al₂O₃

2ω_LΔω = k(4 - 6b²)Ω^2A cos² µ = KΩ^2A cos² µ,

(2)

strands, respectively) of various porosities [2, 3]. It was

where Ω_A is the Leggett frequency of the A phase,

found that the superfluid transition occurs into PdA or

K = k(4 - 6b²), and, in a weak coupling limit, k =

polar phases. On further cooling, transitions from polar

= 1/(3 - 4a²b²). If the transition temperatures of bulk

to PdA, and then to PdB phases were observed. Here

³He (T_c) and of³He in aerogel (T_ca) are close, then

we investigate the ESP phases in a new nematic aerogel

Ω_A(T/T_ca)/Ω_A0(T/T_c) = T_ca/T_c [3], where Ω_A0 is the

with mullite strands. This aerogel is closer to an ideal

Leggett frequency of bulk³He-A, which is known. Then

array of parallel cylinders, as it is more transparent and

measurements of Δω allow to find K and identify the

easily splits along the strands. The mullite aerogel has

phases (in the A phase K = 1/2, in the polar phase

an overall density 150 mg/cm³, porosity ≈ 96 %, and di-

K = 4/3). However, the weak coupling works well only

ameter of strands ≲ 10 nm. We use two samples with

at low pressures, so at high pressure we should use ex-

a cuboid shape of sizes 3-4 mm: undeformed (mullite-

perimentally found K in the polar phase (K_p) which

F) and unidirectionally squeezed by 30 % transversely

decreases from 4/3 at 0 bar to 1.15 at 29.3 bar [3].

to the strands (mullite-S). In particular, we investigate

In the anisotropic 2D LIM state the shifts Δω_∥ (at

how the squeezing changes properties of chiral PdA and

µ = 0) and Δω_⊥ (µ = π/2 and H ∥ ŷ) are given by [5]:

non-chiral polar phases.

Methods. Experiments were done at 7.1-29.3 bar

(

)

2ω_LΔω_∥ = 4

1-b² -b2 ℓ2y

kΩ^2A, µ = 0,

(3)

using cw NMR in fields 139-305 Oe at different angles

(

)

µ between the field H and the strands direction z. In

2ω_LΔω_⊥ = 4b²

1-2 ℓ^2y

kΩ^2A, µ = π/2.

(4)

order to avoid surface solid³He on the strands, they

It follows from Eq. (4) that there is a qualitative dif-

have been covered by ∼ 2.5 atomic layers of⁴He. By

ference between PdA and polar phases: in the polar

¹⁾e-mail: dmitriev@kapitza.ras.ru

phase Δω_⊥ = 0, while in the PdA phase Δω_⊥ = 0.

748

Письма в ЖЭТФ том 110 вып. 11 - 12

2019

Superfluid³He in squeezed nematic aerogel

749

At low pressures (where the weak coupling works) mea-

surements of Δω_∥ and Δω_⊥ allow to determine b² and

ℓ2y . At high pressures for this purpose we can use ex-

perimental value of K_p = 4k_p and assume that, for small

distortions from the polar state, strong coupling correc-

tions do not change qualitatively the dependence of k

on b²; that is, k(P)/k_p(P) = 3/(3 - 4a²b²).

Results. At 29.3 bar the superfluid transition of

³He in mullite-F occurs into the polar phase at T_ca ≈

≈ 0.988 T_c as it is seen from measurements of Δω_∥ (open

circles in Fig. 1a). On further cooling, a second-order

transition into the PdA phase takes place at ≈ 0.95 T_c

as the data deviate from the curve for polar phase. As

it follows from Eq. (2) Δω_⊥ = 0 (filled circles).

In mullite-S the transition to the polar phase occurs

at T_ca ≈ 0.980 T_c, and just below this temperature data

for Δω_∥ (open triangles) follow the curve with the same

slope as for mullite-F. On cooling, the polar phase per-

sists, until the positive shift for µ = π/2 (filled triangles)

appears at T_PdA ≈ 0.915 T_c indicating a transition to

the PdA phase. Using Eqs. (3), (4) and measured Δω_∥

and Δω_⊥ we have calculated b² and ℓ^2y . It was found

that b² increases from 0 to 0.26 on cooling from T_PdA

to 0.5 T_c in agreement with [1], and ℓ^2y levels off at

≈ 0.33 confirming the anisotropy of the 2D LIM state.

At 15.4 bar ℓ^2y

≈ 0.35 and b² increases from 0 to 0.13

Fig. 1. (Color online) (a) - Δω_∥ (open symbols) and Δω_⊥

from T_PdA ≈ 0.83 T_c to 0.65 T_c. The superfluid phase

(filled symbols) versus T in mullite-F (circles) and mullite-

diagram in our samples is shown in Fig. 1b.

S (triangles). Solid and dashed lines are the theory for Δω_∥

It was recently stated that Anderson theorem for s-

in the polar phase with Kp = 1.15 for Tca = 0.988 Tc and

wave superconductors is applicable to superfluid³He in

Tca = 0.980 Tc, respectively. Values of Δω⊥ in mullite-S

nematic aerogel for ideally parallel strands and specu-

are multiplied by 3. (b) - Phase diagram of³He in mullite-

lar reflection of³He quasiparticles [6]. In particular, the

S. Filled circles mark Tca. Open circles mark the transition

between polar and PdA phases. Dashed and short dashed

change of Δω_∥ near T = 0 should be proportional to

lines indicate transitions between normal and polar, polar

-T³ as observed in recent experiments [7]. Our results

and PdA phases, respectively, in mullite-F

agree with another prediction of [6]: the temperature

range of existence of the polar phase is proportional to

Full text of the paper is published in JETP Letters

λ^-1⊥. We also note that the suppression of the super-

journal. DOI: 10.1134/S0021364019230024

fluid transition temperature of³He in mullite-F (with

porosity 96 %) is smaller than in nafen-90 with higher

porosity (97.8 %). It agrees with one more prediction of

1. K. Aoyama and R. Ikeda, Phys. Rev. B 73, 060504

[6] that in the ideal case T_ca ≡ T_c.

(2006).

2. R. Sh. Askhadullin, V. V. Dmitriev, D. A. Krasnikhin,

Conclusions. We have investigated the ESP phases

P. N. Martynov, A. A. Osipov, A. A. Senin, and

of³He in two samples of new (mullite) nematic aero-

A. N. Yudin, JETP Lett. 95, 326 (2012).

gel. In both samples the superfluid transition of³He

3. V. V. Dmitriev, A. A. Senin, A. A. Soldatov, and

occurs into the polar phase with no qualitative differ-

A. N. Yudin, Phys. Rev. Lett. 115, 165304 (2015).

ence in NMR properties. The difference appears in the

4. G. E. Volovik, J. Low Temp. Phys. 150, 453 (2008).

PdA phase in the 2D LIM state, which is anisotropic in

5. R. Sh. Askhadullin, V. V. Dmitriev, P. N. Martynov,

squeezed sample. In the latter case we have determined

A. A. Osipov, A.A. Senin, and A.N. Yudin, JETP Lett.

values of the anisotropy and of the polar distortion. Our

100, 662 (2015).

results provide an additional proof of existence of the

6. I. A. Fomin, JETP 127, 933 (2018).

polar phase and support the application of Anderson

7. V. B. Eltsov, T. Kamppinen, J. Rysti, and G. E. Volovik,

theorem to³He in nematic aerogel.

arXiv:1908.01645 (2019).

This work was supported by the Russian Science

Foundation (project # 18-12-00384).

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2019