научная статья по теме PHENOMENOLOGICAL PHASE DIAGRAM OF SUPERFLUID 3HE IN A STRETCHED AEROGEL Физика

Текст научной статьи на тему «PHENOMENOLOGICAL PHASE DIAGRAM OF SUPERFLUID 3HE IN A STRETCHED AEROGEL»

>K9m 2014, TOM 145, bmii. 5, rap. 871 876

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PHENOMENOLOGICAL PHASE DIAGRAM OF SUPERFLUID 3He IN A STRETCHED AEROGEL

I. A. Fomin*

Kapitza Institute for Physical Problems, Russian Academy of Science 119334, Moscow, Russia

Received January 21, 2014

Highly anisotropic "nematically ordered" aerogel induces global uniaxial anisotropy in superfluid JHo. The anisotropy lowers symmetry of JHo in the aerogel from spherical to axial. As a result, instead of one transition temperature in a state with an orbital moment I = 1 there are two, corresponding to projections I- = 0 and I _ ^ j^jg splitting has a pronounced effect on the phase diagram of superfluid JHo and on the structures of the appearing phases. Possible phase diagrams, obtained phenomenologically on the basis of Landau expansion of the thermodynamic potential in the vicinity of the transition temperature are presented here. The order parameters corresponding to each phase and their temperature dependences are found.

DOI: 10.7868/S0044451014050103

1. INTRODUCTION

At a triplet Cooper pairing, the transition temperature Tc is degenerate with respect to three projections of spin. In superfluid 3He, where Cooper pairs are formed in a state with the orbital angular momentum / = 1, there is additional degeneracy with respect to three projections of the orbital angular momentum. A proper superposition of all components is represented by an order parameter, which is a 3 x 3 matrix of complex amplitudes Atlj. The spin projections are labeled here by the index //. and the orbital ones, by j. The concrete form of the order parameter is determined by minimizing the corresponding thermodynamic potential with respect to Atlj. In the case of superfluid 3He, depending 011 pressure, the stable minima correspond to order parameters describing the Anderson Brinkman Morel (ABM) or Ballian Werthammer (BW) phases [1]. In both cases, the form of the order parameter does not change with temperature, and only the overall amplitude A increases upon cooling. This is eventually a manifestation of the mentioned degeneracy.

Lowering the spherical symmetry of liquid 3He by external fields or oriented impurities can split the transition to the superfluid state and partly separate ele-

E-mail: fbmin'fflkapitza.ras.ru

merits that under cooling evolve together into the corresponding order parameter. For example, the degeneracy of Tc over spin projections is lifted by a magnetic field H^. Its principal effect is described by the Zeeman term in the free energy

■I-» ~ //;,//,

which has to be added to the expansion of the free energy in powers of Atlj. As a result, the transition temperature Tc is split into two, such that the transition temperature for s- = ±1 is higher than that for s- = 0, and in the magnetic field, the ABM phase, which does not include the s- = 0 component, is formed first.

Similarly, the degeneracy of Tc with respect to the orbital projections is lifted by global orbital anisotropy. Such anisotropy can be induced by a deformed aerogel immersed in superfluid 3He [2]. Aoyama and Ikeda [3] theoretically studied the effect of uniaxial global anisotropy on the phase diagram of superfluid 3He. Their argument was based on a model in which global anisotropy is induced by the averaged effect of anisotropic scattering of quasiparticles by oriented impurities. They predicted, in particular, that a uniaxial stretch of aerogel just below the transition temperature would stabilize the polar phase, which on cooling to lower temperatures undergoes a continuous transition to the distorted ABM phase and eventually the BW phase is formed via a first-order transition. These predictions were tested in experiments with the "nematically ordered" aerogel [4], which can be regarded

as being infinitely stretched. The phase diagram found experimentally confirms the predicted sequence of the phase transitions, but other, even qualitative features of the two phase diagrams are different.

In this paper, possible phase diagrams of superfluid 3He in a stretched aerogel are considered phenomeno-logically. It is shown that depending on the values of phenomenological parameters characterizing this system, different routes of development of the order parameter upon cooling from the transition temperature are possible. The orbital anisotropy is formally described by an additional term in the thermodynamic potential

where Kji is a real symmetric tensor, which can be taken traceless. It is assumed to be uniform (i.e., independent of the coordinate). Random local anisotropy is neglected. This approximation is well justified in the present context, when only structures of order parameters of possible phases are considered. On the other hand, random anisotropy can strongly affect the orientation of order parameters of the distorted ABM and of the axi-planar phases, giving rise to a randomly nonuniform Larkin Imry Ma (LIM) state. In the case of a stretched aerogel, it is a two-dimensional LIM state, as discussed in Refs. [5, 6]. In comparing with experiment, in particular, with NMR data, a corresponding averaging over orientations of the order parameter has to be made.

With the global anisotropy taken into account, the standard expansion of the thermodynamic potential in powers of A^j is

= <I>„ + A>//[(r% + sjnA.jA;, + + ^iA^A^AUAI, + ihA^A^ArtAh + + fisA^A^AhAl, + tiiArfAljAriAl, +

+ As Aflj A*,j A A* i)]. (1)

Here,

is the dimensionless temperature, Tc is the transition temperature, defined such that it includes all global isotropic shifts from the corresponding temperature of bulk 3He. The overall coefficient Ncff has the dimension of density of states. Phenomenological coefficients /ii,... , A depend on the pressure and the properties of the aerogel. When anisotropy is uniaxial, in proper axes

Kxx = Kyy = — K. K-_-_ = 2k.

In contrast to a magnetic field, which always favors ,st = ±1 projections, a uniaxial deformation of the aerogel, depending on the sign of k, favors either the /. = ±1 or /. = 0 projection. For a compressed aerogel, k > 0 and the states with /- = ±1 have a higher transition temperature, while for a stretched aerogel, k < 0, a state with /- = 0 is favored.

Stabilization of the polar phase by a stretched aerogel within this approach follows immediately from the explicit form of the second-order terms in the expression for thermodynamic potential (1):

(r + 2k).4/j.,4* . + (r - K)M/(,+ AinjA*iy).

For negative k, the highest transition temperature is r = —2k- For realistic values of the coefficients /i, in particular, if 3i5 < 0 (here and in what follows, conventional shorthand notation for sums of /i coefficients is used, e. g., 3i + A = /?i5. etc.) below r = —2k. the superfluid polar phase is favored [7]. Its order parameter can be written as

A°j = A„expUV K,/»;.

where d^ is a real spin vector and inj is a unit vector in ¿-direction.

The polar phase is stable within the interval of temperatures r ~ k. On further cooling, the suppressed angular momentum projections /- = ±1 come into effect; they change the order parameter symmetry and further phase transitions can occur. While stabilization of the polar phase practically depends only on the sign of k, its stability interval and the sequence of further transitions also depend on the values of the coefficients /ii,... , 35. To avoid the discussion of nonrealis-tic situations, we have to restrict the region of admitted values of /is. Within the BCS theory, their values are proportional to one combination of the parameters, 30 = 7C(3)/87t2T(2:

3i. /„(^1/2.1.1.1.^1).

This set of values of the /is is referred to as the weak coupling limit [1]. In the definition of /i0 C(3) is the Riemann zeta function. The observed thermodynamic properties of bulk superfluid 3He in the vicinity of Tc can be fitted by the 3i. • • • . As that deviate from their weak coupling values by 10 20% [8]. The deviations are smaller at low pressures. For 3He in an aerogel, the situation is less certain. Impurities give rise to corrections to the 3 coefficients of the order of Co/A, where Co is the correlation length of superfluid 3He and A is the mean free path. This ratio is of the order of 1/10. In what follows, we assume that deviations of the /is

for superfluid 3He in a neniatically ordered aerogel from their-weak coupling values are also of the order of 1/10, at least at low pressures.

There is another reason for restricting the present discussion to a region of low pressures (for example, below 10 bar). The diameters of strands in a neniatically ordered aerogel, estimated as d ~ 10 11111 [4], are bigger than in silica aerogels and can be comparable with the correlation length of superfluid 3He, which at pressures above 20 bar is about 20 11111. When d ~ Co. perturbation of the order parameter in a vicinity of a strand is of the order of unity. Well below Tc, the condensate varies over a distance ~ which is smaller than the average distance « 200 11111. In this situation, the condensate is essentially nonuniform and the average order parameter does not properly characterize the state of 3He. The uniform approximation works better at low pressures and in the vicinity of Tc in the region where the Ginzburg and Landau (GL) coherence length £(T) exceeds not only the diameter of a strand but also In this GL region d -C £Q -C £(T), the average order parameter Atlj is a suitable characteristic of the state of superfluid 3He.

Preliminary results of plicnomcnological analysis of the phase diagram of superfluid 3He in a neniatically ordered aerogel were published before [7]. A principal suggestion in that paper was to regard the extra line (ESP2) in the experimentally found phase diagram as evidence of the possible stability (or meta-stability) of the axi-planar phase. Further experiments and their analysis [9] have shown that this suggestion is not correct. Nevertheless, there remains the question of possible stability of the axi-planar phase in an

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