Impact of Strong Anisotropy on Phase Diagram of Superfluid 3 He in Aerogels
aa r X i v : . [ c ond - m a t . o t h e r] J a n Impact of Strong Anisotropy on Phase Diagram of Superfluid He in Aerogels
Tomohiro Hisamitsu, Masaki Tange, and Ryusuke Ikeda
Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan (Dated: February 3, 2020)Recently, one analog of the Anderson’s Theorem for the s -wave superconductor has attractedmuch interest in the context of the p -wave polar pairing state of superfluid He in a model aerogelin the limit of strong uniaxial anisotropy. We discuss to what extent the theorem is satisfied in thepolar phase in real aerogels by examining the normal to polar transition temperature T c and thelow temperature behavior of the superfluid energy gap under an anisotropy of a moderate strengthand comparing the obtained results with experimental data. The situation in which the Anderson’stheorem clearly breaks down is also discussed. PACS numbers:
Recent observations on superfluid He in anisotropicaerogels have clarified profound roles of an anisotropyfor the superfluid phase diagram and properties. Thepolar pairing state [1] has been discovered in nematicaerogels with a nearly one-dimensional structure [2]. Ithas been found that this polar pairing state does notoccur when the magnetic scattering effect due to the solid He localized on the surface of the aerogel structure isactive [3]. This high sensitivity to the type of ”impurity”scatterings of the superfluid phase diagram is not easilyexplained within the original theoretical model assuminga weak global anisotropy of the aerogel structure [1, 4].It has been pointed out that, when the aerogel struc-ture is in the limit of strong anisotropy so that the scat-tering is specular along the anisotropy axis, the normal topolar transition temperature T c ( P ) should be insensitiveto the (nonmagnetic) impurity scattering strength [5].Recently, this result analogous to the Anderson’s Theo-rem in the s -wave superconductor [6] has attracted muchinterest [7, 8] in relation to the low temperature behaviorof the energy gap in the polar phase and to the robust-ness of the p -wave superfluid polar phase in relativelydense nematic aerogels. Previously, various features seenin superfluid He in nematic aerogels [2, 7] have been dis-cussed based on the model assuming the weak anisotropy[1]. Here, the anisotropy is measured by the size of thecorrelation length L z of the random scattering potential.Once taking account of the puzzling result [3] brought bythe magnetic impurities altogether, the approach startingfrom the side of the strong anisotropy may be more ap-propriate. Further, the polar phase has been detected sofar only in the nematic aerogels. Then, one might wonderwhether the polar phase occurs only in the limit of stronganisotropy. However, L z in nematic aerogels seems to befinite if taking account of the fact that splayed strandsand crossings between straight strands are seen in realimages of the nematic aerogels [2, 8].In this communication, consequences of the stronganisotropy in the phase diagram of superfluid He inaerogels with no magnetic scattering effect are studiedin details. It is found that, in the weak-coupling BCSapproximation, the impurity-scattering independent T c is approximately satisfied even in the scattering poten- tial model with a finite correlation length L z along theanisotropy axis, implying that the Anderson’s Theoremis apparently satisfied over a wide range of the strengthsof the anisotropy. Thus, we argue that, consistently withthe original argument [1], the polar phase may be realizedin aerogels with a global anisotropy of a moderate magni-tude. Further, the dependences of the superfluid energygap | ∆( T ) | on the strengths of the impurity scatteringand the anisotropy are also examined, and the T be-havior arising from the horizontal line node of | ∆( T ) | inthe polar pairing symmetry is found to be robust againstchanges of the impurity strength and the anisotropy. Fur-ther, the situation in which T c ( P ) is also reduced so thatthe Anderson’s Theorem is not satisfied will also be dis-cussed.First, let us describe how the Anderson’s Theorem oc-curs in the context of the p -wave superfluid phase in anenvironment with nonmagnetic elastic impurity scatter-ings. The starting model of our analysis to be performedbelow is the BCS Hamiltonian for a spatially uniformequal-spin paired state in zero magnetic field H BCS − µN = X p ,σ (cid:20) ξ p a † p ,σ a p ,σ −
12 (∆ ∗ p a p ,σ a − p ,σ + h . c . ) (cid:21) + g − V | ∆ | , (1)where g is the strength of the attractive interaction, V is the system volume, | ∆ | is the maximum of the quasi-particle energy gap, and ξ p is the quasiparticle energymeasured from the Fermi energy µ .The total Hamiltonian H is the sum of eq.(1) and thenonmagnetic impurity potential term H imp = Z d r u ( r ) n ( r ) , (2)where n ( r ) is the particle density operator. As usual, theimpurity scattering can be modelled by the correlator W ( r ) = 2 πN (0) τ h u ( r ) u (0) i imp , (3)or its Fourier transform w ( k ) = R d r W ( r ) e i k · r , with h u i imp = 0, where h i imp denotes the random average, N (0) is the density of states on the Fermi surface perspin in the normal state, and τ is the relaxation time ofthe normal quasiparticle in the case with no anisotropy.For simplicity, the Born approximation will be used toincorporate the impurity-scattering effect in the Green’sfunctions for the quasiparticles in an equal-spin pairedsuperfluid state. Then, we have a mean field problemfor spin-less Fermions, and solving the corresponding gapequation can be performed in quite the same manner asin the s -wave paired case [9]. The resulting gap equationcan be expressed in the formln (cid:18) TT c ( P ) (cid:19) = πT X ε (cid:20) − | ε | + 3 (cid:28) ∆ p ˜∆ p ∆ − q ˜ ε p + | ˜∆ p | (cid:29) ˆ p (cid:21) , (4)where ε = πT (2 m + 1) with integer m , T c ( P ) is thesuperfluid transition temperature of the bulk liquid, and h i ˆ p denotes the angle average on the unit vector ˆ p overthe Fermi surface. Further, in eq.(4), i ˜ ε p = iε − πN (0) τ Z q w (ˆ p − ˆ q ) G q ( ε ) , ˜∆ p = ∆ p − πN (0) τ Z q w (ˆ p − ˆ q )[ F † q ( ε )] ∗ , (5)and G p ( ε ) = − i˜ ε p − ξ p ˜ ε p + ξ p + | ˜∆ p | , F † p ( ε ) = − ˜∆ ∗ p ˜ ε p + ξ p + | ˜∆ p | (6)are the impurity-averaged Matsubara Green’s functions[9].As a model of the impurity correlator (3) in the pres-ence of a stretched anisotropy favoring the polar phasein which ∆ p = ∆ˆ p z , we will use the following expression W ( r ) = k F δ (2) ( r ⊥ ) exp( −| z | /L z ) × [1 + θ (1 − | δ u | )( | δ u | − / − , (7)where L z is the correlation length defined along theanisotropy axis on the random distribution of the poten-tial u ( r ), and the z -axis is chosen here as the anisotropyaxis or stretched direction. The size of the anisotropyis measured by | δ u | = k L z , while the measure of theimpurity strength is 1 / ( τ T c ) [10], where k F is the Fermiwave number. Then, the Fourier transform w ( k ) of W ( r )becomes w (ˆ k ) = p | δ u | | δ u | ˆ k z (cid:18) | δ u | − / − θ (1 − | δ u | ) (cid:19) . (8)Equation (8) has the following limiting cases. Forthe weak anisotropy, | δ u | <
1, this model reduces tothe expression, w (ˆ k ) ≃ − | δ u | ˆ k z , introduced in Ref.[1].The opposite limit of the infinite | δ u | corresponds to the case with the impurity scattering persistent along thestretched direction. In this case, w (ˆ k ) reduces to w ∞ (ˆ k ) = πk F δ ( k z ) , (9)implying that, as sketched in Fig.1(a), the scattering isspecular along the z -axis. The present model (7) interpo-lating the above-mentioned two limits has been used tostudy how the half-quantum vortex (HQV) pair, whichshould appear in the polar phase, survives in the PdBphase at lower temperatures [11]. For any value of theanisotropy δ u and the impurity strength 1 / ( τ T c ), thepolar to normal transition temperature T c ( P ) and thesuperfluid gap | ∆( T ) | in the polar phase can be numeri-cally obtained using eqs.(4) and (8).Now, it is easy to verify the Anderson’s Theorem forthe polar pairing state with ∆ p = ∆ˆ p z in the limit of thestrong anisotropy. In fact, by applying eq.(9) to eq.(5),any 1 / ( τ T c ) dependence in the last term of eq.(4) iscancelled between the denominator and numerator of theterm, and, as in the s -wave pairing case, the gap equation(4) becomes its expression in clean limit or for the bulkliquid.Next, let us determine the polar to PdB transitiontemperature T PB ( P ). In the present weak-coupling ap-proach, the polar-distorted A (PdA) phase [1, 2, 8] doesnot appear, and, on cooling, the polar phase is trans-formed into the PdB phase through a continuous transi-tion. Since the real polar to PdA transition is also con-tinuous [1, 2], however, the T PB line obtained here is ex-pected to be qualitatively comparable with the polar toPdA transition line. The T PB ( P ) line is easily obtainedaccording to the diagrams sketched in Fig.2 represent-ing the gap equation linearized with respect to the order (cid:82)(cid:82) (cid:19) (cid:10)(cid:67)(cid:11)(cid:10)(cid:68)(cid:11) (cid:92)(cid:64) FIG. 1: (Color online) (a) Specular reflection expressing theconservation of the component along the ˆ z -axis (vertical di-rection) of the momentum. (b) Rough picture of planar scat-tering centers positioned randomly and with their surface per-pendicular to the z -axis. On each center, the component per-pendicular to the z -axis (i.e., parallel to the surface) of themomentum is conserved at each scattering event. parameter of PdB state by using the quantities charac-terizing the polar pairing state. Then, T BP is given bythe temperature T satisfyingln (cid:18) TT c ( P ) (cid:19) = πT X ε (cid:20) − | ε | + 32 (cid:28) − ˆ p z q ˜ ε p + | ˜∆ p | (cid:29) ˆ p (cid:21) . (10)Examples of the T c ( P ) and T PB ( P ) obtained numeri-cally from eqs.(4) and (10) are presented in Fig.3, wherethe experimental data on T c ( P ) [12] were used. As isseen in Fig.3(a) where a moderately large anisotropy | δ u | = 30 is used, T c ( P ) weakly depends on the impu-rity strength τ − . In general, for a stronger anisotropy,the τ − -dependence of T c becomes weaker, while thecorresponding one of T PB becomes stronger. At higherpressures, the pressure dependence of T c /T c ( P ) is quiteweak, reflecting the proximity to the limit of stronganisotropy in which the Anderson’s Theorem is exact,while T c /T c is lowered at low enough P -values becauseof an increase of the dimensionless impurity strength1 / ( τ T c ( P )). In contrast to T c , however, T PB is quitesensitive to the impurity strength and rapidly decreaseswith increasing 1 / ( τ T c ). Thus, the temperature rangeof the polar phase is wider for a lower P .Further, as Fig.3 (b) shows, an increase of theanisotropy extends the region of the polar phase: Withincreasing the anisotropy | δ u | , T c is increased and ap-proaches T c , while T PB decreases and approaches its fi-nite value in the limit of strong anisotropy (see Fig.3 (b)).In any case, the temperature range of the polar phase ata fixed P becomes wider with increasing the anisotropyand/or the impurity strength.The results on T c ( P ) in Fig.3 will be compared withthe corresponding curves determined in experiments[2, 3, 7, 13]. In aerogels, an effective decrease of theporosity leads to an enhancement of the ”impurity” scat-tering via the aerogel structure [3]. In fact, Fig.1(a) andFig.2(a) and (c) in Ref.[13] have shown a slight decreaseof T c and a drastic decrease of the transition line to thePdA phase due to a reduction of the porosity. This ten-dency of the two transition curves is consistent with thefeatures seen with increasing 1 / ( τ T c ) in Fig.3(a). Bycombining this observation with the T c curve under thelarge enough anisotropy, | δ u | = 3 × in Fig.3(b), it is (cid:13) (cid:82) (cid:90) (cid:82) (cid:90) (cid:69) (cid:69) FIG. 2: (Color online) Diagrams expressing the gap equationlinearized with respect to the order parameter of the PdBphase. The parameter c means the order parameter of thePdB phase, and each vertex carries a component perpendic-ular to the z -axis of the momentum. The impurity-averagedGreen’s functions G and F † in eq.(6) are indicated by a linewith a double arrow and that with a left right arrow, respec-tively. PdB Polar (a) P ( ba r) T/T c0 (P) Polar
PdB (b) P ( ba r) T/T c0 (P) FIG. 3: (Color online) (a) Pressure ( P ) v.s. temperature ( T )superfluid phase diagram obtained by changing the parameter τ − measuring the impurity strength under a fixed magnitudeof the anisotropy | δ u | = 30, i.e., k F L z = 5 .
48. The tempera-tures T c ( P ) and T PB ( P ) are represented by the thick and thinsolid lines for (2 πτ ) − (mK) = 0 .
3, the thick and thin dashedlines for (2 πτ ) − = 0 .
5, and the open and closed circles for(2 πτ ) − = 0 .
7, respectively. Note that the variable of the hor-izontal axis is
T /T c ( P ), i.e., the temperature normalized bythe bulk superfluid transition temperature T c at each P . (b)Corresponding ones obtained by changing the anisotropy orthe correlation length L z under the fixed impurity strength(2 πτ ) − (mK) = 0 .
7. The temperatures T c ( P ) and T PB ( P )are represented by the right and left thick solid curves for | δ u | = 4 . | δ u | = 3 × , respectively. The right andleft thin solid curves are the open and closed circles in (a),respectively. natural to regard the nematic aerogels in which the po-lar phase of superfluid He is realized as random mediawith a finite correlation length of the scattering potential.Nevertheless, examining the superfluid polar phase inthe nematic aerogels by starting from the limit of stronganisotropy where the Anderson’s Theorem is satisfied isa proper description.Next, as another quantity related to the Anderson’s
T=T PB T/T c0 =0.5 (a) ∆ ( T ) / ∆ ( ) T/ ∆ (0) T/T c0 =0.5 (b) ∆ ( T ) / ∆ ( ) T/ ∆ (0) FIG. 4: (Color online) Temperature dependence of | ∆ | forthe parameter values | δ u | = 30 and (2 πτ ) − = 1(mK) (a) at30(bar) and (b) at 0(bar). In both cases, the T behavior isnicely seen at least in the range T < . T c (Note that, inthe figures, the temperature is represented in unit of | ∆(0) | in each case). The thick solid curve expresses eq.(11), wherethe a value is 10 . . Theorem, let us examine the temperature dependenceof the energy gap | ∆( T ) | of quasiparticles in the polarphase. As indicated elsewhere [8], the energy gap differ-ence | ∆(0) | − | ∆( T ) | estimated from the NMR frequencydata in the polar phase at 30 (bar) is proportional to T ,reflecting the presence of a line node in | ∆( T ) | . Since therelevant energy scale at low T is not T c but | ∆(0) | , wewill express the T behavior in the form1 − | ∆( T ) || ∆(0) | = a T | ∆(0) | . (11)This relation to be satisfied in the polar phase in aero-gels would indicate that, irrespective of the presence ofthe impurity scattering effect, the line node of | ∆( T ) | in the polar phase remains well defined. According tothe calculation [8] in the weak-coupling approximationand clean limit, the coefficient a takes the value 8 . a -value taken from NMR data in a nematic aerogel at 30 (bar) was 0 . /T c ) [8]. Ac-cording to Ref.[8], this estimated coefficient may becomecomparable [8] with the weak coupling value 8 .
49 in thelimit of strong anisotropy if the strong coupling effect [12]enhancing | ∆(0) | is taken into account. In a stronglyanisotropic case, | δ u | = 3 × , we have obtained thevalue a = 8 .
97 comparable with the weak coupling valuementioned above.On the other hand, as presented in Fig.4, our re-sults for the moderately strong anisotropy, | δ u | = 30,clearly show an effect of the impurity scattering on thecoefficient of the T term. Here, a stronger scatteringstrength (2 πτ ) − = 1(mK) than those used in Fig.3 wasused to obtain a wider polar region at lower tempera-tures. Although the T behavior is still well defined in T < . T c irrespective of the pressure value, the co-efficient a is, as mentioned in Fig.4’s caption, enhancedespecially at lower pressures. If the strong coupling effectis taken into account, according to Ref.[8] the coefficient a ≡ a ( T c / ∆(0)) at 30 (bar) would remarkably decreaseso that the estimated value a = 0 .
38 [8] may be explained.At zero pressure, however, the strong coupling effect isnot effective so that the coefficient a of the T behav-ior at lower pressures may show a large value of orderunity. Examining the T term of the energy gap at lowpressures may become a test for the present theory.It is valuable to point out that the p -wave Ander-son’s Theorem on the superfluid transition temperatureis also satisfied in the case of a normal to (distorted)A phase transition under plane-like defects with no two-dimensional momentum transfer (See Fig.1 (b)) if the l -vector of this A phase is oriented along the normal ofthe plane of the defects. In fact, when the bare pairingvertex is p k ( δ j,k − ˆ z j ˆ z k ), and eq.(9) is replaced by the formproportional to δ (ˆ k x ) δ (ˆ k y ), the superfluid transition tem-perature resulting from eq.(4) becomes T c irrespective ofthe strength of the impurity scattering. In principle, sucha situation can be realized in planar aerogels and wouldresult in an extension of the temperature width of theplanar-distorted A phase region at lower pressures andhence, according to Ref.[15], in a realization of HQVs inthe chiral A phase.As is well known in the context of the dirty s -wavesuperconductors, the Anderson’s Theorem breaks downin systems with a strong enough impurity scattering dueto the impurity effect in the repulsive channels of thequasiparticle interaction [16]. In fact, the T c ( P ) curvereported in Fig.4 of Ref.[3] shows a remarkable devia-tion from T c ( P ). Further, it is possible that the τ − dependence of T c ( P ) obtained under a finite anisotropyof the type seen in Fig.3(a) occurs even in the limit ofstrong anisotropy due to the above-mentioned mecha-nism associated with the Anderson localization, because1 / ( τ T c ( P )) is the pressure-dependent strength of the im-purity scattering. To clarify to what extent this Ander-son localization effect is effective in real systems, furthercomparison between the theoretical results and new datawill be necessary in future.In conclusion, we have investigated to what extent theAnderson’s theorem is satisfied in the polar phase by as-suming the correlation length of the random potential innematic aerogels to be long but finite. It has been foundthat the low temperature behavior of the superfluid en-ergy gap stemming from the presence of the horizontalline node is robust against the impurity scattering and that the resulting phase diagram is qualitatively consis-tent with the available experimental data.One of the authors (R.I.) is grateful to VladimirDmitriev and Bill Halperin for useful discussions. Thepresent work was supported by JSPS KAKENHI (GrantNo.16K05444). [1] K. Aoyama and R. Ikeda, Phys. Rev. B , 060504(R)(2006).[2] V.V. Dmitriev, A.A.Senin, A.A.Soldatov, andA.N.Yudin,et al., Phys. Rev. Lett. , 165304 (2015).[3] V.V. Dmitriev, A.A.Soldatov, and A.N.Yudin, Phys.Rev. Lett. , 075301 (2018).[4] J. A. Sauls, Phys. Rev. B , 214503 (2013); I. A. Fomin,JETP , 765 (2014).[5] I. A. Fomin, JETP , 933 (2018).[6] P. W. Anderson, J. Phys. Chem. Sol. , 26 (1959).[7] V. V. Dmitriev, M. S. Kutuzov, A. A. Soldatov, A. N.Yudin, arXiv:1911.01193.[8] V. B. Eltsov, T. Kamppinen, J. Rysti, and G. E. Volovik,arXiv:1908.01645.[9] A. A. Abrikosov, L. P. Gor’kov, I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, 1963) Sec.39.3.[10] E. V. Thuneberg, S. K. Yip, M. Fogelstrom, J. A. Sauls,Phys. Rev. Lett. , 2861 (1998).[11] M. Tange and R. Ikeda, arXiv:1912.00897.[12] D. Volhardt and P, Wolfle, The Superfluid Phases of He-lium 3 (Taylor and Fransis, 2003).[13] V.V. Dmitriev, A.A.Soldatov, and A.N.Yudin,arXiv:1710.03097.[14] A. M. Zimmerman, M.D.Nguyen, J.W.Scott, and W.P.Halperin, arxiv. 1908.01739.[15] N. Nagamura and R. Ikeda, Phys. Rev. B , 094524(2018).[16] H. Fukuyama, S. Ebisawa, and S. Maekawa, J. Phys. Soc.Jpn. , 3560 (1984); M. V. Sadovskii, Phys. Rep.282