Abstract
A rich variety of Fermi systems condense by forming bound pairs, including hightemperature^{1} and heavyfermion^{2} superconductors, Sr_{2}RuO_{4} (ref. 3), cold atomic gases^{4} and superfluid ^{3}He (ref. 5). Some of these form exotic quantum states with nonzero orbital angular momentum. We have discovered, in the case of ^{3}He, that anisotropic disorder, engineered from highly porous silica aerogel, stabilizes a chiral superfluid state that otherwise would not exist. Furthermore, we find that the chiral axis of this state can be uniquely oriented with the application of a magnetic field perpendicular to the aerogel anisotropy axis. At sufficiently low temperature we observe a sharp transition from a uniformly oriented chiral state to a disordered structure consistent with locally ordered domains, contrary to expectations for a superfluid glass phase^{6}.
Main
Superconducting states with nonzero orbital angular momentum, , are characterized by a competitive, but essential, relationship with magnetism, strong normalstate anisotropy or both^{1,2,3,5}. Moreover, these states are strongly suppressed by disorder, an important consideration for applications^{7} and a signature of their unconventional behaviour^{3,8,9}. Although liquid ^{3}He in its normal phase is perfectly isotropic, it becomes a pwave superfluid at low temperatures with nonzero orbital and spin angular momenta, L=S=1 (ref. 10). One of its two superfluid phases in zero magnetic field is anisotropic with chiral symmetry, where the handedness results from the orbital motion of the bound ^{3}He pairs about an axis . This chiral superfluid, called the A phase or axial state, is stable at high pressure near the normaltosuperfluid transition, Fig. 1a–c, whereas the majority of the phase diagram is the nonchiral B phase, with isotropic physical properties. The stability of the A phase is attributed to strongcoupling effects arising from collisions between ^{3}He quasiparticles^{10}. However, in the presence of isotropic disorder these strongcoupling effects are reduced and the stable chiral phase disappears^{11,12}, Fig. 1a. Here we show that anisotropic disorder can reverse this process and stabilize an anisotropic phase over the entire phase diagram, Fig. 1c.
For many years it was thought to be impossible to introduce disorder into liquid ^{3}He because it is intrinsically chemically and isotopically pure at low temperatures. Then it was discovered^{13,14} that ^{3}He imbibed in ∼98% porosity silica aerogel, Fig. 1d, is a superfluid with a transition temperature that is sharply defined^{12}, but reduced from that of pure ^{3}He. To test predictions that isotropic disorder favours isotropic states and anisotropic disorder favours anisotropic states^{15}, we have grown a 97.5% porosity anisotropic aerogel with growthinduced radial compression^{16}, effectively stretching it along its cylinder axis by 14.3%. Experiments using uncharacterized stretched aerogels have been previously reported^{17,18} and are in disagreement with the work presented here. Silica aerogels, as in Fig. 1d, are formed by silica particles ≈3 nm in diameter, precipitated from a tetramethylorthosilicate solution, and aggregated in a diffusionlimited process. After supercritical drying we obtain a cylinder as shown in Fig. 1d. Our numerical simulation of this process indicates that the gel particles form strands (Fig. 1d). According to theory^{19}, is constrained to point outward from the strand. Stretching an aerogel tends to align the strands, thereby forcing the chiral axis, on average, to be in an easy plane perpendicular to the aerogel anisotropy axis^{6}. On the other hand, for an axially compressed aerogel should be aligned with the compression axis^{6}.
We use pulsed nuclear magnetic resonance (NMR; Methods), to identify the ^{3}He superfluid state and determine the magnitude of its order parameter, Δ. The equilibrium nuclear magnetization is tipped by a pulsed magnetic field to an angle β away from the static applied magnetic field, H, and the Fourier transform of its free precession is the NMR spectrum. The frequency shift of the spectrum, Δω, from the normalstate value centred at the Larmor frequency, ω_{L}, is a direct measure of the temperaturedependent order parameter, Δ(T). For the A and B phases of ^{3}He (ref. 10) the frequency shifts are where Ω_{A,B}^{2} is the square of the longitudinal resonance frequency, which is proportional to Δ_{A,B}^{2} and determined by constants of the normal fluid: gyromagnetic ratio, γ; dimensionless dipole coupling constant, λ_{D}; and singlespin density of states at the Fermi level, N_{F}; as well as the magnetic susceptibilities χ_{A} and χ_{B}. The Aphase susceptibility is temperature independent and equal to the normalstate value, χ_{N}; however, χ_{B} is temperature dependent and suppressed relative to χ_{N}.
For ^{3}He in our anisotropic aerogel we observe a sharp transition to a superfluid state at a temperature, T_{ca}, marked by the onset of NMR frequency shifts for small tip angle, β=8° (Fig. 2a). Our comparison of the data with either of the two pure ^{3}He states based on equations (1) and (2) indicates that the superfluid is an axial pwave state similarly to the A phase. In the Ginzburg–Landau (GL) limit, T≲T_{c}, the initial slope of Δω_{A} is proportional to the square of the order parameter magnitude Δ_{A0}^{2}, where Δ_{A}^{2}=Δ_{A0}^{2}(1−T/T_{c}). We find in our anisotropic aerogel that Δ_{A0}^{2} has a linear pressure dependence, as shown in Fig. 2b, just as is the case for pure ^{3}HeA (refs 20, 21). It is, however, substantially reduced in magnitude and extrapolates to a critical pressure of P_{c}=7.9 bar at T=0, where the order parameter vanishes at a quantum critical point^{22}. We have analysed the transition temperatures T_{ca}(P) using GL theory^{23} (Supplementary Information) to determine the mean free path λ=113 nm and the silica particle correlation length ξ_{a}=39 nm. Our calculated phase diagram compares very well with the data in Fig. 1c. With the same parameters we have also calculated Δ_{A0}^{2} (Supplementary Information), shown in Fig. 2b, and find excellent agreement with the order parameter determined from the frequency shifts, equation (1), consistent with a suppressed A phase. Moreover, extrapolation of Δω(P)→0 is at a pressure that coincides with the critical pressure from the calculated phase diagram, demonstrating important consistency between experiment and theory.
The tipangle dependence of the NMR frequency shift is a fingerprint of a specific pwave state^{10}. Our measurements of Δω(β), Fig. 2c, follow the expected behaviour of the chiral axial state^{24}, equation (1). The magnitude of the shift is reduced when compared with that of pure ^{3}HeA, as shown in Fig. 2b. Furthermore, the axial state belongs to the class of equalspinpairing (ESP) states, with the same susceptibility as the normal Fermi liquid. We have directly demonstrated that this is the case as shown in Fig. 2d.
Last, we have looked for a possible phase transition to some other pwave state by cooling to our lowest temperatures, ≲650 μK, in a substantial magnetic field of 196 mT, and measuring χ and Δω on warming. For example, if there had been a transition to a B phase on cooling we would have observed a discontinuous increase of χ on warming from the B phase to a magnetic fieldinduced A phase, as is characteristic of the firstorder BtoA transition in an isotropic aerogel (Fig. 2d; refs 12). Correspondingly, we would also have observed a discontinuous change in Δω. The absence of these discontinuities demonstrates that an ESP superfluid state in a stretched aerogel is stable down to our lowest temperatures.
Our three independent NMR measurements show that an anisotropic stretched aerogel stabilizes anisotropic superfluidity throughout the phase diagram, Fig. 1c. This is in stark contrast to ^{3}He in the presence of isotropic disorder^{11,12}, Fig. 1a, or for anisotropic disorder established by axial compression, Fig. 1b (ref. 25). In both of these latter cases the B phase is dominant. Understanding the stability of chiral superfluidity, stabilized by anisotropy, is an interesting open problem, in part because topologically nontrivial pairing states have unusual structure.
Our NMR results reported in Fig. 2a–c were restricted to temperatures within ∼20% of T_{ca}. In this hightemperature region there is negligible increase in the NMR linewidth as compared with the normal fluid (Fig. 3), indicating that the chiral axis, , has a welldefined direction orthogonal to both the aerogel anisotropy axis^{6} and the magnetic field. This orientation minimizes the magnetic dipole energy, producing the socalled dipolelocked condition, on which the validity of equation (1) depends^{10}, and with which our tipangle results agree. The combined effects of the dipole energy and aerogel anisotropy define a uniform direction of for the A phase of superfluid ^{3}He in our stretched aerogel.
At lower temperatures, we have discovered a different phase characterized by a significantly broadened NMR line. The data indicate that this superfluid state is no longer the homogeneous A phase with . On warming there is an abrupt increase in the first moment of the NMR spectrum (frequency shift) and a corresponding decrease in the linewidth at a welldefined temperature, T_{d} (Fig. 3). The transition at T_{d}is independent of magnetic field for 31.1<H<196 mT. In addition, the magnetic susceptibility does not change at this temperature; however, T_{d} is pressure dependent, as shown by the open squares in the phase diagram of Fig. 1c. We interpret these results as evidence for a disordered ESP phase with a possible phase transition at T_{d}.
The frequency shift is a measure of Δ_{A}, given in equation (1) for the dipolelocked case where the angle between the chiral axis and the magnetic field is . However, in general the shift also depends on θ, where for small tip angles Δω(θ) reduces to^{26} This enables us to extract the angular distribution, P(θ), of relative to H, from the NMR spectra both above and below T_{d}. The observed spectrum is given by the convolution product of the normalstate lineshape and the frequency distribution P(ω) that corresponds to P(θ) determined by equation (3) (Supplementary Information). Above T_{d} the NMR spectrum (Fig. 4a) can be fitted with an distribution (Fig. 4b) that has a single component centred at θ=90°, as sketched in Fig. 4c, with angular spread Δθ=17.8°±3.6°. Examination of the spectrum below T_{d}, Fig. 4a, indicates that it is bimodal, suggesting that P(θ) has three main components, which we take to be Gaussian functions with adjustable position, width and relative weight. From our fits we determine these parameters and obtain P(θ), Fig. 4b. For the spectra below T_{d}, the disordered state is composed of three domains. Approximately onethird of the sample remains with whereas the majority, ∼2/3 of the distribution, has nearly zero frequency shift with θ=43.8°±0.1°. The theory, equation (3), does not distinguish between angles that are symmetric about θ=90°, and thus we assume that P(θ)=P(180°−θ) as shown in Fig. 4d. Our interpretation of the disordered state in terms of distinct domains is different from the twodimensional ‘orbital glass’ phase defined by a random distribution of confined to a plane perpendicular to the stretching axis^{6}. It is noteworthy that the possible existence of orbital domains, such as we observe here, has been central to the discussion of timereversal symmetry breaking in the unconventional superconducting state in Sr_{2}RuO_{4} (ref. 27).
Methods
In our experiments the external magnetic field H was 31.1, 95.5 or 196 mT and oriented perpendicular to the aerogel cylinder axis. The sample was 4.93 mm long and had a diameter of 3.43 mm. The susceptibility was obtained by numerically integrating the phasecorrected absorption spectrum, whereas the frequency shift and linewidth were determined from the power spectrum. The sample was cooled using adiabatic nuclear demagnetization of PrNi_{5}. NMR on ^{195}Pt was used for thermometry at H=95.5 and 196 mT and calibrated relative to the known phase diagram of pure superfluid ^{3}He from a volume of liquid outside the aerogel sample equal to 30% of the total liquid. For data acquired at H=31.1 mT, temperature was determined from the Curie–Weiss dependence of the solid ^{3}He adsorbed to the aerogel surface^{14,28}.
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Acknowledgements
We are grateful to J. M. Parpia, V. V. Dmitriev, G. E. Volovik, N. Mulders, K. R. Shirer, A. M. Mounce and Y. Lee for discussion and to the National Science Foundation, DMR1103625, DMR0805277 and DMR1106315, for support.
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Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
 J. Pollanen
 , J. I. A. Li
 , C. A. Collett
 , W. J. Gannon
 , W. P. Halperin
 & J. A. Sauls
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Contributions
Experimental work and analysis was principally carried out by J.P. assisted by J.I.A.L. with further support from C.A.C. and W.J.G. Advice and assistance was provided by W.P.H. (experiment) and J.A.S. (theory).
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The authors declare no competing financial interests.
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Correspondence to W. P. Halperin.
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