Half-quantum vortices in the PdA phase

The superfluid phase diagram under confinement by nafen31—a nanostructured material consisting of nearly parallel strands made of Al 2 O 3 , c.f. Fig. 1b—differs from that of the bulk 3He; the critical temperature is suppressed and, more importantly, new superfluid phases—the polar, PdA, and PdB phases—are observed. We refer to the Supplementary Note 1 for a detailed discussion on these phases and their symmetries and focus on our observations regarding the HQVs in the PdA and PdB phases.

The order parameter of the PdA phase can be written as

$$A_{\alpha j} = \sqrt {\frac{{1 + b^2}}{3}} {\mathrm{\Delta }}_{{\mathrm{PdA}}}e^{i\phi }{\hat{\mathbf d}}_\alpha \left( {\widehat {\mathbf{m}}_j + ib\widehat {\mathbf{n}}_j} \right),$$ (1)

where the orbital anisotropy vectors \(\widehat {\mathbf{m}}\) and \(\widehat {\mathbf{n}}\) form an orthogonal triad with the Cooper pair orbital angular momentum axis \(\widehat {\mathbf{l}} = \widehat {\mathbf{m}} \,\times \widehat {\mathbf{n}}\), and \(\widehat {\mathbf{d}}\) is the spin anisotropy vector. Vector \(\widehat {\mathbf{m}}\) is fixed parallel to the nafen strands. The amount of polar distortion is characterized by a dimensionless parameter \(0 \,< \,b \,< \,1\) and \({\mathrm{\Delta }}_{{\mathrm{PdA}}}(T,b)\) is the maximum gap in the PdA phase. The order parameter of the polar phase is obtained for b = 0, while b = 1 produces the order parameter of the conventional A phase.

In our experiments, we use continuous-wave NMR techniques to probe the sample, see Methods for further details. In the superfluid state, the spin-orbit coupling provides a torque acting on the precessing magnetization, which leads to a shift of the resonance from the Larmor value ω L = |γ|H, where γ = −2.04 × 108 s−1 T−1 is the gyromagnetic ratio of 3He. The transverse resonance frequency of the bulk fluid with magnetic field in the direction parallel to the strand orientation, i.e. μ = 0 in Fig. 1a, is31

$${\mathrm{\Delta }}\omega _{{\mathrm{PdA}}} = \omega _{{\mathrm{PdA}}} - \omega _{\mathrm{L}} \approx \frac{{{\mathrm{\Omega }}_{{\mathrm{PdA}}}^2}}{{2\omega _{\mathrm{L}}}},$$ (2)

where Ω PdA is the frequency of the longitudinal resonance in the PdA phase at μ = π/2. The NMR line retains its shape during the second-order phase transition from the polar phase but renormalizes the longitudinal resonance frequency due to appearance of the order-parameter component with b.

Quantized vortices are linear topological defects in the order-parameter field carrying non-zero circulation. In the PdA phase, quantized vortices involve phase winding by \(\phi \to \phi + 2\pi

u\) and possibly some winding of the \(\widehat {\mathbf{d}}\) vector. The typical singly quantized vortices, also known as phase vortices, have \(

u = 1\) and no winding of the \(\widehat {\mathbf{d}}\)-vector, while the HQVs have \(

u = \frac{1}{2}\) and winding of the \(\widehat {\mathbf{d}}\)-vector by π on a loop around the HQV core so that sign changes of \(\widehat {\mathbf{d}}\) and of the phase factor \(e^{i\phi }\) compensate each other. The reorientation of the \(\widehat {\mathbf{d}}\)-vector leads to the formation of \(\widehat {\mathbf{d}}\)-solitons—spin-solitons connecting pairs of HQVs. The soft cores of the \(\widehat {\mathbf{d}}\)-solitons provide trapping potential for standing spin waves39.

Since the \(\widehat {\mathbf{m}}\)-vector is fixed by nafen parallel to the anisotropy axis, the \(\widehat {\mathbf{l}}\)-vector lies on the plane perpendicular to it, prohibiting the formation of continuous vorticity40 like the double-quantum vortex in 3He-A41. Some planar structures in the \(\widehat {\mathbf{l}}\)-vector field, such as domain walls42 or disclinations, remain possible but the effect of the \(\widehat {\mathbf{l}}\)-texture on the trapping potential for spin waves is negligible due to the large polar distortion31 (i.e. for b ≪ 1). Recent theoretical work43 provides arguments why formation of HQVs in the polar phase is preferred compared to the undistorted A phase. Studying whether HQVs are formed in the transition from the normal phase to the PdA phase with finite polar distortion (0 < b < 1) remains a task for the future. In our case, the PdA phase is obtained via the second-order phase transition from the polar phase with preformed HQVs. We already know20 that the maximum tension from the spin-soliton in the polar phase (for μ = π/2) is insufficient to overcome HQV pinning. Thus, survival of HQVs in the PdA phase is expected. Moreover, we note that even for |b| = 1 and in the absence of pinning, a pair of HQVs, once created, should remain stable with finite equilibrium distance corresponding to cancellation of vortex repulsion and tension from the soliton tail18.

In the presence of HQVs, the excitation of standing spin waves localized on the soliton leads to a characteristic NMR satellite peak in transverse (μ = π/2) magnetic field, c.f. Fig. 2, with frequency shift

$${\mathrm{\Delta }}\omega _{{\mathrm{PdAsat}}} = \omega _{{\mathrm{PdAsat}}} - \omega _{\mathrm{L}} \approx \lambda _{{\mathrm{PdA}}}\frac{{{\mathrm{\Omega }}_{{\mathrm{PdA}}}^2}}{{2\omega _{\mathrm{L}}}},$$ (3)

where λ PdA is a dimensionless parameter dependent on the spatial profile (texture) of the order parameter across the soliton. For an infinite 1D \(\widehat {\mathbf{d}}\)-soliton, one has λ PdA = −1, corresponding to the zero-mode of the soliton18,20,44. The measurements in the supercooled PdA phase, Fig. 3a, at temperatures close to the transition to the PdB phase give value λ PdA ≈ −0.9, which is in good agreement with theoretical predictions and earlier measurements in the polar phase with a different sample20. This confirms that the structure of the \(\widehat {\mathbf{d}}\)-solitons connecting the HQVs is similar in polar and PdA phases and the effect of the orbital part to the trapping potential can safely be neglected. Detailed analysis of the satellite frequency shift as a function of magnetic field direction in the PdA phase remains a task for the future.

Fig. 2 Survival of HQVs during phase transitions. The plot shows the measured NMR spectra in transverse (μ = π/2) magnetic field in the presence of HQVs. HQVs were created by rotation with 2.5 rad s−1 during the transition from normal phase to the polar phase. The NMR spectrum includes the response of the bulk liquid and the \({\hat{\mathbf d}}\)-solitons, which appear as a characteristic satellite peak at lower frequency. The satellite intensity in the PdA phase remains unchanged after thermal cycling presented in Fig. 1c. The NMR spectrum in the PdB phase at the same temperature, measured between the two measurements in the PdA phase, is shown for reference Full size image

Fig. 3 NMR spectra and spin-solitons in the polar-distorted phases. a Frequency shift of a characteristic satellite peak in the NMR spectrum expressed via parameter λ as a function of temperature in the PdA and PdB phases. In the PdA phase, the measured values reside slightly above the theoretical prediction for a \({\hat{\mathbf d}}\)-soliton with π winding, shown as the red dashed line. The difference is believed to be caused by disorder introduced by nafen, as in the polar phase20,70. The corresponding values in the PdB phase for the lowest-energy \({\hat{\mathbf d}}\)-soliton (marked “soliton”) and its antisoliton (marked “big soliton”), as well as the combined π-soliton (see text) are shown as dashed blue lines. The π-soliton values turn out to be in the same ratio with respect to the experimental points as in the PdA phase. The error bars are based on the spectral width of the observed feature and denote the uncertainty in the position of the satellite peak, as illustrated by the black bar in the inset of b for the PdB phase. The range marked by an error bar corresponds to the doubled full width at half maximum (FWHM) of the satellite peak in the PdB phase (1 kHz). In the PdA phase, due to improved signal-to-noise ratio, the uncertainty is twice smaller (0.5 kHz). b The plot shows the measured NMR spectrum in the PdB phase at 0.38 T c for different HQV densities, controlled by the angular velocity Ω at the time of crossing the T c . The presence of KLS walls produces characteristic features seen both as widening of the main line (with small positive frequency shift) and as a satellite peak with a characteristic negative frequency shift. The inset shows magnified view of the satellite peak. c The satellite intensity in the PdA phase at 0.60 T c (blue circles) and in the PdB phase multiplied by a factor of 9 (red triangles) at 0.38 T c show the expected \(\sqrt \Omega\)-scaling. The solid black line is a linear fit to the measurements including data from both phases. The non-zero Ω = 0 intersection corresponds to vortices created by the Kibble–Zurek mechanism1,2,20. d The FWHM of the main line, determined from the spectrum in b, gives FWHM ≈3 kHz for 2.5 rad s−1. FWHM for other angular velocities is recalculated from the amplitude of the main NMR line, shown in b, assuming constant area Full size image

Half-quantum vortices in the PdB phase

Since the HQVs are found both in the polar and PdA phases, it is natural to ask what is their fate in the PdB phase? The number of HQVs in the polar and PdA phases can be estimated from the intensity (integrated area) of the NMR satellite, a direct measure of the total volume occupied by the \(\widehat {\mathbf{d}}\)-solitons20. When cooling down to the PdB phase from the PdA phase, one naively expects the HQVs and the related NMR satellite to disappear since isolated HQVs cease to be protected by topology in the PdB phase. However, the measured satellite intensity in the PdA phase before and after visiting the PdB phase remained unchanged, c.f. Fig. 2, which is a strong evidence in favor of the survival of HQVs in the phase transition to the PdB phase. Theoretically, it is possible that HQVs survive in the PdB phase as pairs connected by domain walls, i.e., as walls bounded by strings22. For very short separation between HQVs in a pair and ignoring the order-parameter distortion by confinement, such construction may resemble the broken-symmetry-core single-quantum vortex of the B phase16. In our case, however, the HQV separation in a pair exceeds the core size by 3 orders of magnitude. Let us now consider this composite defect in more detail.

The order parameter of the PdB phase can be written as

$$A_{\alpha j} = \sqrt {\frac{{1 + 2q^2}}{3}} \Delta _{{\mathrm{PdB}}}e^{i\phi }({\hat{\mathbf d}}_\alpha {\hat{\mathbf z}}_j + q_1{\hat{\mathbf e}}_\alpha ^1{\hat{\mathbf x}}_j + q_2{\hat{\mathbf e}}_\alpha ^2{\hat{\mathbf y}}_j){\kern 1pt} ,$$ (4)

where \(|q_1|,|q_2| \in (0,1)\), \(|q_1| = |q_2| \equiv q\) describes the relative gap size in the plane perpendicular to the nafen strands, \({\hat{\mathbf e}}^1\) and \({\hat{\mathbf e}}^2\) are unit vectors in spin-space forming an orthogonal triad with \(\widehat {\mathbf{d}}\), and \(\Delta _{{\mathrm{PdB}}}(T,q)\) is the maximum gap in the PdB phase. For \(q = 0\), one obtains the order parameter of the polar phase, while q = 1 recovers the order parameter of the isotropic B phase. We extract the value for the distortion factor, \(q\sim 0.15\) at the lowest temperatures from the NMR spectra using the method described in ref. 45, see Supplementary Note 6 for the measurements of q in the full temperature range.

In transverse magnetic field H exceeding the dipolar field, the vector \({\hat{\mathbf e}}^2\) becomes locked along the field, while vectors \({\hat{\mathbf d}}\) and \({\hat{\mathbf e}}^1\) are free to rotate around the axis \({\hat{\mathbf y}}\), directed along H, with the angle θ between \({\hat{\mathbf d}}\) and \({\hat{\mathbf z}}\), c.f. Fig. 1b. The order parameter of the PdB phase in the vicinity of an HQV pair has the following properties. The phase ϕ around the HQV core changes by π and the angle θ (and thus vectors \({\hat{\mathbf d}}\) and \({\hat{\mathbf e}}^1\)) winds by π. Consequently, there is a phase jump \(\phi \to \phi + \pi\) and related sign flips of vectors \({\hat{\mathbf d}}\) and \({\hat{\mathbf e}}^1\) along some direction in the plane perpendicular to the HQV core. In the presence of order-parameter components with q > 0, Eq. (4) remains single-valued if, and only if, q 2 also changes sign. We conclude that the resulting domain wall separates the degenerate states with \(q_2 = \pm q\) and together with the bounding HQVs has a structure identical to the domain wall bounded by strings—the KLS wall—proposed by Kibble, Lazarides, and Shafi in refs. 22,28.

The KLS wall and the topological soliton have distinct defining length scales17,33—the KLS wall has a hard core of the order of ξ W ≡ q−1ξ, where ξ is the coherence length, and the soliton has a soft core of the size of the dipole length \(\xi _{\mathrm{D}} \gg \xi _{\mathrm{W}}\). The combination of these two objects may emerge in two different configurations illustrated in Fig. 4. The minimization of the free energy (Supplementary Notes 2 and 3) shows that in the PdB phase, the lowest-energy spin-soliton corresponds to winding of the \(\widehat {\mathbf{d}}\)-vector by π − 2θ 0 , where sinθ 0 = q 2 (2 − 2q 1 )−1, on a cycle around an HQV core. Additionally, the presence of KLS walls results in winding of the \(\widehat {\mathbf{d}}\)-vector by 2θ 0 . These solitons can either extend between different pairs of HQVs, Fig. 4a, while walls with total change Δθ = π are also possible if both solitons are located between the same pair of HQVs, Fig. 4b.

Fig. 4 Kibble–Lazarides–Shafi (KLS) wall configurations in the PdB phase. Each HQV core terminates one soliton—reorientation of the spin part of the order parameter denoted by the angle θ—and one KLS wall. The orientation of the \(\widehat {\mathbf{d}}\)-vector is shown as arrows where their color indicates the angle θ, based on numerical calculations (Supplementary Figure 2). a The KLS wall is bound between a different pair of HQV cores as the soliton. Ignoring the virtual jumps, the angle θ winds by π − 2θ 0 across the soliton and by 2θ 0 across the KLS wall. The order parameter is continuous across the virtual jumps, where ϕ → ϕ + π, θ → θ + π, and q 2 → −q 2 . b The soliton and the KLS wall are bound between the same pair of HQV cores. The total winding of the \(\widehat {\mathbf{d}}\)-vector is π across the structure. In principle, the KLS wall may lie inside or outside the soliton. Here the KLS wall and the soliton are spatially separated for clarity Full size image

The appearance of KLS walls and the associated \(\widehat {\mathbf{d}}\)-solitons has the following consequences for NMR. The frequency shift of the bulk PdB phase in axial field for q < 1/2 is45

$${\mathrm{\Delta }}\omega _{{\mathrm{PdB}},||} = \omega _{{\mathrm{PdB}},||} - \omega _{\mathrm{L}} \approx \left( {1 + \frac{5}{2}q} \right)\frac{{\Omega _{{\mathrm{PdB}}}^2}}{{2\omega _{\mathrm{L}}}},$$ (5)

where Ω PdB is the Leggett frequency of the PdB phase, defined in the Supplementary Note 5. In transverse magnetic field, the bulk line has a positive frequency shift

$${\mathrm{\Delta }}\omega _{{\mathrm{PdB}}, \bot } = \omega _{{\mathrm{PdB}}, \bot } - \omega _{\mathrm{L}} \approx \left( {q - q^2} \right)\frac{{{\mathrm{\Omega }}_{{\mathrm{PdB}}}^2}}{{2\omega _{\mathrm{L}}}},$$ (6)

and winding of the \(\widehat {\mathbf{d}}\)-vector in a soliton leads to a characteristic frequency shift

$$\Delta \omega _{{\mathrm{PdBsat}}} = \omega _{{\mathrm{PdBsat}}} - \omega _{\mathrm{L}} \approx \lambda _{{\mathrm{PdB}}}\frac{{\Omega _{{\mathrm{PdB}}}^2}}{{2\omega _{\mathrm{L}}}},$$ (7)

where the dimensionless parameter λ PdB is characteristic to the defect. Numerical calculations in a 1D soliton model (Supplementary Note 3) for all possible solitons shown in Fig. 3a give the low-temperature values \(\lambda _{{\mathrm{soliton}}}\sim - 0.8\) for π − 2θ 0 -soliton (“soliton”) and \(\lambda _{{\mathrm{big}}}\sim - 1.8\) for its antisoliton, which has π + 2θ 0 winding (“big soliton”). The 2θ 0 -soliton (“KLS soliton”) related to the KLS walls outside spin-solitons gives rise to a frequency shift experimentally indistinguishable from the frequency shift of the bulk line. The last possibility, the “π-soliton” consisting of a KLS soliton and a soliton, c.f. Fig. 4b, gives \(\lambda _\pi \sim - 1.3\) at low temperatures. The measured value, \(\lambda _{{\mathrm{PdB}}}\sim - 1.1\) at the lowest temperatures, as seen in Fig. 3a. The measured values for λ PdB , together with the fact that the total winding of the \(\widehat {\mathbf{d}}\)-vector is also equal to π in the PdA, and polar phases above the transition temperature suggest that the observed soliton structure in the PdB phase corresponds to the π-soliton in the presence of a KLS wall.

In addition, the KLS wall possesses a tension \(\sim \xi q^3{\mathrm{\Delta }}_{{\mathrm{PdB}}}^2N_0\)32,33, where N 0 is the density of states. Thus, the presence of KLS walls applies a force pulling the two HQVs at its ends towards each other. The fact that the number of HQVs remains unchanged in the phase transition signifies that the KLS wall tension does not exceed the maximum pinning force in the studied nafen sample. This observation is in agreement with our estimation of relevant forces (Supplementary Note 4). Strong pinning of single-quantum vortices in B-like phase in silica aerogel has also been observed previously46. An alternative way to remove a KLS wall is to create a hole within it, bounded by a HQV22. Creation of such a hole, however, requires overcoming a large energy barrier related to creation of a HQV with hard core of the size of ξ. Moreover, growth of the HQV ring is prohibited by the strong pinning by the nafen strands. We also note that for larger values of q, there may exist a point at which the KLS wall becomes unstable towards creation of HQV pairs, and as a result, the HQV pairs bounded by KLS walls would eventually shrink to singly quantized vortices. For the discussion of the effect of nafen strands on the KLS walls, see Supplementary Note 4.

Effect of rotation

The density of HQVs created in the polar phase is controlled by the angular velocity Ω of the sample at the time of the phase transition from the normal phase, n HQV = 4Ωκ−1, where κ is the quantum of circulation. The integral of the NMR satellite depends on the total volume occupied by the solitons, whose width is approximately the spin-orbit length and the height is fixed by the sample size 4 mm. The average soliton length is equal to the intervortex distance \(\propto {\mathrm{\Omega }}^{ - 1/2}\). Since the number of solitons is half of the number of HQVs, the satellite intensity scales as \(\propto {\mathrm{\Omega }} \times {\mathrm{\Omega }}^{ - 1/2} = \sqrt {\mathrm{\Omega }}\), which has been previously confirmed by measurements in the polar phase20. Here we observe similar scaling in the PdA and PdB phases, c.f. Fig. 3c.

Although the satellite intensity scales with the vortex density in the same way in both phases, there is one striking difference—the satellite intensity normalized to the total absorption integral in the PdB phase is smaller by a factor of ~9 relative to the PdA phase. Simultaneously, the original satellite intensity in the PdA phase is restored after a thermal cycle shown in Fig. 1b. Our numerical calculations of the soliton structure indicate that neither the PdB phase soliton width nor the oscillator strength would decrease substantially to explain the observed reduction in satellite size and the reason for the observed spectral intensity remains unclear—see Supplementary Note 7 for the calculations.

Another effect of rotation in the PdB phase transverse (μ = π/2) NMR spectrum is observed at the main peak, c.f. Fig. 3b. The full-width-at-half-maximum (FWHM), extracted from the amplitude of the main peak assuming w × h = const, where w is its width and h is height, scales as \(\propto \sqrt \Omega\); Fig. 3d. Increase in the FWHM may indicate that the presence of KLS walls enhances scattering of spin waves and thus results in increased dissipation. Further analysis of this effect is beyond the scope of this article.