Abstract
Symmetries of the physical world have guided formulation of fundamental laws, including relativistic quantum field theory and understanding of possible states of matter. Topological defects (TDs) often control the universal behavior of macroscopic quantum systems, while topology and broken symmetries determine allowed TDs. Taking advantage of the symmetrybreaking patterns in the phase diagram of nanoconfined superfluid ^{3}He, we show that halfquantum vortices (HQVs)—linear topological defects carrying half quantum of circulation—survive transitions from the polar phase to other superfluid phases with polar distortion. In the polardistorted A phase, HQV cores in 2D systems should harbor nonAbelian Majorana modes. In the polardistorted B phase, HQVs form composite defects—walls bounded by strings hypothesized decades ago in cosmology. Our experiments establish the superfluid phases of ^{3}He in nanostructured confinement as a promising topological media for further investigations ranging from topological quantum computing to cosmology and grand unification scenarios.
Introduction
Topological defects generally form in any symmetrybreaking phase transitions. The exact nature of the resulting TDs depends on the symmetries before and after the transition. Our universe has undergone several such phase transitions after the Big Bang. As a consequence, a variety of TDs might have formed during the early evolution of the Universe, where phase transitions lead to unavoidable defect formation via the Kibble–Zurek mechanism^{1,2}. Experimentally accessible energy scales \(\lesssim 1\,{\mathrm{TeV}}\) are currently limited to times \(t \gtrsim 10^{  12}\) s after the Big Bang by the Large Hadron Collider. Theoretical understanding may be extended up to the Grand Unification energy scales \(\lesssim 10^{15}\,{\mathrm{GeV}}\) of the electroweak and strong forces \(\left( {t \gtrsim 10^{  36} \ldots 10^{  32}\,{\mathrm{s}}} \right)\). The nature of the interactions before this epoch remains unknown^{3,4}, but yet unobserved cosmic TDs, the nature of which depends on the Grand Unified Theory (GUT) in question, may help us limit the possibilities. Predictions exist for point defects, such as the t’Hooft–Polyakov magnetic monopole^{5,6}, linear defects or strings^{1}, surface defects or domain walls^{7}, and threedimensional textures^{8}.
Even though cosmic TDs have not been detected, many of their condensedmatter analogs have been reproduced in the laboratory, where they have an enormous impact on the behavior of the materials they reside in^{9}. Examples include vortices in superconductors^{10}, vortices and monopoles in ultracold gases^{11,12}, and skyrmions in chiral magnets^{13}. Superfluid phases of ^{3}He offer an experimentally accessible system to study a variety of TDs and the consequences of symmetrybreaking patterns owing to its rich orderparameter structure resulting from the pwave pairing. Analogs of exotic TDs, such as the Witten string^{14}—the brokensymmetrycore vortex in superfluid ^{3}HeB^{15,16,17}, the skyrmion texture in superfluid ^{3}HeA^{18}, and the Alice string^{19}—halfquantum vortex (HQV) in the polar phase of superfluid ^{3}He^{20}, have been observed.
Of particular interest are composite defects—combinations of TDs and/or nontopological defects of different dimensionality^{21,22,23}. Such defects appear in some GUTs and even in the Standard Model, where the Nambu monopole may terminate an electroweak string^{24,25}. There are two mechanisms for the formation of composite defects: the hierarchy of energy/interaction length scales^{23,26,27}, and the hierarchy (sequential order) of the symmetrybreaking phase transitions^{22,28}. Composite defects originating from the hierarchy of length scales of condensation, magnetic, and spinorbit energies are wellknown in superfluid ^{3}He. For example, the spinmass vortex in ^{3}HeB^{23,29} has a hard core of the coherencelength size, defined by the condensation energy, and a soliton tail with thickness of the much larger spinorbit length. A HQV originally predicted to exist in the chiral superfluid ^{3}HeA^{30} has a similar structure with the soliton tail, which makes these objects energetically unfavorable.
Composite defects related to the hierarchy of symmetrybreaking phase transitions were discussed in the context of the GUT scenarios by Kibble, Lazarides, and Shafi^{22,28}. Here the GUT symmetry, such as Spin(10), is broken into the Pati–Salam group SU(4) × SU(2) × SU(2), which in turn is broken to the Standard Model symmetry group SU(3) × SU(2) × U(1). At the first transition, the linear defects—cosmic strings—become topologically stable, while after the second transition they are no longer supported by topology and form the boundaries of the nontopological domain walls, henceforth referred to as Kibble–Lazarides–Shafi (KLS) walls. To the best of our knowledge, observations of KLS walls bounded by strings have not been reported previously.
In this work, we explore experimentally the composite defects formed by both the hierarchy of energy scales and the hierarchy of symmetrybreaking phase transitions allowed by the phase diagram of superfluid ^{3}He confined in nematically ordered aerogellike material called nafen. In our sample, a sequence of the polar, chiral polardistorted A (PdA) and fully gapped polardistorted B (PdB) phases occurs on cooling from the normal state^{31}, see Fig. 1c. Previously, we established a procedure to form topologically protected HQVs in the polar phase^{20}. At the transition from the polar phase to the PdA phase, we expect the HQVs to acquire spinsoliton tails with the width of the spinorbit length, which is much larger than the coherencelength size of vortex cores. On a subsequent transition to the PdB phase, the symmetry breaks in such a way that HQVs lose topological protection and may exist only as boundaries of the nontopological KLS walls. Simultaneously, the spin solitons between HQVs are preserved in the PdB phase and such an object becomes a doublycomposite defect. Naively, however, one would expect that a much stronger tension of the KLS wall compared to that of the spin soliton, would lead to collapse of an HQV pair, possibly to a singly quantized vortex with an asymmetric core^{15,16,17,32,33}.
Here we report evidence that HQVs do exist in the superfluid PdA and PdB phases of ^{3}He. We create an array of HQVs by rotating the container with the angular velocity Ω in zero magnetic field during the transition from the normal fluid to the polar phase^{20} and proceed by cooling the sample through consecutive transitions to the PdA and PdB phases. The HQVs are identified based on their nuclear magnetic resonance (NMR) signature as a function of temperature and Ω. A characteristic satellite peak present in the NMR spectrum confirms that the HQVs survive in the PdA phase, where they provide experimental access to vortexcorebound Majorana states^{34,35}. Moreover, the HQVs are found to survive the transition to the PdB phase. The observed features of the NMR spectrum in the PdB phase suggest that a KLS wall emerges between a pair of HQVs already connected by the spin soliton. Evidently, the tension of the KLS wall is not sufficient to overcome the pinning of HQVs in nafen. Vortex pinning allows us to study the properties of the outofequilibrium vortex state created during the superfluid phase transitions while suppressing the vortex dynamics. Simultaneously pinning does not affect the symmetrybreaking pattern leading to formation of the KLS walls. Our results show that pinned TDs, once created, may be transferred to new phases of matter with engineered topology^{36,37,38}.
Results
Halfquantum vortices in the PdA phase
The superfluid phase diagram under confinement by nafen^{31}—a nanostructured material consisting of nearly parallel strands made of Al_{2}O_{3}, c.f. Fig. 1b—differs from that of the bulk ^{3}He; 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
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 continuouswave NMR techniques to probe the sample, see Methods for further details. In the superfluid state, the spinorbit coupling provides a torque acting on the precessing magnetization, which leads to a shift of the resonance from the Larmor value ω_{L} = γH, where γ = −2.04 × 10^{8} s^{−1} T^{−1} is the gyromagnetic ratio of ^{3}He. 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, is^{31}
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 secondorder phase transition from the polar phase but renormalizes the longitudinal resonance frequency due to appearance of the orderparameter component with b.
Quantized vortices are linear topological defects in the orderparameter field carrying nonzero circulation. In the PdA phase, quantized vortices involve phase winding by \(\phi \to \phi + 2\pi \nu\) and possibly some winding of the \(\widehat {\mathbf{d}}\) vector. The typical singly quantized vortices, also known as phase vortices, have \(\nu = 1\) and no winding of the \(\widehat {\mathbf{d}}\)vector, while the HQVs have \(\nu = \frac{1}{2}\) 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—spinsolitons connecting pairs of HQVs. The soft cores of the \(\widehat {\mathbf{d}}\)solitons provide trapping potential for standing spin waves^{39}.
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 vorticity^{40} like the doublequantum vortex in ^{3}HeA^{41}. Some planar structures in the \(\widehat {\mathbf{l}}\)vector field, such as domain walls^{42} 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 distortion^{31} (i.e. for b ≪ 1). Recent theoretical work^{43} 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 secondorder phase transition from the polar phase with preformed HQVs. We already know^{20} that the maximum tension from the spinsoliton 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 tail^{18}.
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
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 zeromode of the soliton^{18,20,44}. 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 sample^{20}. 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.
Halfquantum 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 \(\widehat {\mathbf{d}}\)solitons^{20}. 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 strings^{22}. For very short separation between HQVs in a pair and ignoring the orderparameter distortion by confinement, such construction may resemble the brokensymmetrycore singlequantum vortex of the B phase^{16}. 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
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 spinspace 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 orderparameter components with q > 0, Eq. (4) remains singlevalued 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 scales^{17,33}—the KLS wall has a hard core of the order of ξ_{W} ≡ q^{−1}ξ, 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 lowestenergy spinsoliton 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.
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 is^{45}
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
and winding of the \(\widehat {\mathbf{d}}\)vector in a soliton leads to a characteristic frequency shift
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 lowtemperature 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 spinsolitons gives rise to a frequency shift experimentally indistinguishable from the frequency shift of the bulk line. The last possibility, the “πsoliton” 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 πsoliton 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 singlequantum vortices in Blike phase in silica aerogel has also been observed previously^{46}. An alternative way to remove a KLS wall is to create a hole within it, bounded by a HQV^{22}. 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 spinorbit 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 phase^{20}. 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 fullwidthathalfmaximum (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.
Discussion
To summarize, we have found that HQVs, created in the polar phase of ^{3}He in a nanostructured material called nafen, survive phase transitions to the PdA and PdB phases. Previously, HQVs have been reported in the polar phase^{20}, at the grain boundaries of dwave cuprate superconductors^{47}, in chiral superconductor rings^{48}, and in Bose condensates^{49,50}. Of these systems, only the polar phase contains vortexcorebound fermion states as others are either Bose systems or lack the physical vortex core altogether. The domain walls with the sign change of a single gap component in ^{3}HeB were suggested to interpret the experimental observations in bulk samples^{51,52} (q = 1) and in the slab geometry^{53}. Such walls, however, differ from those reported here as they are not bounded by strings but rather terminate at container walls. In the slab geometry, such walls are additionally topologically protected by a \({\Bbb Z}_2\) symmetry due to pinning of the \(\widehat {\mathbf{l}}\) vector by the slab.
The survival of HQVs in the PdA and PdB phases has several important implications. First, HQVs in twodimensional (2D) p_{x} + ip_{y} topological superconductors (such as the A or PdA phases) are particularly interesting since their cores have been suggested to harbor nonAbelian Majorana modes, which can be utilized for topological quantum computation^{54}. This fact has attracted considerable interest in practical realization of such states in various candidate systems^{55,56,57,58}. While the PdA phase has the correct p_{x} + ip_{y} type order parameter, scaling the sample down to effective 2D remains a challenge for future. However, the presence of the nafen strands, smaller in diameter than the coherence length, increases the separation of the zeroenergy Majorana mode from other vortexcorelocalized fermion states to a significant fraction of the superfluid energy gap, making it easier to reach relevant temperatures (\(k_{\mathrm{B}}T \lesssim\) energy separation of corebound states) in experiments^{59,60}.
Second, we have shown how in the PdB phase, the HQVs, although topologically unstable as isolated defects, survive as composite defects known as “walls bounded by strings” (here KLS walls bounded by a pair of HQVs)—first discussed decades ago by Kibble, Lazarides, and Shafi in the context of cosmology^{22}. Although the present existence of KLS walls in the context of the Standard Model is shown to be unacceptable, as they either dominate the current energy density (firstorder phase transition) or disappeared during the early evolution of the Universe (secondorder phase transition), they occur in some GUTs and beyondtheStandardModel scenarios, especially in ones involving axion dark matter^{61,62,63}. Any sign of similar defects in cosmological context would thus immediately limit the number of viable GUTs. Under our experimental conditions, the transition from the PdA phase to the PdB phase is weakly firstorder (\(q \ll 1\) at transition), but in principle, the order parameter allows a secondorder phase transition to the PdB phase directly from the polar phase. Such a phase transition may be realized in future, e.g., by tuning confinement parameters. Studying the parameters affecting the amount of supercooling of the metastable PdA state (“false vacuum”) before it collapses to the lowestenergy PdB state (“true vacuum”) may also give insight on the nature of phase transitions in the evolution of the early Universe.
In conclusion, we have shown that the creation and stabilization of HQVs in different superfluid phases with controlled and tunable orderparameter structure is possible in the presence of strong pinning by the confinement. The survival of HQVs opens up a wide range of experimental and theoretical avenues ranging from nonAbelian statistics and topological quantum computing to studies of cosmology and GUT extensions of the Standard Model. Additionally, our results pave way for the study of a variety of further problems, such as different fermionic and bosonic excitations living in the HQV cores and within the KLS walls, and the interplay of topology and disorder provided by the confining matrix^{64}. A fascinating prospect is to stabilize new topological objects in novel superfluid phases by tuning the confinement geometry^{36,37,38}, temperature, pressure, magnetic field, or scattering conditions^{65}.
Methods
Sample geometry and thermometry
The ^{3}He sample is confined within a 4mmlong cylindrical container with ∅4 mm inner diameter, made from Stycast 1266 epoxy; see Fig. 1a for illustration. The experimental volume is connected to another volume of bulk B phase, used for thermometry and coupling to nuclear demagnetization stage. This volume contains a commercial quartz tuning fork with 32 kHz resonance frequency, commonly used for thermometry in ^{3}He^{66,67}. The fork is calibrated close to T_{c} against NMR signal from bulk ^{3}HeB surrounding the nafenfilled volume. At lower temperatures, we use a selfcalibration scheme^{68} by determining the onset of the ballistic regime from the fork’s behavior^{69}.
Sample preparation
To avoid paramagnetic solid ^{3}He on the surfaces, the sample is preplated with ~2.5 atomic layers of ^{4}He^{65}. The HQVs are created by rotating the sample in zero magnetic field with angular velocity Ω while cooling the sample from the normal phase to the polar phase. Then the rotation is stopped since, based on our observations, the HQVs remain pinned (and no new HQVs are created) over all relevant time scales, at least for 2 weeks after stopping the rotation. The typical cooldown rate close to the critical temperature was of the order of 0.01 T_{c} per hour to reduce the amount of vortices created by the Kibble–Zurek mechanism. Once the state had been prepared, the temperature was kept below the polar phase critical temperature until the end of the measurement.
NMR spectroscopy
Static magnetic field of 12–27 mT corresponding to NMR frequencies of 409–841 kHz is created using two coils oriented along and perpendicular to the axis of rotation. The magnetic field can be oriented at an arbitrary angle in the plane determined by the two main coils. Special gradient coils are used to minimize the field gradients along the directions of the main magnets. The magnetic field inhomogeneity along the rotation axis is \({\mathrm{\Delta }}H_{{\mathrm{ax}}}/H_{{\mathrm{ax}}}\sim 10^{  4}\) and in the transverse direction an order of magnitude larger, \({\mathrm{\Delta }}H_{{\mathrm{tra}}}/H_{{\mathrm{tra}}}\sim 10^{  3}\). The NMR pickup coil, oriented perpendicular to both main magnets, is a part of a tuned tank circuit with quality factor Q ~ 140. Frequency tuning is provided by a switchable capacitance circuit, thermalized to the mixing chamber of the dilution refrigerator. We use a cold preamplifier, thermalized to a bath of liquid helium, to improve the signaltonoise ratio in the measurements.
Rotation
The sample can be rotated about the vertical axis with angular velocities up to 3 rad s^{−1}, and cooled down to \(\sim 150\,\mu {\mathrm{K}}\) using ROTA nuclear demagnetization refrigerator. The refrigerator is well balanced and suspended against vibrational noise. The earth’s magnetic field is compensated using two saddleshaped coils installed around the refrigerator to avoid parasitic heating of the nuclear stage. In rotation, the total heat leak to the sample remains below 20 pW^{67}.
Data availability
All the data supporting the findings are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank V.V. Zavyalov and V.P. Mineev for useful discussions and related work on spinsolitons and HQVs. This work has been supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 694248) and by the Academy of Finland (grant nos. 298451 and 318546). The work was carried out in the Low Temperature Laboratory, which is part of the OtaNano research infrastructure of Aalto University.
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The experiments were conducted by J.T.M. and J.R.; the sample was prepared by V.V.D. and A.N.Y.; the theoretical analysis was carried out by J.T.M., V.V.D., J.N., G.E.V., A.N.Y. and V.B.E.; numerical calculations were performed by J.N. and K.Z.; V.B.E. supervised the project; and the paper was written by J.T.M., J.N., G.E.V. and V.B.E., with contributions from all authors.
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Mäkinen, J.T., Dmitriev, V.V., Nissinen, J. et al. Halfquantum vortices and walls bounded by strings in the polardistorted phases of topological superfluid ^{3}He. Nat Commun 10, 237 (2019). https://doi.org/10.1038/s41467018082048
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