Suppressing the Kibble-Zurek mechanism by a symmetry-violating bias
J. Rysti J. T. Mäkinen S. Autti T. Kamppinen G. E. Volovik V. B. Eltsov






Department of Applied Physics, Aalto University, POB 15100, FI-00076 AALTO, Finland


[]jere.makinen@aalto.fi
Department of Applied Physics, Aalto University, POB 15100, FI-00076 AALTO, Finland


Department of Applied Physics, Aalto University, POB 15100, FI-00076 AALTO, Finland
Department of Physics, Lancaster University, Lancaster LA1 4YB, UK.


Department of Applied Physics, Aalto University, POB 15100, FI-00076 AALTO, Finland


Department of Applied Physics, Aalto University, POB 15100, FI-00076 AALTO, Finland
L.D. Landau Institute for Theoretical Physics, Moscow, Russia


Department of Applied Physics, Aalto University, POB 15100, FI-00076 AALTO, Finland





The formation of topological defects in continuous phase transitions is driven by the Kibble-Zurek mechanism. Here we study the formation of single- and half-quantum vortices during transition to the polar phase of ^3He in the presence of a symmetry-breaking bias provided by the applied magnetic field. We find that vortex formation is suppressed exponentially when the length scale associated with the bias field becomes smaller than the Kibble-Zurek length. We thus demontrate an experimentally feasible shortcut to adiabaticity – an important aspect for further understanding of phase transitions as well as for engineering applications such as quantum computers or simulators.




Suppressing the Kibble-Zurek mechanism by a symmetry-violating bias
    V.B. Eltsov
    December 30, 2023
===================================================================



In continuous phase transitions, random local choice of the symmetry-breaking order parameter value leads to the formation of topological defects, such as quantized vortices. Originally a speculation in high-energy physics and cosmology <cit.>, this mechanism is now known as the Kibble-Zurek mechanism (KZM) <cit.>, and it is a cornerstone of out-of-equilibrium condensed matter physics. KZM has been observed in a wide range of systems such as superfluids, superconductors, and Bose condensates <cit.>. In the KZM scenario the transition takes place independently in various locations, and the characteristic size of these regions depends on the rate at which the transition is crossed. Each region inherits a random realization of the broken-symmetry feature of the new phase, such as the phase of the order parameter in a superfluid transition. When the expanding regions of the broken-symmetry phase merge, topological defects such as quantized vortices are formed. The predicted power-law dependence of the defect density on the quench rate has been confirmed in superfluid helium <cit.> as well as other systems (see e.g. reviews <cit.>). 

In the theory of broken-symmetry phase transitions, a symmetry-violating bias field plays an important role, initiating the choice between the different degenerate states beyond the critical point <cit.>. Bias can in particular be applied to non-adiabatic thermodynamic <cit.> or quantum <cit.> phase transitions that result in the formation of topological defects via the KZM. It has been proposed that if the applied symmetry-breaking bias is sufficiently large, the adiabatic (defect-free) regime is restored <cit.>. The crossover from the non-adiabatic Kibble-Zurek regime to the adiabatic regime occurs at the characteristic value of the bias, which depends on the rate at which the critical point is crossed. Such crossover has been analyzed theoretically in a quantum phase transition in the Ising chain, see Refs. <cit.> and in its classical counterpart in the Supplemental Material. Generally speaking, the KZM is expected to be modified in the presence of various external factors, such as inhomogeneities <cit.>, or a propagating front of the phase transition <cit.>. Applying a bias, however, allows for the external control of the magnitude of the KZM directly. Controlled restoration of the adiabatic regime by a symmetry-breaking bias can be utilized in applications requiring delicate and fast control of engineered quantum systems <cit.>.

In this Letter we probe experimentally the use of an external bias for suppressing the formation of single quantum vortices (SQV) and half-quantum vortices (HQV)<cit.> produced by the Kibble-Zurek mechanism in the phase transition from normal ^3He to the polar phase <cit.> of ^3He. We report three central observations: (i) For HQVs the threshold bias for the onset of suppression is set by matching the characteristic length of the applied symmetry-breaking field to the Kibble-Zurek length set by the speed at which the transition is crossed. (ii) Beyond the onset, the suppression takes over exponentially, with the onset threshold normalizing the bias field in the exponent. (iii) The creation of SQVs is similarly suppressed for increasing bias fields while the threshold value is different from that for HQVs.










The rich spectrum of topological defects in the polar phase, and the related bias fields one can apply, are understood in terms of the order parameter of the polar phase

    A_j β=Δ_ Pd̂_j m̂_β e^iΦ .

Here Δ_ P is the maximum gap in the quasiparticle energy spectrum and Φ is the superfluid phase. The unit vector d̂ determines the direction of the easy plane of the magnetic anisotropy and m̂ that of the orbital anisotropy. The anisotropy originates from p-wave Cooper pairing with the orbital momentum and spin of a pair equal to one. The polar phase is stabilized within the confining nanomaterial, which consists of nearly parallel solid strands, and m̂ is pinned along the strand direction, Fig. <ref>(a). The direction of d̂ is set by the competition between the magnetic anisotropy energy χ (d̂· H)^2/2 in the magnetic field H, and the spin-orbit interaction energy g_ so (d̂·m̂)^2, where  χ is the magnetic susceptibility and g_ so is the spin-orbit coupling. 

In large magnetic fields H^2≫ H_ so^2=g_ soχ^-1, the d̂ vector is kept in the plane perpendicular to H and the spin-orbit interaction takes the form

    F_ so=g_ sosin^2μsin^2α ,

where μ is the angle between the magnetic field and the m̂ vector and α is the azimuthal angle of d̂ in the plane perpendicular to the magnetic field. For μ≠ 0, the spin-orbit interaction in Eq. (<ref>)  lifts the degeneracy over α, and thus it plays the role of the symmetry-violating bias with the magnitude g_ sosin^2μ. 

We first study entering the superfluid state from the normal state while applying a symmetry-breaking bias field via the spin-orbit coupling. For a magnetic field oriented along the strands, μ=0, the spin-orbit bias is absent and the symmetry-breaking scheme (ignoring SO(2) orbital rotations about m̂) is

    G=U(1)× SO(2) →Υ=Z_2 .

Here G describes the symmetries of normal ^3He, U(1) is the symmetry under the phase transformation, and SO(2) is the symmetry under rotation in spin space about the axis of the magnetic field. Υ denotes the symmetry of the polar phase order parameter, where Z_2 is the spin rotation by π (corresponding to the change d̂→ -d̂) accompanied by the phase change by π. Since the homotopy group π_1(G/Υ)=Z× Z × Z_2, this symmetry-breaking scheme leads to three types of topological defects: SQVs in the Φ-field, spin vortices in the α-field, and HQVs, where both Φ and α change by π.

When the spin-orbit interaction is turned on (μ≠ 0), the SO(2) symmetry in Eq.(<ref>) is explicitly violated, and one obtains the following symmetry-breaking scheme:

    G̃=U(1) →Υ̃=1 .

Now the homotopy group is π_1(G̃/Υ̃)=Z, which means that only one of the three types of topological defects remain stable: the SQVs, which are not influenced by the spin-orbit interaction. Spin vortices and HQVs become termination lines of topological solitons <cit.>, illustrated in Fig. <ref>(b). Assuming only HQVs are present, 𝐝̂-solitons connect pairs of HQVs of the opposite 𝐝̂-winding.

The presence of the solitons can be detected and their total volume in the sample measured using the nuclear magnetic resonance (NMR) techniques.The bulk of the sample forms the main peak in the continuous-wave NMR spectrum at the frequency ω_ main. The 𝐝̂-soliton provides a trapping potential for standing spin waves, seen as a satellite peak in the NMR spectrum at the frequency ω_ sat <cit.>. The relative sizes of the main peak and the satellite are determined by the volume occupied by the 𝐝̂-solitons in the sample. Measuring the initial density of KZ defects has traditionally been a complicated task due to the fast annihilation of non-equilibrium defects at temperatures close to the phase transition <cit.>. In our experiments the confining strands pin vortices in place once they are formed <cit.>, which provides the observer a frozen window to the out-of-equilibrium physics of the second-order phase transition and a direct measurement of the KZ vortex density.

We calibrate the size of the satellite peak by preparing a state by a very slow cooldown through the critical temperature T_ c in zero magnetic field while the sample is in stable rotation. Under these conditions we create HQVs with aerial density given by n_ v = 4Ωκ^-1, where Ω is the angular velocity, κ=h/(2 m_3) is the quantum of circulation, h is the Planck constant, and m_3 is the ^3He atom mass. The calibration gives the relative satellite size I_sat=I_0 √(Ω), where I_0=0.090 s^1/2 rad^-1/2 (see Supplemental Material). The inter-vortex distance assuming a square lattice is L=n_ v^-1/2=1/2√(κ) I_0 I_sat^-1. We then use this relation to estimate the HQV density and intervortex distance for HQVs created purely by the KZM (i.e. for Ω = 0). The combined effect of rotation and KZM is discussed in the Supplemental Material.







We control the spin-orbit bias by applying a fixed magnetic field of H=11 mT with chosen transverse component H_⊥=Hsinμ during the cooldown through T_ c. We repeat cooldowns with different values of H_⊥ and different cooldown rates, Fig. <ref>. We observe a constant satellite size for small H_⊥ and its gradual suppression for larger values of H_⊥. We suggest that the threshold field H_⊥ t where the suppression of the formation of half-quantum vortices starts is determined by comparing the Kibble-Zurek length l_ KZ= a ξ_0(τ_ Q/τ_0)^1/4 with the characteristic length of the bias, ξ_ bias, given by the thickness of the d̂ solitons. Here a ∼ 1 fixes the exact length scale for the defect formation (in our measurements a ≈ 2.3, see Supplemental Material), the quench rate is τ_ Q^-1=-d(T/T_ c)/dt|_T=T_ c, T is temperature, t is time, ξ_0 is the superfluid coherence length at low temperature, τ_0 = ξ_0 v_F^-1∼ 1ns is the order parameter relaxation time, v_F is the Fermi velocity, ξ_ bias∼ξ_ so/sinμ, and ξ_ so = 17 μm is the dipole length <cit.>. 

Taking L | _H_⊥= 0 for transitions without bias as an estimate for l_ KZ <cit.>, and equating it with ξ_ bias gives the following threshold bias for the suppression of HQV creation

    H_⊥ t =  ξ_ so/l_ KZ H  .

In the spirit of Ref. <cit.> we propose that the defect density ∝ I_ sat^2 decays exponentially after the transition field. In terms of the satellite intensity, this reads

    I_ sat =
    
      I_ sat0   for  H_⊥ < H_⊥ t
    
          I_ sat0exp( 1-H_⊥/H_⊥ t)    for  H_⊥≥ H_⊥ t ,

where I_ sat0 is the initial satellite intensity. We note that for this model ∫_0^∞ I_ sat dH_⊥ = 2 I_ sat0 H_⊥ t and the numerical integral of the measured I_ sat can be used to determine H_⊥ t without fitting.

Our experiments, Fig. <ref>, confirm the validity of the model (<ref>) up to the point when the inter-vortex distance becomes comparable to the sample size. We emphasize that the threshold field H_⊥ t (which also normalizes the exponent) is determined by integration of the experimental data without fitting procedure. The result agrees well with the conjecture ξ_ t≡ξ_ bias(H_⊥ t) = l_ KZ.

The result for the slowest quench rate deviates, however, from this dependence. In the presence of a thermal gradient, the phase transitioncompared proceeds via a propagating front, where ordering of the low-temperature phase lags behind the temperature front where T = T_ c by distance l_ F. The KZM operates in the band of width l_ F and is generally modified in comparison to the homogeneous cooling scenario <cit.>. As τ_ Q increases, l_ F decreases and l_ KZ increases. We suggest that the smaller of the two characteristic lengths, l_ KZ and l_ F, determines the threshold bias ξ_ t. We estimate that in our measurement l_ F < l_ KZ only for the slowest quench rate (black diamonds in Fig. <ref>) for which l_ F∼ 210 μm, matching the observed value at ξ_ t (see Supplemental Materials for details).

Alternatively to the spin-orbit bias we can apply a direct field bias with a weak magnetic field oriented perpendicular to the orbital vector m̂. The small magnetic field H_⊥<H_ so violates the symmetry under rotation about m̂, which leads to the formation of solitons, absent at zero magnetic field, with the soliton thickness now determined by the magnetic field directly, ξ_ bias=ξ_ H=ξ_ so H_ so/H. Equating ξ_ bias with l_KZ yields a criterion for the threshold field similar to that in Eq. (<ref>) but with H replaced by H_ so. The expected decrease of the threshold field in this case is H_ so/H = Ω_ P/ω_ main≈√(2 (1- ω_ sat / ω_ main))≈ 0.17 <cit.>, where Ω_ P is the polar phase Leggett frequency. It is confirmed experimentally by the blue squares in Fig. <ref>. Here the ratio of the threshold field relative to the red circles, which correspond to the spin-orbit bias with similar quench rate, is ≈ 0.16.







Finally, we study the fate of SQVs under the symmetry-breaking bias created by tilting the magnetic field. Due to the absence of the interconnecting topological solitons one would naïvely expect that such a bias has no effect on the KZM for SQVs. Without the solitons, SQVs are not seen in continuous-wave NMR, but they were found to increase the relaxation rate of a magnon BEC <cit.>. Independent measurements with SQVs created by rotation drive indicate that the relaxation rate increases monotonically when the SQV density grows <cit.>. In our measurements we create both HQVs and SQVs by the KZM and subtract the effect of HQVs by using a calibration of the relaxation rate of the magnon BEC with respect to the satellite intensity, see Fig. 3b from Ref. <cit.>. The remaining contribution to the relaxation we attribute to SQVs. This contribution shows a characteristic dependence with a threshold and exponential suppression akin to Eq. (<ref>), Fig. <ref>. A possible explanation for this behavior is that an applied bias influences the structure of the vortex core, which in p-wave superfluids can be complicated. For example in the bulk B phase, the SO(2) symmetry of the SQV core is spontaneously broken at low temperatures and the core transforms into a pair of tightly-bound half-quantum cores <cit.>. In the polar phase, the spin-orbit interaction and magnetic anisotropy may play the role of the symmetry-violating bias for the phase transitions inside the vortex core, but the detailed investigation of this phenomenon remains a task for the future.

In conclusion, we report a crossover from the Kibble-Zurek regime of half-quantum vortex creation to the adiabatic regime, where vortex formation is rapidly suppressed by a symmetry-violating bias. We thus demonstrate an experimentally feasible shortcut to adiabaticity, where the adiabatic regime can be reached without an infinitely slow transition rate. In our experiments the symmetry-violating bias is provided either by the spin-orbit interaction or directly by external magnetic field. The crossover to the adiabatic regime takes place when the characteristic length scale of the bias, given by the thickness of the topological solitons connecting neighboring HQVs, becomes smaller than the KZ length determined by the transition rate or the thickness of the transition front in the case of slow inhomogeneous cooling. Beyond the onset, the suppression of the Kibble-Zurek mechanism takes place exponentially. We also report similar suppression of SQV formation by the KZM and provide preliminary evidence indicating that there may be a symmetry-breaking core transition in the SQV cores, sensitive to the symmetry-violating external bias.

The symmetry-breaking aspect of the bias field is essential for the suppression of the KZM, which otherwise is very robust. As an example, presented in the supplementary material, we show that adding an array of half-quantum vortices created by rotating the sample has no effect on the KZ mechanism even when the characteristic length scale of the added lattice becomes smaller than the KZ length. We also note that HQVs are composite defects, whose KZM formation is rarely studied experimentally, as well as analogs of Alice strings <cit.>. The KZM formation of HQVs studied here may shed light to defect formation across phase transitions in theories considering such systems.

Our results can be generalised to the bias-induced restoration of adiabadicity in various phase transitions including quantum phase transitions, which could provide applications in technologies such as quantum simulators and computers <cit.>. On a more speculative note, it is not excluded that bias plays a role in the so-called collapse of the wave function in quantum mechanics. In principle, the latter can be seen as “phase transition” occuring in the continuous spectrum of an infinite system <cit.>. One of the many quantum states participating in a given quantum superposition is perhaps then selected by the infinitesimal bias unavoidably present in any experiment.
 

We thank V. V. Dmitriev for providing the NAFEN sample. 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 Academy of Finland project No. 332964. S.A. acknowledges support from the Jenny and Antti Wihuri Foundation via the Council of Finnish Foundations and T. K. acknowledges support from the Finnish Cultural Foundation. This research made use of the OtaNano – Low Temperature Laboratory infrastructure of Aalto University, that is part of the European Microkelvin Platform (European Union's Horizon 2020 Grant No. 824109).

J. R. and J. T. M. contributed equally to this work.









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Sequation



Sfigure







§ SUPPLEMENTAL MATERIAL








 §.§ Experimental setup


We prepared the superfluid state by cooling the sample from above T_ c to the polar phase using the ROTA nuclear demagnetization cryostat <cit.>. The entire cryostat can be rotated up to 3 rad s^-1, and the axis of rotation is parallel to the confining strands, see Fig. <ref>(a). For confinement we used 94% open NAFEN material (Al_2O_3, density 243 mg/cm^3) produced by AFN technology Ltd in Estonia. The Larmor frequency, defined by the magnetic field, was f_ L=ω_ L/2π=γ H/2π=363 kHz, where γ = 2 π× 32.435 MHz T^-1 is the absolute value of the gyromagnetic ratio of ^3He. The measurements were done at 7 bar pressure at which the critical temperature of the bulk superfluid is T_ c≈ 1.6 mK and the polar phase critical temperature within the aerogel sample was found to be T_ ca≈ 0.972 T_ c. Temperature was measured from the NMR spectrum of bulk B phase at high temperatures using the known B-phase Leggett frequency <cit.>, and from the NMR line in the polar phase in the axial (along NAFEN strands) field at low temperatures <cit.>. As a secondary thermometer we used a quartz tuning fork <cit.> located in bulk ^3He. The container walls and NAFEN strands were preplated with a few atomic layers of ^4He in order to order to stabilize the polar phase <cit.> and to suppress the paramagnetic signal from solid ^3He otherwise formed on all surfaces. The magnetic field could be freely rotated in the plane perpendicular to the NMR coils' axis from parallel to the strands (μ=0) to transverse to them (μ=π/2). Fast quenches through T_ ca are done by heating the sample with a resistive heater above T_ c and then switching the heater off. Slower transitions are done by controlling the temperature directly with the nuclear demagnetization stage. The quench rate τ_ Q is determined from the fork reading. During the transition magnetic field is applied at an angle and magnitude as described in the main text. To avoid formation of spin vortices, the NMR excitation is switched off during the cooldown. The NMR measurements are performed in transverse (μ = π/2) field at T = 0.5 T_ c.



 §.§ Ising chain under symmetry-breaking bias


The restoration of adiabatic regime by a symmetry-violating bias was considered in Ref. <cit.> building on toy model known as quantum Ising chain. Let us discuss the role of a symmetry-violating bias using its classical counterpart. The Hamiltonian of such a system, in terms of parameter β, may be written as

    F= -1/2cos^2β - g_⊥sinβ   - g_∥cosβ ,

where the term g_∥cosβ plays the role of a symmetry-violating bias. For g_∥ = 0 there is a discrete Z_2 symmetry cosβ→ -cosβ, and at g_⊥=1 there is a broken-symmetry phase transition between a paramagnetic and a ferromagnetic state. The minima of the free energy are located at

    cosβ=0    for   g_⊥ >1 ,   g_∥=0    and
       cosβ=±√(1-g_⊥^2)   for   g_⊥ <1 ,   g_∥=0

for the paramagnetic and ferromagnetic phases, respectively. For g_∥≠ 0 the discrete Z_2 symmetry is explicitly broken, and the phase transition transforms into a smooth (adiabatic) crossover. This effect competes with the non-adiabatic Kibble-Zurek effect for the creation of topological defects: domain walls between two stable values of β. For each τ_ Q there exists a characteristic value of the symmetry-violating  bias, g_∥ t, which separates two regimes: the Kibble-Zurek regime of creation of domain walls at small bias g_∥ < g_∥ t and the adiabatic regime at large bias g_∥ > g_∥ t, where the creation of the Kibble-Zurek domain walls begins to suppress.




 §.§ Robustness of the KZ mechanism







As an example of the robustness of the Kibble-Zurek mechanism (KZM), we consider the combined effect of the KZM and rotation drive which creates an additional density of the very same vortices. We observe that the KZ mechanism is robust against such direct external driving forces, a scenario touched theoretically in Ref. <cit.>. That is, the vortex density created by the KZM is additive with the vortex density created by rotation.

In the KZM the average inter-vortex distance is given by the Kibble-Zurek length l_ KZ = a ξ_0 (τ_ Q/τ_0)^1/4, where a ∼ 1 fixes the exact length scale for the defect formation, τ_ Q^-1=. - d(T/T_ c)/ d t|_T=T_ c is the cooldown rate at T_ c, ξ_0 = ξ(T=0) is the coherence length at zero temperature, and the order-parameter relaxation time τ_0 = ξ_0 v_F^-1∼ 1ns, where v_F is the Fermi velocity. The vortices created by the KZM are randomly oriented, which in the case of half-quantum vortices (HQVs) complicates tracing the measured soliton volume back to vortex density as neighboring vortices are not aligned. In our case the vortex density is nevertheless very low, and the overwhelming majority of soliton volume therefore simply defined by the distance to the nearest vortex <cit.>. In the low-density limit the relative intensity of the satellite signal becomes

    I_ KZ = 1/2ξ_ so/l_ KZ∝τ_ Q^-1/4

where l_ KZ= a ξ_0(τ_ Q/τ_0)^1/4.

HQVs can also be created by rotation drive. The resulting satellite peak intensity I_ sat depends on the angular velocity Ω applied during the phase transition <cit.>:
 
    I_Ω = Ω^1/2ξ_ D g_ s b/2 ≡ I_0 Ω^1/2 .

 Here g_ s∼ 1.7 is a numerical factor calculated in the supplemental material of Ref. <cit.>, and b ∼1 describes the vortex lattice. In experiments this dependence is realized in the limit of vanishing cooldown rate.

 
 

 
In general, the ways the KZ vortex density and that created by the rotation drive combine can be expected to lie between two limiting scenarios: (a) The two mechanisms may be independent. In this case the vortex densities are additive and the satellite intensity becomes 


    I_ sat= √(I_ KZ^2 + I_Ω^2  ) = √(I_ KZ^2 + I_0^2 Ω),

where I_ KZ and I_Ω stand for the satellite sizes created by either mechanism independently. (b) The rotation drive may merely act to polarize the KZ vortices until the KZ vortex density is exceeded by that required by the rotation applied. In this case the satellite intensity becomes I_ sat=   max[I_ KZ , I_Ω ]. 

The measured satellite intensity as a function of rotation velocity and cooldown rate is plotted in Fig. <ref>. There are two main observations: (i) The KZM is independent of applied rotation drive. Comparison of the measurements with the theoretical prediction for I_ sat assuming rotation bias and KZM act in complete independence (Eq. (<ref>)) shows good quantitative agreement. The independence is emphasized especially by the fast-cooldown data for which I_Ω≈ I_ KZ, therefore testing Eq. (<ref>) in the most relevant range of parameters. The asymptotic dependence on Ω is confirmed by the data from slow cooldowns. The fitted value of I_0 is I_0=0.090 √(s rad^-1), while the theoretical prediction with ξ_ D = 17 μm is I_0=0.11 √(s/rad) <cit.>. The fitted averaged soliton length (vortex separation) of KZ vortices corresponds to a = 2.3, which is of expected order of magnitude <cit.>. (ii) Spin vortices can be created by applying a strong rf field on resonance during the phase transition. They are seen as additional satellite intensity. This additional vortex density does not change the effect of the rotation drive, which parallels the conclusion drawn above from observation (i). This confirms that the determination of vortex density from the soliton volume works as expected for independently formed topological objects. It should be noted that both these observations are likely to be valid only in the limit of low vortex density. At typical vortex densities in the experiments, the hard-core HQVs occupy no more than a few ppb of the sample volume, and even the soft-core spin vortices no more than a few ppm.



 §.§ KZM in the wake of a thermal front


In a realistic experimental scenario, the transition temperature is not crossed simultaneously in the whole sample volume. Any temperature gradient would result in a propagating thermal front, and as a result the symmetry-breaking phase transitions occurs in the wake of the propagating front<cit.>.

The volume containing the sample of the polar phase is open from one side to the heat exchanger volume and closed elsewhere, resulting in a thermal gradient across the sample. The temperature difference across the volume may be estimated by

    | Δ T/Δ x| = Q̇/𝒦 d^2 ,

where Q̇ is the heat leak to the sample volume, d = 4 mm is the length of one side, and 𝒦 is thermal conductivity. We estimate the total heat leak to the sample container to be ∼ 20 pW from which we estimate the order of magnitude for the heat leak to the sample to be ∼ 1 pW. Just above T_ c the thermal conductivity of normal ^3He at 7 bar pressure is 𝒦∼ 10^4 erg (cm·s·K)^-1 <cit.>. We estimate the thermal conductivity in our sample in the direction perpendicular to NAFEN strands to be suppressed relative to the direction along the strands (assumed to be about that in bulk ^3He) roughly by the same amount as spin diffusion is suppressed, i.e. by a factor of ∼ 8 <cit.>. Under these assumptions an estimate of the temperature gradient across the sample is Δ T/Δ x ∼ 5 × 10^-6 mK mm^-1. The resulting thermal front propagation velocity in our experiments is in the range v_ T≡ (T_ c / τ_ Q )/(Δ T/Δ x) ∼ 40-900 mm s^-1, where smaller values correspond to slower quenches.

The propagation velocity of the thermal front should be compared with the equilibration rate of the order parameter behind the front, which gives an estimate for the lowest front propagation velocity that still realizes the original KZM scenario <cit.>

    v_ Tc∼ v_ F( τ_0/τ_ Q) ^1/4 ,

where v_ F is the Fermi velocity. For our experimental conditions, v_ Tc lies in the range 30-60 mm s^-1. We note that in this model, for the slowest quench rate in the experiments presented in the main text, v_ T∼ v_ Tc and the initial vortex density is suppressed with respect to the original KZM scenario. This effect can be seen for solid black diamonds in Fig. 2 in the main text (same as the solid green diamond with the highest τ_ Q in Fig. <ref>).

We suggest that in addition to the lower initial vortex density, the slowly propagating front of the phase transition has another consequence. Unlike in the original KZ scenario, the vortices are now formed within a “homogeneous” region behind the front outside which the superfluid phase is already well defined. The thickness of this region becomes the new limiting length scale. According to Ref. <cit.> the thickness of this layer scales as l_ F≈ v_ TΔ t, where

    Δ t ≈τ_ Qv_ T^2/v_ F^2 .

For the slowest quench rate in Fig. <ref> (black diamonds), this gives l_ F≈ 210 μm. We note that this is very close to the estimated ξ_ bias(H_⊥ t) ≈ 170  μm while the KZ length extracted from the satellite intensity is l_ KZ≈ 310 μm. We thus propose that the relevant length scale for the KZM for determination of the threshold bias is the smaller one of l_ KZ and l_ F.




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