[1,2]Mark Hindmarsh

1] Department of Physics and Astronomy, University of Sussex, Falmer, Brighton
BN1 9QH, UK 2] Department of Physics and Helsinki Institute of Physics, PL 64,
FI-00014 University of Helsinki, Finland 3] Hearne Institute of Theoretical
Physics, Department of Physics & Astronomy, Louisiana State University, Baton
Rouge LA 70803, USA 4] Department of Physics, Lancaster University, Lancaster
LA1 4YB, UK 5] Department of Physics, Royal Holloway, University of London,
Egham TW20 0EX, UK 6] Department of Physics, University of the Basque Country
UPV/EHU, 48080 Bilbao, Spain 7] EHU Quantum Center, University of Basque
Country, UPV/EHU

# A-B transition in superfluid 3He and cosmological phase transitions

<EMAIL_ADDRESS>J. A. Sauls  Kuang Zhang  S. Autti  Richard P.
Haley  Petri J. Heikkinen  Stephan J. Huber  Lev V. Levitin  Asier Lopez-
Eiguren  Adam J. Mayer  Kari Rummukainen  John Saunders  Dmitry Zmeev  [  [  [
[  [  [  [

###### Abstract

First order phase transitions in the very early universe are a prediction of
many extensions of the Standard Model of particle physics and could provide
the departure from equilibrium needed for a dynamical explanation of the
baryon asymmetry of the Universe. They could also produce gravitational waves
of a frequency observable by future space-based detectors such as the Laser
Interferometer Space Antenna (LISA). All calculations of the gravitational
wave power spectrum rely on a relativistic version of the classical nucleation
theory of Cahn-Hilliard and Langer, due to Coleman and Linde. The high purity
and precise control of pressure and temperature achievable in the laboratory
made the first-order A to B transition of superfluid 3He ideal for test of
classical nucleation theory. As Leggett and others have noted the theory fails
dramatically. The lifetime of the metastable A phase is measurable, typically
of order minutes to hours, far faster than classical nucleation theory
predicts. If the nucleation of B phase from the supercooled A phase is due to
a new, rapid intrinsic mechanism that would have implications for first-order
cosmological phase transitions as well as predictions for gravitational wave
(GW) production in the early universe. Here we discuss studies of the A-B
phase transition dynamics in 3He, both experimental and theoretical, and show
how the computational technology for cosmological phase transition can be used
to simulate the dynamics of the A-B transition, support the experimental
investigations of the A-B transition in the QUEST-DMC collaboration with the
goal of identifying and quantifying the mechanism(s) responsible for
nucleation of stable phases in ultra-pure metastable quantum phases.

###### keywords:

helium 3; phase transitions; time-dependent Ginzburg-Landau equation;
cosmology; early universe; gravitational waves

##  1 Introduction

First order phase transitions are predicted to occur during cooling in the
early universe as a signal of physics beyond the Standard Model. The
transition is expected to proceed by the nucleation of bubbles of the stable
phase by quantum or thermal fluctuations after supercoooling. These bubbles
grow rapidly due to the pressure difference between inside and outside, and
subsequently merge. The “fizz” generates pressure waves, which on collision
produce shear stresses with non-vanishing quadrupole moment, and hence
gravitational waves [1, 2] (see [3] for a review).

The power spectrum of the gravitational waves depends on a relatively small
number of equilibrium and near-equilibrium properties of the phase transition,
at least in the case where the super-cooled transition happens fairly close to
the critical temperature. Two of the most important are the temperature at
which bubbles nucleate, and the duration of the transition. Both can be
computed from the bubble nucleation rate per unit volume as a function of
temperature. In the standard approach, the nucleation rate density is computed
in homogeneous nucleation theory [4, 5] adapted to relativistic quantum field
theory [6, 7].

Given the importance of cosmological nucleation theory for the prediction of
GWs in the early universe, and its close relation to nucleation theory
developed for laboratory systems, it is important to test the theory against
experiment. Bubble nucleation observed in nature and in standard laboratory
systems usually happens around seeds: small particles in suspension or surface
irregularities. Testing nucleation theory in the bulk requires extremely pure
systems in containers with very smooth walls. Thus, the ideal system for such
a test is superfluid 3He, which in zero magnetic field has two phases
distinguished by their residual symmetries, and a first order transition
between them. The transition from a normal Fermi liquid to the superfluid
takes place between 1–2.5 mK, at which temperatures superfluid 3He is
essentially pure. Even 4He is insoluble in the limit $T\rightarrow 0$, with a
solubility of order $X_{4}\sim 10^{-6}$ at $T=2\,\mbox{mK}$ [8].

Figure 1: Phase diagram of bulk 3He at low temperatures and zero magnetic
field.

At zero magnetic field, A phase is stable in a small wedge in the plane of
temperature $T$ and pressure $P$ below the superfluid critical temperature
(see Fig. 1), but above the polycritical pressure $P_{\text{PCP}}\simeq 21$
bar. As the system is cooled further, A phase becomes metastable below a
temperature $T_{\text{AB}}$. Homogeneous nucleation theory predicts that the
lifetime of the A phase at any temperature or pressure is enormous, far longer
than the age of the universe. Yet the transition is observed to happen within
in a few hours, even in cells with smooth walls [9, 10]. Thus homogeneous
nucleation theory fails dramatically, raising questions as to its extension to
cosmological phase transitions.

The QUEST-DMC (Quantum-Enhanced Superfluid Technologies for Dark Matter and
Cosmology) collaboration was set up under UK Quantum Technology for
Fundamental Physics programme to tackle the nucleation problem using new
techniques, both experimental and theoretical, which have become available
since the experiments performed in the 1990s. In this article we report on
progress in building simulation code to investigate the dynamics of the order
parameter of the system, using established field theory of superfluid 3He, and
tools and techniques borrowed from cosmological simulations of phase
transitions (see e.g. Ref. [11] on time-dependent Ginzburg-Landau (TDGL)
theory, Ref. [12] on cosmological phase transitions, and Ref. [13, 14] on
topological defects.

##  2 Cosmological phase transitions and GWs

At early times in its evolution the Universe was very close to thermal
equilibrium, as the perfect blackbody spectrum of the Cosmic Microwave
Background radiation demonstrates [15]. The earlier the time, the higher the
temperature, and at very high temperatures the state of matter in the early
Universe must change. At temperatures (thermal energies) higher than around
100 MeV, reached at about 0.1 ms after the Big Bang, the theory of the strong
interactions - Quantum Chromodynamics (QCD) - predicts that quarks and gluons
inside nucleons are liberated. At still higher temperatures, around 100 GeV,
the average value of the Higgs field is predicted to vanish. Elementary
particles are then massless, and the electroweak symmetry between the photons,
W and Z bosons restored.

If an early Universe phase transition were of first order the consequences
would be very interesting. The development of homogeneous nucleation theory in
quantum field theory [6, 7], combined with relativistic combustion theory [16,
17] builds a picture of a transition proceeding by nucleation of bubbles of
the low temperature phase by thermal or quantum fluctuations, followed by
rapid expansion, which transfers some of the latent heat of the system into
motion of the plasma. The resulting shear stresses generate gravitational
waves [1, 2]. One can estimate that these gravitational waves would be of
frequencies observable by pulsar timing arrays (QCD transition, nHz) and
space-based interferometers with million-km baselines (electroweak transition,
mHz). Another important consequence of a first order electroweak phase
transition is that it supplies one of the prerequisites for the dynamical
generation of the baryon (matter-antimatter) asymmetry in the Universe: namely
departure from thermal equilibrium [18, 19].

The first calculations of the free energy of the Standard Model (SM) [20, 21]
indicated that gauge field theories like the electroweak theory have a first
order transition, albeit with rather small latent heat [22]. Early
calculations with lattice quantum field theory also showed that gluons confine
in a first order phase transition [23, 24].

Further investigation of the Standard Model transitions has shown that they
are both cross-overs [25, 26, 27]. All thermodynamic quantities evolve
smoothly with temperature, and while there are peaks in various
susceptibilities, there is no possibility that the Universe became stuck in a
metastable state. The Universe of the Standard Model stays very close to
thermal equilibrium, as the particle scattering rate is many orders of
magnitude higher than the rate of change of temperature due to expansion.

This is perhaps disappointing. However, the majority of particle physicists
are convinced that the Standard Model is not the ultimate description of
matter and interactions. Apart from lacking a description of gravity, the SM
has no explanation for dark matter or the baryon asymmetry (see e.g. [28, 29]
for pedagogical reviews). Moreover, calculations of the vacuum fluctuations of
the Standard Model particles indicate that the current magnitude of the Higgs
field, 174 GeV, is not the lowest energy state [30, 31]. A related puzzle is
what determines the magnitude of the Higgs field, and why it is so different
from the fundamental mass scale set by gravity, $10^{19}$ GeV.

The problems with the Standard Model motivate extending it. The study of
extensions to the Standard Model is known as BSM (beyond the Standard Model)
physics and it emerges that a first order phase transition in the early
Universe is very much a possibility in Standard Model extensions [32]. The
search for gravitational waves from the early Universe then becomes a search
for BSM physics. There follow some major questions: how to observe these
gravitational waves, and how to calculate their spectrum. Both have become
very active areas in the intersection between particle physics and cosmology,
and now ultra-low temperature physics.

The simulations and modelling of first order transitions in the early Universe
(see [3] for a review) have shown that the gravitational wave spectrum depends
mainly on a handful of thermodynamic quantities: the critical temperature, the
transition rate, the latent heat, the phase boundary terminal velocity, and
the sound speeds in the two phases. The most difficult to calculate accurately
are the transition rate $\beta$ (the inverse lifetime of the metastable phase)
and the phase boundary speed $v_{\text{w}}$, as they are both non-equilibrium
quantities. The amplitude and shape of the gravitational wave power spectrum
are quite sensitive to these parameters. For example, the peak frequency is
proportional to the ratio $v_{\text{w}}/\beta$ and the transition temperature.

Calculations of the transition rate $\beta$ are based on the homogeneous
nucleation theory of Langer [5], a formalisation of the Cahn-Hilliard theory
[4]. It was introduced into quantum field theory at zero temperature by
Coleman [6], and at non-zero temperature by Linde [7] (see [33] for a
discussion of the theory). There has been recent progress with perturbative
calculations of the rate parameter $\beta$ [34], and it has also been
calculated non-perturbatively using numerical lattice simulations of Standard
Model-like gauge-Higgs system [35, 36]. Calculations of the wall speed are
based on modelling of the plasma in terms of quasi-stable particles and the
Boltzmann equation [37, 38]. Here, too, progress can be made with numerical
methods [39, 40]. Both of these frameworks have direct analogies in superfluid
3He, which offers the opportunity to test and further develop the theory
behind the gravitational wave calculations.

##  3 The A-B nucleation puzzle

According to the homogeneous nucleation theory of Cahn and Hilliard [4] and
Langer [5], a metastable system makes the transition to the stable state with
lower free energy via in small region, which is nevertheless large enough for
the pressure difference between the interior and exterior to overcome the
surface tension of the boundary between. In the bulk, such a region with
lowest energy is spherical: the critical droplet or bubble. The critical
bubble provides the route through the space of order parameter configurations
to the ground state.

The thermal activation rate per unit volume is

$\Gamma(T,P)=\lambda_{\text{a}}n_{\text{a}}e^{-E_{\text{c}}/k_{\text{B}}T},$
(1)

where $\lambda_{\text{a}}$ is the attempt frequency, the rate at which the
system tries to get over the barrier between the metastable state and ground
state, $n_{\text{a}}$ is the number density of regions in which an attempt can
be made, $E_{\text{c}}$ is the energy barrier, and $T$ is the bath
temperature. The attempt frequency and density are set by microscopic dynamics
of the order parameter and are difficult to calculate. This calculation has
recently been automated for a single-component order parameter [41], but the
rate evaluation for the 18-component order parameter of 3He is far more of a
challenging. However, the pair correlation length, $\xi$, provides an estimate
for the maximum density, $n_{\text{a}}\sim\xi^{-3}$. The attempt rate is
governed by a combination of inertial dynamics and diffusion over the barrier
[5, 42, 33].

In the case of superfluid 3He, the two phases in question are the A and B
phases separated by the first-order transition line $T_{AB}(p)$ (see Fig. 1).
In zero magnetic field, above the polycritical point pressure
$P_{\text{PCP}}\simeq 21$ bar, the A phase is stable below the superfluid
critical temperature $T_{\text{c}}$ and $T_{\text{AB}}$, where the B phase
takes over as the phase with lower free energy. Both phases belong to the
spin-triplet ($S=1$), p-wave ($L=1$) manifold of pairing states defined by the
macroscopic amplitude of fermion pairs,
$\langle\psi_{\textbf{p},a}\psi_{-\textbf{p},b}\rangle$, where p is the
relative momentum of the pair of orbiting 3He fermions while $a$ and $b$ are
the spin projections ($\uparrow$ or $\downarrow$) of the fermions comprising
the pair. The corresponding mean-field pairing self energy,
$\Delta_{ab}(\textbf{p})=g\,\langle\psi_{\textbf{p},a}\psi_{-\textbf{p},b}\rangle$,
where $g$ is the attractive pairing interaction in the spin-triplet, p-wave
Cooper channel. These amplitudes are the elements of a symmetric $2\times 2$
matrix order parameter,
$\widehat{\Delta}(\textbf{p})=i\vec{\sigma}\sigma_{y}\cdot\vec{d}(\textbf{p})$,
where $i\vec{\sigma}\sigma_{y}$ are the symmetric Pauli matrices, and the
three-component spin vector, $\vec{d}(\textbf{p})$, is in general a linear
superposition of the p-wave basis functions for momenta restricted to the
Fermi surface. Thus,

$\widehat{\Delta}(\textbf{p})=(i\sigma_{\alpha}\sigma_{y})\,A_{\alpha
i}\,(\hat{\textbf{p}})_{i}\,,$ (2)

is parametrized by a $3\times 3$ complex matrix, $A_{\alpha i}$, that
transforms as a vector with respect to index $\alpha$ under spin rotations,
and, separately, as a vector with respect to index $i$ under orbital
rotations.111We follow the notation of Ref. [43] for the form of the spin-
triplet, p-wave order parameter and Ginzburg-Landau (GL) functional. See also
Vollhardt and Wölfle [44] for a pedagogical development of the same. This
representation for the order parameter provides us with an $S=1$, $L=1$ basis
for an irreducible representation of the maximal symmetry group of normal 3He,

$G=SO(3)_{\mathbf{L}}\times SO(3)_{\mathbf{S}}\times U(1)_{\mathbf{N}}\times
T\times C\times P,$ (3)

which includes $SO(3)_{\mathbf{L}}$ rotations in three-dimensional space,
$SO(3)_{\mathbf{S}}$ rotations in the spin space, and the global phase
transformation group $U(1)_{\mathbf{N}}$, as well as discrete symmetries $T$,
$C$ and $P$, where $T$ is time-reversal symmetry, $C$ is the particle-hole
symmetry, and $P$ is parity symmetry. The subscripts $\mathbf{L}$,
$\mathbf{S}$ refer to the generators for the rotation groups while
$\mathbf{N}$ is the number operator which is the generator for changes in
phase.

The bulk B-phase, which minimizes the bulk free energy over most of the
pressure-temperature plane below $T_{c}$, is the “isotropic” state defined by

$A_{\alpha i}^{\text{B}}=\frac{1}{\sqrt{3}}\Delta_{\text{B}}\,\delta_{\alpha
i}\,,$ (4)

with residual symmetry $H_{\text{B}}=SO(3)_{\mathbf{L}+\mathbf{S}}\times T$,
i.e. the B-phase is time-reversal invariant and invariant under joint
rotations of spin and orbital components of the order parameter. This state
was shown by Balian and Werthamer to be the ground state for spin-triplet,
p-wave pairing in the weak-coupling limit [45]. The high degree of symmetry of
the B-phase implies a large continuous degeneracy space,

$R_{\text{B}}=U(1)_{\text{N}}\times SO(3)_{\mathbf{L}-\mathbf{S}}\,,$ (5)

corresponding to the choice of phase of the order parameter defined by the
elements of $U(1)_{\text{N}}$, as well as the relative orientation of the spin
and orbital coordinates of the spin-triplet, p-wave Cooper pairs, represented
by $SO(3)_{\mathbf{L}-\mathbf{S}}$. As a result the class of degenerate
B-phase order parameters is,

$A_{\alpha
i}^{\text{B}}=\frac{1}{\sqrt{3}}\Delta_{\text{B}}\,e^{i\varphi}\,R_{\alpha
i}[\vartheta\hat{\textbf{n}}]\,,$ (6)

where $\varphi$ is the global phase and $R_{\alpha
i}[\vartheta\hat{\textbf{n}}]$ is an orthogonal matrix defining a rotation of
the spin and orbital coordinates by angle $\vartheta$ about the direction
$\hat{\textbf{n}}$. Thus, there are 4 continuous degeneracy parameters, and
hence 4 gapless Nambu-Goldstone (NG) modes. The phase mode is realized as
collisionless sound in superfluid 3He-B and plays a central role in
observations of the spectrum of Higgs modes [46, 47]. There are three spin-
orbit modes that are key signatures of spin-triplet pairing in NMR
spectroscopy of 3He [48]. Nuclear dipolar interactions and Zeeman energies in
an external magnetic field partially lift the degeneracy of these modes,
opening small gaps providing a novel example of the Light Higgs scenario [49].

The stability of the A-phase at high pressure and temperatures relatively near
to $T_{c}$ results from corrections to weak-coupling BCS theory that become
sufficiently large at high pressures to stabilize an equal-spin pairing (ESP)
state that also spontaneously breaks time-reversal symmetry with an order
parameter of the form,

$A_{\alpha
i}^{\text{A}}=\Delta_{\text{A}}\,\hat{\textbf{d}}_{\alpha}\,\left(\hat{\textbf{m}}_{i}+i\,\hat{\textbf{n}}_{i}\right)/\sqrt{2}\,,$
(7)

where $\hat{\textbf{d}}$ is a real unit vector in spin space along which the
A-phase Cooper pairs have zero spin projection. The orthonormal unit vectors
$\hat{\textbf{m}}$ and $\hat{\textbf{n}}$ combined with the relative phase of
$\pi/2$ define orbital motion of the A-phase Cooper pairs with orbital angular
momentum $+\hbar$ per Cooper pair along the axis
$\hat{\textbf{l}}=\hat{\textbf{m}}\times\hat{\textbf{n}}$. This axis is chiral
and it highlights both broken mirror symmetry and broken time-reversal
symmetry by the A-phase. The latter allows for a macroscopic ground state
angular momentum predicted to be, $\mathcal{L}_{z}={N}\hbar/{2}$ for a system
with $N$ 3He atoms. 222See Ref. [50] for a discussion of the connection
between the topological edge states, the edge currents and $\mathcal{L}_{z}$
as well as review of theoretical literature on the ground state angular
momentum. The corresponding residual symmetry group is then,
$H_{\text{A}}=SO(2)_{\textbf{d}}\times U(1)_{\mathbf{L}_{z}-\mathbf{N}}\times
Z_{2}$, where $SO(2)_{\textbf{d}}$ is the group of rotations in spin space
about the axis $\hat{\textbf{d}}$. The A phase breaks orbital rotation
symmetry as well as global gauge symmetry, however a rotation by any angle
about the chiral axis can be undone with an appropriately chose element of
$U(1)_{\mathbf{N}}$ which leads to the residual gauge-rotation symmetry of the
A-phase defined by $U(1)_{\mathbf{L}_{z}-\mathbf{N}}$. The residual discrete
$Z_{2}$ symmetry results from the combination of time-reversal and mirror
reflection in a plane containing the chiral axis. This symmetry allows for the
remarkable transport properties of the A phase including the anomalous Hall
effect that was reported for electrons moving in 3He-A driven by an electric
field perpendicular to the chiral axis [51, 52].

The A phase is also endowed with a large degeneracy space, in this case by the
combined degeneracy in the orientation of the spin direction
$\hat{\textbf{d}}$ on the surface of a unit sphere, $S^{2}$, and the
orientation of the orbital triad,
$\\{\hat{\textbf{m}},\hat{\textbf{n}},\hat{\textbf{l}}\\}$, which is the group
of rotations in $3$-space, $SO(3)$. However, the combined transformations:
$\hat{\textbf{d}}\rightarrow-\hat{\textbf{d}}$ and
$\left(\hat{\textbf{m}}+i\,\hat{\textbf{n}}\right)\rightarrow-\left(\hat{\textbf{m}}+i\,\hat{\textbf{n}}\right)$
is a discrete symmetry of the A-phase ($Z_{2}^{{}^{\prime}}$), and thus the
degeneracy space excludes these combined changes of sign such that [53]

$R_{\text{A}}=S^{2}\times SO(3)/Z_{2}^{{}^{\prime}}\,.$ (8)

The continuous degeneracy space implies the existence of 5 Nambu-Goldstone
modes, 2 spin wave modes, 2 orbital wave modes and the sound mode. It is also
worth noting that the $Z_{2}^{{}^{\prime}}$ symmetry is directly related to
topologically stable half quantum vortices originally predicted by Volovik and
Mineev for ESP states [54, 55], and which were recently discovered in NMR
spectroscopy of the ESP polar phase of 3He under rotation [56].

In simulating nonequilibrium dynamics and nucleation processes the low energy
excitations - Goldstone and pseudo-Goldstone modes - are expected to play an
important role in transporting mass, energy and magnetization. Another notable
fact is that the residual symmetry group of the B phase is not a sub-group of
$H_{\text{A}}$, and thus the phase transition is necessarily first order.
Thus, to nucleate the B phase from the homogeneous A-phase requires deviations
or fluctuations of the order parameter from the local equilibrium A phase that
incur an energy barrier, inhibiting nucleation and allowing for supercooling
of the A phase below $T_{\text{AB}}$. Cooling below $T_{\text{AB}}$ at
pressures above $P_{\text{PCP}}$ puts the superfluid into a metastable state
with free energy excess $\Delta f_{\text{AB}}=f_{\text{A}}-f_{\text{B}}$,
where $f_{\text{A}}$ and $f_{\text{B}}$ are the condensation energy densities
of the A- and B-phases, which can be determined by integrating the measured
specific heats from $T_{c}$ to the relevant temperature below $T_{c}$.

The path in order parameter space that minimizes the energy cost of a domain
wall (DW) separating the A and B phases, $A_{\alpha i}^{\text{A}}\rightarrow
A_{\alpha i}^{\text{DW}}(x)\rightarrow A_{\alpha i}^{\text{B}}$, for a bubble
of radius $R$ of 3He-B embedded in 3He-A plays a key role in the theory of
nucleation of the -phase in supercooled A phase. The surface energy of the A-B
interface was measured at high pressure by Osheroff and Cross [57], and at low
pressure and high magnetic field by Bartkowiak et al [58]. Theoretical
calculations using GL theory [59, 60, 61] give $\sigma_{\text{AB}}\propto\xi
f_{\text{B}}$, where $\xi$ is the Ginzburg-Landau coherence length, with
proportionality constant close to 1 in good agreement with experiment.

To estimate the energy and radius of the critical bubble in the thin-wall
approximation ($R\gg\xi$) we express the total energy of a bubble of B-phase
embedded in metastable A-phase as a sum of the gain in condensation energy
proportional to $-\Delta f_{\text{AB}}$, and the cost in surface energy,
proportional to $\sigma_{\text{AB}}$. Both depend on temperature and pressure.
Thus,

$E(R)=4\pi R^{2}\sigma_{\text{AB}}-\frac{4\pi}{3}R^{3}\Delta f_{\text{AB}}\,.$
(9)

The critical bubble is determined by the condition $E^{\prime}(R)=0$,
representing a spherical bubble poised between expansion and contraction. The
radius and energy of the critical bubble are then

$R_{\text{c}}=\frac{2\sigma_{\text{AB}}}{\Delta f_{\text{AB}}},\quad
E_{\text{c}}=\frac{16\pi}{3}\frac{\sigma_{\text{AB}}^{3}}{\Delta
f_{\text{AB}}^{2}},$ (10)

and the critical bubble free energy is

$E_{\text{c}}\simeq\epsilon|f_{\text{B}}|\xi^{3}\left(\Delta
f_{\text{AB}}/|f_{\text{B}}|\right)^{-2},$ (11)

where $\epsilon\simeq 10$. The Ginzburg-Landau coherence length near the
critical temperature is $\xi=\xi_{\text{GL}}(1-T/T_{\text{c}})^{-1/2}$, where
$\xi_{\text{GL}}=\sqrt{7\zeta(3)/20}\,\xi_{0}$ and $\xi_{0}=\hbar
v_{\text{F}}/2\pi k_{\text{B}}T_{\text{c}}$ is the Cooper pair correlation
length in the ballistic limit [62]. The order of magnitude of the condensation
energy density is set by the density of states at the Fermi surface
$N(0)=m^{*}k_{f}/(2\pi^{2}\hbar^{2})$ and the critical temperature according
to
$f_{\text{B}}\sim-N(0)(k_{\text{B}}T_{\text{c}})^{2}(1-T/T_{\text{c}})^{2}$.
Hence

$E_{\text{c}}/k_{\text{B}}T\sim\text{Gi}\,({T_{\text{c}}}/{T})\left(1-T/T_{\text{c}}\right)^{1/2}\left(\Delta
f_{\text{AB}}/|f_{\text{B}}|\right)^{-2}.$ (12)

where $\text{Gi}=N(0)\xi_{0}^{3}(k_{\text{B}}T_{\text{c}})$ is the Ginzburg
number, which takes values between 800 at zero pressure and 2900 at melting
pressure, about $34$ bar. A plausible estimate for the order of magnitude of
the length scale in the attempt density is the Cooper pair correlation length
$\xi_{0}$, which takes values in the range 16 nm to 77 nm. An upper bound the
attempt rate is $v_{\text{F}}/\xi_{0}$, where the Fermi velocity
$v_{\text{F}}$ is in the range $30\,\text{m}\,\text{s}^{-1}$ to
$60\,\text{m}\,\text{s}^{-1}$.

The last factor in Eq. (12) diverges as $(1-T/T_{\text{AB}})^{-2}$ as the A-B
equilibrium line is approached, but even without the divergent factor the size
of the Ginzburg number already ensures that the exponential
$e^{-E_{c}/k_{\text{B}}T}$ in the nucleation rate, Eq. (1), overwhelms the
attempt frequency in an experimentally accessible volume, leading to an
estimate of the lifetime of the metastable A-phase that vastly exceeds the
current age of the universe. Thus, as pointed out early after the discovery of
the superfluid phases [59, 63] classical nucleation theory predicts that the
superfluid 3He should remain in the metastable A-phase indefinitely in any
experiment that can be realistically conceived. However, experimental
investigations of supercooled 3He-A all show that the B phase nucleates on
timescales of seconds to hours suggesting another mechanism is responsible for
the nucleation of 3He-B [64, 65, 9, 10, 10, 66, 67, 68, 69].

##  4  Explanations for the nucleation puzzle

Proposed explanations for the puzzle are many and varied [63, 70, 71, 66, 72,
73, 74, 75, 69]. The leading contenders consider that nucleation in bulk,
metastable superfluid 3He-A is caused by energy injection by cosmic-ray muons
or another energetic particle. In the “Baked Alaska” scenario proposed by
Leggett [63] the energy deposition breaks Cooper pairs creating a local region
of “hot” quasiparticles surrounded by the cold metastable A phase. A shell of
energetic quasiparticles expands, driving the system locally normal, behind
which the superfluid returns to a temperature below $T_{c}$, allowing the
B-phase to nucleate with measurable probability. At this point the size of the
shell must be larger than the critical bubble size, otherwise the surface
tension of the A-B phase boundary will overwhelm the pressure difference and
the B-phase bubble will collapse. Support for such a scenario of local heating
nucleating the B-phase is reported by Schiffer et al. where it was shown that
764 keV neutrons as well as MeV $\gamma$ rays from 60Co stimulate the A-B
transition [9].

A particularly interesting consequence of a second-order symmetry-breaking
phase transition such as the normal-superfluid transition in 3He, was pointed
out by Kibble in the context of phase transitions in the early Universe: the
generation of topological defects [76]. He showed how to predict the type of
defect on the basis of the topology of the manifold of equilibrium states and
the defect density on the basis of estimates of the correlation length as the
universe cools through a continuous phase transition. The density estimate was
later updated by Zurek who made the explicit link to defect formation in rapid
quenches in superfluid 4He [77, 78]. Experiments to investigate vortex
formation in superfluid 4He [79, 80] proved inconclusive, but experiments in
the B phase of superfluid 3He [81, 82] demonstrated spontaneous vortex
generation in rapid quenches consistent with the Kibble-Zurek mechanism.

The “Cosmological Scenario” for A-B nucleation proposed by Volovik and Kibble
[72, 83] envisages that energy is transported rapidly out of the injection
region by thermal diffusion, and that the front where the quasiparticle
temperature goes below $T_{\text{c}}$ is swiftly followed by another front
where it goes below $T_{\text{AB}}$. The rapidity of the quench inside the
energy deposition region suggests that it contains causally disconnected
regions of local order, which evolve either into the A or B phases according
to the Kibble-Zurek scenario. In this case a complex region of multiple
phases, separated by domain walls, emerges. If a large enough B phase region
has formed, it will expand and eventually take over the condensate. Moreover,
different types of topological defects such as domain walls between B-phases
[84], as well as vortices can be generated in the energy deposition region.
Indeed vortices have been detected in the Helsinki group’s experiments
injecting energy with neutrons [82]. There has been discussion between the
authors of the competing models [85, 86], but as yet no consensus.

It has also been pointed out that nucleation may be seeded by complex order
parameter configurations at the boundaries of an experimental cell, either by
surface roughness, or by topological singularities [87, 71]. Nucleation at
rough boundaries has been argued to fit the data of Hakonen et al. [65, 74],
where nucleation occurred close to a “catastrophe” line in the $(T,P)$ plane
with a characteristically peaked temperature distribution. The catastrophe
line would be different for each experiment, as it would depend on the details
of the boundaries, particularly in the heat exchanger, where complex surfaces
of many square metres in area are found. Recently, it has been shown that the
position of the catastrophe line also depends on the path in the $(T,P)$ plane
taken when cooling through the A phase into the metastability region [69].
This was explained in terms of a model of complex order parameter
configurations acting as B-phase seeds in small cavities, principally the heat
exchanger.

More exotic explanations have been put forward. Non-topological order
parameter configurations known as Q-balls [88] have been proposed as an
alternative to the critical bubble as the route from the A phase to the B
phase [70]. Such objects have been experimentally detected and studied [89].
It has also been proposed that resonant tunnelling, a quantum-mechanical
phenomenon where quantum tunnelling can proceed via a classically allowed
intermediate state of the same energy, could be at play in superfluid 3He as
well [75]. However, such classically allowed states were shown not to exist in
the quantum field theory of a single scalar field [90], so the existence of
resonant tunnelling for the multicomponent order parameter of 3He is not
clear.

##  5 New experiments on A-B nucleation

In order to study nucleation in the bulk superfluid, one would like to
eliminate or control the effect of the container walls. The first imperative
is to isolate the metastable superfluid from the heat exchanger, which
contains a large rough area in contact with the superfluid. This can be done
in two ways. One method is to utilize the Zeeman energy of the ESP A phase in
which case its equilibrium bulk free energy is reduced in a magnetic field.
Above about 0.6 T, the A phase is the stable superfluid phase over the whole
$(T,P)$ plane [91, 92]. By placing two opposing magnets close to each other,
it is possible to create a region of low magnetic field, where the superfluid
is in the metastable A phase, surrounded by higher field, where the A phase is
stable [93]. In this configuration the phase transition happens in the low-
field region, well away from the walls of the container.

A new set of experiments use engineered cells and surfaces to confine 3He into
multiple chambers of different heights within a single experimental cell to
study A-B nucleation. Container walls generally lead to pair breaking and
distortion of the order parameter near the wall. The magnitude of pair-
breaking depends on the atomic scale properties of the wall [94, 95, 96]. In
general pairing of states with orbital angular momentum in the plane of a
surface are suppressed. If the wall is smooth, e.g. by pre-plating with
superfluid 4He, quasiparticles reflect specularly, and the in-plane orbital
states are unaffected. This is the case for the A phase with the chiral axis
aligned normal to the wall; pair-breaking is suppressed and thus the the
A-phase order parameter survives all the way to the wall. The B phase, on the
other hand, is modified by pair-breaking of the orbital component normal to
the wall. Its order parameter is distorted towards the planar phase at the
wall. The planar distorted B phase extends into the bulk over a distance of a
few coherence lengths. In the case of an atomically rough surface pair-
breaking occurs for all orbital components, in which case both the A and B
phase are suppressed near the wall by diffuse scattering. Thus, the ideal
geometry is a slab that stabilizes the A phase with minimal pair-breaking,
i.e. by 4He pre-plating, below $T_{c}$, but is thick enough to support the
B-phase at lower temperatures as shown for example in Fig. 3 of Ref. [96].
Indeed, experiments in thin nanofabricated cavities show that A phase can be
stabilised at any pressure over a range of temperatures [97, 98], and is the
stable superfluid phase that onsets at the critical temperature.

These facts motivated the construction of the experimental cell used in the
QUEST-DMC experiments which consists of 5 superfluid “lakes” of 3He, each
being approximately 7 $\mu$m in depth, surrounded by shallower regions of
$\sim$70 nm, in which the 3He is forced to be either in the normal phase
(diffuse scattering) or only the A phase (specular scattering). This design
allows for the study of the A-B transition in multiple regions of metastable A
phase in lakes of various volumes. The results of the intital studies in this
geometry are reported elsewhere [99]. The ultimate goal is study intrinsic A-B
nucleation by understanding, controlling and potentially eliminating extrinsic
nucleation related to boundaries, defects, and particles depositing energy in
3He.

##  6  Simulations of nonequilibrium phase transitions

In order to develop a deeper understanding of nucleation mechanisms for phase
transitions, the QUEST-DMC theory effort is developing computational tools to
simulate nonequilibrium dynamics for the A-B transition of metastable 3He-A.
This is a technically challenging problem involving the dynamics of a bosonic
field defined on a multi-dimensional order parameter space coupled to
excitations of the underlying fermionic vacuum. In many cases the dynamics
occurs under conditions that are far from equilibrium, particularly for
nucleation generated by localized energy deposition. Here we discuss
simulations based on dynamics described by TDGL theory based on the extension
of the strong-coupling GL functional by Wiman and Sauls [100, 62] that
captures the A-B transition and extends the GL theory to temperatures below
$T_{\text{AB}}$ [101].

###  6.1 Time-dependent Ginzburg-Landau theory

Time-dependent Ginzburg-Landau (TDGL) equations have long been studied in the
context of the nonequilibrium dynamics of superconductors, particularly for
superconductors in the “dirty” limit, $\hbar/\tau\gg\Delta$, where $\tau$ is
the mean scattering time for unbound fermionic quasiparticles (see e.g.
Kopnin’s review [102]). These equations have been also been studied in the
context of Kibble-Zurek quench dynamics and normal-superfluid boundary
propagation for $U(1)$ superfluids by several authors [103, 104, 105]. TDGL
equations for superconductors and superfluids in the clean limit were
developed early on by Abrahams and Tsuneto for a $U(1)$ superconductor
starting from an expansion of the nonequilibrium mean-field equations for the
order parameter [106] (see also Ref. [107]), and for superfluid 3He by
Kleinert [108]. Below we formulate the dynamics as a bosonic field theory for
superfluid 3He with dissipation from the excitations of the underlying
fermionic vacuum.

The space-time evolution of the bosonic field describing Cooper pairs in bulk
3He, $A_{\alpha i}(\textbf{r},t)$, is governed by field equations obtained
from the TDGL Lagrangian, $L=K-U$, where

$K=\int dV\,\tau_{0}\,\mbox{Tr}\left(\dot{A}{\dot{A}}^{\dagger}\right)\,,$
(13)

is the kinetic energy associated with temporal fluctuations of the field
$A(\textbf{r},t)$ with $\dot{A}=\partial{A}/\partial t$ and $\tau_{0}$ the
inertia of the field which determines the dispersion of the bosonic modes of
the superfluid phases [11]. The potential energy functional is defined by the
GL free energy functional, which includes a second-order invariant,

$U_{2}[A]=\int dV\,\alpha\,\mbox{Tr}\left(A\,A^{{\dagger}}\right)\,,$ (14)

that controls the phase transition to the broken symmetry phase. For
$\alpha>0$ the equilibrium state is the symmetric normal Fermi-liquid, while
for $\alpha<0$ the equilibrium state spontaneously breaks the symmetry, with a
finite bosonic amplitude $A$. The broken symmetry equilibrium state is
determined by the fourth-order interactions of the bosonic field,

$U_{4}[A]=\int dV\,\sum_{p=1}^{5}\beta_{p\,}\,u_{p}(A)\,,$ (15)

where the five linearly independent fourth-order invariants of the maximal
symmetry group $G$ for 3He in Eq. (3) are

$\displaystyle u_{1}$ $\displaystyle=$
$\displaystyle|\mbox{Tr}\left(AA^{T}\right)|^{2}\,,\quad
u_{2}=\mbox{Tr}\left(AA^{\dagger}\right)^{2}$ $\displaystyle u_{3}$
$\displaystyle=$
$\displaystyle\mbox{Tr}\left(AA^{T}A^{*}A^{\dagger}\right)\,,\quad
u_{4}=\mbox{Tr}\left(AA^{\dagger}AA^{\dagger}\right)\,,\quad
u_{5}=\mbox{Tr}\left(AA^{\dagger}A^{*}A^{T}\right)\,.$ (16)

Spatial gradients of the bosonic field also play a central role in the
dynamics and contribute to the effective potential in the form of
supercurrents, textural bending energies and deformations of the order
parameter near the cores of singular topological defects and boundaries,

$U_{\partial}[A]=\int dV\,\sum_{m=1}^{3}\,K_{m\,}\,v_{m}(\partial A)\,,$ (17)

where the three linearly independent leading order gradient energies are

$\displaystyle v_{1}$ $\displaystyle=$ $\displaystyle\partial_{k}A_{\alpha
j}\partial_{k}A^{*}_{\alpha j}\,,\quad v_{2}=\partial_{j}A_{\alpha
j}\partial_{k}A^{*}_{\alpha k}\,,\quad v_{3}=\partial_{k}A_{\alpha
j}\partial_{j}A^{*}_{\alpha k}\,.$ (18)

Thus, the Lagrangian density for the bosonic fields takes the form,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\tau_{0}\,\mbox{Tr}\left(\dot{A}\dot{A}^{{\dagger}}\right)-\alpha\,\mbox{Tr}\left(A\,A^{{\dagger}}\right)-\sum_{p=1}^{5}\beta_{p\,}u_{p}(A)-\sum_{m=1}^{3}\,K_{m}\,v_{m}(\partial
A)\,,$ (19)

where $\tau_{0}$ is the effective inertia for Cooper pair fluctuations. This
Lagrangian respects the maximal symmetry group $G$ of 3He in Eq. (3). Weak
violation of particle-hole symmetry by the parent Fermi-liquid allows for an
additional invariant in the Lagrangian that is first-order in $\partial_{t}A$,

$K_{\Gamma}=i\Gamma\left[\mbox{Tr}\left(\dot{A}A^{\dagger}\right)-\mbox{Tr}\left(A\dot{A}^{\dagger}\right)\right]\,.$
(20)

However, this C-violating term is expected to be small; thus we neglect it in
the dynamical simulations that follow. In contrast we retain the dissipative
term that is first order in $\partial_{t}A$ that arises from coupling of the
thermal bath of fermionic excitations to nonequilibrium states of the bosonic
field as discussed in Sec. 6.2.

The Lagrangian (19) generates the dynamical equations for the bosonic
excitations of superfluid 3He [11]. The bosonic field theory is significant as
it provides an understanding of fundamental dynamical features resulting from
spontaneous symmetry breaking in condensed matter and quantum field theories.
A good example is Nambu’s fermion-boson mass relations [109] for the broad
class of Nambu/Jona-Lasinio field theories [110, 111], which includes 3He, for
mass generation by spontaneous symmetry breaking [11, 112].

The parameters $\alpha$, $\beta_{p}$ and $K_{m}$ that define the effective
potential are temperature- and pressure-dependent, and can be calculated from
the microscopic theory of superfluid 3He [113, 114, 115, 100, 116], have the
values

$\alpha(T)=\frac{1}{3}N(0)(T/T_{\text{c}}-1),$ (21)
$\beta_{p}=\beta_{0}\left(b^{\text{wc}}_{p}+\frac{T}{T_{\text{c}}}b^{\text{sc}}_{p}\right)\,,\quad
p\in\\{1,\ldots,5\\}$ (22)

where the $\beta_{p}$ parameters in the weak-coupling limit are determined by
pressure-independent ratios and an overall scale set by

$\beta_{0}=\frac{7\zeta(3)}{80\pi^{2}}\frac{N(0)}{3(k_{\text{B}}T_{\text{c}})^{2}},\quad\\{b^{\text{wc}}_{p}\\}=(-1,2,2,2,-2)\,.$
(23)

The strong-coupling corrections to the $\beta_{p}$ parameters,
$b_{p}^{\text{sc}}$, are calculated based on the leading order corrections to
weak-coupling BCS theory as formulated by Rainer and Serene [113]. A key
result is that the strong-coupling corrections to the GL functional are
determined by the scattering amplitude for normal-state quasiparticles with
energies and momenta confined to the Fermi surface. This scattering amplitude
also determines the normal-state thermodynamic and transport properties of the
normal Fermi liquid phase of 3He. This allows us to solve the inverse problem
to determine the scattering amplitude from the experimental data for the
normal Fermi liquid phase of 3He and the heat capacity jumps for the A- and
B-phases at $T_{c}$. This program was carried out and shown to predict the
stability of the A-phase above the polycritical pressure as well as the
temperature dependence of the gap, thermodynamic potential and heat capacity
of the B-phase at low temperatures [114, 115, 116]. The calculated A-B
transition line is in excellent agreement with experimental results as shown
in Fig. 8.6 of Ref. [116].

The other development in strong-coupling theory for 3He is the recognition of
the importance of the temperature-dependent scaling of the strong-coupling
$\beta$ parameters below $T_{c}$ shown in Eq. (22). This scaling is based on
the microscopic strong-coupling theory and developed in Refs. [100, 62]. The
temperature and pressure dependence of the strong-coupling corrections
captures the A-B transition transition line to good accuracy, and extends the
predictive capabilities of the GL theory to temperatures below $T_{\text{AB}}$
[101] for pressures above the polycritical pressure. This is essential for
developing TDGL theory to study order parameter dynamics in the metastable A
phase. In what follows we use the results for $b^{\text{sc}}_{i}(p)$ tabulated
in Ref. [101]; these values are slightly different than the more accurate
results reported in Ref. [116], but both sets are comparable in their
magnitude and pressure dependences.333A review of strong-coupling theory,
including the effective interactions in liquid 3He that give rise to the
stability of 3He-A and deviations from weak-coupling BCS theory for the
thermodynamic properties, will be published in a separate report.

Figure 2: Phase diagram of bulk 3He showing the experimental superfluid
critical temperature $T_{\text{c}}$ (black solid line), the A-B equilibrium
line $T_{\text{AB}}$ (black dashed line) [117] and the theoretical result for
$T_{\text{AB}}$ based on the linear temperature scaling and pressure-
dependence of the strong-coupling $\beta$ parameters below $T_{\text{c}}$
[101].

The spatial derivative terms in Eq. (17) determine the energy cost of
deformations of the order parameter from its homogenous equilibrium value. All
three stiffness coefficients are positive and can be calculated to good
approximation in weak-coupling theory,

$K_{1}=K_{2}=K_{3}=\frac{7\zeta(3)}{60}N(0)\xi_{0}^{2}\,,$ (24)

where $\xi_{0}=\hbar v_{\text{F}}/2\pi k_{B}T_{c}$ is the zero-temperature
Cooper pair correlation length.

The counterpart to the space derivative terms given in Eq. (13) is the kinetic
energy which is second-order in time-derivatives. The stiffness to temporal
fluctuations of the order parameter in the weak-coupling limit is given by

$\tau_{0}=\hbar^{2}\frac{7\zeta(3)}{48}\frac{N(0)}{\pi^{2}k_{\text{B}}^{2}T_{c}^{2}}\,.$
(25)

This result is equal in magnitude to the fourth-order contribution to the
linear combination of $\beta$ parameters that determine the gap amplitude for
the B-phase, i.e. $\tau_{0}=\beta_{\text{B}}=\beta_{12}+\beta_{345}/3$ in
weak-coupling theory. This result guarantees that the $J=0^{+}$ Higgs mode has
the mass of two fermions at the continuum edge,
$M_{0^{+}}=2\Delta_{\text{B}}$. Based on symmetry grounds it is argued that
the relation $\tau_{0}=\beta_{\text{B}}$ is preserved to leading order in
strong-coupling theory in [11].

For static, but in general homogeneous superfluid phases, the effective
potential, $U_{2}+U_{4}+U_{\partial}$, defines the GL free energy functional,
$\mathcal{F}_{\text{GL}}[A]=\int dV\,f_{s}[A]$, with

$f_{s}[A]=\alpha\,\mbox{Tr}\left(A\,A^{{\dagger}}\right)+\sum_{p=1}^{5}\beta_{p\,}u_{p}(A)+\sum_{m=1}^{3}\,K_{m}\,v_{m}(\partial
A)\,.$ (26)

The GL functional is supplemented by boundary conditions for the order
parameter $A$ that depend on the geometry and atomic scale properties of the
boundary. In the simulations discussed later in this report we compare energy
densites relative to that of the homogeneous B-phase,
$f_{\text{B}}=-\alpha^{2}/4\beta_{\text{B}}$ and we normalize to the energy
density scale

$f_{0}=\frac{1}{3}N(0)(k_{\text{B}}T_{c})^{2}.$ (27)

It is worth noting that our formulation of TDGL theory for the space-time
dynamics of pure spin-triplet, p-wave superfluid 3He can be extended to
include attractive, but sub-dominant, spin-triplet, f-wave Bosonic
excitations, including the predicted $S=1$, $L=3$, $J=4^{-}$ Higgs mode [118]
for which there is experimental evidence from acoustic Faraday rotation of
transverse sound [119].

The general form of the $S=1$, $L=3$ order parameter is
$F_{\alpha;ijk}(\textbf{r},t)$ which transforms as a vector under $SO(3)_{S}$
for the index $\alpha$ and as a rank 3 symmetric, traceless tensor under
orbital rotations ($SO(3)_{L}$) for the indices $i,j,k$. The leading order
contribution to the effective potential is then
$\alpha_{f}(T)\,F_{\alpha;ijk}F^{*}_{\alpha;ijk}$, where
$\alpha_{f}(T)=\frac{1}{3}\,N(0)\,\left(T/T_{c_{f}}-1\right)$ where
$T_{c_{f}}$ is the f-wave pairing instability temperature which is a direct
measure of the f-wave pairing interaction. For sub-dominant f-wave pairing we
have $0<T_{c_{f}}<T_{c}$. There are many new invariants that contribute the
extended TDGL functional which can be enumerated using group representation
theory. Whether or not there is an f-wave condensate depends on the material
parameters of the new invariants. It may be possible that such a condensate
exists in the cores of o-vortices in the B phase, analogous to the p-wave
condensate in the core of an Abrikosov (s-wave) vortex [55].

Finally we note that symmetry breaking perturbations from the nuclear Zeeman
energy in an external magnetic field and the nuclear magnetic dipole-dipole
energy can be included in this framework, c.f. Refs. [120, 100], but here we
focus on A-B transition in the absence of magnetic field and neglect the weak
nuclear dipole-dipole energy. These effects of these perturbations on A-B
transition will be discussed in a future work.

###  6.2 TDGL equations with dissipation by the fermionic bath

The Euler-Lagrange equations obtained from Eq. (19) generate the non-
dissipative coupled dynamical equations for the $3\times 3$ complex matrix
order parameter, $A_{\alpha i}(\textbf{r},t)$, for superfluid 3He,

$\begin{array}[]{rcl}\tau_{0}\ddot{A}_{\alpha i}&+&\alpha A_{\alpha
i}-K_{1}\partial^{2}A_{\alpha
i}-(K_{2}+K_{3})\partial_{i}\partial_{j}A_{\alpha j}\\\
&&+2\left[\beta_{1}A^{*}_{\alpha
i}\mbox{Tr}\left(AA^{T}\right)+\beta_{2}A_{\alpha
i}\mbox{Tr}\left(AA^{\dagger}\right)\right.\\\
&&\qquad+\left.\beta_{3}(AA^{T}A^{*})_{\alpha
i}+\beta_{4}(AA^{\dagger}A)_{\alpha i}+\beta_{5}(A^{*}A^{T}A)_{\alpha
i}\right]=0\,.\end{array}$ (28)

These coupled equations involve only the bosonic degrees of freedom with
parameters corresponding to the fermionic vacuum in local equilibrium. We
include thermal fluctuations of the fermionic vacuum via a Langevin source
term that couples the thermal fluctuations locally to the bosonic field.
Several authors have formulated the dynamics of a bosonic field theory coupled
to a thermal bath in terms of a stochastic Langevin source [121, 103]. We add
to the right-hand side of Eqs. (28) a Gaussian noise source $\zeta_{\alpha
i}(\textbf{r},t)$ with intermediate-time averages $\langle\zeta_{\alpha
i}(\textbf{r},t)\rangle=0$ and 444Our formulation is similar that of Ref.
[121].

$\langle\zeta_{\alpha i}(\textbf{r},t)\zeta_{\beta
j}(\textbf{r}^{\prime},t^{\prime})\rangle=2\gamma\,k_{\text{B}}T\,\delta_{\alpha\beta}\delta_{ij}\delta(\textbf{r}-\textbf{r}^{\prime})\delta(t-t^{\prime})\,.$
(29)

The parameter $\gamma$ plays an important role as it leads, via the
fluctuation-dissipation theorem, to damping of space-time fluctuations of the
bosonic field via an additional dissipative time-derivative term,
$\gamma\dot{A}_{\alpha i}$, on the left-side of Eq. (28) that is
characteristic of Langevin dynamics. Thus, the set of dynamical equations
including the dampling and Langevin noise source terms are,

$\begin{array}[]{rcl}\tau_{0}\ddot{A}_{\alpha i}&+&\gamma\dot{A}_{\alpha
i}+\alpha A_{\alpha i}-K_{1}\partial^{2}A_{\alpha
i}-(K_{2}+K_{3})\partial_{i}\partial_{j}A_{\alpha j}\\\
&&+2\left[\beta_{1}A^{*}_{\alpha
i}\mbox{Tr}\left(AA^{T}\right)+\beta_{2}A_{\alpha
i}\mbox{Tr}\left(AA^{\dagger}\right)\right.\\\
&&\qquad+\left.\beta_{3}(AA^{T}A^{*})_{\alpha
i}+\beta_{4}(AA^{\dagger}A)_{\alpha i}+\beta_{5}(A^{*}A^{T}A)_{\alpha
i}\right]=\zeta_{\alpha i}(\textbf{r},t)\,.\end{array}$ (30)

For temperatures very close to $T_{c}$, i.e. the “gapless region” where
$|\Delta(T)|\ll\pi k_{\text{B}}T_{c}$, the damping by the fermionic bath is
given by [122]

$\gamma=\hbar\frac{\pi N(0)}{48k_{\text{B}}T_{c}}\,.$ (31)

However, $\gamma$ decreases rapidly below $T_{c}$ as the mean field order
parameter, and excitation gap, become established. At low temperatures the
temporal dynamics is dominated by the inertial term defined by Eq. (13).
However, at intermediate temperatures we retain both the inertial and damping
terms. On dimensional grounds It is convenient to express
$\gamma=\hbar^{2}\tilde{\gamma}(\sqrt{35\zeta(3)/60})N(0)/6\pi(k_{\text{B}}T_{\text{c}})^{2}$,
where $\tilde{\gamma}$ has dimensions of frequency.

###  6.3 Dynamic lattice field theory simulations

Simulating order parameters living on a high dimensional manifold in 3+1
dimensions requires high-performance computational resources, both in terms of
parallel floating point performance and effective input and output (I/O) for
visualisation of the results. In the last two decades, high performance
computing (HPC) technologies have improved remarkably both in hardware and
software branches. One of goals of QUEST-DMC project is developing libraries
to solve and analyze static and time-dependent GL equations easily, in the way
of distributed parallelism and parallel I/O. We solve the TDGL equations in
Eq. (30) with finite difference discretization and explicit time
discretization [123]. In order to write Eq. (30) in dimensionless form, the
order parameter $A_{\alpha i}$ is expressed in units of
$k_{\text{B}}T_{\text{c}}$, the length unit is the zero-temperature limit of
the GL coherence length $\xi_{\text{GL}}=\sqrt{7\zeta(3)/20}\,\xi_{0}$, and
the time unit is $t_{\text{GL}}=\sqrt{5/3}\,(\xi_{\text{GL}}/v_{\text{F}})$.
Then, the discretized equations are evolved in time for each point on a
Cartesian grid in three space dimensions. To utilize HPC systems with
distributed memory, we use the lattice field theory library HILA as our
framework [124, 125]. HILA offers a uniform grid, on which the number of sites
along each Cartesian direction can be chosen separately, and a series of pre-
defined class templates to easily handle scalars, vectors and matrices as
physical fields on the lattice. It also has excellent scaling with number of
lattice sites.

Another significant aspect of simulating dynamics with TDGL theory are the
boundary conditions and initial conditions. We adopt periodic boundary
conditions and surface scattering boundary conditions to model different
experimental situations [94, 116, 62]. When we test scenarios, e.g., Baked
Alaska or the Cosmological scenario, which rely on physical processes in the
bulk of sample periodic boundary conditions can be used. On the other hand,
for heterogeneous environments e.g., order parameter distortion or textural
singularities near physical boundaries, a wide range of boundary conditions
that take into account different levels of atomic scale roughness have been
developed [126, 127, 96, 128].

We set the initial conditions for the order parameter configuration on the
3-dimensional spatial lattice for various homogeneous equilibrium states such
as A, B or normal phase. Different levels of noise can be introduced to
simulate the fluctuating forces from the thermal excitations. In the rest of
Sec. 6, we discuss our simulation results with this types of initial states
and let those generated from other situations to be discussed in future works.

###  6.4 Preliminary results for a highly disordered initial state

Here we discuss a test of the Cosmological scenario using the numerical
technology which we introduced in Sec. 6.3. In order to gain a general
understanding of our numerical tool kit, as well as the features of physical
system built upon it, we set up a statistically homogeneous noisy initial
state, with material parameters, $\alpha(T)$ and $\beta_{p}(T)$, with uniform
and fixed temperature and pressure. This can be thought of as modelling a
quench with cooling rate $1/\tau_{Q}\rightarrow\infty$ over the simulation
grid with non-equilibrium order parameter configuration [77, 72].

The spatial grid was $512\times 256\times 256$ sites with a lattice spacing
$0.5\,\xi_{\text{GL}}$. We chose to simulate dynamics at $p=25$ bar and
$T=1.228$ mK, for which $\xi_{\text{GL}}=12.0\,\text{nm}$, making the larger
side length 6.2 $\mu$m. The temperature-dependent Ginzburg-Landau coherence
length at the given temperature is $\xi_{\text{GL}}(T)=17.6\,\text{nm}$. The
total simulation time was $10^{3}t_{\text{GL}}$, corresponding to $665$ ns
based on $t_{\text{GL}}=0.65$ ns at this pressure and temperature. The damping
parameter was set to $\tilde{\gamma}=0.02t_{\text{GL}}^{-1}=30.05$ MHz. This
choice is somewhat larger than the value inferred from the absorption of sound
near the $J=2^{-}$ Higgs mode in 3He-B [129, 46]. The values based on
experimental measurements are order $0.1-5$ MHz.

The spatial boundary conditions were chosen to be periodic in all three
directions. This boundary condition allows us to focus on the possible
nucleation of B-phase during a simulation in the absence of confining
boundaries. The initial configuration of $A_{\alpha i}$ was chosen from a
Gaussian distribution at every site without spatial correlations, but with
mean order parameter corresponding to the equilibrium A phase.

Figure 3: Snapshots of distributions of stationary free energy $f_{\text{s}}$
in Eq. (26) and $\sqrt{\mbox{Tr}\left(AA^{\dagger}\right)}$ at running time
$t=91.7$ ns. (a) Distribution of stationary free energy $f_{s}$ over
simulation grid as described in Sec. 6.3. We calibrated $f_{\text{s}}$ against
bulk free energy $f_{\text{B}}$ of equilibrium B-phase, then the dark blue
corresponds of bulk B-phase, while cyan corresponds to bulk A-phase. The
domain wall has higher stationary energy because of the gradient energy. (b)
Profiles of $\sqrt{\mbox{Tr}\left(AA^{\dagger}\right)}$ between $3.4$ and
$3.6$, corresponding to the range found in domain wall structures at the
temperature and pressure of the simualtion ($1.228$ mK, $25$ bar). The domain
walls visible are either between A- and B-phases or between different
B-phases. The latter are non-topological domain walls, which require specific
symmetry between two domains [84]. The coexistence of A-B domain wall and B-B
domain walls can be understood as defects nucleated during fast quench, which
is a natural and expected result in cosmological scenario [72].

With the sufficient onsite noise, which creates a “glass-like” order
parameter, we obtain B-phase bubbles that appear after a run time $t_{n}\simeq
50$ ns, which is significantly longer than the characteristic pair formation
time $\hbar/\sqrt{\mbox{Tr}\left(AA^{\dagger}\right)}$ $\sim 1$ ns.
Interestingly, increasing the onsite noise amplitude does not necessarily lead
to an increased likelihood of the nucleation of B-phase. We found that only
noise amplitudes in the range $0.4\Delta_{A}-0.5\Delta_{A}$, where
$\Delta_{A}$ is equilibrium A-phase gap, triggered nucleation of B-phase in
this particular simulation. Figure 3 shows the B-phase bubble (dark blue) that
have nucleated in the surrounding metastable A-phase (cyan), and the
corresponding A-B domain walls (red) after time $t=91.7$ ns. Specifically,
Fig. 3(a) shows the distribution of locally stationary free energy density
$f_{\text{s}}$ defined in Eq. (26), which is calibrated against the magnitude
of the bulk free energy density $f_{B}$ of B-phase. The metastable A-phase
free energy density is depicted in cyan, while A-B domain walls, which appear
in light yellow and red, have much higher energy density contributed in part
by gradient energy.

In addition to B-phase bubbles and A-B domain walls, we also find domain walls
separating degenerate, but symmetry inequivalent, B-phases, as shown in Fig.
3(a) and (b). Such walls and their relevance to Cosmology have been first
considered in Ref. [84]. In isotropic pure 3He [130] these domain walls are
non-topological, but can exist when a certain symmetry is present between two
different B-phase domains [84, 131]. The specific symmetry group involved in
this situation consists of $\pi$-rotation of global $U(1)$ phase and global
$\pi$-rotations of $SO(3)_{\mathbf{S}}$ in spin space. This interesting
coexistence of A-B domain walls and B-B domain walls is to be expected in the
Cosmological scenario based on the degeneracy space of the bulk B-phase.

By contrast, symmetry-breaking fields, such as that imposed on 3He by nematic
aerogel, lead to topological protection [132, 133, 134, 135, 136]], analogous
to Cosmological Lazarides-Shafi domain walls. In 3He confined to slab
geometry, such domain walls are at the heart of the putative crystalline
superfluid phase with spontaneously broken translation symmetry [137, 62, 138,
139] (see also [140, 141, 142] for transport experiments with stepped
confinement). Here the mechanism of nucleation of the domain walls is a major
outstanding question, that may be related to the A-B transition puzzle. In
addition to the “hard” [131] domain walls discussed above, “soft” textural
domain walls can be stabilised and manipulated by a combination of
confinement, magnetic field and nuclear dipolar energy [143].

These preliminary studies with a homogeneous quench and infinite cooling rate
do not allow us yet to draw firm conclusions about the nucleation of B phase
bubbles in metastable A-phase. However, we note that the order parameter
remains non-zero everywhere throughout the evolution, suggesting that the
bubbles of B phase can appear without the system entering the normal phase
following energy injection. Simulations based on more realistic temperature
profiles, together with physically realistic cooling dynamics, are ongoing and
will be discussed in future reports.

##  7 Summary and outlook

In the construction of models of fundamental physics beyond the Standard Model
to account for dark matter and the baryon asymmetry, amongst other puzzles, a
common prediction is a first order phase transition in the early universe.
Such phase transitions would produce an isotropic stochastic background of
gravitational waves. If the phase transition took place at temperatures at or
above the electroweak symmetry breaking scale – where many models predict new
particles and interactions – the gravitational waves would be potentially
observable at future space-based gravitational wave observatories such as LISA
[32].

Computations of the expected signal depend, amongst other things, on a
relativistic version of the homogeneous nucleation theory of Cahn and Hilliard
[4] and Langer [5]. It is therefore important to test the theory in the
laboratory. Superfluid 3He has a first order phase transition between the A
and B phases, for which the theory predicts that at pressures, temperatures
and magnetic fields where the A phase is metastable, the system should remain
in the A phase in the course of any conceivable experiment. Yet the transition
usually happens within in a few hours [66]. The QUEST-DMC collaboration aims
to resolve this puzzle, and to decide whether the experimental observations
point to a new rapid bulk nucleation mechanism, or to explanations based on
external sources of excitation energy such as high energy cosmic ray particles
or radioactive decay products from labaoratory materials [63, 73].

The QUEST-DMC collaboration involves new experiments to control and eliminate
boundary effects in two ways: nanofabricated cells taking advantage of
confinement stabilization of phases, and utilizing magnetic fields to isolate
metastable A-phase from experimental boundaries. Careful choice of
nanotechnology makes walls atomically smooth, while shaped magnetic field
distributions ensure that the A phase is metastable only in a portion of the
experiment not in contact with the walls. In this way surface nucleation sites
are eliminated. Future experiments will use the advances in a parallel strand
of work on superfluid 3He as a dark matter detector to understand fluxes of
energetic particles and to eliminate them by building experimental facilities
underground [144, 145].

At the same time we are building simulation algorithms and numerical codes to
investigate the space-time dynamics of the 18-component order parameter of the
simulation system, superfluid 3He. We can simulate the evolution of this
system disturbed by energy injection, or investigate the distortion of the
order parameter around boundaries in complex geometries. In our first set of
numerical experiments using the new code, we have shown that regions of
B-phase are nucleated following a spatially uniform random disturbance of the
metastable A phase. Sufficiently large fluctuations can produce sufficiently
large regions of B phase to overcome the surface tension of the phase
boundary, and thus grow into a stable B-phase. We plan more detailed
simulations of existing nucleation scenarios, including the Baked Alaska model
and the Cosmological nucleation scenario, where the disturbance in the order
parameter is localised.

#### Acknowledgments.

We thank Grigory E. Volovik and Vladimir Eltsov for comments.

#### Author contributions.

The manuscript was written by MH, JAS and KZ, with contributions from all
authors. The simulation code was written by AE-L with contributions from KZ.
Visualisation code was written by KZ. The parallel framework code was written
by KR. The project was supervised by MH, RPH and JS.

#### Funding sources.

This work was supported by: STFC grants ST/T006773/1, ST/T006749/1, and
ST/T00682X/1; Research Council of Finland grants 320123 and 333609; CSC – IT
Centre for Science, Finland; and the Helsinki Institute of Physics. AL-E
acknowledges support from Eusko Jaurlaritza (IT1628-22) and by the
PID2021-123703NB-C21 grant funded by MCIN/AEI/10.13039/501100011033/ and by
ERDF; “A way of making Europe”. JAS acknowledges past support from the US
National Science Foundation, Grant DMR-1508730: “Nonequilibrium States of
Topological Quantum Fluids and Unconventional Superconductors” and current
support from the U.S. Department of Energy, Office of Science, National
Quantum Information Science Research Centers, Superconducting Quantum
Materials and Systems Center (SQMS) under contract number DE-AC02-07CH11359.

#### Competing interests.

The authors declare no competing interests.

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