# The origin and evolution of magnetic white dwarfs in close binary stars

Matthias R. Schreiber1,2,∗ Diogo Belloni3,1,∗ Boris T. Gänsicke4 Steven G.
Parsons5 Monica Zorotovic6

###### Abstract

The origin of magnetic fields in white dwarfs remains a fundamental unresolved
problem in stellar astrophysics. In particular, the very different fractions
of strongly ($\stackrel{{\scriptstyle>}}{{\scriptstyle\sim}}$ 1 MG) magnetic
white dwarfs in evolutionarily linked populations of close white dwarf binary
stars cannot be reproduced by any scenario suggested so far. Strongly magnetic
white dwarfs are absent among detached white dwarf binary stars that are
younger than approximately 1 Gyr. In contrast, in semi-detached cataclysmic
variables in which the white dwarf accretes from a low-mass star companion,
more than one third host a strongly magnetic white dwarf[palaetal20-1]. Here
we present binary star evolutionary models that include the spin evolution of
accreting white dwarfs and crystallization of their cores, as well as magnetic
field interactions between both stars. We show that a crystallization- and
rotation-driven dynamo similar to those working in planets and low-mass
stars[christensenetal09-1] can generate strong magnetic fields in the white
dwarfs in cataclysmic variables which explains their large fraction among the
observed population. When the magnetic field generated in the white dwarfs
connects with that of the secondary stars, synchronization torques and reduced
angular momentum loss cause the binary to detach for a relatively short period
of time. The few known strongly magnetic white dwarfs in detached binaries,
including AR Sco[marshetal16-1], are in this detached phase.

Departamento de Física, Universidad Técnica Federico Santa María, Av. España
1680, Valparaíso, Chile

Millennium Nucleus for Planet Formation (NPF), Valparaíso, Chile

National Institute for Space Research, Av. dos Astronautas, 1758, 12227-010,
São José dos Campos, SP, Brazil

Department of Physics, University of Warwick, Coventry CV4 7AL, UK

Department of Physics and Astronomy, University of Sheffield, Sheffield S3
7RH, UK

Instituto de Física y Astronomía, Universidad de Valparaíso, Av. Gran Bretaña
1111, Valparaíso, Chile
∗These authors contributed equally to this work

The vast majority of close binary stars containing at least one white dwarf
form through common envelope evolution[ivanovaetal13-1]. The emerging detached
post common envelope binary stars evolve towards shorter orbital periods
driven by angular momentum loss[schreiber+gaensicke03-1] and eventually become
semi-detached cataclysmic variable stars (CVs)[kniggeetal11-1]. Despite this
clear evolutionary link, the fraction of strongly magnetic white dwarfs
differs drastically between both types of close white dwarf binaries.

We know more than 160 CVs with a strongly (larger than $1$ MG) magnetic white
dwarf[ferrario15-1]. In most of these systems the white dwarf accretes from
its Roche-lobe filling low-mass star companion along the magnetic field lines.
Due to interactions of the magnetic fields of both stars, the rotation of the
white dwarf is synchronised with the orbital motion of the secondary star,
such systems are called polars. In a smaller but still important fraction of
CVs, the magnetic field of the white dwarf disrupts the inner parts of the
accretion disk but the magnetic fields of both stars do not connect. In these
so-called intermediate polars the rotation of the white dwarf and the orbital
motion are therefore not synchronised[nortonetal04-1]. A recent study of a
volume-limited sample has shown that over a third of all CVs contain a
magnetic white dwarf, with polars and intermediate polars making up
approximately $29$ and $7$ per cent of the total population[palaetal20-1].

The situation is very different among the detached systems that are believed
to be the progenitors of CVs. Only 15 strongly magnetic white dwarfs in close
detached binaries are currently known[schwopeetal09-1, parsonsetal20-1] and
they make up less than two per cent of the over one thousand known
systems[schreiberetal10-1, rebassa-mansergasetal12-1]. All but one of these
detached magnetic white dwarf binaries have been identified due to a peculiar
emission line[reimersetal99-1], which turned out to be the third harmonic of a
cyclotron fundamental emitted by a low-density plasma. This observational
finding is convincingly interpreted as resulting from wind accretion onto a
strongly magnetic white dwarf in a close but detached binary[schmidtetal05-1].
Because angular momentum loss will eventually drive them into a semi-detached
CV configuration, these systems have been termed pre-polars[schwopeetal09-1].
The only exception is AR Sco, which was discovered because of the optical and
radio pulses of its rapidly ($1.97$ min) rotating white dwarf[marshetal16-1].
AR Sco and most pre-polars are located in the orbital period range between 3
and 5 hours, their secondary stars are close to filling their Roche-lobes
($R/R_{\mathrm{L}}$ typically exceeding $80$ per cent)[parsonsetal20-1], and
the magnetic white dwarfs are cold with effective temperatures typically below
10,000 K which implies that they formed at least one Gyr ago. Among younger
close detached white dwarf binaries, not a single strongly magnetic white
dwarf has been found[liebertetal05-1, belloni+schreiber20-1]. This puzzling
situation, a large fraction (approximately $35$ per cent) of strongly magnetic
white dwarfs in CVs and the much smaller fraction of magnetic white dwarfs
among the detached systems that are believed to be CV progenitors, was aptly
expressed by Liebert[liebertetal05-1] who asked Where Are the Magnetic White
Dwarfs with Detached, Non-degenerate Companions? This fundamental question of
stellar evolution has remained without an answer for 15 years and the solution
must link the evolution of white dwarf binaries and magnetic field generation
in white dwarfs.

Several hypotheses have been put forward to explain magnetic field generation
in white dwarfs in the last decades. In their present form, the fossil field
scenario[braithwaite+spruit04-1], a dynamo operating during common envelope
evolution[toutetal08-1], or coalescing double white dwarfs[garcia-
berroetal12-1] face one serious problem when considered as the main formation
mechanism for close magnetic white dwarf binaries. In all these cases the
magnetic fields are generated during the formation process of the white dwarf,
i.e. according to the current versions of these theories also detached post
common envelope binaries containing hot white dwarfs should host strong
magnetic fields which is clearly not the case[belloni+schreiber20-1].

As the only mechanism that depends on the age of the white dwarf, a dynamo
similar to those operating in low-mass main-sequence stars and
planets[christensenetal09-1] has been suggested to operate if the white dwarf
is rotating rapidly when the core of the white dwarf is
crystallizing[isernetal17-1]. As a white dwarf with a Carbon/Oxygen core
cools, the ions in the core begin to freeze in a lattice structure. During
crystallization the chemical potential, temperature, and pressure of the
liquid and solid phases must remain equal. As a consequence, in crystallizing
white dwarfs the solid phase becomes richer in oxygen and sinks while the
carbon excess mixes with the outer liquid envelope which is redistributed by
Rayleigh–Taylor instabilities. This configuration is similar to that found in
the core of the Earth, where the light element release associated with the
inner core growth is a primary driver of the dynamo generating the Earth’s
magnetic field[lister+buffet95-1].

The predicted field strengths generated by this dynamo in white dwarfs depend
entirely on the scaling law that is assumed. Applying the same scaling law
that is used for planetary magnetic fields leads to field strengths in white
dwarfs of up to approximately $1$ MG[isernetal17-1]. However, taking into
account a likely dependence of the dynamo efficiency on the magnetic Prandtl
number and that in white dwarfs this number is orders of magnitudes larger
than in planets, one could expect this dynamo to be able to generated fields
much stronger than $1$ MG (for more details see Methods).

Assuming that the crystallization and rotation driven dynamo can generate
strong magnetic fields we computed evolutionary tracks starting with non-
magnetic post common envelope binaries. We incorporated several physical
mechanisms in the stellar evolution code MESA[paxtonetal19-1]. The spin-up of
the white dwarf due to the accretion of angular momentum[kingetal91-1] was
included as well as crystallization of the white dwarf[bedardetal20-1]. We
furthermore incorporated the reduction in angular momentum loss through
magnetic braking when the magnetic fields of both stars
connect[bellonietal20-1], a condition for synchronisation[hameuryetal87-1],
and a prescription for angular momentum transfer from the spin of the white
dwarf to the orbit during synchronisation (see Methods for more details).

Figure 1 shows an exemplary evolutionary track resulting from our simulations
and a schematic illustration of the different stages the system passes
through. In our model the white dwarf is born without a strong magnetic field
which is in agreement with the absence of strongly magnetic white dwarfs in
young close detached post common envelope binaries (stage 1). The non-magnetic
detached binary star then evolves slowly towards shorter orbital periods and
the white dwarf cools. When the secondary fills its Roche-lobe the binary
becomes a CV, accretion spins up the white dwarf[kingetal91-1] and the core
may be crystallizing (stage 2). If this happens, both conditions for the
crystallization and rotation driven dynamo are met and a strong magnetic field
is generated (stage 3). If the magnetic field is strong enough to connect with
the field of the secondary star the latter provides a synchronising torque on
the white dwarf spin. The spin angular momentum transferred to the orbital
motion due to the synchronising torque causes the secondary to detach from its
Roche-lobe. This effect is supported by the fact that the connection of the
two magnetospheres reduces angular momentum loss due to magnetic wind braking
because part of the gas escaping from the secondary is accreted by the white
dwarf along the connected field lines, instead of leaving the
system[bellonietal20-1]. At the beginning of synchronisation the system
becomes a radio pulsing white dwarf binary, such as AR Sco, in which a rapidly
rotating white dwarf spins down on a relatively short time scale (stage 4) of
approximately $1-10$ Myr. When the white dwarf spin is synchronised with the
orbital motion of the secondary star, the binary star has evolved into the
pre-polar stage in which wind accretion generates the observed cyclotron
emission (stage 5). Angular momentum loss through reduced magnetic braking and
gravitational radiation brings the stars closer together until the secondary
again fills its Roche-lobe and finally a polar is formed (stage 6). The
detached phase (stage 3-5) occurs preferentially for orbital periods shorter
than roughly $5$ hr as the reduction of magnetic braking and the transfer of
spin angular momentum to the orbit have less impact at longer periods.
Detached phases occur less frequently at orbital periods shorter than
approximately $3$ hr because the conditions for magnetic field generation are
typically met at early stages of CV evolution (see Methods for a detailed
comparison of observations and model predictions).

While a different mechanism such as common envelope merger[toutetal08-1]
and/or double white dwarf merger[garcia-berroetal12-1] is probably required to
explain most single white dwarfs with magnetic field strengths exceeding 1 MG
(see Supplementary Material for details), the new evolutionary sequence
predicted by our model is in excellent agreement with the observations of
close binary stars containing strongly magnetic white dwarfs. As the
conditions for the dynamo are relatively frequently met in CVs (see
Supplementary Material for a discussion of the predicted relative numbers),
our model successfully reproduces the large fraction of CVs containing
magnetic white dwarfs. The few known detached binaries containing magnetic
white dwarfs are not progenitors of CVs but instead systems that have been
non-magnetic CVs in the past and where mass transfer stopped because the field
was generated. This naturally explains why there is not one single hot and
young strongly magnetic white dwarf in a detached binary. Instead our model
predicts that all of them are older than 1 Gyr with crystallizing cores as is
observed (Fig. 2) and also explains why all 15 currently known close detached
binaries containing magnetic white dwarfs are close to Roche-lobe filling
(Supplementary Figure 1). Furthermore, the observed clustering of pre-polars
in the orbital period range of $3-5$ hr is well reproduced in our simulations
as the detached phase typically occurs in this range. Finally, our
evolutionary sequence predicts short phases of rapidly rotating and strongly
magnetic white dwarfs in detached binaries thereby offering an explanation for
the existence of the radio pulsating system AR Sco. We conclude that a dynamo-
driven generation of magnetic fields in white dwarfs resolves several long
standing problems at once.

Fig. 1. Spin period, Roche-lobe filling factor, and mass transfer rate as a
function of orbital period. The slowly rotating white dwarf (stage 1) in a
non-magnetic post common envelope binary is spun-up due to accretion (stage
2). In case the white dwarf is crystallizing a strong magnetic field is
generated by a dynamo (stage 3) and as soon as it connects with the field of
the secondary the system becomes detached. At first it resembles the radio
pulsing white dwarf binary AR Sco (stage 4). When the spin of the white dwarf
and the orbital motion are synchronised (stage 5) the system becomes a pre-
polar before mass transfer starts again (stage 6). When the secondary becomes
fully convective, a second short detached phase (between $\approx 2.07$ and
$\approx 2.09$ Gyr) occurs which represents a short version of the classical
orbital period gap (hatched region).

Fig. 2 Observed white dwarf effective temperatures versus their masses for
non-magnetic post common envelope binaries from the Sloan Digital Sky Survey
(light gray)[nebot-gomezetal11-1, rebassa-mansergasetal12-1], pre-polars
(red)[parsonsetal20-1], and the first radio pulsing white dwarf binary AR Sco
(blue)[marshetal16-1, Garnavich2020]. The errors on the observations
correspond to 1 $\sigma$ and arrows indicate upper limits. The lines
represents the mass fraction of crystallized matter in the white dwarf
interiors. Solid, dashed, dash-dotted and dotted lines show the temperature
for each white dwarf mass corresponding to the onset of crystallization, 10
per cent of mass is crystallized, 50 per cent and 80 per cent,
respectively[bedardetal20-1]. As predicted by our scenario, magnetic systems
host on average much older and more massive white dwarfs than their non-
magnetic counterparts. The trend towards larger masses in magnetic systems is
totally consistent with the observed white dwarf mass distribution in
cataclysmic variables[zorotovicetal11-1, schreiberetal16-1] which fits with
the model prediction that these systems have been CVs in the past. In
addition, the white dwarfs in AR Sco and all pre-polars are consistent with
having crystallizing cores.

## Methods

###  0.1 The convective dynamo and scaling laws.

The magnetic fields of Earth and Jupiter, as well as those of rapidly rotating
low-mass stars, are generated by convection-driven dynamos. The basic
ingredients for these dynamos to work are a strong density stratification, and
an extended convection zone. To derive the magnetic field strength from
fundamental properties of a given planet or star the use of scaling laws is
required. In general it is neither clear which quantities need to be
considered in these scaling laws nor whether a single scaling law exists for
all planets and stars whose magnetic fields are generated by the convective
dynamo. In the last decades, several scaling laws have been proposed, most of
them based on the assumption of a balance between Coriolis force and Lorentz
force[christensen10-1]. In recent years a scaling law derived from geo-dynamo
models[christensen+aubert06-1] which assumes the energy flux to be the
dominant parameter has become popular. This scaling law has been shown to
successfully explain the field strengths of Earth and Jupiter as well as those
of rapidly rotating fully convective stars2.

In the first application of the convective dynamo mechanism to white dwarfs
this energy flux scaling law was assumed20. The vast majority of white dwarfs
consist of Carbon and Oxygen. As a white dwarf cools, the ions in the core
begin to freeze in a lattice structure[vanhorn68-1]. The phase diagram of the
carbon–oxygen mixture is of the spindle form[segretain+chabrier93-1,
horowitzetal10-1], therefore, as soon as the white dwarf has sufficiently
cooled and starts to crystallize, the solid phase becomes richer in oxygen and
sinks while the carbon excess mixes with the outer liquid envelope. As a
consequence, an oxygen-rich core forms which is significantly denser than the
carbon-rich liquid which is redistributed by the Rayleigh–Taylor
instability[isernetal97-1, isernetal00-1]. Such a strong density
stratification in combination with convection represent the key ingredients
for the dynamo generating the Earth’s magnetic fields and those of low-mass
stars. Assuming the same scaling law that has been successfully applied to the
Earth, Jupiter, and rapidly rotating fully convective stars for crystallizing
white dwarfs predicts field strengths of at most a few MG20 which, if true,
would exclude the dynamo from being the mechanism generating the much stronger
$1-200$ MG fields found in white dwarfs in close binary stars.

Here we argue that this scaling law is most likely not applicable to white
dwarfs. In the magneto hydrodynamic simulations that led to the energy flux
scaling law, the magnetic Prandtl number ($P_{\mathrm{m}}$, defined as the
ratio between viscous and magnetic diffusivity) is assumed to be equal to one.
However, in both fully convective stars as well as the Earth this number is
significantly smaller ($10^{-4}-10^{-5}$ and $10^{-6}$, respectively). To
quote the discovery paper[christensen+aubert06-1] of the energy flux scaling
law _“we cannot rule out an additional dependence on other parameters, in
particular the magnetic Prandtl number. Although the suggested dependence is
weak, it poses a serious problem. Given the large range of extrapolation over
five orders of magnitude from our models to planetary values of
$P_{\mathrm{m}}$, the results obtained from the scaling laws with or without a
dependence on $P_{\rm{m}}$ differ substantially.”_ This early statement has
been confirmed more recently by Brandenburg[brandenburg14-1] who wrote: _“This
leads to a scaling law that has been verified over many orders of magnitude
(Christensen et al. 2009). As we have seen, this scaling law must be affected
by $P_{m}$…”_. Given that the magnetic Prandtl number for crystallizing white
dwarfs has been estimated to be $0.58^{20}$, i.e. roughly six orders of
magnitude larger than that of the Earth and still $4-5$ orders of magnitude
larger than that of fully convective low-mass stars, the dynamo could be more
efficient in white dwarfs.

In fact, the comparison between predicted and observed field strengths of
single white dwarfs presented by Isern et al.20 indeed indicates that observed
fields in crystallizing single white dwarfs exceed those predicted by the
energy flux scaling law. The energy flux scaling law is only applicable in the
saturated regime of the dynamo, i.e. it only works for rapidly rotating
objects with Rossby numbers below $\sim 0.1$. For white dwarfs, this implies
spin periods shorter than $\sim 90$ s20. In white dwarfs that rotate
substantially slower, the generated fields should be smaller than those
predicted by the energy flux scaling law. All the observed magnetic single
white dwarfs within 20 pc (Table 1 of Isern et al.20) rotate with periods
ranging from hours to days. Despite rotating orders of magnitudes too slow for
the energy flux scaling law to be applicable (which is applicable in the
saturated regime), the field strengths of these white dwarfs even slightly
exceed those predicted by the energy flux scaling law. This implies that, if a
convection driven dynamo works in crystallizing and rotating white dwarfs, the
energy flux scaling law underestimates the field strengths of slowly rotating
single white dwarfs. The field strengths of rapidly rotating white dwarfs in
the saturated regime should therefore as well largely exceed those predicted
by the energy flux scaling law. A suitably revised scaling law that
incorporates the dependence on the magnetic Prandtl number is therefore likely
to predict fields much stronger than $\sim 1$ MG for white dwarfs that rotate
in the saturated regime of the dynamo.

Based on the above arguments, we here assumed that a rapidly rotating white
dwarf ($P_{\mathrm{WD}}\simeq 20-30$ s) with a fractional crystallized mass
between 10 and 80 percent can drive a dynamo that generates the large magnetic
field strengths observed in white dwarfs in cataclysmic variables (up to
several 100 MG). We calculate the evolution of close detached white dwarf
binaries and cataclysmic variables (CVs) using MESA22, taking into account the
spin-up due to accretion and crystallization due to cooling of the white dwarf
as well as magnetic field generation if a crystallizing white dwarf rotates
rapidly.

###  0.2 Spin-up of the white dwarf.

If a white dwarf accretes from an accretion disk in a CV its angular momentum
and therefore its angular velocity increases, while during nova eruptions the
white dwarf looses mass and angular momentum. According to King et al.23, the
resulting angular momentum balance for an accreting white dwarf is given by

$I\,\frac{{\rm d}\omega}{{\rm d}t}\ =\
\alpha\,(\,-\dot{M}_{2}\,)\,\left(G\,M_{\rm WD}\,R_{\rm WD}\,\right)^{1/2}\ +\
\left(1+\epsilon\right)\left(\dot{M}_{2}\,\eta\,R_{\rm
WD}^{2}\,\omega\right),$ (1)

where $\omega$ is the white dwarf spin, $I$ its moment of inertia, $G$ the
gravitational constant, $\dot{M}_{2}$ the mass transfer rate averaged over
nova cycles, and $M_{\rm WD}$ and $R_{\rm WD}$ are the white dwarf mass and
radius.

The first term in the right-hand part of the equation corresponds to the spin-
up due to accretion, while the second term represents the spin-down due to
material leaving the white dwarf during nova eruptions. The parameter
$0\leq\alpha\leq 1$ represents the spin-up efficiency, which might be smaller
than one due to mass loss processes not considered in equation Eq. 1 (e.g.
winds from the inner accretion disk or jets from the boundary layer/white
dwarf). The parameter $0\leq\epsilon=\dot{M}_{\rm WD}/\dot{M}_{2}\leq 1$
relates the mass loss of the white dwarf to the mass transfer rate. The range
of possible values reflects the fact that the white dwarf mass might decrease
over a nova cycle, i.e. more mass might be expelled during a nova eruption
than was accreted between two eruptions. The parameter $0\leq\eta\leq 1$
represents the square of the radius of gyration of the ejected envelope and
depends on the geometry of nova eruptions.

By solving the non-homogeneous differential equation 1, the WD spin as a
function of the donor mass is given by

$\omega=\frac{C_{1}}{C_{2}}+C_{3}\,\exp\,{\left(\,C_{2}\,M_{\rm
2}\,\right)}~{},$ (2)

where $M_{\rm 2}$ is the donor mass in solar masses and the terms $C_{1}$,
$C_{2}$ and $C_{3}$ are given by

$C_{1}=\alpha\,\frac{A}{k^{2}}\,\frac{\omega_{\rm bk,0}}{M_{\rm WD,0}}$ (3)
$C_{2}=\frac{\eta}{k^{2}\,M_{\rm
WD}}\,\left(\,1+\varepsilon\,\right)+B\,\varepsilon$ (4)
$C_{3}=\left(\,\omega_{0}-\frac{C_{1}}{C_{2}}\,\right)\,\exp{\left(-C_{2}\,M_{\rm
2,0}\right)}$ (5)

where $\omega_{\rm bk}$ is the break-up white dwarf spin, given by

$\omega_{\rm bk}=\sqrt{\frac{GM_{\rm WD}}{R_{\rm
WD}^{3}}}=2\,\pi\,\sqrt{\left(\frac{M_{\rm WD}}{{\rm
M}_{\odot}}\right)\left(\frac{R_{\rm WD}}{\rm au}\right)^{-3}}~{}~{}~{}~{}{\rm
yr}^{-1}~{}~{}.$ (6)

The moment of inertia $I$ is $k^{2}M_{\rm WD}R_{\rm WD}^{2}$, with $k$ given
by ${0.452+0.0853\log_{10}\left(1-M_{\rm WD}/1.44{\rm M}_{\odot}\right)}$, if
${M_{\rm WD}\lesssim 1.368}$ M⊙, and $\approx 0.275$, if ${M_{\rm WD}\gtrsim
1.368}$ M⊙. The constants $A$ and $B$ are given by

$A=\left[\frac{1-0.61\,\left(\,M_{\rm WD,0}/{\rm
M}_{\odot}\,\right)^{4/3}}{1-0.61\,\left(\,M_{\rm WD}/{\rm
M}_{\odot}\,\right)^{4/3}}\right]^{3/4}~{}~{},$ (7)
$B=\frac{3.05\,\left(\,M_{\rm WD}/{\rm
M}_{\odot}\,\right)^{4/3}-1}{3\,\left[1-0.61\left(\,M_{\rm WD}/{\rm
M}_{\odot}\,\right)^{4/3}\right]}\ +\ \frac{0.051\,\left(\,M_{\rm WD}/{\rm
M}_{\odot}\,\right)}{k\,\left[\,1-0.69\left(\,M_{\rm WD}/{\rm
M}_{\odot}\,\right)\right]}~{}~{}.$ (8)

The sub-index $0$ in some quantities indicates their initial values just after
the common-envelope evolution.

###  0.3 Synchronisation of the white dwarf.

In accreting high-field magnetic white dwarf binaries the magnetic field
dominates the accretion process. For sufficiently strong white dwarf magnetic
moments, the overflowing material is channelled along the white dwarf’s
magnetic field lines as soon as the magnetic pressure dominates the gas ram
pressure. For such strong white dwarf magnetic fields, the interaction of the
magnetospheres of the white dwarf and the donor cause a synchronizing torque
on the white dwarf[campbell85-1].

We assume here that the onset of synchronization occurs when the synchronizing
torque is greater than the accretion torque (assuming that the magnetospheres
of the white dwarf and the donor are connected)26, i.e.

${\textstyle\frac{\mu_{\rm WD}}{10^{33}\,{\rm
G\,cm^{3}}}>1.11\,\left(\frac{-\dot{M}_{2}}{10^{16}\,{\rm
g/s}}\right)\,\left(\frac{M_{\rm tot}}{{\rm
M}_{\odot}}\right)^{1/3}\,\left(\frac{M_{2}}{{\rm
M}_{\odot}}\right)^{-1}\,\left(\frac{P_{\rm orb}}{{\rm
hr}}\right)^{1/3}\,\left(\frac{B_{\rm 2}}{100\,{\rm
G}}\right)^{-1}\,\left[0.5-0.277\,\log_{10}\,\left(\frac{M_{2}}{M_{\rm
WD}}\right)\right]^{2}}$ (9)

where $\mu_{\rm WD}$ is the white dwarf magnetic moment, $M_{2}$ is the donor
mass, $M_{\rm tot}=M_{\rm WD}+M_{2}$ is the binary total mass, $P_{\rm orb}$
is the orbital period and $B_{2}$ is the strength of the magnetic field of the
secondary star.

Once the condition provided by Equation 9 is satisfied, the spin-down process
starts. For simplicity, we assume that the white dwarf spin decreases
exponentially on a given time-scale, i.e.

$\omega(t)=\Omega+Ce^{-t/\tau}$ (10)

where $\tau$ is the synchronisation time-scale, $\Omega$ the orbital angular
velocity, $\omega$ the WD spin, and $C=\omega_{0}-\Omega$ is a constant
(because variations in $\Omega$ are negligible during the spin-down phase).
This assumption gives

$\frac{{\rm d}\omega}{{\rm d}t}=\frac{(\Omega-\omega)}{\tau}.$ (11)

This equation of course represents a very crude approximation of a very
complicated process. The torque on the primary as a function of asynchronism
has been calculated previously in great detail[campbell10-1], but only for
rather small asynchronism. Performing such detailed calculations of the
magnetic field interaction for the large asynchronism in our systems, is
beyond the scope of this paper. For the purpose of this paper, we assume that
the total angular momentum loss by the white dwarf is transferred to the
secondary, incorporated in the orbital motion, and that this occurs on the
time-scale $\tau$, i.e.

$\dot{J}_{\rm orb}=-\frac{\rm d}{{\rm d}t}(\omega
I)=I\frac{(\omega-\Omega)}{\tau}>0.$ (12)

In this simple model the torque is always decreasing with increasing
synchronism which might not be correct[campbell10-1]. However, this should not
affect the validity of our model as only the amount of spin angular momentum
transferred to the orbit and the time scale on which this happen are important
for the secular evolution of a given system.

###  0.4 Reduced magnetic braking.

For secondary stars still containing a radiative core, angular momentum loss
through magnetic wind braking drives mass transfer in CVs6,11. The strength of
magnetic braking depends significantly on the extension of wind zones compared
to dead zones. The latter correspond to regions where gas is prevented from
escaping the star due to the pressure of closed magnetic field loops, i.e. the
magnetic pressure is greater than the thermal pressure[Mestel_1968]. If the
white dwarf in a CV is strongly magnetic and the magnetospheres of both stars
connect, the wind zone is reduced which leads to a decrease of orbital angular
momentum loss due to magnetic braking[WW02, LWW94]. For short orbital periods
and/or strong magnetic fields, angular momentum loss through magnetic braking
can be completely suppressed.

The reduction of magnetic braking for a given $\mu_{\rm WD}$ can be
parametrized with the fraction of open field lines $(\Phi)$. In the extreme
case of no wind zone, i.e. $\mu_{\rm WD}$ is strong enough to prevent any
magnetic braking, ${\Phi=0.0}$. On the other hand, in case the impact of the
magnetic moment is negligible, the second dead zone does not exist and
magnetic braking is not reduced, leading to ${\Phi=0.258}$. This prescription
for the reduction of MB depends on the binary parameters and $\mu_{\rm WD}$,
and therefore allows us to incorporate the impact of strong WD magnetic fields
in binary evolution codes[WW02],25.

The resulting angular momentum loss due to magnetic braking in the presence of
a strongly magnetic white dwarf $(\dot{J}_{\rm MB,mag})$ can be written as

$\dot{J}_{\rm MB,mag}\ =\ \dot{J}_{\rm MB,non-
mag}\,\left(\frac{\Phi}{0.258}\right)^{5/3}\,,$ (13)

where $\dot{J}_{\rm MB,non-mag}$ is the angular momentum loss due to magnetic
braking in non-magnetic CVs. In this work we used the widely adopted RVJ
prescription[RVJ], with $\gamma=3$, i.e.

$\displaystyle\dot{J}_{\rm MB,non-mag}\ =\ -3.8\times
10^{-30}\,\left(\frac{M_{2}}{\rm
g}\right)\,R_{\odot}^{4}\,\left(\frac{R_{2}}{{\rm
R}_{\odot}}\right)^{3}\,\left(\frac{\Omega_{2}}{\rm s^{-1}}\right)^{3},$ (14)

where $R_{2}$ is the donor radius and $\Omega_{2}$ is the donor spin, which is
the same as the orbital angular velocity. This reduced magnetic braking model
has been incorporated in binary population models of magnetic CVs and has been
shown to convincingly reproduce the orbital period distribution of magnetic
CVs25 and we here implemented the same prescription in MESA.

###  0.5 Modelling magnetic CV evolution with MESA.

The above described effects were incorporated in the stellar evolution code
Modules for Experiments in Stellar Astrophysics (MESA) (version
r10108)[Paxton2011, Paxton2013, Paxton2015, Paxton2018],22. The MESA equation
of state is a blend of the OPAL[Rogers2002], SCVH[Saumon1995], PTEH[Pols1995],
HELM[Timmes2000], and PC[Potekhin2010] equation of states. Radiative opacities
are primarily from OPAL[Iglesias1993, Iglesias1996], being the regime low-
temperature dominated[Ferguson2005] or high-temperature, Compton-scattering
dominated[Buchler1976]. Electron conduction opacities are also
included[Cassisi2007]. Nuclear reaction rates are a combination of rates from
NACRE[Angulo1999], JINA REACLIB[Cyburt2010], plus additional tabulated weak
reaction rates[Fuller1985, Oda1994, Langanke2000]. Screening is
included[Chugunov2007] as well as thermal neutrino loss[Itoh1996]. Roche lobe
radii in binary systems are computed using Eggleton’s fit[Eggleton1983] and
mass transfer rates in Roche lobe overflowing binary systems are determined
following Ritter’s prescription[Ritter1988].

The parameters of our standard model were set as follows. During the whole CV
evolution we assumed that all material accreted by the WD is lost through nova
eruptions. In Equation 1, this corresponds to setting $\epsilon=0$. For the
geometry of nova shells, we assumed that mass leaves the WD with some
asymmetry (i.e. neither completely spherical nor radial), which implies values
of $\eta$ in Eq. 1 between $2/3$ and $1$. We set ${\eta=2.5/3}$. Additional
mass loss from the system[matthews15-1, coppejans20-1] was taken into account
by using $\alpha=0.75$ in Equation 1. The donor magnetic field was assumed to
be ${B_{2}=1}$ kG, which is consistent with recent measurements of field
strengths of low-mass stars[reiners07-1]. We fixed the WD mass to $0.8$ M⊙,
which corresponds to the average value found among observed
CVs29,[mcallister19-1]. For the white dwarf magnetic field, we assume ${B_{\rm
WD}=60}$ MG, which is slightly stronger than the average field strength of
polars7, but corresponds to the average field strength found in pre-polars9.
We assumed that the dynamo is efficient only if the core of the white dwarf is
crystallizing (between 10 and 80 per cent crystallized) and the spin period of
the white dwarf is $\simeq 20-30$ s. Finally, for the synchronisation time-
scale, we set ${\tau=1}$ Myr, which is consistent with the expected rapid
spin-down in close binaries harbouring a rapidly rotating magnetic white dwarf
such as AR Sco3.

Using this set of parameters our model can reproduce the observations of
magnetic detached and semi-detached close white dwarf binary stars
(Supplementary Figure 1). According to this model, pre-polars and AR Sco have
been CVs in the past and at that time the strong magnetic fields of their
white dwarfs were generated. This prediction is in perfect agreement with the
cool white dwarf temperatures of these detached systems, with them being close
to Roche-lobe filling, and with their preferred occurrence in the 3-5 hr
orbital period range.

We emphasise that no fine-tuning was required to obtain this agreement. In
contrast, the model naturally explains the observations and variations of the
key parameters do not affect the general predictions (Supplementary Figures 2,
3 and 4). For instance, by changing the spin-up parameters (i.e. $\alpha$,
$\eta$ and $\epsilon$), the only effect is to accelerate or delay the spin-up
of the white dwarf and slightly change the minimum spin period that can be
reached23. For all reasonable values of these parameters the white dwarf will
manage to spin-up to sufficiently large spin rates so that the conditions for
generating strong magnetic fields can eventually be met.

Similarly, varying the synchronisation time-scale $\tau$ affects the
synchronising torque but does not change the evolutionary sequence
(Supplementary Figure 2). The shorter $\tau$, the stronger the spin-down
torque and therefore the longer the detached phase. Only for $\tau\gtrsim 100$
Myr, the detached phase can be fully prevented if, in addition, the magnetic
field strength of the white dwarf does not exceed $\sim 200$ MG. The latter
condition arises as for very strong fields reduced magnetic braking alone can
cause the system to detach in full analogy to the mechanism producing the
classic orbital period gap.

A slightly stronger constraint on the synchronisation time scale results from
the fact that it should not exceed the duration of the generated detached
phase as otherwise the observed synchronised pre-polars would remain
unexplained. For white dwarfs rotating near break-up, in order to have a
synchronised system after the detached phase, time scales cannot exceed
$30-40$ Myr. This maximum time scale is definitely not a problem for our
model, since the synchronisation time scales for the range of magnetic field
strengths we consider here, i.e. those of polars, pre-polars and AR Sco, are
expected to be well below $30-40$ Myr[stiller18-1].

The overall evolution does also not sensitively depend on the assumed white
dwarf and secondary star magnetic fields. Varying these field strengths
changes only slightly the condition for synchronisation (Equation 9). For
weaker/stronger $B_{2}$, this condition becomes more/less restrictive.
Increasing or decreasing $B_{\rm WD}$ affects somewhat the orbital period at
which synchronism can be obtained, which is longer for stronger fields
(Supplementary Figure 3). The white dwarf magnetic field strengths also
affects the duration of the detached phase. The stronger the white dwarf
magnetic field, the more angular momentum loss due to magnetic braking is
reduced and the longer it takes the system to evolve back into a semi-detached
configuration.

The largest and most critical assumption in our model certainly are the
conditions we assume for magnetic field generation which represent a strong
but reasonable simplification. According to the most popular scaling law, the
dynamo saturates for fast rotation (in white dwarfs for spin periods below 90
s20) and the produced field strength depends then solely on the convective
energy flux. However, as outlined above, the efficiency of the dynamo is also
expected to significantly depend on the magnetic Prandtl number which is not
considered in any currently available scaling law. As long as a reliable
scaling law taking into account this dependency is not available, describing
the evolution of close magnetic white dwarf binaries needs to be based on
simple assumptions such as those made in this paper. As soon as realistic
scaling laws for the dynamo operating in crystallizing white dwarfs are
available, we plan to incorporate those together with detailed white dwarf
models into our simulations.

The aim of this paper is to show how the evolution of close white dwarf
binaries is affected if, and that is a reasonable assumption, crystallization
and fast rotation can generate a strong magnetic field. We would expect the
generated field strength in these models to depend on the rotation rate and
the energy flux in the convection zone. Given the large range of orbital
periods at the end of common envelope evolution, and the therefore large
variety of evolutionary time scales towards the CV phase, we expect that such
a more detailed model can explain the large range of field strength observed
in magnetic CVs including those of intermediate polars. In case of the latter
the field is simply not strong enough to synchronise the white dwarf spin with
the orbital motion and the system does not enter a detached phase apart from
the usual $2-3$ hr orbital period gap.

In the Supplementary Information we present a detailed discussion of different
evolutionary channels predicted by our model and estimate the fraction of
magnetic and non-magnetic white dwarfs in CVs taking into account important
parameters for binary star evolution[RVJ, zorotovicetal10-1, zorotovic14-1,
bellonietal18-1, MD17] as well as potential accretion heating and chemical
impurities of the core of the white dwarf[townsley+bildsten04-1,
epelstainetal07-1, isernetal98-1]. We also elaborate on the implications of
our scenario for the generation of magnetic fields in hot single white
dwarfs[charpinetetal09-1] as well as magnetic white dwarfs in wide
binaries[landstreet+bagnulo20-1]. We find that these magnetic fields are most
likely generated by alternative mechanisms such as fossil fields resulting
from main sequence star mergers[Ferrario2009, schneideretal19-1], double white
dwarf mergers[garcia-berroetal12-1], or common envelope mergers[toutetal08-1,
Wickramasinghe2014, Briggs2015].

Correspondence and requests for materials should be addressed to M.R.S.
(email: matthias.schreiber@usm.cl).

We thank the referee Chris Tout and two anonymous referees for their helpful
comments. M.R.S. acknowledges support from the Millennium Nucleus for Planet
Formation (NPF) and Fondecyt (grant 1181404). D.B. was supported by the grant
#2017/14289-3, São Paulo Research Foundation (FAPESP) and ESO/Gobierno de
Chile. B.T.G. was supported by the UK STFC grant ST/P000495. S.G.P
acknowledges the support of a STFC Ernest Rutherford Fellowship. M.Z.
acknowledges support from CONICYT PAI (Concurso Nacional de Inserción en la
Academia 2017, Folio 79170121) and CONICYT/FONDECYT (Programa de Iniciación,
Folio 11170559)

All authors contributed to the discussion and writing of this article. M.R.S.
and D.B. developed the idea and carried out the simulations. S.G.P. and B.T.G
provided an observational overview of magnetic white dwarfs and recent
observations of pre-polars. M.Z. provided crystallization temperatures for
different white dwarf masses based on published white dwarf evolutionary
sequences and estimated relative numbers of magnetic and non-magnetic CVs
using binary population models.

The authors declare that they have no competing interests.

The data presented in this work is available upon request. In addition, the
MESA files required to reproduce our simulations will be publicly available at
the MESA Zenodo community (https://zenodo.org/communities/mesa/).

MESA is publicly available (http://mesa.sourceforge.net/).

## Supplementary Discussion and Figures

###  0.6 The fraction of magnetic white dwarfs in CVs.

We have shown that a crystallization and rotation driven dynamo can explain
the properties of all close and strongly magnetic white dwarf binaries and the
absence of magnetic white dwarfs in young detached white dwarf binary stars.
In what follows we discuss under which conditions our model is likely to also
reproduce the observed fraction of strongly magnetic white dwarfs in CVs. The
measured fraction of strongly magnetic white dwarfs in CVs is more than one
third (15 out of 42 systems within 150 pc1). If the briefly detached pre-
polars and AR Sco are included in this sample (the four pre-polars SDSS
0303+0054, WX LMi, SDSS J1212+0136, SDSS J1452+2045 and AR Sco are within 150
pc3,10), the fraction of systems with magnetic white dwarfs further increases
to 20 of 47 systems corresponding to $43\pm 6$ per cent.

At first glance, our model might seem to produce too many magnetic white
dwarfs in CVs as every white dwarf in a non-magnetic CV accretes angular
momentum and is therefore expected to become rapidly rotating in a short
period of time. In addition, if the white dwarf continuously cools it should
start to crystallize and one might expect that our model predicts every white
dwarf in a CV to eventually develop a strong magnetic field. However,
accretion can increase the core temperature of white dwarfs in CVs77 and
prevent or delay crystallization. In addition, the generation of a strong
magnetic field through the rotational and crystallization driven dynamo only
works if fast rotation and a crystallizing core with an outer convection zone
are present at the same time. The strength of the magnetic field generated by
this dynamo depends on the convective energy flux2 and most likely also on the
magnetic Prantdl number. However, in the absence of a scaling relation
including the dependence on the latter, it remains unclear at which stages of
crystallization the conditions are suitable for generating strong magnetic
fields. In our model we assumed that in addition to fast rotation at least 10
per cent and not more than 80 per cent of the core need to be crystallized for
the dynamo to be able to generate a strong magnetic field but this assumption
is rather arbitrary. In what follows we describe the three different types of
CVs that are non-magnetic according to our model and explain how the critical
core mass ratios and other important parameters impact the predicted relative
numbers of magnetic and non-magnetic CVs.

First (channel 1), our model predicts the white dwarfs in CVs that did not yet
accrete enough angular momentum remain non-magnetic. How long this phase
takes, and therefore also the relative number of CVs that are expected to be
found in this stage in observed samples, depends on the efficiency of the
accretion of angular momentum and the mass transfer rate (see equation 1 in
the main text) which in turn depends on the angular momentum loss. For
instance, CVs born with fully convective secondaries are expected to be driven
by gravitational radiation only and typically have mass transfer rates 1-2
orders of magnitude lower than systems with earlier secondary stars.
Therefore, the white dwarfs born with fully convective secondary stars will
take longer to spin up to near-break velocities than those with earlier
secondary stars that start mass transfer at longer orbital periods. However,
as the spin up time scale is in general short compared to the secular time
scale of CVs, we expect binary population synthesis based on our model to
predict rather few CVs to be in this non-magnetic CV state.

Second (channel 2), CVs with white dwarfs that did not yet start the
crystallization process in their cores remain non-magnetic CVs until the
required condition is reached. How long this takes determines how many of such
systems are predicted to exist in observed samples and depends on the fraction
of crystallized core-mass that is assumed to be necessary to make the dynamo
work (we assumed 10 per cent), on the time it takes the system to evolve from
the common envelope to the CV phase, and whether accretion can heat up the
core77,78 to avoid or delay crystallization. Assuming core heating is
inefficient, according to recent white dwarf models24 the time needed for a
0.6, 0.8 and 1.0 M⊙white dwarf to have 10 per cent of its interior
crystallized is approximately 3.2, 1.9 and 1.1 Gyr. This time span might be
substantially increased for smaller white dwarf masses as the magnetic field
needs to diffuse through a radiative zone which sits on top of the convective
mantle. Only for massive white dwarfs the convective mantle extends close to
the surface20. This implies that detached binaries that emerged from common
envelope evolution with short orbital periods, i.e. close to the start of mass
transfer, can remain non-magnetic CVs for
$\raisebox{-1.72218pt}{$\stackrel{{\scriptstyle>}}{{\scriptstyle\sim}}$}\,3$
Gyr. If core heating due to accretion is efficient and prevents the white
dwarf from crystallizing during the CV phase, these CVs will even remain non-
magnetic forever. Therefore a non-negligible number of non-magnetic CVs
descending from short period post common envelope binaries is predicted to
exist in observed samples of CVs.

Finally (channel 3), CVs containing white dwarfs that significantly
crystallized before being spun up to fast rotation due to accretion, will not
develop a strong magnetic field. In other words, if the white dwarf had enough
time too cool so that a large fraction of its core is crystallized before the
secondary fills its Roche-lobe, the convective energy flux in the outer core
will not be sufficient to generate a strong magnetic field. These systems will
remain non-magnetic forever under the assumption that core heating due to
accretion is inefficient. Even if accretion heating can melt a significant
fraction of the core, this process is supposed to take several Gyr78, i.e. the
system would remain non-magnetic for a large fraction of its CV lifetime.
Thus, whether a given system becomes a non-magnetic CV formed through this
channel depends mostly on the time it spends between the white dwarf
formation, i.e. the common envelope phase, and the onset of accretion. How
many systems of a predicted CV population spend enough time in this detached
white dwarf binary phase depends sensitively on the crystallized mass fraction
(we assumed 80 per cent) at which the dynamo becomes unable to generate a
strong magnetic field, on the inter-correlated initial orbital period, mass
ratio, and eccentricity distributions of the progenitor systems, on the common
envelope efficiency, and on the strength of angular momentum loss due to
magnetic braking.

While the relative number of non-magnetic CVs predicted by our model through
channel 1 is most likely very small, we expect that the other two groups of
systems can easily explain the $\sim\,50-70\,$ per cent of non-magnetic
systems in the observed CV sample. Which of the two scenarios contributes more
to the fraction of non-magnetic CVs depends on several assumptions. For
example, for a large common envelope efficiency, relatively inefficient
magnetic braking, and a critical crystallized mass fraction below 80 per cent,
the majority of systems will evolve through channel 3 and will remain non-
magnetic for at least several Gyr. In contrast, for a small common envelope
efficiency, efficient magnetic braking, a large value for the critical
fraction of crystallized core mass, and inefficient core heating, most CVs
will eventually become magnetic but many of them may spend several Gyr as non-
magnetic systems before the dynamo starts operating (channel 2).

The parameters (strengths of magnetic braking, critical crystallized mass
fractions, common envelope efficiency, efficiency of core heating) are
essential for predicting the evolution of close white dwarf binaries and for
determining the fraction of strongly magnetic CVs predicted by our model.
Unfortunately, several of these parameters are currently neither theoretically
nor observationally well constrained. The common envelope efficiency is most
likely relatively small as indicated by observational surveys of detached
white dwarf binaries73 and heating of the core is likely to play an important
role77,78. In contrast, the strength of magnetic braking is unknown6 and the
same is true for the critical mass fractions (as detailed simulations for the
dynamo we propose here do not yet exist). In addition, the initial binary
parameter distributions usually assumed in CV population models are
uncorrelated which is inconsistent with observational surveys performed in the
past decades76. These surveys show that the orbital period, primary mass, mass
ratio and eccentricity distributions are inter-correlated. This is important
because the combination of these parameters determines whether a given binary
system will survive or not the common-envelope phase and which of those that
survive and become CVs within the Hubble time, have relatively long or short
detached lifetimes (possibly becoming slightly or highly crystallized) before
the onset of mass transfer.

Based on binary population models we performed in the past74,75, using
standard parameters for CV evolution (common envelope efficiency of 0.25,
magnetic braking according to Rappaport et al.44, assuming the critical
crystallized mass fractions used in this paper (i.e. 10 and 80 per cent), and
ignoring core heating of the white dwarf through accretion, we estimate the
predicted fraction of magnetic systems to be roughly 50 per cent. While this
value slightly exceeds the currently observed number of magnetic CVs,
relatively small changes in the prescription for magnetic braking and/or the
critical mass fractions of crystallized matter in the core of the white dwarf
can easily bring into agreement the observed fractions and the predictions of
our model. For the same assumptions but including efficient core heating, the
predicted fraction of magnetic CVs decreases to $\sim\,20$ per cent. This
estimate is below the measured fraction but our conclusion remains unchanged
as relatively small changes in uncertain parameters such as critical core mass
and magnetic braking could increase the number of magnetic systems to match
the observed fraction.

For the sake of completeness, as an additional unknown parameter, we emphasize
that also the metallicity of the white dwarf progenitor star could play a role
since the settling of neutronized impurities like 22Ne and 56Fe caused by
solidification can release more energy than the separation of the C/O mixture
which significantly affects the cooling of the white dwarf and the
crystallization process79.

As a final remark, we state that the small number of observationally
discovered pre-polars and the even smaller number of AR Sco-like systems are
in agreement with the relatively short time scale that our model predicts for
these systems.

###  0.7 Implications for single white dwarfs and white dwarfs in wide
binaries.

The model presented in this paper describes a new evolutionary sequence for
close white dwarf binaries which explains the origin and the relative numbers
of strongly magnetic white dwarfs in these systems. While solving the puzzle
for the occurrence of strongly magnetic white dwarfs in close binaries
represents a key result on its own, we here briefly mention the implications
for single white dwarfs and white dwarfs in wide binary stars.

It is clear that the dynamo proposed here to operate in CVs cannot explain all
strongly magnetic single white dwarfs because it requires fast rotation and
crystallization. A non-negligible number of strongly magnetic single white
dwarfs7 are hot and therefore most likely not in the process of
crystallization and, even worse, several of the highest field single white
dwarfs are known to have rather long spin periods${}^{7},80$. While field
strengths of the order of $0.1-1$ MG might be generated in crystallizing
single white dwarfs with rotation periods of the order of hours or days,
strong magnetic fields in single white dwarfs must be generated at least
partly by a different mechanism. The same conclusion holds for the recently
discovered strongly magnetic white dwarfs in wide binaries81. In most cases
the separation between the two stars in these binaries exceeds several hundred
astronomical units and therefore a previous phase of mass transfer and spin-up
of the white dwarf can be excluded.

Similarly, the main mechanism for generating these strong magnetic fields in
young single white dwarfs and wide binary stars must not be operating in close
binaries as we would otherwise observe strongly magnetic white dwarfs in young
detached close white dwarf binaries. The candidate scenarios that may fulfill
this conditions are white dwarf mergers19, fossil fields17, or the common
envelope merger scenario18,84,85. Given that the most likely progenitors of
magnetic white dwarfs according to the fossil field scenario are main sequence
star mergers82,83, all three scenarios require single magnetic white dwarfs to
form from binary stars. This implies that all scenarios require initial triple
star configurations to produce strongly magnetic white dwarfs in binaries. As
triples are intrinsically less common than binary stars, all three models may
produce more strongly magnetic single white dwarfs than strongly magnetic
white dwarfs in close and wide binaries. Which of the mechanisms is most
suitable to explain magnetic single white dwarfs and magnetic white dwarfs in
wide binaries depends on the relatively poorly constraint distribution of
initial triple star configurations.

We conclude that the crystallization and rotation driven dynamo proposed here
explains the large number of magnetic white dwarfs in CVs, as well as the
existence and characteristics of AR Sco and the pre-polars, and might be
complemented by e.g. the double white dwarf merger and/or the common envelope
merger scenario which are good candidates for explaining strongly magnetic
single white dwarfs.

Supplementary Figure 1. Comparison of different evolutionary tracks with
observations. The detached phases (dashed lines) explain the existence of only
cool magnetic white dwarfs in detached binaries that are close to Roche-lobe
filling because the magnetic field was generated during the CV phase when the
white dwarf was old and crystallizing. The model also offers an explanation
for AR Sco which in fact is a progenitor of the pre-polars. In AR Sco, nicely
reproduced by the blue track, the secondary detached from its Roche-lobe but
spin and orbital motion are not yet synchronised. The subsequent detached
synchronised phase explains the orbital periods, Roche-lobe filling factors,
and white dwarf temperatures of pre-polars. To generate the tracks we only
changed the initial orbital period ($P_{\mathrm{i}}$) and initial secondary
mass ($M_{2}$), as well as the white dwarf spin period ($P_{\mathrm{WD}}$)
when the magnetic field is generated. The parameters we used were the
following $P_{\mathrm{i}}=0.14$ days, $M_{2}=0.17$ M⊙, $P_{\mathrm{WD}}=30$ s
(purple), $P_{\mathrm{i}}=0.68$ days, $M_{2}=0.55$ M⊙, $P_{\mathrm{WD}}=15$ s
(blue), $P_{\mathrm{i}}=0.925$ days, $M_{2}=0.75$ M⊙, $P_{\mathrm{WD}}=20$ s
(green), $P_{\mathrm{i}}=1.544$ days, $M_{2}=1.2$ M⊙, $P_{\mathrm{WD}}=30$ s
(orange). In all tracks the white dwarf mass was assumed to be $0.8$ M⊙29 and
the generated magnetic field strengths was assumed to be $60$ MG. The
errorbars on the observations correspond to 1$\sigma$.

Supplementary Figure 2. Evolutionary sequences highlighting the impact of
different choices of the synchronisation time scale. The shorter the time
scale, the more pronounced and the longer the detached phase. For the longest
time scales (blue, purple), accretion starts before the system synchronised.
Time scales exceeding $1$ Myr are most likely not realistic.

Supplementary Figure 3. Evolutionary sequences highlighting the impact of the
white dwarf magnetic field strength. The stronger the generated field, the
longer the detached phase as angular momentum loss due to magnetic braking is
more reduced. For the weakest field strengths (brown, red), a less pronounced
standard gap still occurs when the secondary becomes fully convective.

Supplementary Figure 4. Evolutionary sequences highlighting the impact of
different choices of the white dwarf spin period ($P_{\rm WD}$) at onset of
magnetic field generation. The hatched region indicates the locus of the
standard orbital period gap. In each track, solid and dashed lines correspond
to semi-detached and detached phases respectively. With the exception of the
case ${P_{\rm WD}=14}$ s (brown curve), a detached phase is generated above
the standard orbital period gap, at an orbital period $\approx 3.75$ hr. For
$P_{\rm WD}$ at least twice the break-up period, magnetic field generation
starts at the same period and the slower the spin, the shorter the detached
phase. To spin-up the white dwarf to velocities close to the break-up period
(brown track), the spin-up phase is much longer and therefore the pronounced
detached phase occurs at a significantly shorter orbital period.

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