mainReferences methodsReferences for Methods

# Observations of a black hole X-ray binary indicate formation of a
magnetically arrested disk

Bei You1,∗, Xinwu Cao2,∗, Zhen Yan3,∗, Jean-Marie Hameury4,
Bozena Czerny5, Yue Wu1,6,7, Tianyu Xia1,8,9, Marek Sikora10,
Shuang-Nan Zhang11,12, Pu Du11, Piotr T. Zycki 10

1: Department of Astronomy, School of Physics and Technology, Wuhan
University, Wuhan 430072, China
2: Institute for Astronomy, School of Physics, Zhejiang University, Hangzhou
310058, China
3: Shanghai Astronomical Observatory, Chinese Academy of Sciences (CAS),
Shanghai 200030, China
4: Observatoire Astronomique de Strasbourg, Université de Strasbourg and
Centre National de la Recherche Scientifique, Unite Mixte de Recherche 7550,
67000 Strasbourg, France
5: Center for Theoretical Physics, Polish Academy of Sciences, Warsaw 02-668,
Poland
6: School of Astronomy and Space Science, Nanjing University, Nanjing 210023,
China
7: Key laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, Nanjing 210023, China
8: CAS Key laboratory for Research in Galaxies and Cosmology, Department of
Astronomy, University of Science and Technology of China, Hefei 230026, China
9: School of Astronomy and Space Sciences, University of Science and
Technology of China, Hefei 230026, China
10: Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences,
Warsaw 00-716, Poland
11: Key Laboratory for Particle Astrophysics, Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China
12: University of Chinese Academy of Sciences, Chinese Academy of Sciences,
Beijing 100049, China

Corresponding authors
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
> Accretion of material onto a black hole drags any magnetic fields present
> inwards, increasing their strength. Theory predicts that sufficiently strong
> magnetic fields can halt the accretion flow, producing a magnetically
> arrested disk (MAD). We analyze archival multi-wavelength observations of an
> outburst from the black hole X-ray binary MAXI J1820+070 in 2018. The radio
> and optical fluxes are delayed by about 8 and 17 days respectively, compared
> to the X-ray flux. We interpret this as evidence for the formation of a MAD.
> In this scenario, the magnetic field is amplified by an expanding corona,
> forming a MAD around the time of the radio peak. The optical delay is then
> due to thermal viscous instability in the outer disk.

A black hole X-ray binary (BHXRB) is a binary system that consists of a black
hole (BH) and a normal star, which emits large amounts of radiation in the
X-ray band. The gas of the star drawn by the strong gravity flows towards the
BH and forms an accretion disk (namely accretion flow). In the disk, matter
moves inwards under the effect of viscosity. Relativistic jets can be launched
near a BH due to magnetic fields carried by the accretion flow in the vicinity
of the BH (?, ?). Numerical simulations have shown that any large-scale
magnetic field present in the accreting material can be dragged inwards
through the accretion flow, increasing its strength. If the radial magnetic
force becomes sufficiently high that it equals the gravitational force of the
BH, the accretion flow halts and a magnetically arrested disk (MAD) is formed
(?, ?).

Sufficiently strong magnetic field strengths to produce a MAD have been
observationally inferred in accretion systems with supermassive BHs (?, ?, ?,
?). MADs are also predicted in BHXRBs, though there is little observational
evidence for these (?). The formation process of a MAD has not been observed
in either type of BH (supermassive or stellar-mass). Accretion flows onto
stellar-mass BHs can evolve much faster than those around supermassive BHs.
For example, outbursts in BHXRBs last for months to years, potentially
allowing multi-wavelength observations to probe the formation of a MAD.

The outbursts of BHXRB are well explained by the disk-instability model (?).
In quiescence, mass accumulates in a cold, geometrically thin accretion disk
(i.e. thin disk) until a thermal-viscous instability is triggered when,
somewhere in the disk, the temperature reaches the hydrogen ionization
temperature. An outburst then starts and the disk empties. A BHXRB usually
experiences two very different spectral states during an outburst, i.e. a hard
state and a soft state, with the X-ray emission dominated by either hard
($>$10 keV) or soft ($<$ 10 keV) X-rays. The timing, and radio emission
properties also differ between the two states (?, ?). The period between the
two states is called state transition.

In the hard state, the hot gas near the BH forms the corona which is modeled
as an advection-dominated accretion flow (ADAF), since a significant fraction
of the viscously released gravitational energy is advected with the flow and
enters the event horizon (?). Hard states are observed during the rise and
decay of an outburst, and are referred to as the rising and decaying hard
state in the following. The soft-to-hard state transition and the decaying
hard state are accompanied by a hard X-ray flare. The flare is also detected
in the optical, infrared, and radio bands (?, ?), which indicates the
formation of a jet. However, the detailed mechanisms and processes are still
unclear. Hard X-rays during the flare are usually thought to be emitted by
inverse Compton scattering in the jet and/or in the ADAF (?).

### An outburst of MAXI J1820+070 in 2018

MAXI J1820+070 (also known as ASASSN-18ey) is an X-ray binary consisting of a
low-mass star orbiting a stellar-mass BH (BH mass $M_{\rm BH}\simeq
8.5M_{\odot}$), with an orbital period of 16.5 hours (?). It was discovered
during an X-ray outburst that occurred on 2018 March 11 (?). It is located at
a distance of $2.96\pm 0.33$ kiloparsecs (kpc) (?). The outburst was observed
by several X-ray missions (?, ?, ?, ?). We analyze archival observations by
the Insight-Hard X-ray Modulation Telescope [Insight-HXMT, (?)]. The Insight-
HXMT data cover a broad energy band of 1–250 keV, from 2018 March 14 to 2018
October 21 corresponding to modified Julian date (MJD) 58191-58412 (?, ?).
MAXI J1820+070 was also an optically bright transient (?), so was monitored by
the American Association of Variable Star Observers (AAVSO) which produced a
light curve (brightness as a function of time) from 2018 March 12 to 2018
December 21 (MJD 58188-59184) (?). Radio emission from MAXI J1820+070 was also
detected by multiple telescopes, indicating the presence of a jet. We use the
radio data from Arcminute Microkelvin Imager Large Array (AMI-LA) (?). These
datasets provide high-cadence monitoring of the 2018 outburst of MAXI
J1820+070 over a broad range of wavelengths. The proximity of the source and
its low interstellar extinction of E(B-V)=0.20–0.26 (?) made it sufficiently
bright to investigate the accretion process of the magnetized flow near a BH.

Fig. 1 shows the Insight-HXMT light curves from the low energy (LE, 1–10 keV),
medium energy (ME, 10–30 keV), and high energy (HE, 27–150 keV) instruments,
between MJD 58191 and 58412. We focus our analysis on the radio, optical, and
X-ray observations of MAXI J1820+070 during the hard X-ray flare including the
soft-to-hard state transition and decaying hard state periods (MJD
58380–58412).

### Lags between flares of different wavelengths

Fig. 2A shows the X-ray luminosity of MAXI J1820+070 during the flare, which
we determined from the Insight-HXMT data (?). The luminosities of the disk and
Comptonization components derived from X-ray spectral fitting are also shown.
Fig. 2B shows the radio brightness at 15.5 GHz derived from AMI-LA
observations, which exhibits a radio flare. The optical ($V$-band) data from
AAVSO are shown in Fig. 2C, which also exhibits a flare superimposed on the
exponential decay of the underlying outburst.

We performed an interpolated cross-correlation function (ICCF) analysis (?) to
determine the time lag between the radio and X-ray light curves, finding a
radio lag of $8.06^{+0.88}_{-0.56}$ days (Fig. S1A). ICCF analysis comparing
the radio and the $V$-band shows an optical lag of $8.61^{+1.06}_{-0.83}$ days
with respect to the radio, indicating negligible contribution from the jet to
the $V$-band flux (see Supplementary Text). ICCF analysis comparing the X-ray
and the $V$-band shows an optical lag of $17.61^{+0.50}_{-0.44}$ days with
respect to the X-ray (Fig. S1B). This lag is far longer than the time taken
for light to travel across the binary, which is several seconds for an orbital
period of 16.5 hours, so cannot be due to reprocessed X-ray illumination of
the outer disk or the companion star. The uncertainties of the ICCF analysis
above are quoted at the 1-$\sigma$ level. Besides the cross-correlation
analysis of those light curves shown above, the correlations between the X-ray
and radio/V-band luminosities are also used to study the origins of the
emission (?).

## Interpretation of the radio time lag

The hard X-ray emission during this flare comes from soft X-ray photons
inverse Compton scattered by hot electrons (i.e., Comptonization) within the
jet or the ADAF. Given that (i) the radio emission originates from the jet
(?), and (ii) the radio flare lags the X-ray by $\sim$ 8 days, we rule out the
possibility that X-rays are emitted by the jet, so suggest that the hard
X-rays were most likely emitted by the ADAF.

In the hard state of BHXRB, a geometrically thick ADAF is thought to locate
close to BH, extending outwards and connecting to a thin disk with a truncated
inner radius $R_{\rm tr}$ (equals to the ADAF radial extent) (?, ?). We assume
the truncated radius was $R_{\rm tr}=R_{\rm ISCO}$ at $t_{0}$ = MJD 58381 when
the soft state ends, where $R_{\rm ISCO}$ is the innermost stable circular
orbit (ISCO). Based on the derived disk luminosity which is estimated from the
spectral fitting (?), we find that the truncation radius increases to $R_{\rm
tr}=6.79R_{\rm ISCO}$ when the hard X-ray emission peaks at $t_{1}$ = MJD
58389, and further increases to $R_{\rm tr}=32.83R_{\rm ISCO}$ when the radio
emission peaks on $t_{2}$ = MJD 58397 (?). $R_{\rm tr}$ therefore increases
with decreasing mass accretion rate. In this case, the fraction of the total
gravitational power dissipated in the whole ADAF increases with $R_{\rm tr}$,
while the power in the very inner part of ADAF, where the hard X-ray mainly
comes from, decreases with decreasing accretion rate [ (?), Equation S5].
These two mechanisms compete with each other, so a peak in the hard X-ray
emission occurs when the latter mechanism becomes dominant, around $t_{1}$ =
MJD 58389.

Previous studies have suggested that a weak and coherent external magnetic
field can be dragged inwards and amplified by an ADAF, due to its high radial
velocity (?, ?). The external magnetic field is expected to be more strongly
amplified by larger ADAFs (?). We therefore calculated the advection and
diffusion of magnetic fields in the ADAF of MAXI J1820+070 (?). We find the
magnetic field at $R_{\rm ISCO}$ is $B\simeq 2\times 10^{6}~{}\rm G$ at
$t_{0}$ = MJD 58381 (the onset of the flare), $B\simeq 4.5\times 10^{7}~{}\rm
G$ at $t_{1}$ = MJD 58389 (the peak of the hard X-ray emission), saturates at
$B\simeq 7\times 10^{7}~{}\rm G$ around $t_{2}$ = MJD 58397, and then
declines. The jet power increases with the magnetic field strength near the
BH, so its radio emission also increases until $t_{2}$ = MJD 58397, explaining
the observed delay of $\sim 8$ days with respect to the X-ray emission. The
jet power at the radio peak is estimated as $10^{37}$ erg s-1, consistent with
the theoretical prediction from the magnetic field of $7\times 10^{7}~{}\rm
Gauss$ (?), which supports our interpretation that the magnetic field is
transported and amplified by the expanding ADAF.

### Formation of a magnetically arrested disk

In our interpretation, the increasingly strong magnetic field influences the
subsequent accretion flow, because the magnetic pressure acts against the
gravity of the BH. According to our calculations of the radius-dependent
magnetic fields within the ADAF (?), the BH gravity still dominates over
magnetic pressure at the hard X-ray peak ($t_{1}$ = MJD 58389). After the hard
X-ray peak, the magnetic field inside the expanding ADAF continues to be
amplified, eventually becoming dominant at the inner edge, so a MAD forms at
the time of the radio peak ($t_{2}$ = MJD 58397) when the magnetic pressure
becomes equal to gravity force [ (?), Fig. S5].

We conclude that the increase of the truncated radius during the flare in MAXI
J1820+70 leads to the observed 8-day radio lag relative to the hard X-ray
emission. The MAD is then gradually established in the ADAF as it continues to
extend radially. Fig. 3 illustrates the physical process associated with the
X-ray/radio flare and the formation of the MAD in our proposed scenario. An
animated version is provided in Movie S1.

### Interpretation of the optical time lag

We consider whether the long delay time of the optical flare can be produced
by a viscously heated thin disk. In this scenario, the X-ray flare heats up
the outer thin disk, reviving the thermal-viscous instability in the thin
disk, which produces a delayed optical emission. We use a disk instability
code (?) to determine the time-dependent evolution of the thin disk. The
results are shown in Fig. 4. We find a small optical flare is produced,
starting around MJD 58390 but fading away after a few days, which is much
shorter than the observed lag of $\sim$17 days. Therefore, the standard disk
instability model (DIM) cannot explain this optical lag.

A thin disk is expected to launch disk winds, which has previously been
studied using observations and simulations (?, ?). Previous work has found
that the disk wind is present throughout the outburst of MAXI J1820+070 (?).
We therefore postulate that the disk wind is launched during the flare. Disk
winds affect the profile of the outburst light curves of BHXRBs (?, ?),
because the disk wind removes angular momentum from the disk, which is
equivalent to increasing the viscosity. As a result, a brighter and more
delayed optical peak is produced comparing to the case of no disk wind. Using
a simple parametrization of the disk wind, we performed additional simulations
of the DIM (?). We find that the resulting $V$-band flare is consistent with
the observed light curve, and lags the hard X-rays by more than 15 days (Fig.
4).

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## Acknowledgments

We thank Wen-Fei Yu, Jian-Feng Wu, Fu-Guo Xie, Wei-Min Gu and Mou-Yuan Sun for
the discussions. BY thanks Yong-Kang Zhang from Jinan University for the help
in producing the animation. Funding: B.Y. is supported by the National Program
on Key Research and Development Project 2021YFA0718500; by NSFC grants
12273026 and U1931203; by Natural Science Foundation of Hubei Province of
China 2022CFB167; by the Fundamental Research Funds for the Central
Universities 2042022rc0002; by Xiaomi Foundation / Xiaomi Young Talents
Program. X.C. is supported by the NSFC grants 11833007, 12073023, 12233007,
and 12147103; by the science research grants from the China Manned Space
Project with No. CMS-CSST- 2021-A06; by the fundamental research fund for
Chinese central universities (Zhejiang University). Z.Y. is supported by the
Natural Science Foundation of China grants U1838203 and U1938114; the Youth
Innovation Promotion Association of CAS id 2020265; by the funds for key
programs of Shanghai astronomical observatory. S.-N.Z. is supported by the
National Program on Key Research and Development Project Grant No.
2016YFA0400802; by International Partnership Program of Chinese Academy of
Sciences (Grant No.113111KYSB20190020). P.D. is supported by NSFC grants
12022301 and 11991051. Author contributions: B.Y. initiated the project,
performed the data analysis and model interpretation, and led the writing of
the text. X.C. led the interpretation of the MAD and the writing of the
relevant text. Z.Y. contributed to the data analysis, model discussion, and
the writing of the text. J.-M.H. provided the DIM code and contributed to the
model discussion and the writing of the text. B.C. contributed to the
discussion of the DIM. Y.W. and T.X. contributed to the simulation of the DIM
and the data analysis. M.S., S.-N.Z., P.D. and P.Z. contributed to the model
discussion and the writing of the text. Competing interests: The authors
declare no competing interests. Data and materials availability: All Insight-
HXMT data used in this work (Proposal ID: P0114661) are publicly available and
can be downloaded from the official website: http://archive.hxmt.cn/. The
radio luminosities are taken from source data which are publicly available
(?), but are corrected for the source distance $d=2.96$ kpc in this work. All
the reduced data (including LE/ME/HE light curves and X-ray/Radio/$V$-band
luminosities), the scripts which perform the ICCF analysis and plot the
figures, and the numerical codes for computing the magnetic field strength
(MAD) and the disk instability (DIM), are available at https://github.com/Bei-
You-BH/MAD, which are archived at Zenodo (?). The algorithm linmix we used for
fitting the correlations is available at https://github.com/jmeyers314/linmix.

Figure 1:  Insight-HXMT light curves of MAXI J1820+070 during the 2018
outburst. The orange, black and skyblue points, correspond to the light curves
by HE (27–150 keV), ME (10–30 keV), and LE (1–10 keV), respectively. The
unshaded region, the light purple region, and the light grey region indicate
the periods in which the source was in the rising hard state, the hard-to-soft
transition state, and the soft state, respectively. The flare, covering the
soft-to-hard transition state and the decaying hard state, is marked by the
light orange region. The error bars, which are too small to be seen for most
data, correspond to a 1-$\sigma$ confidence interval.  Figure 2:  Multi-
wavelength light curves of the flare. A distance of $d=2.96$ kpc (?) is
adopted in estimating the observed luminosity. (A) The thermal luminosity from
the thin disk, the Comptonization luminosity from the ADAF, and the total
luminosity, in the 0.1-100 keV band, as determined by our decomposition of the
Insight-HXMT data (?). (B) The radio emission at 15.5 GHz measured by AMI-LA.
(C) The extinction-corrected $V$-band luminosities provided by AAVSO are
represented by the gray points. The dashed cyan line is an exponential model
fitted to the gray points between MJD 58380-58390. The orange points are the
$V$-band luminosity between MJD 58395-58440 after subtracting the exponential
model. The uncertainties, which are too small to be seen, are quoted at the
1-$\sigma$ level.  Figure 3:  Schematic diagram of our proposed
interpretation. The outer thin disk is truncated at a radius $R_{\rm tr}$,
within which is the ADAF. (A) The configuration at the onset of the flare,
when the thin disk is assumed to extend to ISCO and the ADAF starts to expand
its size with increasing $R_{\rm tr}$. (B) Configuration during the peak hard
X-ray emission, when the outer thin disk brings both mass and magnetic field
into the inner ADAF. (C) Configuration at the time of peak radio emission,
when the sufficient magnetic field is archived through amplification by the
ADAF. A MAD forms within the magnetospheric radius $R_{\rm m}$ where the
magnetic pressure becomes equal to gravity force (?). Movie S1 shows an
animated version of this figure. Figure 4:  Observed $V$-band flare compared
to our simulations. The $V$-band luminosity is from Fig. 2C (gray points). The
red curve is the simulated $V$-band luminosity from the DIM simulation
including the disk wind. The dashed orange line is the same equivalent
simulation without the disk wind.

Supplementary Materials for

Observations of a black hole X-ray binary indicate formation of a magnetically
arrested disk

Bei You∗, Xinwu Cao∗, Zhen Yan∗, Jean-Marie Hameury,

Bozena Czerny, Yue Wu, Tianyu Xia, Marek Sikora,

Shuang-Nan Zhang, Pu Du, Piotr T. Zycki

∗ Corresponding author.

Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
This PDF file includes:

Materials and Methods
Supplementary Text
Figures S1 - S8
Caption for Movies S1
Reference 41-97

Other Supplementary Materials for this manuscript :

Movies S1

##  S1 Materials and Methods

###  S1.1 Observations and data reduction

The archival observation of MAXI J1820+070 by the Insight-HXMT was performed
from 2018 March 14 (MJD 58191) to 2018 October 21 (MJD 58412). The energy
spectra from LE, ME and HE instruments are extracted following the same
procedure as in the ref (?). The broad (2-150 keV) Insight-HXMT energy spectra
during the flare (MJD 58380–58412) are well fitted with a model
tbabs*(diskbb+nthcomp) in XSPEC (?), where tbabs is the absorption component
with a fixed column density $N_{\rm H}=0.15\times 10^{22}$ cm-2, diskbb is the
multiple blackbody component from the thin disk, and nthcomp is the
Comptonization component from the ADAF. As an example, the X-ray spectrum and
the corresponding best-fitting model during the hard X-ray peak at MJD 58389
are plotted in Fig. S2. We exclude the energy range 21–24 keV, since the
fluorescence lines around 21–24 keV due to the photoelectric effect of
electrons in Silver K-shell are detected by the Si-PIN detectors of ME. Then,
we use the cflux in XSPEC to estimate the unabsorbed disk and Comptonization
luminosity, and the total luminosity in 0.1-100 keV during the flare. The disk
luminosity continuously decreases with time, while the Comptonization
luminosity peaks on MJD 58389 (Fig. 2A). The energy spectrum, i.e.,
$N(E)\propto E^{-\Gamma}$, becomes harder as the photon index $\Gamma$
decreases from about 2.4 to 1.6.

The reflection component is not included in the spectral fitting, since it is
too weak to be detected by the Insight-HXMT during the flare. There are three
observations during this flare by Nuclear Spectroscopic Telescope Array
(NuSTAR) with much better sensitivity (?). The reflection component in all the
three NuSTAR spectra is roughly 5-20 times weaker than the Comptonization
component.

The radio flux (15.5 GHz) is directly obtained from the public work (?), which
is reduced from the observations by AMI-LA. We just use the new distance
measurement of 2.96 kpc to recalculate the radio luminosity. The radio
luminosity $L_{\mathrm{R}}$ during the flare is plotted in Fig. 2B.

The archival optical light curve is from the AAVSO. We select the $V$-band
data with uncertainties $<$ 0.02 mag. We first converted the Johnson
magnitudes to AB magnitudes (?), and calculated the monochromatic flux
according to $m_{\mathrm{AB}}=-2.5\ln f_{\mathrm{V}}-48.6$ (?). We use
$E(B-V)=0.23$ (?) to obtain the extinction correction flux. In order to study
the flare on a daily basis, we rebined the $V$-band light curve with a bin
size of 1 day, and the uncertainties are calculated by error propagation. The
light curve in the $V$-band during the flare is plotted in Fig. 2C.

###  S1.2 Determination of time lags using the interpolated cross-correlation
function

The time lags between the radio/optical/X-ray light curves are measured using
the ICCF which has previously been used in the time-series analysis of
reverberation mapping of AGNs (?, ?). The correlation coefficient $r$
describes the strength of the correlation. The peak delays are obtained from
the location with the maximum correlation coefficient $r_{\rm max}$, and the
centroid delays are measured from the centroid of ICCF above a threshold
($r>0.8r_{\rm max}$). The uncertainties of lags are determined by the flux
randomization/random subset sampling (FR/RSS) method (?), which takes into
account both the measurement uncertainties and the uncertainties resulting
from the sampling cadence. In general, the FR/RSS method generates a subsample
of the light curves by randomly selecting the points from the light curves and
dithering the points according to their error bars. The centroid and peak lag
distributions can be obtained from the ICCF of the subsamples. The confidence
intervals of the lags can be determined using the centroid and peak lag
distributions.

The $V$-band light curve contains the contribution of the underlying
exponential decay from the main outburst as well as that of the flare. To
subtract the contribution of the flux level from the underlying decay to this
flare between MJD 58395 and 58440, we fit an exponential function between MJD
58380 and MJD 58390, then subtract it from the observed luminosity after MJD
58395. We use the subtracted light curve to estimate the lag between the X-ray
and $V$-band flares.

We apply the ICCF to the multi-wavelength light curves in Fig. 2, and the
results are plotted in Fig. S1. We find that: i) the radio emission lags the
X-ray emission by the peak time delay $\tau_{\rm RX}^{\rm
p}=8.06^{+0.88}_{-0.56}$ days and the centroid delay $\tau_{\rm RX}^{\rm
c}=8.32^{+1.31}_{-0.48}$ days; ii) the optical $V$-band emission lags the
X-ray emission by the peak time delay $\tau_{\rm OX}^{\rm
p}=17.61^{+0.50}_{-0.44}$ days and the centroid delay $\tau_{\rm OX}^{\rm
c}=17.30^{+0.41}_{-0.56}$ days; iii) the optical $V$-band emission lags the
radio emission by the peak time delay $\tau_{\rm OR}^{\rm
p}=8.61^{+1.06}_{-0.83}$ days and the centroid delay $\tau_{\rm OR}^{\rm
c}=8.80^{+0.64}_{-0.94}$ days. The peak and centroid time delays are
consistent within uncertainties. We report the peak time delay of $\tau_{\rm
RX}^{\rm p}=8.06^{+0.88}_{-0.56}$ days, $\tau_{\rm OX}^{\rm
p}=17.61^{+0.50}_{-0.44}$ days and $\tau_{\rm OR}^{\rm
p}=8.61^{+1.06}_{-0.83}$ days in the main text.

###  S1.3 Formation of the MAD within the expanding ADAF

The LE X-ray light curve during the soft state between MJD $\sim$ 58330-58380
is used to estimate the mass accretion rate of the gas fed to the thin disk
over time, i.e., $\dot{M}_{\rm d}(t)$. We fitted the LE X-ray light curve
during this soft state with an exponential function
$\propto\exp[-(t-t_{\bullet})/\tau]$ (?), finding $t_{\bullet}=58330.72$ and
$\tau=57.61$ days. During the soft state, the thin disk approaches the ISCO of
the BH, which indicates the mass accretion rate also decays exponentially with
the same $t_{\bullet}$ and $\tau$ (?).

We assume that the disk luminosity $L_{\rm d}(t_{0})=\eta\dot{M}_{\rm
d}(t_{0})c^{2}$ at $t_{0}$ = MJD 58381, where the radiation efficiency
$\eta=$0.1, a typical value adopted for a thin disk (?), and $c$ is the speed
of light. Then we estimate the Eddington-scaled accretion rate of the thin
disk $\dot{m}_{\rm d}=\dot{M}_{\rm d}/\dot{M}_{\rm Edd}$ (Eddington accretion
rate $\dot{M}_{\rm Edd}=L_{\rm Edd}/\eta c^{2}$, where Eddington luminosity
$L_{\rm Edd}=1.3\times 10^{38}M_{\rm BH}/M_{\odot}~{}\rm erg~{}s^{-1}$), which
gives $\dot{m}_{\rm d}(t_{0})=0.0474$. We assume $\dot{m}_{\rm d}(t)$ during
the flare follows the same exponential decay as in the soft state (?, ?), and
then determine the mass accretion rates of the thin disk, $\dot{m}_{\rm
d}(t_{1})=0.041$ and $\dot{m}_{\rm d}(t_{2})=0.036$.

With the mass accretion rate derived above and the observed disk luminosity
provided in Section S1.1, the truncation radius $R_{\rm tr}$ of the outer thin
disk can be estimated. The luminosity of a truncated disk is

$L_{\rm d}(t)\propto{\frac{\dot{m}_{\rm d}(t)}{R_{\rm tr}(t)}}.$ (S1)

At $t=t_{0}$, we assume the disk extends to the ISCO $R_{\rm tr}(t_{0})=R_{\rm
ISCO}$. The truncated radius ${R_{\rm tr}(t)}$ for $t>t_{0}$ is estimated from
the observed disk luminosity using

${\frac{R_{\rm tr}(t)}{R_{\rm ISCO}}}\simeq{\frac{\dot{m}_{\rm d}(t)L_{\rm
d}(t_{0})}{\dot{m}_{\rm d}(t_{0})L_{\rm d}(t)}}.$ (S2)

We find that $R_{\rm tr}(t_{1})/R_{\rm ISCO}\simeq 6.79$, and $R_{\rm
tr}(t_{2})/R_{\rm ISCO}\simeq 32.83$. The values of the truncation radius
$R_{\rm tr}$ during the flare are plotted in Fig. S3.

The Comptonization component predominantly originates from the inner hot ADAF,
which we use to set constraints on the properties of the ADAF. Outflows are
thought to be driven from the hot ADAF (?, ?), and the mass accretion rate
varies with radius; we model the Eddington-scaled accretion rate of the ADAF
$\dot{m}_{\rm ADAF}$ as

$\dot{m}_{\rm ADAF}(R)=\dot{m}_{\rm d}\left({\frac{R}{R_{\rm
tr}}}\right)^{p_{\rm w}},$ (S3)

where a typical value of $p_{\rm w}=0.3$ is adopted to describe the radial
dependence of the accretion rate of the ADAF due to the outflow (?). Since the
mass accretion rate $\dot{m}_{\rm d}(t_{2})$ at the time when the radio
emission peaked ($t=t_{2}$) is only slightly lower than $\dot{m}_{\rm
d}(t_{0})$, the radiation efficiency of the ADAF is roughly the same as that
of a thin disk at $t=t_{0}$ (?). We therefore adopt a constant radiation
efficiency. This implies that the Comptonization component emission is partly
regulated by the outflows that reduce the mass accretion rate in the inner
region of the ADAF, where most Compton scattered photons are emitted.

A fraction of gravitational energy released in the ADAF is carried away by the
outflows. In the presence of a strong large-scale magnetic field threading the
ADAF, we assume that such outflows are predominantly driven by the magnetic
field. In this case, the magnetically driven outflows not only carry away the
energy but also the angular momentum of the ADAF, which leads to an increase
of the radial velocity of the accretion flow $v_{\rm R}$ by a factor $f_{\rm
m}$,

$v_{\rm R}=(1+f_{\rm m})v_{\rm R,\rm vis},$ (S4)

where $v_{\rm R,vis}$ is the velocity of the pure viscously driven ADAF [ (?),
their equation 16]. Only $\sim 1/(1+f_{\rm m})$ of the kinetic energy
dissipated in the accretion flow is radiated out, while a fraction $\sim
f_{\rm m}/(1+f_{\rm m})$ of the kinetic energy of the ADAF is tapped to
accelerate the outflows (?). Thus, the Comptonization luminosity normalized to
the Eddington value, $\lambda_{\rm C}$ is approximated by

$\lambda_{\rm C}(t)\equiv{\frac{L_{\rm C}(t)}{L_{\rm
Edd}}}\sim{\frac{\dot{m}_{\rm d}(t)-\lambda_{\rm d}(t)}{1+f_{\rm
m}}}\left[{\frac{2.25R_{\rm ISCO}}{R_{\rm tr}(t)}}\right]^{p_{\rm w}},$ (S5)

where $L_{\rm C}$ is the Comptonization luminosity, $\lambda_{\rm d}\equiv
L_{\rm d}/L_{\rm Edd}$ is the observed Eddington ratio of the disk luminosity,
and Equation S3 is used. $\dot{m}_{\rm d}=\lambda_{\rm d}$ applies only in the
condition of the disk not being truncated, i.e. $R_{\rm tr}=R_{\rm ISCO}$. In
our scenario, the truncation radius increases with time, so $\dot{m}_{\rm
d}>\lambda_{\rm d}$. The radiation from the region with $R\sim 2.25R_{\rm
ISCO}$ contributes to most of the accretion flow luminosity (?). Using the
derived $\dot{m}(t_{2})=0.036$, $R_{\rm tr}(t_{2})/R_{\rm ISCO}\simeq 32.83$,
and the observed $L_{\rm C}(t_{2})$, Equation S5 gives $f_{\rm m}\simeq 0.92$.
From Equation S1 and S5, the Comptonization luminosity $L_{\rm C}$ is
approximately proportional to $\dot{m}_{\rm d}(1-R_{\rm ISCO}/R_{\rm
tr})R_{\rm tr}^{-p_{\rm w}}$, which depends on $\dot{m}_{\rm d}$ and the
relevant $R_{\rm tr}$. As $R_{\rm tr}$ increases with time, the Comptonization
luminosity $L_{\rm C}$ increases because of $1-R_{\rm ISCO}/R_{\rm tr}$, while
it decreases due to $R_{\rm tr}^{-p_{\rm w}}$ instead. Therefore, the
dependence of the Comptonization luminosity on $\dot{m}_{\rm d}$ and $R_{\rm
tr}$ leads to the observed hard X-ray flare.

The gas and magnetic field are fed into the inner ADAF from the outer thin
disk (?, ?, ?). The delayed radio flare implies an increase in the jet power
in this source after the X-ray flare. For X-ray binaries, the magnetic field
of the donor star could be the source of the external magnetic field, as has
been proposed in the high-mass X-ray binary Cyg X-3 (?). That system contains
a Wolf-Rayet star, which could have a strong magnetic field with several
hundred Gauss at its surface (?). However, the magnetic field strength of the
K5 companion star in MAXI J1820$+$070 is likely a few Gauss (?), which is too
low for advection in the ADAF (?). Therefore, the external magnetic field must
be maintained by the outer thin disk.

We consider two mechanisms for generating a coherent magnetic field in the
outer thin disk. The first one is the inverse-cascade of the dynamo process in
a thin disk (?). If the whole magnetic field loop generated in the outer disk
is dragged by the ADAF, then no MAD would be formed near the BH, because no
net flux is added to the ADAF. However, the size of magnetic field loops
formed by this mechanism is about the disk radius (?), so it is possible that
the field line threading the inner ADAF connects to the outer thin disk as a
whole loop. The field line threading the ADAF can then be advected to the BH,
while the other end of the field line connects to the outer thin disk (or even
diffuses out). A similar phenomenon has been observed in some numerical
simulations (?). The second potential mechanism is based on numerical
simulations (?). Those simulations found that the turbulence in a radially
extended thick disk can generate large-scale poloidal magnetic flux in situ
with a relatively strong initial toroidal magnetic field, which then
accumulates to form a MAD state near the BH by accretion. The simulations were
carried out for a thick disk; the magnetic field generation in thin disks is
not as efficient as their thick counterparts (?).

No quantitative relation between the disk properties and the large-scale
magnetic field strength has been determined in numerical simulations for a
thin disk. We therefore assume that a weak coherent magnetic field is
generated through the inverse-cascade of the dynamo process in the outer disk.
Thus, the magnetic field advection/diffusion in the ADAF can be calculated.
The magnetic field strength $B_{\rm tr}$ in the region near $R\sim R_{\rm tr}$
of the outer thin disk is estimated as [ (?), their equation 27]

$B_{\rm tr}\sim 2.48\times 10^{8}\alpha^{-1/20}m^{-11/20}\dot{m}_{\rm
d}^{3/5}R_{\rm tr}^{-49/40}~{}{\rm Gauss},$ (S6)

where $m=M_{\rm BH}/M_{\odot}$, and $\alpha$ is the viscosity parameter with a
typical value $\sim 0.1$–0.4 (?). The ratio of gas to magnetic pressure is

$\beta={\frac{8\pi p_{\rm gas}}{B_{\rm tr}^{2}}}=2.533\times
10^{3}\alpha^{-4/5}m^{-2/5}\dot{m}_{\rm d}^{1/5}R_{\rm tr}^{-1/10},$ (S7)

where $p_{\rm gas}$ is the gas pressure. Using the derived values of the mass
accretion rate in the disk, we estimate $\beta\sim 2.6\times 10^{3}$ at
$t_{1}$, and $\beta\sim 2.1\times 10^{3}$ at $t_{2}$ for $\alpha=0.1$. This
implies that the field generated in the outer thin disk is weak.

In principle, we can calculate the magnetic field advection/diffusion based on
a global solution of the ADAF with magnetically driven outflows (?, ?). The
dynamics of the ADAF can be described by a self-similar solution (?), from
which the radial velocity of a pure viscously driven ADAF $v_{R,\rm
vis}\simeq-3\nu/2R$, where $\nu$ is the turbulent viscosity. The relative disk
thickness $H/R$ of the ADAF is $\sim 1$ (?), which is adopted to calculate
$v_{R,\rm vis}$ and then the radial velocity of the ADAF with magnetic
outflows using Equation S4. With the derived radial velocity of the ADAF, the
field advection and diffusion in the ADAF can be calculated when the value of
the Prandtl number and the external coherent magnetic field strength are
specified (?, ?). The magnetic Prandtl number $P_{\rm m}\equiv\nu/\eta_{\rm
m}$ ($\eta_{\rm m}$ is the magnetic diffusivity) is found to be around unity,
or $\sim 2-5$ with shearing box simulations (?, ?, ?, ?).

In the inner region of the ADAF, the magnetic field needs to be enhanced until
it dominates the dynamics, to produce the MAD (?, ?). We estimate the magnetic
field strength $B_{\rm MAD}$ required for a MAD state (?),

$B_{\rm MAD}\sim 1.5\times
10^{9}(1-f_{\Omega})^{1/2}\epsilon^{-1/2}m^{-1/2}\dot{m}^{1/2}_{\rm
ADAF}(R)R^{-5/4}~{}~{}{\rm Gauss},$ (S8)

where $\epsilon\equiv v_{\rm R}/v_{\rm K}\sim 10^{-2}$ to $0.1$ (?, ?, ?) and
$v_{\rm K}$ is the Keplerian velocity; $f_{\Omega}=\Omega/\Omega_{\rm K}$
($\Omega$ is the angular velocity and $\Omega_{\rm K}$ is the Keplerian one).
In this case, we assume that the ADAF with magnetic outflows is arrested in
the inner region (interior to a magnetospheric radius $R_{\rm m}$, where the
magnetic pressure becomes equal to gravity force (?)) of the ADAF, of which
the radial velocity is described by

$v_{R}=\left\\{\begin{array}[]{l}(1+f_{\rm m})v_{R,\rm
vis}=-{\frac{3}{2}}(1+f_{\rm m})\alpha c_{\rm s}{\frac{H}{R}},~{}~{}~{}{\rm
if}~{}~{}R>R_{\rm m};\\\ -\epsilon v_{\rm K},~{}~{}~{}{\rm if}~{}~{}R\leq
R_{\rm m},\end{array}\right.$ (S9)

where $c_{\rm s}$ is the sound speed. We calculate the field dragged inwards
by the ADAF with the estimated value of $f_{\rm m}$, when the values of $H/R$,
$\epsilon$ and $R_{\rm m}$ are specified (?). The external coherent field
strength is given by Equation S6, and we tune the value of $R_{\rm m}$ until
the inner MAD satisfies the field strength required for a MAD (Equation S8),
if the magnetic field near the BH is sufficiently strong to form a MAD (see
the blue solid line in Fig. S4). We note that, in most cases, the field
strength is always lower than the MAD condition (Equation S8) no matter what
value of $R_{\rm m}$ is adopted, due to the weak external field generated in
the outer thin disk or/and the low amplification of the field in the inner
ADAF, which corresponds to an ADAF without entering a MAD state (see the red
solid line in Fig. S4).

Fig. S4 shows the magnetic field strength for $t_{1}$ and $t_{2}$
respectively. The Prandtl number $P_{\rm m}=2$ is assumed in our calculations.
The rotational velocity of the gas in the MAD is found to be sub-Keplerian in
the range of 0.5-1.0 (?), and we simply adopt $f_{\Omega}=0.5$ in the
calculations. The parameters, $H/R=0.8$, and $\epsilon=10^{-2}$ are adopted in
the calculations (?, ?). Our calculations show that no MAD state is attained
at $t_{1}$ (because the magnetic field strength of the ADAF is substantially
lower than the MAD criterion). The strength of the magnetic field dragged
inwards by the ADAF reaches $\sim 7\times 10^{7}$ Gauss near the BH at $t_{2}$
(the radio peak), and the inner ADAF ($R_{\rm m}=1.17R_{\rm ISCO}$) is
magnetically arrested at $t_{2}$ (Fig. S4).

The formation of a MAD in this flare is mainly caused by the substantially
expanding ADAF (i.e., the receding outer thin disk) when the mass accretion
rate declines by $\sim$24%, although the detailed physical process triggering
such an increase of $R_{\rm tr}$ by a factor of $\sim$ 30 is still unclear.

The delay between the radio and hard X-ray indicates a continuous expansion of
the ADAF after the hard X-ray peak. This allows for further amplification of
the magnetic field in the ADAF. If the truncated radius changes marginally
after the hard X-ray peak, the amplification of the field strength between
$R_{\rm tr}$ and $R_{\rm ISCO}$ in the ADAF is almost unchanged, and therefore
the field strength near the BH is roughly $\propto\dot{m}_{\rm d}^{3/5}$ (see
Equation S6). The hard X-ray emission from the inner ADAF also depends on the
mass accretion rate. In such a case, this produces nearly simultaneous
variations in X-ray and radio bands for BHXRBs.

###  S1.4 Jet power

The jet power of MAXI J1820+070 during the rising hard state has been
estimated as $\sim 10^{38}$ erg s-1 by previous work (?, ?). The radio
luminosity of the jet at a given frequency correlates with the jet power as
$L_{\nu}\propto P_{\rm jet}^{\frac{17}{12}-\frac{2}{3}\varsigma}$ (?, ?),
where the $\varsigma$ is the radio spectral index. We apply this correlation
to estimate the jet power as $\sim 4\times 10^{37}$ erg s-1 at the radio peak
of flare by using the observed radio luminosity at 15.5 GHz and assuming a
flat radio spectrum ($\varsigma\sim$0).

The powering of jets by the electromagnetic extraction of the rotating energy
from the BH is called the Blandford-Znajek (BZ) mechanism (?). The BZ power
for a moderately spinning BH with $B_{\rm h}=7\times 10^{7}$ Gauss is
estimated to be $10^{37}$ erg s-1 [ (?), their Equation 4] (see Fig. S6),
which is consistent with the jet power of MAXI J1820+070 at the radio peak
(see above). This is also consistent with our interpretation that the
expanding ADAF during the flare amplifies and transports the magnetic field
from the outer disk, and eventually reaches the MAD level.

Our model focuses on the increase of magnetic field strength to form the MAD.
The jet power we used is estimated from observation and consistent with BZ
power from the magnetic field of the MAD. The model cannot determine how and
when the jet is launched, so we cannot compare it to the onset of the radio
emission.

###  S1.5 Disk instability model with the disk wind

The ICCF analysis between the X-ray/$V$-band flare indicates that, during the
flare, the optical luminosity is due to viscous heating in the outer disk and
not to reprocessing of X-rays. The hard X-ray flare illuminates the truncated
disk, triggering the revival of the instability which propagates outwards in
the outer disk and thus leads to the $V$-band flare.

We use a disk instability code (?) to simulate the time-dependent evolution of
the thin disk. The input parameters are: orbital period $P=16.45$ hours (?);
primary and secondary masses $M_{\rm BH}=8.5M_{\odot}$, and $M_{\rm
star}=0.6M_{\odot}$, respectively; mass transfer rate from the donor star
$\dot{M}_{\rm tr}=5\times 10^{16}\rm g\,s^{-1}$. The models with constant
values of Shakura-Sunyaev viscosity $\alpha$ cannot reproduce the outburst
light curves of dwarf novae (?). Instead, two different viscosities for the
thin disk’s thermal equilibrium at a given radius, i.e., $\alpha_{\rm h}$ for
the hot regime, $\alpha_{\rm c}$ for the cold regime, were introduced in the
DIM model (?, ?), which is verified by the magnetohydrodynamic simulation (?).
We assume $\alpha_{\rm h}=0.3$ and $\alpha_{\rm c}=0.02$ (?, ?). We assume
that the thin disk is truncated at inner region due to evaporation at a rate
of (?, ?)

$\dot{M}_{\rm eva}(R)=0.008\dot{M}_{\rm Edd}\left[\left(\frac{R}{R_{\rm
Sch}}\right)^{1/4}+40\left(\frac{R}{800R_{\rm Sch}}\right)^{2}\right]^{-1},$
(S10)

where $R_{\rm Sch}$ is the Schwarzschild radius. The truncation radius is
determined by requiring that the evaporation rate is equal to the accretion
rate, which varies with time. We set a lower limit of $5\times 10^{8}\rm cm$
($\sim 70R_{\rm Sch}$) (?) to avoid excessive computing time while keeping the
timescales in the inner disk region much shorter than in the outer regions. In
principle, the disk instability is responsible for the full outburst and
should account for the entire light curve starting from MJD 58192 to MJD
58440, including the flare. However, for simplicity, we do not generate the
entire light curve, but instead use the DIM simulation to explore the disk
evolution during the flare when a cooling front (appearing as a depression in
the surface density profile) associated with the main outburst (before MJD
58380) has swept inwards over the thin disk.

In the DIM, the disk evolution is governed by the vertically integrated radial
equations for mass, angular momentum, and energy conservation (?). For the
energy conservation, the surface heating and cooling rate are determined using
a pre-calculated grid of vertical structures in the ($\Omega$, $\Sigma$,
$T_{\rm c}$ and $T_{\rm irr}$) space (?, ?). $\Omega$, $\Sigma$, $T_{\rm c}$
and $T_{\rm irr}$ are the angular velocity, local surface density at radius
$R$, midplane temperature, and irradiation temperature respectively. $T_{\rm
irr}$ is determined from the irradiation luminosity $L_{\rm irr}$ as $\sigma
T_{\rm irr}^{4}=L_{\rm irr}/4\pi R^{2}$, where $\sigma$ is the Stefan-
Boltzmann constant. In this approach, irradiation increases the surface disk
temperature $T_{\rm s}$, according to $\sigma T_{\rm s}^{4}=Q^{+}+\sigma
T_{\rm irr}^{4}$ where $Q^{+}$ is the viscous heating flux; this assumes that
irradiated X-rays do not penetrate deep into the disk. We do not take into
account other effects such as the evaporation of the disk or the formation of
outflows (?, ?). Instead, they are approximated separately either via an
explicit evaporation term in the inner disk (Equation S10), or via an explicit
disk wind term (see below). Although not fully consistent, this simplified
approach can model the outburst of BHXRB (?).

The irradiation luminosity is assumed to be related to the mass accretion rate
at the inner disk radius $\dot{M}_{\rm in}$, as $L_{\rm irr}=\zeta_{\rm
d}\dot{M}_{\rm in}c^{2}$, where $\zeta_{\rm d}$ includes both the radiative
efficiency and irradiation efficiency. However, during the flare, the observed
hard X-ray emission originating from the ADAF is the major contributor to
X-rays illuminating the outer thin disk, so that $L_{\rm irr}=\zeta_{c}L_{\rm
X}$. This additional illumination heats the disk region behind the cooling
front to be hot enough so that hydrogen begins to be reionized (referred to as
a ’hot state’). Thus, the thin disk switches back to the hot state with the
revival of the instability. In this case, the resulting spike in the density
profile and the steep gradients of the temperature profile (heating front)
propagates outwards, which then leads to a lagged optical ($V$-band) light
curve (?). Therefore, in our simulation, the irradiation luminosity is $L_{\rm
irr}=\rm max(\zeta_{c}L_{\rm X},\zeta_{d}\dot{M}_{\rm in}c^{2})$. The time-
dependent $L_{\rm X}$ is estimated by fitting the observed Comptonization
X-ray luminosity (Fig. 2B) with a Gaussian function. The irradiation
efficiency represents the fraction of the X-ray luminosity irradiating the
outer thin disk, which depends on the geometry of the inner ADAF and of the
outer thin disk, the X-ray albedo, and the X-ray spectrum. The irradiation
efficiency has to be tuned to trigger the instability at a certain radius of
the thin disk. For simplicity, in our DIM simulation, we assume a constant
efficiency, $\zeta_{d}=1\times 10^{-4}$ and $\zeta_{c}=7.5\times 10^{-5}$
following previous work (?).

We introduce the disk wind into the disk instability model following a
previously described method (?). The continuity equation and specific angular
momentum equation are:

$\frac{\partial\Sigma}{\partial t}-\frac{1}{2\pi
R}\frac{\partial\dot{M}}{\partial R}=-\dot{\Sigma}_{\mathrm{w}},$ (S11)

$\Sigma\frac{\partial R^{2}\Omega}{\partial t}-\frac{\dot{M}}{2\pi
R}\frac{\partial R^{2}\Omega}{\partial R}=\frac{1}{R}\frac{\partial}{\partial
R}\left(R^{3}\nu\Sigma\frac{\partial\Omega}{\partial
R}\right)-j\dot{\Sigma}_{\mathrm{w}},$ (S12)

where $\dot{M}$ is the mass accretion rate at radius $R$, and $\Omega$ is
assumed to be Keplerian in our simulations. $\nu$ is the kinematic viscosity
coefficient, which is determined by the Shakura-Sunyaev prescription
$\nu=\alpha c_{\rm s}H$ where $c_{\rm s}$ is the sound speed, $H$ is the disk
scale height and $\alpha$ is a coefficient equal to $\alpha_{\rm c}$ on the
cold branch when hydrogen is neutral and to $\alpha_{\rm h}$ on the hot branch
when hydrogen is ionized. $\dot{\Sigma}_{\mathrm{w}}$ is the loss rate of the
local surface density due to the disk wind, which is assumed to be related to
the local surface density $\Sigma$ via a factor $f$ as
$\dot{\Sigma}_{\mathrm{w}}=f\Sigma$, where $f$ is in units of $\rm s^{-1}$.
$j$ is the excess specific angular momentum carried away by the disk wind,
which is assumed to be proportional to the Keplerian specific angular
momentum, i.e. $j=g\sqrt{GM_{\rm BH}R}$, where $g$ is the factor of setting
the excess specific angular momentum carried away by the disk wind. We assume
the factor also depends on the radius as $g=g^{{}^{\prime}}R_{9}$
[$R_{9}\equiv R/(10^{9}~{}\rm cm)$], where $g^{{}^{\prime}}$ is a scaling
factor.

In the case of the thermal disk winds, no excess angular momentum is carried
by the disk winds so $g=0$ (?), while in the case of magnetic disk winds,
$g>1$ is expected in the global ideal magnetohydrodynamics (MHD) simulations
of thin disks (?). In our simulations, the wind mass loss rate
$\dot{\Sigma}_{\mathrm{w}}=\partial\Sigma/\partial t$ is $\sim 10^{-4}~{}\rm
g\mathrm{~{}cm}^{-2}\mathrm{~{}s}^{-1}$ and $\Sigma\sim
10^{2}\mathrm{~{}g}\mathrm{~{}cm}^{-2}$. Setting
$\dot{\Sigma}_{\mathrm{w}}=f\Sigma$, the factor $f$ is $\sim 10^{-6}~{}~{}\rm
s^{-1}$.

During the flare, no spectroscopic data are available to put constraints on
the disk wind loss and its evolution. Therefore, we parameterize the disk wind
in terms of the mass loss (i.e. the factor $f$) and the removal of the angular
momentum (i.e. the factor $g$). We assume that the mass loss due to the disk
wind is time- and radius-dependent as

$f(t,R)=f^{{}^{\prime}}(t)R_{9}\\!^{-1}\frac{\alpha(R)-\alpha_{\rm
c}}{\alpha_{\rm h}-\alpha_{\rm c}}~{}\rm s^{-1},$ (S13)

where the term [$\alpha(\rm R)-\alpha_{\rm c}]/[\alpha_{\rm h}-\alpha_{\rm
c}]$ ensures that the disk wind is driven in the hot branch only. The $R_{9}$
dependence means that the disk wind characteristic timescale $1/f$ increases
with radius. We assume that the disk wind starts at $t_{\rm start}$ with
$f^{{}^{\prime}}=0$, and mass loss rapidly increases until $t_{\rm end}$,
following

$f^{{}^{\prime}}(t)=\left\\{\begin{aligned} 0&,\quad\text{if $t\leq t_{\rm
start}$}.\\\ \frac{f^{{}^{\prime}}_{\rm max}}{t_{\rm end}-t_{\rm
start}}\left(t-t_{\rm start}\right)&,\quad\text{if $t_{\rm start}\leq t<t_{\rm
end}$}.\\\ f^{{}^{\prime}}_{\rm max}&,\quad\text{if $t\geq t_{\rm
end}$.}\end{aligned}\right.$ (S14)

The linear dependence of both $f$ and $g$ on the radius is not a unique
solution to the disk wind for the optical flare. According to the definitions
above, the loss rate of the angular momentum per unit area via the disk wind
is determined by the term $fg\Sigma R^{1/2}\propto fgR^{-1/4}$ where
$\Sigma\propto R^{-3/4}$ for a Shakura & Sunyaev disk. If the term $fg$ is
independent of radius, the loss rate of the angular momentum will be low at a
large radius. We also try other options with different scaling forms, e.g.
$f\propto 1/R_{9}^{2}$, which can also reproduce the observed light curve. We
adopt the linear radial dependence of $g$ in the following simulations.

Using the above parameterization of the disk wind, the DIM simulations show
that the revival of the instability is triggered at $R=5\times 10^{10}$ cm
($\sim 3.9\times 10^{4}R_{\rm g}$, where $R_{\rm g}=GM_{\rm BH}/c^{2}$ is the
gravitational radius), the heating front reached a maximum distance of
$R=7\times 10^{10}$ cm ($\sim 5.6\times 10^{4}R_{\rm g}$), and the predicted
$V$-band light curve roughly matches the observed one (Fig. 4), when $t_{\rm
start}$ = MJD 58391, $f^{{}^{\prime}}_{\rm max}=2.7\times 10^{-7}~{}\rm
s^{-1}$ (i.e., 1/42.9 $\rm day^{-1}$) at $t_{\rm end}$ = MJD 58395 and
$g^{{}^{\prime}}=3.167$. The simulated $V$-band light curve lags the X-ray
light curve by more than 15 days, in agreement with the observations.

The disk wind can explain the delay of the optical peak. The peak position of
the optical emission in the light curve depends on how far away the heating
front can reach. For a normal outburst, in which the heating front reaches the
outer edge of the thin disk, the initial rise from quiescence and outburst
peak are unaffected by the existence of disk winds [ (?), their Fig. 3]. By
contrast, in our DIM simulations, the propagation of the instability during
the flare starts from the soft state, when the cooling front is already
propagating inwards over the disk. Due to the illumination by the hard X-ray
flare of the truncated thin disk, the disk region behind the cooling front is
heated until it is hot enough for the hydrogen to be reionized. Thus, the thin
disk switches back to the hot state with a revival of the instability
propagating outwards in the outer thin disk.

We postulate that the disk wind is launched in regions where the thin disk is
in the hot state during the flare. The disk wind carries angular momentum ($g$
is non-zero) which is equivalent to an increase in the viscosity. Because the
disk wind is launched only in the hot regions of the thin disk, the increase
is in $\alpha_{\rm h}$, not $\alpha_{\rm c}$. An increased $\alpha_{\rm h}$
leads to a decrease in the minimum density required for the hot state at a
given radius, so the heating front propagates further out, i.e. the optical
luminosity increases for a longer period.

To illustrate the role of the disk wind in moving the heating front further
out, we carry out a simulation without disk wind, i.e., setting $f=0$. A flare
still appears, starting at around MJD 58390 (Fig. 4), but the heating front is
not strong enough to propagate at very large distances in the outer disk, and
the resulting $V$-band flare fades away in a few days. This does not match the
observations.

The wind launched from the thin disk has been studied numerically in previous
work (?, ?). Observations have shown that there are disk winds in the soft
state (?) and in the hard state (?, ?) for accreting systems. The thin disk
wind was present throughout the outburst of MAXI J1820+070 in 2018, as
indicated by infrared spectroscopy (?), although only two spectra are
available during the flare (at MJD 58390 and 58391). We included the disk wind
during the flare only, ignoring the role of a pre-existing disk wind in the
rising hard state and soft state. A pre-existing disk wind can be approximated
in the model using a viscosity parameter larger than the pure
magnetorotational instability (MRI) value (?) - which is not well determined
yet. The mass loss rate via the disk wind and its time evolution cannot be
determined from the available observational data (?). Therefore, we cannot
fully include the disk wind in the DIM simulation and do not explicitly
include the disk wind prior to the flare. We do require, however, that the
wind mass loss rate increases during the flare.

In another BHXRB GRO J1655-40, an optical rise occurred a few days before the
X-rays (?), due to the truncation of the thin disk. Modeling indicated that
the instability developed in a truncated disk, and the inner disk builds up on
viscous timescale to emit X-rays, leading to the lagged X-ray emission in GRO
J1655-40 (?). By contrast, in our interpretation of the onset of the soft-to-
hard transition of MAXI J1820+070, the thin disk has been built up and
extended to the ISCO. As the truncation radius of the outer thin disk recedes
with decreasing mass accretion rate, the ADAF gradually builds up, leading to
the hard X-ray flare occurring first. This X-ray flare illuminates the
truncated thin disk, triggering the revival of the instability which
propagates outwards in the outer disk, producing the lagged optical flare.

###  S1.6 Correlations between radio/$V$-band and X-ray

####  S1.6.1 Correlation between radio and X-ray

There is a universal radio-X-ray (R-X) correlation $L_{\mathrm{R}}\propto
L_{\mathrm{X}}^{\Gamma}$ with $\Gamma\sim 0.6$ in the BHXRBs (?), which
indicates strong accretion-jet coupling. Previous R-X correlation analysis
usually used the 1–10 keV (or 3–9 keV) X-ray luminosity. In this work, we
investigate the R-X correlation by using the Comptonization luminosity
$L_{\mathrm{C}}$ instead.

$L_{\mathrm{R}}$ and $L_{\mathrm{C}}$ show no time delay during the rising
hard state (MJD 58191-58283), according to the ICCF analysis. Note that, after
the peak of the rising hard state (MJD 58200), the X-ray corona is contracting
over time while the truncation radius of the thin disk is close to ISCO and
remains unchanged (?). The same conclusion is also drawn from the follow-up
detailed spectral fits of NuSTAR (?), Neutron star Interior Composition
Explorer (NICER) (?) and Insight-HXMT (?) observations. As discussed in
Section S1.3, the increase of the truncation radius during the flare (after
MJD 58380) leads to the observed 8-day radio lag relative to the hard X-ray
emission. Therefore, the constant truncation radius of the thin disk during
the rising hard state may be responsible for the observed zero delay between
the X-ray emission and the radio emission.

We then find that the simultaneous $L_{\mathrm{R}}$ and $L_{\mathrm{C}}$ (time
difference less than 1 day) follow a correlation $L_{\mathrm{R}}\propto
L_{\mathrm{C}}^{0.53\pm 0.06}$ during the rising hard state (blue dashed line
in Fig. S7), which is consistent with that in ref. (?). We use the linmix
algorithm (?) implemented in Python to determine the above correlation by
fitting the data. During the flare, the data during the soft-to-hard state
transition apparently deviate from the above correlation, and then roughly
follow it after the transition to the decaying hard state (see Fig. S7).

####  S1.6.2 Correlation between X-ray and $V$-band

There is also a correlation between X-ray and optical luminosities in the hard
states of BHXRBs, which has been used to infer the origin of the optical
emission (?). We first investigate the $L_{\mathrm{V}}$ and $L_{\mathrm{C}}$
during the rising hard state. There is no time delay detected between them,
according to the ICCF analysis. The Spearman coefficient of the simultaneous
$L_{\mathrm{V}}$ and $L_{\mathrm{C}}$ (time difference less than 1 day) is
$0.92\pm 0.01$ at a significance of 10.98 $\sigma$. We then fit the
correlation with a linear function on the logarithmic scale and find
$L_{\mathrm{V}}\propto L_{\mathrm{C}}^{1.00\pm 0.09}$. This slope is steeper
than the universal $L_{\mathrm{V}}$-$L_{\mathrm{X}}$ correlation in the hard
state of BHXRBs ($\sim$0.6) (?).

We then investigate the correlation between $L_{\mathrm{V}}$ and
$L_{\mathrm{C}}$ during the flare. The Spearman coefficient is $-0.33\pm 0.02$
at a significance 1.7 $\sigma$, which indicates there is no correlation
between $L_{\mathrm{V}}$ and $L_{\mathrm{C}}$. Apparently, $L_{\mathrm{V}}$
and $L_{\mathrm{C}}$ during the flare (both in the soft-to-hard state
transition and decaying hard state) do not follow the same positive
correlation as that during the rising hard state (see Fig. S8).

##  S2 Supplementary text

###  S2.1 Radio and optical delay due to jet traveling

We also notice that the optical fast variability lags behind the X-ray by
roughly 165 ms in MAXI J1820+070 [ (?), see also the time delay of radio
variability in ref. (?)]. Moreover, the delay increases with wavelength as
3.25 $\mu{\rm s}~{}\AA^{-1}$ (?). Since the shorter wavelength emission is
expected from the inner part of the jet (?), this time delay of fast
variability and the wavelength dependence can be well explained by the
propagation of perturbations in the jet and gives the jet bulk Lorentz factor
as $6.8$ (?). If we linearly extrapolated the wavelength to 15.5 GHz, the time
lag is $\sim 10$ minutes, which is much smaller than the derived radio lag of
$\sim$ 8 days in our work. Therefore, our extremely long optical/radio delays
cannot be associated with the jet traveling time.

###  S2.2 MAD of BHXRB in quiescent

According to Equation S8, the magnetic field strength for a MAD depends on the
mass accretion rate $\dot{m}$ and truncation radius $R_{\rm tr}$, which were
both estimated from the observed disk component in our model (Equation S2 and
Equation S3). The time variations of the magnetic field strength at the inner
edge of the ADAF are shown in Fig. S5. We find that the magnetic field
strength decreases after $t=t_{2}$, which is caused by the decreasing MAD
field strength due to the decrease of the mass accretion rate over time (see
Equation S8). Nevertheless, based on the physical quantities $\dot{m}_{\rm d}$
and $R_{\rm tr}$ derived from the X-ray light curves (the disk and
Comptonization component emissions), our model calculations show that the
magnetic field strength in the ADAF peaks at a time corresponding to the
maximal radio emission, and then declines. The disk emission is too weak to be
detected after MJD 58400 when radio emission was still decreasing. This
prevents us from determining $\dot{m}$, $R_{\rm tr}$, and therefore the
magnetic field strength (Fig. S5), using the methods above (?). But, it was
also shown in previous work that the MAD would exist in the quiescent state
(?).

###  S2.3 Standard and normal evolution (SANE) accretion flow

Previous work has argued that the broadband X-ray spectrum of a MAD is quite
similar to that of a SANE accretion flow (?), which proposes that these two
different types of accretion flows are indistinguishable using only X-ray
spectroscopy. However, for the MAD with very low accretion rate
$\dot{m}<10^{-5}$, magnetic reconnections heat up the thermal/nonthermal
electron, and the Thomson optical depth of MADs is too low for X-rays to be
emitted by Comptonization. Thus, it was proposed that a MAD emits nonthermal
synchrotron emission which dominates over Comptonization, and produces a hard
and power-law spectrum in X- and gamma-ray bands (?). In contrast, the SANE-
mode accretion flow can emit hard X-ray emissions via Comptonization.
Therefore, the SANE-mode accretion flow leads to soft spectra in the hard
X-ray range, while MADs produce power-law hard X-ray spectra owing to the
nonthermal electrons. In addition, the radio luminosity of MADs may be
different from that of SANE-mode accretion flows because of the difference in
magnetic fields (?).

Figure S1:  Cross-correlation analysis between the (A) radio and the X-ray
luminosities and (B) $V$-band and the X-ray luminosities, both after MJD =
58380. In both panels, the red line shows the cross-correlation function (left
axis) for each time lag tested. The dashed and dotted blue histograms are the
cross-correlation centroid lag distribution and the peak lag distribution
respectively (right axis). See text for details.  Figure S2:  Example unfolded
energy spectrum of MAXI J1820+070 (black data points) fitted with the model
tbabs*(diskbb+nthcomp), during the hard X-ray peak at MJD = 58389. tbabs is
the absorption component with a fixed column density $N_{\rm H}=0.15\times
10^{22}$ cm-2, diskbb is the multiple blackbody component from the thin disk,
and nthcomp is the Comptonization component from the ADAF. The total model is
plotted in red, which is the total of the diskbb in orange and the nthcomp in
blue. We exclude the spectrum in the range of 21-24 keV due to contaminating
atomic lines. The error bars of the spectrum correspond to a 68% confidence
interval.  Figure S3: The truncation radii derived from the disk component of
the X-ray emission with Equation S2. The red and blue stars represent the
truncation radii at $t=t_{1}$ and $t=t_{2}$ (?), respectively. The same method
is also applied to other epochs of Insight-HXMT observation to estimate the
truncation radii during the flare (black points). Figure S4: Strength of the
magnetic field advected by the ADAF in the simulations as a function of radius
(solid lines). The dashed lines are the required strength to satisfy the MAD
criterion as a function of radius, calculated with $\epsilon=0.01$. The red
and blue lines correspond to the calculations at $t=t_{1}$ and $t=t_{2}$,
respectively. The size of the MAD is $R_{\rm m}=1.17R_{\rm ISCO}$, when
$t=t_{2}$. Figure S5: The strength of the magnetic field of the ADAF at
$R_{\rm ISCO}$ varies as a function of time. The star points indicate the
simulated magnetic strength advected by the ADAF, and the circle points are
the MAD criterion. The red and blue colors represent the epochs corresponding
to the hard X-ray peak and radio peak, at $t=t_{1}$ and $t=t_{2}$,
respectively. The black points correspond to other epochs. Figure S6:
Calculated BZ power, $P_{\rm BZ}$, as a function of the BH spin parameter $a$
for a BH with 8.5 solar mass and the field strength at the horizon $B_{\rm
h}=7\times 10^{7}$ Gauss is adopted. Figure S7:  The correlation between radio
and X-ray Comptonization luminosity. The blue dots, blue triangles, and orange
triangles represent the data from rising hard state, decaying hard state, and
soft-to-hard state transition periods, respectively. The blue dashed line is
the best-fitting result of the rising hard state data (blue dots)
$L_{\mathrm{R}}\propto L_{\mathrm{X}}^{0.53}$. The arrow indicates the
evolution of this source in the plane of $L_{\rm R}$-$L_{\rm C}$, during the
soft-to-hard state transition. The uncertainties in $L_{\mathrm{C}}$ and
$L_{V}$ are quoted at the 1-$\sigma$ level.  Figure S8:  The correlation
between the optical $V$-band and X-ray Comptonization luminosity. The symbols
here are the same as the ones in Fig. S7. The blue dashed line is the best-
fitting result of the rising hard state data (blue dots) $L_{V}\propto
L_{\mathrm{C}}^{1.00}$. The uncertainties in $L_{\mathrm{C}}$ and $L_{V}$ are
quoted at the 1-$\sigma$ level.

Caption for Movie S1.
Formation of a MAD in MAXI J1820+070. (A) Animated version of Fig. 3. (B) The
hard X-ray flare arising from the ADAF, the data points in which are estimated
from the spectral fits of Insight-HXMT observation (?), is attributed to the
decrease of the mass accretion rate (panel D) and the increase of the ADAF
radial extent (panel A). (C) The radio flare originating from the jet, which
is observed by AMI-LA, arises from the magnetic field being transported and
amplified by the expanding ADAF (panel A). (D) The exponential decay of
Eddington-scaled mass accretion rate during the flare, which is predicted
based on the disk luminosity and the LE light curve (?).