11institutetext: Departament de Física Quàntica i Astrofísica, Institut de
Ciències del Cosmos, Universitat de Barcelona, IEEC-UB, Martí i Franquès 1,
08028, Barcelona, Spain

# Signatures of extended radio emission from escaping electrons in the
Lighthouse Nebula

P. Bordas 11 X. Zhang 11 V. Bosch-Ramon 11 J.M. Paredes 11

(Received — ; accepted —)

Several supersonic runaway pulsar wind nebulae (sPWNe) with jet-like extended
structures have been recently discovered in X-rays. If these structures are
the product of electrons escaping the system and diffusing into the
surrounding interstellar medium, they can produce a radio halo extending for
several arcmins around the source. We model the expected radio emission in
this scenario in the Lighthouse Nebula sPWN. We assume a constant particle
injection rate during the source lifetime, and isotropic diffusion into the
surrounding medium. Our predictions strongly depend on the low- and high-
energy cutoffs given in the particle distribution. Our results indicate that
extended radio emission can be detected from the Lighthouse Nebula without the
need to invoke extreme values for the model parameters. We provide synthetic
synchrotron maps that can be used to constrain these results with observations
by current highly sensitive radio instruments.

###### Key Words.:

Radio: pulsars – pulsars: jets

##  1 IGR J11014–6103, the Lighthouse Nebula

Pulsar wind nebulae (PWNe) interacting with the surrounding interstellar
medium (ISM) can give rise to distinct extended morphological features. Torus-
like structures and bipolar jets are observed in pulsars displaying slow
proper motions, as well as bow-shaped shocks and extended cometary tails in
pulsars moving at high velocities (see e.g. Gaensler & Slane, 2006). In the
X-ray band, these structures have been resolved in detail in a number of cases
(Kargaltsev & Pavlov, 2008). A small subset of supersonically moving PWN (sPWN
from now on) have, in addition, been observed to display unusually long X-ray
outflows. Some of the most prominent examples of such outflows are found in
the Guitar Nebula (Hui et al., 2012) and in the Lighthouse Nebula (Pavan et
al., 2014), but the number of large-scale filaments from sPWNe is gradually
increasing, see for example the recent findings in PSR J1135-6055 (Bordas &
Zhang, 2020), J1809–1917 (Klingler et al., 2020), PSR J2030+4415 (de Vries &
Romani, 2020), PSR J2055+2539 (Marelli et al., 2019), or the filaments found
in the Galactic Center region such as G0.13-0.11 (Zhang et al., 2020), likely
associated to a sPWN (see also previous findings in Kargaltsev et al., 2017).
The origin of these extended jet-like structures is unknown, but their length,
almost rectilinear geometry, and misaligned orientation with respect to the
pulsar proper motion are difficult to be accommodated for in the framework of
canonical pulsar jet formation theories, where jets are produced by the
magnetic collimation of the plasma downstream of the pulsar wind termination
shock (see e.g. Lyubarsky, 2002; Komissarov & Lyubarsky, 2004; Porth et al.,
2014 and references therein). This led to alternative interpretations,
including a scenario in which particles accelerated at the pulsar wind/medium
shock interface escape the PWN, and diffused into the ambient medium magnetic
field, yielding synchrotron emission which is then observed in the X-ray band
as quasi-rectilinear jet-like structures (Bandiera, 2008). Recent numerical
simulations seem to support such a scenario, explaining also the asymmetric
morphologies observed at opposite sides of the pulsar (Barkov et al., 2019b;
Bucciantini et al., 2020).

In this paper we model the radio emission produced by high-energy electrons
escaping the sPWN IGR J11014–6103, the Lighthouse Nebula, into the ISM, where
they radiate in the ISM magnetic field. After a brief description of the
source, in Section 2 we outline the model used to describe the emitting
electron population and their diffusion into the ISM. Section 3 provides
particle and radiation distributions, as well as synthetic maps at radio
wavelengths used to predict the expected signal for dedicated observations of
the source by highly sensitive facilities. Section 4 discusses the potential
of radio observations to constrain the origin of the jet-like structures
observed from sPWNe.

Figure 1: Chandra/ACIS exposure-corrected broadband (0.5 – 7.0 keV) flux image
of IGR J11014–6103 (the “Lighthouse Nebula”). The image has been produced
using all Chandra archival observations of the Lighthouse Nebula , for a total
of $\sim 365$ ks, merged using CIAO’s merge_obs tool, and smoothed using a 2D
Gaussian kernel with radius on a pixel size of 0.492$\arcsec$ and has been
smoothed with a 2D-Gaussian kernel with radius = 1.7$\arcsec$ and
$\sigma=0.7$″.

Originally discovered by INTEGRAL as an unidentified hard X-ray source, IGR
J11014–6103 (also known as the “Lighthouse Nebula”; Pavan et al., 2014) is
located $\sim$11 arcmins southwest of its likely progenitor supernova remnant
SNR MSH 11-61A, at a distance of $\sim$7 kpc from the Sun (see García et al.,
2012). The Lighthouse Nebula is powered by the pulsar PSR J1101$-$6101,
displaying a period of $P=62.8$ ms and an estimated spin-down energy of
$L_{\rm sd}=1.36\times 10^{36}\leavevmode\nobreak\ \mathrm{erg\ s^{-1}}$
(Halpern et al., 2014). Observations in the X-ray band have revealed a complex
system morphology, including a trailing PWN, shaped in a narrow cone elongated
toward the parent SNR, and prominent X-ray jet-like structures oriented nearly
perpendicular to the PWN axis (see Pavan et al., 2016). At radio energies,
extended emission has been detected with ATCA at 2 GHz from the IGR
J11014–6103 tail (Pavan et al., 2014). This emission is not spatially
coincident with that observed at X-rays, with the peak of the radio emission
being separated further away along the tail by about 22$\arcsec$ with respect
to the X-ray maximum.

The jets observed in the Lighthouse Nebula display a rectilinear structure
extending for more than 10 pc into the surrounding medium (see Fig. 1). Their
X-ray luminosity is unusually high, at the level of that retrieved from the
pulsar and the tail structures, altogether representing a few percent of the
pulsar spin-down power, $L_{\rm sd}\sim 10^{36}$ erg s-1. The jet X-ray
spectrum is moderately hard, with photon indices $\Gamma_{\rm X,jets}\approx
1.6$, whereas the tail region displays $\Gamma_{\rm X,tail}\approx 1.9$. Both
the jet spectral hardness and its high X-ray efficiency may be expected if
these are magnetic jets carrying a hard electron distribution, and there is an
amplified magnetic field along the jet structures (see e.g. Bykov et al.,
2017). However, accounting for the high-speed velocity of the source, $v_{\rm
psr}\gtrsim 1\,000\leavevmode\nobreak\ \mathrm{km\leavevmode\nobreak\
s^{-1}}$, the absence of any significant bending along their $\gtrsim 10$ pc
is striking prompts for alternative mechanisms for the production of such jet-
like structures. Other sPWNe have been interpreted in a diffusion scenario
(Bandiera, 2008), and rectilinear structures or “filaments” have also been
recently reported from the Galactic Center region (MeerKAT Collaboration,
2018), and discussed as “kinetic jets” of escaping particles diffusing into
the ambient magnetic field (Barkov & Lyutikov, 2019). It is currently
uncertain whether the jet-like structures in the Lighthouse Nebula are
ballistic jets carrying an amplified magnetic field or filaments of escaping
high-energy electrons from the sPWN into the ISM.

##  2 Modeling the radio emission in a diffusion scenario

The pulsar PSR J1101-6101 in the Lighthouse Nebula displays a spin-down power
of $1.36\times 10^{36}$ erg/s (Halpern et al., 2014). Observations with the
Chandra satellite report a 2–10 keV X-ray flux from the jets at the level of
6$\times 10^{-13}$ erg cm-2 s-1, whereas spectral analysis suggests a
synchrotron origin rather than a thermal emission scenario (Pavan et al.,
2014). Particles accelerated at the Lighthouse Nebula that are responsible for
this emission could also emit in the radio band. Under these premises, we
consider the presence of a nonthermal population of relativistic electrons
injected close to the sPWN bow shock, carrying a total kinetic luminosity
$L_{\rm inj}\approx L_{\rm X}\times\,t_{\rm sync}/t_{\rm dyn}$. Here $L_{\rm
X}$ is the jet X-ray luminosity derived from Chandra observations (we assume a
distance to the Lighthouse Nebula of $d_{\rm LN}=7$ kpc), $t_{\rm sync}$ is
the synchrotron cooling time for electrons emitting at the characteristic
frequency $h\nu_{\rm c}=2$ keV in a given magnetic field $B$, $t_{\rm
sync}\approx 460\,(B/30\rm{\mu G})^{-1/2}$ yr, and $t_{\rm dyn}$ is the
dynamical timescale to travel along the jet-like feature, $t_{\rm dyn}\sim
l_{\rm jet}/v_{\rm dyn}$; with $l_{\rm jet}\sim 10$ pc and $v_{\rm dyn}\sim
c$, $t_{\rm dyn}\sim 32$ yr. A total power $L_{\rm inj}=3\times 10^{34}$ erg
s-1 is therefore injected, which is distributed in energy following a power-
law with index $\Gamma=2$ and an exponential cutoff at both the low- and the
high-energy ends of the particle spectrum at $\gamma_{\rm min}^{\rm cut}$ and
$\gamma_{\rm max}^{\rm cut}$, respectively. A single power-law model has been
assumed for simplicity in our model, although we note that PWNe could inject a
more complex particle distribution displaying a spectral break following the
presence of separate electron populations, for example those injected within
the pulsar’s light cylinder, and electrons accelerated at the shock between
the pulsar wind and the external medium (see e.g. Atoyan & Aharonian, 1996;
Venter & de Jager, 2007; Torres et al., 2014). The low-energy cutoff is an
assumption in our model, and it is treated as a free parameter, with two cases
explored in this report:$\gamma_{\rm min}^{\rm cut}=10^{3}$ and $10^{5}$. Such
a low-energy cutoff may be expected in case an energy selection mechanism is
in place in IGR J11014–6103 for particles able to escape the PWN. We use
$\gamma_{\rm max}=10^{8}$ for the high-energy particle distribution cutoff,
which should be attainable in the bow shock of a runaway pulsar such as IGR
J11014–6103 (compatible with the jet X-rays; see e.g. Bykov et al., 2017). As
for the spectral index, we note that the value $\Gamma=2$ is on the other hand
adopted for simplicity, but also because it predicts potentially detectable
fluxes. If the electron distribution were much harder, then little emission
would be produced in the radio band.

Figure 2: Energy distribution of non-thermal particles injected into the JET
and ISM regions for different assumed values of the low-energy cutoff
$\gamma_{\rm min}^{cut}$. Particle injection is discretized as occurring every
$t_{\rm inj}=1$ kyr (dotted lines). The final cumulative distribution at
present time accounts for radiation cooling due to IC on the CMB and
synchrotron losses. Distributions are shown for particles injected recently
($t_{\rm inj}\leq 1$ kyr) that diffuse through the JET region (blue lines),
and particles injected at $t_{\rm inj}>1$ kyr diffusing into the ISM region
(in red lines). In both cases, a magnetic field of $B_{\rm ISM}=B_{\rm
JET}=5\mu$G is assumed. A low cutoff for the particle distribution at
injection is considered, adopting two different values $\gamma_{\rm
min}^{cut}=10^{3}$ and $\gamma_{\rm min}^{cut}=10^{5}$ (solid and dashed
lines, respectively). A high energy cutoff $\gamma_{\rm max}^{cut}=10^{8}$ is
also assumed in all cases. The distribution normalization is computed
accounting for the source observed X-ray luminosity (see text for details).

Figure 3:  Left: Spectral energy distribution of the emission produced by
electrons injected in the ISM (in red) and JET regions (in blue). A magnetic
field $B_{\rm ISM}=5\mu$G is assumed for the ISM, whereas a range in the value
of $B_{\rm JET}=5,10,15,20$ and $30\mu$G is probed for the JET region.
Particle injection is considered up to $t_{\rm inj}=10^{3}$ yr for the JET
region and from $t_{\rm inj}=10^{3}$ yr until $t_{\rm inj}=20$ kyr for the ISM
region (red-dotted lines). Particles are assumed to follow a power-law
distribution with a low and a high energy cutoff $\gamma_{\rm
min}^{cut}=10^{3}$ and $\gamma_{\rm max}^{cut}=10^{8}$, respectively. Green
and orange vertical bands denote Chandra and MeerKAT frequency coverage.
Right: Same as in the left panel but for a particle spectrum with a low-energy
cutoff $\gamma_{\rm min}^{cut}=10^{5}$.

Particles injected close to the bow shock are assumed to escape the immediate
sPWN region and diffuse into the surrounding medium, where they evolve with
time due to radiation losses, both synchrotron and inverse Compton (IC)
emission, the latter accounting for the surrounding cosmic microwave
background (CMB) and infrared radiation fields ($U_{\rm CMB}\approx 0.25$ eV
cm-3, $U_{\rm IR}\approx 0.2$ eV cm-3, Popescu et al., 2017). We evaluate the
resulting cooled particle distribution at the present time, $t_{\rm LN}=20$
kyr (Halpern et al., 2014). A sufficient number of particle injection events
are considered, which are equally spaced along the source trajectory during
this time, with $\Delta t$ shorter than the typical synchrotro and IC cooling
timescales for particle Lorentz factors $\gamma_{\rm e}\lesssim\gamma_{\rm
max}$. After injection, we consider electron diffusion in the surrounding ISM.
A diffusion coefficient $D_{\rm diff}(E)=3\times
10^{27}\,\left(\frac{E/\rm{GeV}}{B/\rm{3}\mu\rm{G}}\right)^{1/2}$ is assumed
(Gabici et al., 2007). For simplicity, we assume that diffusion is isotropic
from the injection point up to distances $l_{\rm diff}\sim(D_{\rm
diff}\,t\,)^{1/2}$ (the medium is considered uniform, and thus adiabatic
losses are not relevant).

Freshly injected particles, with injection times $t_{\rm inj}\leq 1$ kyr, are
presumably responsible for the X-ray emission observed from the jet-like
features in the Lighthouse Nebula . For this injection period we consider
distinct conditions of the ISM, which we label as the “JET region”,
characterized by a magnetic field $B_{\rm JET}=5$–30 $\mu$G that can be
significantly higher than $B_{\rm ISM}\sim 5\mu$G under which electrons evolve
after $t_{\rm inj}>1$ kyr. The emission from particles injected until $t\sim
1$ kyr diffusing into this JET region, and the emission resulting from their
later diffusion in the ISM region (for $t\gtrsim 1$ kyr), was computed
separately. We emphasize however that the injected electron particle
population is the same in the JET and ISM regions, only the value of the
corresponding magnetic field for each region varies. The former are evolved
until $t_{\rm inj}=1$ kyr only, whereas the latter evolves from $t_{\rm
inj}=1$ kyr up to $t_{\rm inj}=20$ kyr. The different conditions from the JET
and ISM regions follow the assumption that the JET region corresponds to a
temporary feature that relocates as the pulsar moves, and to which we assign a
typical crossing timescale of $\sim 1$ kyr based on the JET apparent width and
the pulsar velocity. The origin of the JET feature could be related to the
pulsar motion pulling the threaded ISM magnetic field and, as the pulsar
propagates, new ISM magnetic lines become entangled at the front of the PWN
whereas the “older” ones relax and turn back into normal ISM magnetic field
conditions.

Synchrotron emission is computed by employing the formulae in Pacholczyk
(1970) together with the analytical approximations in Aharonian et al. (2010).
The cumulative radio emission is evaluated throughout the area surrounding the
source. Synthetic synchrotron emission maps are computed, which are then
convolved with the beam of a radio telescope at GHz frequencies in order to
obtain concrete observational predictions.

In this study we aim at predicting the radio emission produced in the
surroundings of the sPWN, and therefore we do not attempt to model in detail
the observed X-ray emission from the jet, including its morphological
properties. In this regard, we only consider isotropic diffusion of particles
injected in the JET and ISM regions. The effects that may result from
anisotropies generated for example by the structured geometry of the
underlying magnetic field, are therefore disregarded. Our estimates are based
on assumed values for the injected kinetic energy, the energy distribution of
the emitting particles, and on the ISM and JET regions’ magnetic field values.
We do not consider higher-energy radiation channels, although we note that
gamma-ray halos around PWN have long been discussed (see e.g. Aharonian,
1995). In this regard, extended TeV emission has been recently reported from
the PWN in Geminga (Abeysekara et al., 2017), which is consistent with free
particle propagation in the ISM (see discussion in Giacinti et al., 2020).
Finally, we focus in this study we focus on the outcomes that may be expected
in the case of IGR J11014–6103. A general model extendable to any other sPWN
is not explored in this paper. We note also that the values adopted for the
parameters describing the injected particle distribution can be seen as rather
optimistic (e.g. $\Gamma$ could be harder than the adopted value of 2 and the
low-energy cutoff $\gamma_{\rm min}^{\rm cut}$ could be more restrictive than
$10^{3}$). These, however, are relatively unconstrained parameters for the
case of the source under study. At the same time, the adopted values are such
that they can provide estimates of the expected radio emission that can then
be contrasted with dedicated observations.

##  3 Results

Figure 4: Synthetic synchrotron specific flux maps displaying the expected
radio emission from IGR J11014–6103 in the 0.9–1.7 GHz band in which MeerKAT
operates, convolved with an instrument beam of $15\arcsec\times 15\arcsec$
(Jonas & MeerKAT Team, 2016). The contribution from particles injected all
along the IGR J11014–6103’s lifetime, which escaped the nebula and diffused
into the ISM ($B_{\rm ISM}=5\mu$G), is displayed in the left panel, and that
from the JET region ($B_{\rm ISM}=30\mu$G) is shown in the right panel. A
white star indicates the position of the pulsar, which is assumed to move
leftwards along the horizontal dashed-line.

###  3.1 Injected particle distribution

The energy distribution of particles injected along IGR J11014–6103’s lifetime
is shown in Fig. 2, which displays the two cases described in Sect. 2 for the
value of $\gamma_{\rm min}=10^{3}$ and $10^{5}$. For the high-energy cutoff, a
value of $\gamma_{\rm max}^{\rm cut}=10^{8}$ is assumed in the two cases.
Particles diffusing through the JET and the ISM regions are shown separately
in Fig. 2. For the ISM region, individual single-injection events every 1 kyr
(dotted lines) are considered. The particles evolve under synchrotron and IC
losses while accumulating, forming the final particle distribution (solid
lines). The older the injections are, the lower their contribution is at the
high-energy end of the final distribution. Freshly accelerated particles are
considered only for the JET region (dashed lines in Fig. 2).

###  3.2 Estimated synchrotron fluxes

The JET and the ISM regions differ in the assumed value for the magnetic
field. We used $B_{\rm ISM}=5\mu$G to compute the synchrotron emission by each
injection event and their total final contribution in the ISM. For the JET
region, the value of $B_{\rm JET}$ ranges between that of the ISM and the
value inferred to produce the observed X-ray emission in the Lighthouse Nebula
(for electrons at a maximal energy $\gamma_{\rm max}\approx 10^{8}$) of
$B_{\rm ISM}=30\mu$G (Pavan et al., 2014). The energy flux of the predicted
synchrotron emission is displayed in Fig. 3 for the cases $\gamma_{\rm
min}^{\rm cut}=10^{3}$ and $10^{5}$. Individual contributions and total
emission in the ISM are displayed in red with dashed and solid lines,
respectively. The energy flux from the JET region is in blue, for five
different values of $B_{\rm JET}$: 5, 10, 15, 20 and 30$\mu$G.

The radio emission for a low-energy cutoff $\gamma_{\rm min}^{\rm cut}=10^{3}$
reaches values of a few $\times 10^{29}$ erg s-1, which should be at levels
detectable by existing facilities sensitive in this band. Radio emission is
strongly suppressed for $\gamma_{\rm min}^{\rm cut}=10^{5}$, which instead
provides a larger number of particles at higher energies. Current X-ray
satellites can detect the JET region in X-rays even for modest magnetic fields
$\sim 15\mu$G. The total radio emission from the ISM region exceeds that of
the JET region by a factor 2–10, depending on the value of $B_{\rm JET}$ and
the adopted value of $\gamma_{\rm min}$. The JET region is however brighter
than any single injection event in the ISM when taking a similar injection
time lapse $\sim 1$ kyr. We include in Fig. 3 the observation bands of the
radio telescope MeerKAT and the X-ray satellite Chandra, noted by orange and
green bands, respectively.

###  3.3 Synthetic radio maps

The cooled particle distribution injected in each event producing the
synchrotron emission described above is distributed isotropically from its
injection point. The synchrotron maps displayed in Fig. 4 reflect this
isotropy in the estimated spatial extension of the emission. The injection
locations along the pulsar trajectory produce a “trail” of radio emission that
is brighter along this direction, whereas the sum of the single events display
an overall elliptical geometry encompassing the sPWN position. For the JET
region, a single injection produces a radially symmetric extended source in
our simplified model. In Fig. 4 the maps display the angular distribution of
the predicted synchrotron fluxes, which has been convolved with the beam of a
radio telescope of $15\arcsec\times 15\arcsec$, such as that of the recently
operative MeerKAT telescope (Jonas & MeerKAT Team, 2016). We report only the
case $\gamma_{\rm min}^{\rm cut}=10^{3}$ in Fig. 4, for which radio emission
can reach detectable levels, above $\sim 50\leavevmode\nobreak\ \mu$Jy beam-1.
For $\gamma_{\rm min}^{\rm cut}=10^{5}$ radio emission would be undetectable.
The spatial distribution in our model provides evidence that the JET region is
the brightest spot around IGR J11014–6103, but it is concentrated in a
relatively small angular region of a few arcmins. We did not attempt to
introduce further complexity with a nonisotropic particle distribution, which
may produce extended structures in the radio maps, for example rectilinear
jet-like features such as the ones reported in X-rays.

##  4 Discussion

The origin of the extended jet-like features observed in the Lighthouse Nebula
and other sPWNs is unclear. The high velocity of these systems and the
apparent rectilinear geometry of their associated outflows, with no
appreciable signature of bending as expected due to the ambient medium ram
pressure, suggest alternative mechanisms responsible for such jet-like
features. Bandiera (2008) proposed a scenario in which the jets could be
produced by high-energy electrons escaping the parent PWN and diffusing into
the ambient magnetic field. Recent 3D magnetohydrodynamic (MHD) numerical
simulations seem to support such a possibility (Barkov et al., 2019b;
Bucciantini et al., 2020). Here we provide further observational predictions
to probe this hypothesis in the context of the Lighthouse Nebula, modeling the
radio emission from the immediate surroundings of the source.

The results reported in Sect. 3 indicate that, for suitable parameters
describing the injected particle energy distributions escaping the sPWN,
substantial radio emission around IGR J11014–6103 could be produced by these
particles when diffusing into the ISM. We note that the value assumed for
these parameters (e.g. $\Gamma=2$ and $\gamma_{\rm min}^{\rm cut}=10^{3}$)
were chosen so as to account for cases in which detectable radio fluxes may be
expected, whereas in others (e.g. $\gamma_{\rm min}^{\rm cut}=10^{5}$) such
emission would be suppressed. For the former case, the JET region provides the
larger contribution, since $B_{\rm JET}$ is larger than $B_{\rm ISM}$. The
fact that $B_{\rm JET}>B_{\rm ISM}$ could follow the presence of streaming
instabilities generated in the flow of relativistic electrons, generating a
turbulent component of the magnetic field responsible for the field
amplification, and for the confinement of the radiating electrons themselves
(Bykov et al., 2017). The total cumulative radio emission from the more
diffuse ISM region, on the other hand, can be a factor of a few to $\sim$20
times higher than that in the JET region. According to our model, observations
should therefore reveal a bright concentration toward the jet region, and a
broader, dimmer diffuse radio nebular emission surrounding the source.
Compared to the radio emission from the PWN tail reported in (Pavan et al.,
2014), the emission from the ISM region in our model displays somewhat softer
fluxes. The nebular emission in the ISM region reported here might be
intrinsically softer than that within the PWN in IGR J11014–6103. Otherwise,
the presence of several populations of accelerated electrons in the PWN
particle injection spectrum (see e.g. Atoyan & Aharonian, 1996), could also
make the radio fluxes harder. For the JET region, although we do not aim to
describe the morphological and spectral properties of the extended jet-like
features observed in the Lighthouse Nebula in X-rays, our model predicts
fluxes compatible with the ones observed at this band, at a level $L_{\rm
2-10keV}\gtrsim 10^{33}$ erg s-1, when assuming previously derived values of
$B_{\rm JET}$ of a few $\times$ 10 $\mu$G (Pavan et al., 2014). At higher
energies, in the gamma-ray band, it is worth noting that, given the value of
$U_{\rm IR}$ the produced IC gamma-ray luminosity by the electrons responsible
for the JET X-rays should be a factor $\sim U_{\rm IR}/(B_{\rm
JET}^{2}/8\pi)\sim 0.3$ times that observed in X-rays, $L_{\rm 2-10keV}$ (a
detailed treatment is left for future work).

As stated above the radio emission predicted in our model strongly depends on
the assumed limits in the energy distribution of the emitting particles. As
displayed in Fig. 3, right panel, a strong low-energy cutof of $\gamma_{\rm
min}^{\rm cut}\gtrsim 10^{5}$ would render these structures too dim to be
detected. On the contrary, the synchrotron radio flux maps displayed in Fig. 4
for $\gamma_{\rm min}^{\rm cut}\gtrsim 10^{3}$ indicate that extended
structures should be resolvable by current radio telescopes capable of
reaching sensitivities at the level of $\gtrsim 50\mu$Jy/beam. The value of
$\gamma_{\rm min}^{\rm cut}$ depends on the mechanisms through which
accelerated particles are able to escape the PWN region. For this to occur,
the diffusion distance within the bow-shock region should be larger than the
shocked wind layer thickness. Given that this is a several times smaller than
the bow-shock size, the escape energy is just a factor order of 10 smaller
than that given by the Hillas limit, which for the Lighthouse Nebula has an
associated Lorentz factor of $\gtrsim 10^{8}$. Nevertheless, particles with
lower energies may still be able to escape the PWN along the expanding PWN
tail where the magnetic field density decreases and/or the geometry of the
shock interface with the ambient ISM becomes more favorable for electrons to
diffuse away. In this regard, it is worth noting that for the case of the
Lighthouse Nebula, the base of the main jet-like feature seems to be shifted
to a position behind the pulsar with respect to the incoming medium (see e.g.
Pavan et al., 2016; see also the inset in Fig. 9 in Kargaltsev et al., 2017),
where the PWN conditions could imply lower values for the particles’ Lorentz
factor to escape the nebula. Alternatively, particle escape can follow
magnetic reconnection between the sPWN and the ISM magnetic fields (see e.g.
Barkov et al., 2019a; Bucciantini et al., 2020), as suggested in the case of
the rectilinear structures detected in J1809-1917, for which the gyroradius of
the X-ray emitting electrons is by far shorter than the typical size of the
nebula (Klingler et al., 2020).

Finally, although in this report we have focused on the Lighthouse Nebula , we
note that other sPWNe may also display extended radio emission from electrons
escaping their nebulae. If confirmed, the detection of these radio structures
would represent a footprint for an efficient particle escape mechanism taking
place in these systems, allowing in turn the formation of extended,
rectilinear, and misaligned jet-like structures as observed in some of the
most extreme sPWNe.

###### Acknowledgements.

PB, VBR, JMP and XZ acknowledge the financial support by the Spanish
Ministerio de Economía, Industria y Competitividad (MINEICO/FEDER, UE) under
grant AYA2016-76012-C3-1-P, from the State Agency for Research of the Spanish
Ministry of Science and Innovation under grant PID2019-105510GB-C31 and
through the “Unit of Excellence María de Maeztu 2020-2023” award to the
Institute of Cosmos Sciences (CEX2019-000918-M), and by the Catalan DEC grant
2017 SGR 643.

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# Scaling of out-of-time ordered correlators in a non-Hermitian kicked rotor
model

Wen-Lei Zhao<EMAIL_ADDRESS>School of Science, Jiangxi University of
Science and Technology, Ganzhou 341000, China  Ru-Ru Wang  School of Science,
Jiangxi University of Science and Technology, Ganzhou 341000, China

###### Abstract

We investigate the dynamics of the out-of-time-ordered correlators (OTOCs) via
a non-Hermitian extension of the quantum kicked rotor model, where the kicking
potential satisfies $\mathcal{PT}$-symmetry. The spontaneous
$\cal{PT}$-symmetry breaking emerges when the strength of the imaginary part
of the kicking potential exceeds a threshold value. We find, both analytically
and numerically, that in the broken phase of $\cal{PT}$ symmetry, the OTOCs
rapidly saturate with time evolution. Interestingly, the late-time saturation
value scales as a pow-law in the system size. The mechanism of such scaling
law results from the interplay between the effects of nonlocal operator in
OTOCs and the time reversal induced by non-Hermitian driven potential.

##  I Introduction

In recent years, the out-of-time-ordered correlators (OTOCs)
$C=-\langle[A(t),B]^{2}\rangle$ have attracted extensive attentions in diverse
fields of physics, ranging from quantum chaos Maldacena16 ; Rozenbaum17 ,
quantum information Harris22  to black holes physics Hayden07 . A fundamental
concept in these fields is the information scrambling, namely, the spread of
information encoding in local degrees of freedoms over entire system so as to
inaccessible by local measurement Zanardi21 ; Touil21 ; Prakash20 . This
progress is quantified by the growth of local operators with time evolution,
due to which it will be no longer commutable with other operators separated
long distance TianciZhou20 ; SKZhao22 . The operator growth is dominated by
the classical chaos in a way the rate of exponential growth of OTOCs is
proportional to the classical Lyapunov exponent Yin21 ; Moudgalya19 .
Nowadays, the OTOCs have been widely utilized to diagnose the many-body
localization RHFan17 ; Smith19 , quantum entanglement Garttner18 ;
Keyserlingk18 ; Lerose20 , quantum thermalization Lewis-Swan19 ; Zhu22 ;
Balachandran21 , and many-body chaos Kobrin21 ; Borgonovi19 ; Shenglong22 ,
hence receiving intensive investigations in the field of many-body physics
Heyl2018 . Interestingly, experimental advances have observed both the quantum
information scrambling and quantum phase transition by measuring the OTOCs in
the system of quantum circuit Mi21  and a nuclear magnetic resonance quantum
simulator Nie20 .

For $\cal{PT}$-symmetric systems, the dynamics of OTOCs signals the Yang-Lee
edge singularity LJZhai20  of phase transition and displays the quantized
response to external driven potential WlZhao22 . It is now widely accepted
that the non-Hermiticity is a fundamental modification to conventional quantum
mechanics Berry2004 ; Ashida20 ; Bender1998 ; Bender2002 ; Zhaoxm2021 ; Yu2021
; Zhaoxm20211 , since many systems such as optics propagation in “gain-or-
loss” medium Javier22 ; Ganainy18 , the electronics transport in dissipative
circuit Xiao19 ; Chitsazi17 ; Deyuan21 , and cold atoms in tailored magneto-
optical trap Kreibich16 ; Keller97 ; Li19 ; YongmeiXue ; Zejian22 , are
described by non-Hermitian theory. The extension of Floquet systems to non-
Hermitian regime even unveils rich physics Longwen22 ; Longwen21 ; Longwen21B
; DaJian19 ; Longhi17 . For example, the scaling of the spontaneous
$\cal{PT}$-symmetry breaking and its relation with classical chaos are
uncovered in a non-Hermitian chaotic system West10 . The ballistic energy
diffusion Longhi2017  and quantized acceleration of momentum current Zhao19
are reported in a $\cal{PT}$-symmetric kicked rotor (PTKR) model. The quantum
kicked rotor (QKR) and its variants provide ideal platforms for investigating
fundamental problems, such as the quantum transport in momentum-space lattice
Santhanam22 ; Ho12 , the quantum-to-classical transition of chaotic systems
Gadway13 ; Huang21 , as well as quantum thermalization in many-body systems
Vuatelet21 . The operator growth and chaotic information scrambling in
different variations of QKR are still open issue and deserve urgent
investigations.

In this context, we investigate, both analytically and numerically, the
dynamics of OTOCs in a PTKR model, with focus on the broken phase of
$\cal{PT}$ symmetry. We find that the OTOCs rapidly saturate with time
evolution. Interestingly, the saturation value is the power-law function of
the dimension of the system, which demonstrates a kind of scaling-law of the
OTOCs with the system size. The mechanism of such scaling law results from two
aspects. One is that the action of a nonlocal operators constructing the OTOCs
on the state leads to a power-law decayed distribution in momentum space. The
another is that the non-Hermitian kicking potential induces the perfect time
reversal of the quantum state in momentum space. By utilizing the power-law
decayed quantum state, we analytically obtain the scaling of OTOCs with the
size of momentum space, for which the OTOCs is the power-law function of the
dimension of the system. This demonstrates the OTOCs unboundedly increases
with the system size, revealing a kind of fast scrambling Belyansky20 ;
Kuwahara21 . Our result sheds light on the Floquet engineering of the fast
scramblers in non-Hermitian map systems.

The paper is organized as follows. In Sec. II, we show our model and the
scaling-law of OTOCs. In Sec. III, we present the theoretical analysis of the
scaling law. Sec. IV contains the conclusion and discussion.

##  II model and results

The Hamiltonian of a PTKR reads

$H=\frac{p^{2}}{2}+V_{K}(\theta)\sum_{j=0}^{\infty}\delta(t-t_{n})\;,$ (1)

with the kicking potential

$V_{K}(\theta)=K\left[\cos(\theta)+i\lambda\sin(\theta)\right]\;,$ (2)

which satisfies the $\cal{PT}$ symmetry $V_{K}(\theta)=V_{K}^{*}(-\theta)$.
Here $p=-i\hbar_{\rm{eff}}\partial/\partial\theta$ is the angular momentum
operator and $\theta$ is the angle coordinate, which obeys the communication
relation $[\theta,p]=i\hbar_{\text{eff}}$ with $\hbar_{\text{eff}}$ the
effective Planck constant. The parameters $K$ and $\lambda$ control the
strength of the real and imaginary part of the kicking potential,
respectively. The time $t_{n}$ is integer, i.e., $t_{n}=1,2\ldots$, indicating
the kicking number. All variables are properly scaled and thus in
dimensionless units. The eigenequation of angular momentum operator is
$p|\phi_{n}\rangle=p_{n}|\phi_{n}\rangle$ with eigenstate
$\langle\theta|\phi_{n}\rangle=e^{in\theta}/\sqrt{2\pi}$ and eigenvalue
$p_{n}=n\hbar_{\rm{eff}}$. On the basis of $|\phi_{n}\rangle$, an arbitrary
quantum state can be expanded as
$|\psi\rangle=\sum_{n}\psi_{n}|\phi_{n}\rangle$. The evolution of the quantum
state from $t_{j}$ to $t_{j+1}$ is given by
$|\psi(t_{j+1})\rangle=U|\psi(t_{j})\rangle$, where the Floquet operator $U$
takes the form

$U=\exp\left(-i\frac{p^{2}}{2\hbar_{\rm{eff}}}\right)\exp\left[-i\frac{V_{K}(\theta)}{\hbar_{\rm{eff}}}\right]\;.$
(3)

The OTOCs are defined as the average of the squared commutator, i.e.,
$C(t)=-\langle[A(t),B]^{2}\rangle$, where both operators
$A(t)=U^{\dagger}(t)AU(t)$ and $B$ are evaluated in the Heisenberg picture,
and the $\langle\cdots\rangle=\langle\psi(t_{0})|\cdots|\psi(t_{0})\rangle$
indicates the expectation value taken over the initial state
$|\psi(t_{0})\rangle$ Heyl2018 . It usually uses the thermal states for taking
the average in investigation of OTOCs of lattice systems. For the Floquet-
driven system, however, there is no the well-defined thermal states, as the
temperature trends to be infinity as time evolves DAlessio14 . Without loss of
generality, we choose a Gaussian wavepacket as an initial state, i.e.,
$\psi{(\theta,0)}=(\sigma/\pi)^{1/4}\exp(-\sigma\theta^{2}/2)$ with
$\sigma=10$. We consider the case with $A=\theta$ and $B=p^{m}$
($m\in\mathbb{N}$), hence $C(t)=-\langle[\theta(t),p^{m}]^{2}\rangle$.

Our main result is the scaling law of the OTOCs

$C{(t)}\sim N^{2m-1}\theta^{2}_{c}\;,$ (4)

where $N$ is the dimension of the momentum space of the PTKR model, and
$\theta_{c}=\pi/2$. The above prediction is verified by numerical results in
Fig. 1. As an illustration, we consider $m=1$, 2, and 3 in numerical
simulations. Figure 1(a) shows that for a specific $m$, the $C(t)$ saturates
rapidly as time evolves, which is in perfect agreement with our theoretical
prediction in Eq. (4). In order to further confirm the scaling law of $C(t)$,
we numerically investigate the $C$ at a specific time for different $N$.
Figure 1(b) shows that for $t=t_{10}$, the value of $C$ increases in the
power-law of $N$, which coincides with the theoretical prediction in Eq. (4).
The scaling of $C(t)$ with dimension of the system demonstrates that it
diverges as $N\rightarrow\infty$, which is of interest in the study of fast
scrambling Kuwahara21 . We would like to mention that we previously found the
scaling law for the OTOCs constructed by $A=\theta$ and $B=p$, in a Gross-
Pitaevski map system Zhao21 . Our present work explores the scaling law for
$B=p^{m}$ with arbitrary integer $m$, moreover extends the investigation to
non-Hermitian systems, which is obviously an significant advance in the fields
of operator growth in chaotic systems.

Figure 1: (a) Time dependence of $C(t)$ with $B=p$ (squares), $p^{2}$
(circles), and $p^{3}$ (triangles) with $N=2^{13}$. (b) The $C(t)$ at the time
$t=t_{10}$ versus $N$. Solid lines in (a) and (b) denote our theoretical
prediction in Eq.(2). The parameters are $K=2\pi$, $\lambda=0.9$, and
$\hbar_{\rm{eff}}=0.1$.

##  III THEORETICAL ANALYSIS

###  III.1 Analysis of the $C_{1}(t)$

Straightforward derivation yields the expression of OTOCs

$C{(t)}=C_{1}(t)+C_{2}(t)-2\text{Re}[C_{3}(t)]\;,$ (5)

where the three terms in right side are defined by

$C_{1}(t)=\langle\psi_{R}(t_{0})|p^{2m}|\psi_{R}(t_{0})\rangle\;,$ (6)
$C_{2}(t)=\langle\varphi_{R}(t_{0})|\varphi_{R}(t_{0})\rangle\;,$ (7)

and

$C_{3}(t)=\langle\psi_{R}(t_{0})|p^{m}|\varphi_{R}(t_{0})\rangle\;,$ (8)

with $|\psi_{R}(t_{0})\rangle=U^{\dagger}(t_{n},t_{0})\theta
U(t_{n},t_{0})|\psi(t_{0})\rangle$ and
$|\varphi_{R}(t_{0})\rangle=U^{\dagger}(t_{n},t_{0})\theta
U(t_{n},t_{0})p^{m}|\psi(t_{0})\rangle$.

To get the state $|\psi_{R}(t_{0})\rangle$, one needs three steps: i). forward
evolution $t_{0}\rightarrow t_{n}$, i.e.,
$|\psi(t_{n})\rangle=U(t_{0},t_{n})|\psi(t_{0})$, ii). action of the operator
$\theta$ on $|\psi(t_{n})\rangle$, i.e.,
$|\tilde{\psi}(t_{n})\rangle=p|\psi(t_{n})\rangle$, and iii). backward
evolution $t_{n}\rightarrow t_{0}$, i.e.,
$|\psi_{R}(t_{0})\rangle={U}^{\dagger}(t_{0},t_{n})|\tilde{\psi}(t_{n})\rangle$.
The $C_{1}(t_{n})$ [see Eq. (6)] is just expectation value of the $p^{2m}$
taken over the state $|\psi_{R}(t_{0})\rangle$. For the numerical calculation
of the state $|\varphi_{R}(t_{0})\rangle$, one should first construct the
operation of $p^{m}$ on the initial state $|\psi(t_{0})\rangle$, i.e.,
$|\varphi(t_{0})\rangle=p|\psi(t_{0})\rangle$. Then, forward evolution from
$t_{0}$ to $t_{n}$ yields the state
$|\varphi(t_{n})\rangle=U(t_{0},t_{n})|\varphi(t_{0})\rangle$. At the time
$t=t_{n}$, the action of $p^{m}$ on the state $|\varphi(t_{n})\rangle$ results
in a new state $|\tilde{\varphi}(t_{n})\rangle=p|\varphi(t_{n})\rangle$,
starting from which the time reversal process $t_{n}\rightarrow t_{0}$ yields
the state
$|\varphi_{R}(t_{0})\rangle={U}^{\dagger}(t_{0},t_{n})|\varphi(t_{0})\rangle$.
The norm of the $|\varphi_{R}(t_{0})\rangle$ is just the $C_{2}(t_{n})$ [see
Eq. (7)]. As the two states $|\psi_{R}(t_{0})\rangle$ and
$|\varphi_{R}(t_{0})\rangle$ are available at the end of time reversal, one
can calculate the $C_{3}(t_{n})$ according to Eq. (8)].

It is known that in the $\cal{PT}$-symmetry breaking phase, the norm of
quantum state
$\mathcal{N}_{\psi}(t_{n})=\langle\psi(t_{n})|\psi(t_{n})\rangle$
exponentially increases for both the forward and backward time evolution. To
eliminate the contribution of norm to OTOCs, it is necessary to take the
normalization for the time-evolved state. Specifically, for the forward
evolution $t_{0}\rightarrow t_{n}$, we set the norm of the quantum state
equals to that of the initial state, i.e.,
$\mathcal{N}_{\psi}(t_{j})=\langle\psi(t_{0})|\psi(t_{0})\rangle$ with $0\leq
j\leq n$. The backward evolution starts from the time $t=t_{n}$ with the state
$|\tilde{\psi}(t_{n})\rangle$, whose norm
$\mathcal{N}_{\tilde{\psi}}(t_{n})=\langle\psi(t_{n})|\theta^{2}|\psi(t_{n})\rangle$
is expectation value of $\theta^{2}$ with the state $|\psi(t_{n})\rangle$. It
is apparent that the value of $\mathcal{N}_{\tilde{\psi}}(t_{n})$ is an
important information encoded by the operation of $\theta$ on the state
$|\psi(t_{n})\rangle$. Based on this, we take the normalization of the quantum
state in the backward evolution $t_{n}\rightarrow t_{0}$ in such a way that
its norm equals to $\mathcal{N}_{\tilde{\psi}}(t_{n})$, i.e.,
$\mathcal{N}_{\psi_{R}}(t_{j})=\mathcal{N}_{\tilde{\psi}}(t_{n})$. One can
find that for both the forward and backward evolution, the norm of a time-
evolved state always equals to that of the state which the time evolution
starts from. The the same procedure of normalization is applied in calculating
$C_{2}(t_{n})$. Therefore, we have the equivalence
$\mathcal{N}_{\varphi}(t_{j})=\langle\varphi(t_{0})|\varphi(t_{0})\rangle$ and
$\mathcal{N}_{\varphi_{R}}(t_{j})=\langle\tilde{\varphi}(t_{n})|\tilde{\varphi}(t_{n})\rangle$
($0\leq j\leq n$) for the forward evolution and time reversal, respectively.

Figure 2: Time evolution of $\mathcal{N}$ (a), $\langle\theta\rangle$ (a), and
$\langle p\rangle$ (b) with $t=t_{10}$. In (a): Solid and dash-dotted lines
indicate $\mathcal{N}=\theta^{2}_{c}$ and
$\langle\theta\rangle=\theta_{c}(=\pi/2)$, respectively. In (b): Solid line
indicates $\langle p\rangle=Kt$. Green dashed lines in (a) and (b) are
auxiliary lines. The parameters are the same as in Fig. 1.

We rewrite the $C_{1}$ as

$C_{1}(t)=\langle\psi_{R}(t_{0})|p^{2m}|\psi_{R}(t_{0})\rangle=\langle
p^{2m}(t_{0})\rangle_{R}\mathcal{N}_{\psi_{R}}(t_{0})\;,$ (9)

where
$\mathcal{N}_{\psi_{R}}(t_{0})=\langle\psi_{R}(t_{0})|\psi_{R}(t_{0})\rangle$
is the norm of the quantum state $|\psi_{R}(t_{0})\rangle$ and $\langle
p^{2m}(t_{0})\rangle_{R}=\langle\psi_{R}(t_{0})|{p}^{2m}|\psi_{R}(t_{0})\rangle/\mathcal{N}_{\psi_{R}}(t_{0})$
indicates the exception value of ${p}^{2m}$ of the state
$|\psi_{R}(t_{0})\rangle$ with the division of its norm. We numerically
investigate both the forward and backward evolution of the norm $\mathcal{N}$,
and the mean values $\langle\theta\rangle$ and $\langle p\rangle$ for a
specific time, e.g., $t=t_{10}$. Note that we define the expectation value of
observable $Q$ as $\langle
Q\rangle=\langle\psi(t)|Q|\psi(t)\rangle/\mathcal{N}(t)$ with
$\mathcal{N}(t)=\langle\psi(t)|\psi(t)\rangle$. It is apparently that such
kind of definition eliminates the contribution of norm to mean value. Figure
2(a) shows that the norm is unity during the forward time evolution (i.e.,
$t_{0}\rightarrow t_{10})$, and remains at a fixed value, i.e., i.e.,
$\mathcal{N}_{\psi_{R}}(t_{0})\approx\theta^{2}_{c}$ during the backward
evolution (i.e., $t_{10}\rightarrow t_{0}$). For $t_{0}\rightarrow t_{10}$,
the value of norm equals to that of the normalized initial state, so
$\mathcal{N}(t_{j})=1$. For the time reversal $t_{10}\rightarrow t_{0}$, our
normalization procedure results in the equivalence
$\mathcal{N}_{\psi_{R}}(t_{j})=\langle\psi(t_{n})|\theta^{2}|\psi(t_{n})\rangle$.
Interestingly, our numerical investigations in Figs. 3(a), (c) and (e) show
that for the forward evolution, the initially Gaussian wavepacket rapidly
moves to the position $\theta_{c}=\pi/2$. During the time reversal, it remains
localized at $\theta_{c}$ with the width of distribution being much smaller
than that of the state of forward evolution. Correspondingly, the mean value
$\langle\theta\rangle$ has very slight differences with $\theta_{c}$ [see Fig.
2(a)]. Since the quantum state is extremely localized at $\theta_{c}$, one can
get the approximation

$\mathcal{N}_{\psi_{R}}(t_{0})=\langle\psi(t_{n})|\theta^{2}|\psi(t_{n})\rangle\approx\theta^{2}_{c}\;.$
(10)

Figure 3: Distributions in real (left panels) and momentum (right panels)
space. In (a)-(d): Black and red lines indicate the distribution of the states
at the forward $|\psi(t_{j})\rangle$ and backward $|\psi_{R}(t_{j})\rangle$
evolution, respectively, with $t=t_{0}$ (top panels), $t=t_{5}$ (middle
panels), and $t=t_{10}$ (bottom panels). In (e)-(f): Red and black lines
indicate the distribution of the states $|\psi(t_{10})\rangle$ and
$|\tilde{\psi}(t_{10})\rangle=\theta|\psi(t_{10})\rangle$. Blue dashed lines
indicate the power-law decay $|\psi|^{2}\propto(p-p_{c})^{-2}$. The parameters
are the same as in Fig. 1.

Figures 3(b), (d) and (f) show the momentum distribution of the state during
both forward and backward evolution. For the forward evolution, the quantum
state behaves like a soliton which moves to positive direction in momentum
space, resulting in the linear increase of mean momentum, i.e., $\langle
p\rangle=Kt$ [see Fig. 2(b)]. The mechanism of the directed acceleration has
been unveiled in our previous investigation in Ref. Zhao19 ; WlZhao22 .
Intriguingly, at the time $t=t_{10}$, the action of $\theta$ yields a state
with a power-decayed shape, i.e.,
$|\psi_{R}(p,t_{0})|^{2}\propto(p-p_{c})^{-2}$ [see Fig. 3(f)]. Most
importantly, during backward evolution, the quantum state still retains the
power-law decayed shape, for which the center $p_{c}$ decreases with time and
almost overlaps with that of the state of forward evolution. This clearly
demonstrates a kind of time reversal of transport behavior in momentum space.

In the aspect of the mean momentum $\langle p\rangle$, we find that the value
of $\langle p\rangle$ linearly decreases during the backward evolution and is
in perfect symmetry with that of forward evolution, which is a solid evidence
of time reversal. In the end of the backward evolution, the quantum state
$|\psi_{R}(t_{0})\rangle$ is localized at the point $p=0$ [see Fig. 3(b)]. By
using the power-law distribution $|\psi_{R}(p,t_{0})|^{2}\sim p^{-2}$, it is
straightforwardly to get the estimation of the expectation value of $p^{2m}$,
i.e., $\langle
p^{2m}\rangle_{\psi_{R}}=\int_{p_{-N/2}}^{p_{N/2}}{p^{2m}|\psi_{R}(p,t_{0})|^{2}dp}\propto
N^{2m-1}$. Taking both the $\langle p^{2m}\rangle_{\psi_{R}}$ and
$\mathcal{N}_{\psi_{R}}(t_{0})$ in Eq. (10) into Eq. (9) yields the relation

$C_{1}(t)\propto N^{2m-1}\theta_{c}^{2}\;,$ (11)

which is verified by our numerical results in Fig. 4. As an illustration, we
consider the cases with $m=1$, 2 and 3. Our numerical results of the time
dependence of $C_{1}$ is in good agreement with Eq. (11). It is now clear that
the scaling of $C(t)$ with $N$ originates from the power-law decay of the
state $|\psi_{R}(t_{0})\rangle$. The reason for the formation of power-law
decayed wavefuction has been uncovered in Ref. Zhao21 .

Figure 4: The $C_{1}$ (squares), $C_{2}$ (circles), and
$\left|\text{Re}[C_{3}]\right|$ (triangles) versus time with $B=p$ (a),
$p^{2}$ (b), and $p^{3}$ (c). Dash-dotted, solid, and dashed lines indicate
our theoretical prediction in Eq. (11) for $C_{1}$, Eq. (13) for $C_{2}$, and
Eq. (15) for $C_{3}$, respectively. The parameters are the same as in Fig. 1.

###  III.2 Analysis of $C_{2}(t)$

We proceed to evaluate the time dependence of $C_{2}(t)$ in Eq. (7), which is
just the norm of the state $|\varphi_{R}(t_{0})\rangle$ at the end of backward
evolution. According to our normalization procedure, the value of $C_{2}$
equals to the norm of the state
$|\tilde{\varphi}(t_{n})\rangle=\theta|\varphi(t_{n})\rangle$, hence

$C_{2}=\langle\varphi(t_{n})|\theta^{2}|\varphi(t_{n})\rangle=\langle\theta^{2}\rangle\mathcal{N}_{\varphi}(t_{n})\;,$
(12)

with
$\mathcal{N}_{\varphi}(t_{n})=\langle\varphi(t_{n})|\varphi(t_{n})\rangle$ and
$\langle\theta^{2}\rangle=\langle\varphi(t_{n})|\theta^{2}|\varphi(t_{n})\rangle/\mathcal{N}(t_{n})$.
We numerically find that the state $|\varphi(t_{n})\rangle$ is extremely
localized at the position $\theta_{c}$ during the forward evolution [see Fig.
5]. Then a rough estimation yields
$\langle\theta^{2}\rangle\sim\theta_{c}^{2}$. The norm
$\mathcal{N}_{\varphi}(t_{n})$ equals to that of the initial state
$|\varphi(t_{0})\rangle=p^{m}|\psi(t_{0})\rangle$. By using the initially
Gaussian wavepacket
$\psi(p,t_{0})=(1/\sigma\hbar_{\text{eff}}^{2}\pi)^{1/4}\exp(-p^{2}/2\sigma\hbar_{\text{eff}}^{2})$,
one can straightforwardly get

$\mathcal{N}_{\varphi}(t_{n})=\int_{-\infty}^{\infty}p^{2m}|\psi(p,t_{0})|^{2}dp=\frac{(2m-1)!!}{2^{m}\alpha^{m}}\;,$

where $\alpha=1/(\sigma\hbar^{2})$ and $(...)!!$ denotes a double factorial.
Taking both the $\langle\theta\rangle$ and $\mathcal{N}_{\varphi}(t_{n})$ into
Eq. (12) yields

$C_{2}(t)\sim\theta^{2}_{c}\frac{(2m-1)!!}{2^{m}\alpha^{m}}\;,$ (13)

which is in good agreement with our numerical results in Fig. 4.

Figure 5:  Probability density distributions in real space at the time
$t=t_{0}$ (squares), $t_{5}$ (triangles), and $t_{1}0$ (circles) with $B=p$
(a), $p^{2}$ (b), and $p^{3}$ (c). The parameters are the same as in Fig. 1.

###  III.3 Analysis of $C_{3}(t)$

The value of $C_{3}(t)$ depends on both the states $|\psi_{R}(t_{0})\rangle$
and $|\varphi_{R}(t_{0})\rangle$ [see Eq. (8)]. Figure 6 shows the probability
density distributions of the two states in both the real space and momentum
space. For comparison, the two states are normalized to unity. One can find
the perfect consistence between $|\psi_{R}(t_{0})\rangle$ and
$|\varphi_{R}(t_{0})\rangle$. Then, we roughly regard the $C_{3}$ as the
expectation value of the $p^{m}$ taking over the state $\psi_{R}(t_{0})$ or
$\varphi_{R}(t_{0})$, i.e., $C_{3}(t)\approx\langle
p^{m}(t_{0})\rangle_{\psi_{R}}\sqrt{\mathcal{N}_{\psi_{R}}(t_{0})}\sqrt{\mathcal{N}_{\varphi_{R}}(t_{0})}$,
where according to above derivations
$\mathcal{N}_{\psi_{R}}(t_{0})=\theta^{2}_{c}$ and
$\mathcal{N}_{\varphi_{R}}(t_{0})=C_{2}(t)$. By using the power-law decayed
wavepacket $|\psi_{R}(t_{0})|^{2}\propto p^{-2}$, one can get the estimation

$\displaystyle\langle p^{m}(t_{0})\rangle_{\psi_{R}}$
$\displaystyle\approx\int_{p_{-N/2}}^{p_{N/2}}{p^{m}|\psi_{R}(p,t_{0})|^{2}dp}$
(14) $\displaystyle\sim\begin{cases}0&\text{for odd}\;m\;,\\\
N^{m-1}&\text{for even}\;m\;.\end{cases}$

Accordingly, the $C_{3}$ is approximated as

$C_{3}(t)\sim\begin{cases}0&\text{for odd}\;m\;,\\\ \eta N^{m-1}&\text{for
even}\;m\;.\end{cases}$ (15)

with the prefactor
$\eta\propto\theta_{c}^{2}\left[{(2m-1)!!}/{2^{m}\alpha^{m}}\right]^{\frac{1}{2}}$.

Figure 6: Comparison of the distribution of states $|\psi_{R}(t_{0})\rangle$
(solid lines) and $|\varphi_{R}(t_{0})\rangle$ (dashed lines) in real (left
panels) and momentum space (right panels) with $B=p$ (top panels), $p^{2}$
(middle panels), and $p^{3}$ (bottom panels). Blue dashed lines in (b), (d)
and (f) indicate the power-law decay
$|\psi_{R}(t_{0})|^{2}(|\varphi_{R}(t_{0})|^{2})\propto p^{-2}$. The
parameters are the same as in Fig. 1.

We numerically calculate the absolute value of the real part of $C_{3}$.
Interestingly, our numerical results of $|\text{Re}[C_{3}]|$ is in good
agreement with the analytical prediction in Eq. (15) [see Fig. 4], which
proves the validity of our theoretical analysis. We further numerically
investigate the $|\text{Re}[C_{3}(t)]|$ at a specific time for different $N$.
Figure 7 shows that for $B=p$, the value of $|\text{Re}[C_{3}(t)]|$ is nearly
zero with varying $N$, which is consistent with our theoretical prediction in
Eq. (15). For $B=p^{3}$, the value of $|\text{Re}[C_{3}(t)]|$ has slight
difference with zero for large $N$, signaling the derivations with Eq. (15).
This is due to the fact the quantum state $|\psi_{R}(t_{0})|^{2}$ is not
exactly symmetric around $p$. Interestingly, for $B=p^{2}$, the value of
$|\text{Re}[C_{3}(t)]|$ increases linearly with increasing $N$, which is a
clear evidence of the validity of our theoretical prediction.

Figure 7: The $|\text{Re}[C_{3}(t)]|$ at the time $t=t_{10}$ versus $N$ with
$B=p$ (squares), $p^{2}$ (circles), and $p^{3}$ (triangles). Red solid line
indicates our theoretical prediction in Eq. (15) with $\eta=6.05\times
10^{-7}$ for $B=p^{2}$. The parameters are the same as in Fig. 1.

##  IV CONCLUSION AND DISCUSSION

In the present work, we investigate the dynamics of the
$C(t)=-\langle[\theta(t),p^{m}]\rangle$ in a PTKR model. The spontaneous
$\cal{PT}$ symmetry breaking is assured by the condition
$\lambda>\lambda_{c}$. In the broken phase of $\cal{PT}$-symmetry, we find,
both analytically and numerically, the scaling law of $C(t)$ with the
dimension of the momentum space, i.e., $C(t)\sim N^{2m-1}\theta_{c}^{2}$. This
demonstrates that the value of $C$ increases unboundedly with $N$, which
implies that the local perturbation can spread to the entire system very
rapidly. In order to reveal the mechanism of the scaling, we make detailed
investigations on both the forward and backward evolution of the quantum
state. Our investigations show that the action of $\theta$ on a quantum state
leads to the formation of the power-law decayed momentum distribution
$|\psi(p)|^{2}\propto(p-p_{c})^{-2}$. Interestingly, such a shape retains
during the time reversal, besides the decrease of $p_{c}$ to almost zero.
Based on the power-law decayed state, we analytically derive the time
dependence of the three parts of the $C$, which is confirmed by numerical
results.

In recent years, fruitful physics of quantum many-body systems, such as
dynamical phase transition, many-body localization, and thermalization have
received extensive studies. It is found that the energy conservation of
chaotic systems leads to the scaling law of OTOCs, for which the late-time
saturation of OTOCs scales as the inverse polynomial with the system size
Huang19 . For chaotic systems with long-range interaction, the time-dependence
of OTOCs obeys the dynamical scaling law near the phase transition point Wei19
. Accordingly, our finding of the power-law scaling of OTOCs with the system
size of the PTKR model serves as a new element of the quantum information
scrambling in non-Hermitian map systems.

## ACKNOWLEDGMENTS

Wen-Lei Zhao and Ru-Ru Wang are supported by the National Natural Science
Foundation of China (Grant No. 12065009) and the Science and Technology
Planning Project of Jiangxi province (Grant No. 20224ACB201006).

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11institutetext: Institute for High Energy Physics NRC Kurchatov Institute,
142281 Protvino, Russia 22institutetext: Pacific Quantum Center, Far Eastern
Federal University, 690950 Vladivostok, Russia 33institutetext: Institute of
Theoretical and Experimental Physics NRC Kurchatov Institute, 117218 Moscow,
Russia 44institutetext: Institut Denis Poisson CNRS/UMR 7013, UniversitÃ de
Tours, 37200 France

# Decomposition of the static potential in the Maximal Abelian gauge

Vitaly Bornyakov 112233<EMAIL_ADDRESS>Vladimir Goy 2244 Ilya Kudrov
33 Roman Rogalyov 11

###### Abstract

Decomposition of SU(2) gauge field into the monopole and monopoleless
components is studied in the Maximal Abelian gauge using Monte-Carlo
simulations in lattice SU(2) gluodynamics as well as in two-color QCD with
both zero and nonzero quark chemical potential. The interaction potential
between static charges is calculated for each component and their sum is
compared with the non-Abelian static potential. A good agreement is found in
the confinement phase. Implications of this result are discussed.

##  1 Introduction

We study the decomposition of the non-Abelian gauge field in the Maximal
Abelian gauge (MAG) Kronfeld:1987ri ; tHooft:1981bkw  into the sum of the
monopole component and the monopoleless component. For this purpose we employ
the SU(2) lattice gauge theory.

In terms of vector potentials $A_{\mu}(x)$, the decomposition has the form

$A_{\mu}(x)=A_{\mu}^{mod}(x)+A_{\mu}^{mon}(x)$ (1)

where $A_{\mu}^{mon}(x)$ is the monopole component and $A_{\mu}^{mod}(x)$ is
the monopoleless component defined below and referred to as the modified gauge
field.

In the MAG, the Abelian dominance for the string tension has long been known
suzuki1 ; suzuki2 ; bbms ; bm ; Sakumichi:2014xpa  (for a review see e.g.
Chernodub:1997ay ; Haymaker:1998cw ). Moreover, it was found suzuki3 ; stack ;
bbms  that the properly determined monopole component of the gauge field
produces the string tension close to its exact value in agreement with
conjecture that the monopole degrees of freedom are responsible for
confinement tHooft:1975pu ; Mandelstam:1974pi .

In Refs. miyamura ; Kitahara:1998sj  it was shown that the topological charge,
chiral condensate and effects of chiral symmetry breaking in quenched light
hadron spectrum disappear after removal of the monopole contribution from the
relevant operators. Similar computations were made within the scope of the
$Z_{2}$ projection studies deforcrand . In particular it was shown that after
removal of P-vortices the confinement property disappears. We perform a
similar removal of monopoles. We consider the following types of the static
potential: $V_{mod}(r)$ obtained from the Wilson loops of the modified gauge
field $U_{\mu}^{mod}(x)$, $V_{mon}(r)$ obtained from the Wilson loops of the
monopole gauge field $u_{\mu}^{mon}(x)$ and the sum of these two static
potentials.

For one value of lattice spacing it was shown Bornyakov:2005hf  that
$V_{mod}(r)$ can be well approximated by purely Coulomb fit function and the
sum $V_{mod}(r)+V_{mon}(r)$ provides a good approximation to the original non-
Abelian static potential $V(r)$ at all distances.

Here we study this phenomenon at three lattice spacings using the Wilson
action and thus we can draw conclusions about the continuum limit. We also
present the results for one lattice spacing obtained with the improved lattice
field action thus checking the universality. Furthermore, we present results
for the SU(2) theory with dynamical quarks, i.e. for QC2D. These results were
partially presented in Bornyakov:2021enf .

##  2 Details of simulations

We study the SU(2) lattice gauge theory. Vector potentials of the gauge field
can be defined in terms of the link variables by the formula

$A_{\mu}(x)={1\over 2iag}\;\Big{(}U_{x\mu}-U_{x\mu}^{\dagger}\Big{)}\;,$ (2)

where $a$ is the lattice spacing. Up to terms of the order
$\underline{O}(a^{2})$, the decomposition (1) can be rearranged to the form

$U_{\mu}(x)=U_{\mu}^{mod}(x)u_{\mu}^{mon}(x)\,$ (3)

which furnishes the subject of our research.

To fix the MAG we use the simulated annealing algorithm bbms  with one gauge
copy per configuration. Usually after fixing MAG the following decomposition
of the non-Abelian lattice gauge field $U_{\mu}(x)$ is made

$U_{\mu}(x)=C_{\mu}(x)u_{\mu}(x)\quad,$ (4)

where $u_{\mu}(x)$ is the Abelian field and $C_{\mu}(x)$ is the non-Abelian
coset field. The Abelian gauge field $u_{\mu}(x)$ is further decomposed svs
into the monopole (singular) part $u^{mon}_{\mu}(x)$ and the photon (regular)
part $u^{ph}_{\mu}(x)$:

$u_{\mu}(x)=u^{mon}_{\mu}(x)u^{ph}_{\mu}(x)\ .$ (5)

Then it follows from eq. (1) that

$U_{\mu}^{mod}(x)=C_{\mu}(x)u^{ph}_{\mu}(x).$ (6)

Note that $u^{ph}(x)$ is the Abelian projection of $U_{\mu}^{mod}(x)$ and
involves no monopoles.

We need to compute the usual Wilson loops

$W(C)=\frac{1}{2}\mbox{Tr}{\cal{W}}(C)\,,\quad{\cal{W}}(C)=\left(\prod_{l\in
C}U(l)\right)\quad,$ (7)

the monopole Wilson loops

$W_{mon}(C)=\frac{1}{2}\mbox{Tr}\left(\prod_{l\in C}u^{mon}(l)\right)\quad,$
(8)

and the non-Abelian Wilson loops with removed monopole contribution

$W_{mod}(C)=\frac{1}{2}\mbox{Tr}{\cal{W}}_{mod}(C)\,,\quad{\cal{W}}_{mod}(C)=\left(\prod_{l\in
C}U^{mod}(l)\right)\quad,$ (9)

It is known that MAG fixing leaves $U(1)$ gauge symmetry unbroken. The
monopole Wilson loop $W_{mon}(C)$ is invariant under respective residual gauge
transformations. This is not true for $W_{mod}(C)$ Bornyakov:2005hf . Thus we
need to fix the Landau $U(1)$ gauge by finding the maximum of the gauge-fixing
functional,

$\max_{\omega}\sum_{x,\mu}\mathrm{Re}(\omega(x)u_{\mu}(x)\omega^{\dagger}(x+a\hat{\mu}))\,,$
(10)

where $\omega\in U(1)$ is the gauge transformation. To imrove the noise-to-
signal ratio for the static potential we use the APE smearing ape  in
computations of the Wilson loops.

Figure 1: Non-Abelian static potential $V(r)$ (filled symbols) is compared
with the sum $V_{mon}(r)+V_{mod}(r)$ (empty symbols) for Wilson action at
$\beta=2.4$ (squares), $\beta=2.5$ (circles), $\beta=2.6$ (triangles).

We generate 100 statistically independent gauge field configurations with the
Wilson lattice action at $\beta=2.4,2.5$ on the $24^{4}$ lattice and at
$\beta=2.6$ on the $32^{4}$ lattice. The respective values of lattice spacing
are $a=0.118,\ 0.085$ and $0.062$ fm, which are determined from a fit to the
lattice data Fingberg:1992ju  on the string tension, whose “experimental”
value is set to $\sqrt{\sigma}=440$ MeV. As a check of universality the
computations were done with the tadpole improved action at $\beta=3.4$ on
$24^{4}$ lattices. Additionally we present our results Bornyakov:2017txe
obtained in QC2D on $32^{4}$ lattice at zero and nonzero quark chemical
potential $\mu_{q}$.

It is worth to note that another decomposition, namely eq. (4) was
investigated in Ref. Sakumichi:2014xpa  in the case of SU(3) gluodynamics.
Good agreement between the static potenial $V(r)$ and the sum
$V_{abel}(r)+V_{off}(r)$ was found. We believe that this decomposition also
deserves further study.

##  3 Results

Figure 2: The relative deviation $\Delta(r)$ vs. distance $r$ for three values
of the lattice spacing. Figure 3: Non-Abelian static potential $V(r)$ (filled
symbols) is compared with the sum of the monopole and modified potentials
$V_{sum}(r)$ (empty symbols) for the improved action at $\beta=3.4$.

In Fig. 1 we show the usual non-Abelian static potential $V(r)$ denoted as
’full’ and compare it with the sum $V_{sum}(r)=V_{mon}(r)+V_{mod}(r)$. One can
see that approximate equality

$V(r)\approx V_{sum}(r)$ (11)

is satisfied for all three lattice spacings and the approximation improves
toward the continuum limit. To give further support to this statement we plot
in Fig. 2 the relative deviation $\Delta(r)$ defined as

$\Delta(r)=\frac{V(r)-V_{sum}(r)}{V(r)}\;.$ (12)

It is clear that $\Delta(r)$ decreases with decreasing lattice spacing.

Figure 4: The non-Abelian static potential $V(r)$ (filled squares) is compared
with the sum of the monopole and modified potentials $V_{sum}(r)$ (filled
circles) for QC2D at zero chemical potential. The potentials $V_{mon}(r)$
(empty circles) and $V_{mod}(r)$ (empty squares) are also shown. Figure 5: The
non-Abelian static potential $V(r)$ (filled squares) is compared with the sum
of the monopole and modified potentials $V_{sum}(r)$ (filled circles) in QC2D
at $a\mu_{q}=0.19$. The potentials $V_{mon}(r)$ (empty circles) and
$V_{mod}(r)$ (empty squares) are also shown.

We also studied the universality of the decomposition of the static potential
eq. (11). The simulations were made with the tadpole improved action at
$\beta=3.4$ with lattice spacing approximately equal to that of the Wilson
action at $\beta=2.5$. The results of our computations are presented in Fig.
3. It is seen that the agreement between $V(r)$ and $V_{sum}(r)$ is nearly as
good as in Fig. 1 for $\beta=2.5$.

Furthermore, we completed the same computations in QC2D on $32^{4}$ lattices
with small lattice spacing at zero and nonzero quark chemical potential
$\mu_{q}$ (details of simulations in QC2D can be found, e.g., in
Bornyakov:2017txe ). The results of these computations are presented in Fig. 4
for $\mu_{q}=0$ and in Fig. 5 for $a\mu_{q}=0.19$. It can be seen clearly that
approximate decomposition (11) is fulfilled with rather high both at zero and
nonzero $\mu_{q}$.

##  4 Conclusions

We have studied the decomposition of the static potential into the linear term
produced by the monopole (Abelian) gauge field $U_{mon}(x)$ and the Coulomb
term produced by the monopoleless non-Abelian gauge field $U_{mod}(x)$. In the
case of Wilson action we have presented the results (Figs. 1 and 2) for three
values of lattice spacing to demonstrate that the agreement becomes better
with a decrease of lattice spacing. Thus our results imply that the relation
(1) becomes exact in the continuum limit. Further work is needed to provide
more evidence for this conclusion. Next, we have demonstrated that the
decomposition (1) holds true also in the case of improved lattice action (see
Fig. 3). Furthermore, we have found that it works also in QC2D for both zero
and nonzero quark chemical potential. It should be noticed that in Ref.
Bornyakov:2021enf  we also presented results for the static potential in the
adjoint representation and found that the decomposition (1) works quite well
in this case although not so well as in the case of the fundamental
representation. It is of course interesting to verify how decomposition (1)
works for other observables, e.g. action density and energy density of the
flux tube. Also the decomposition should be checked in the case of SU(3) gauge
group.

One can draw the following conclusions from the decomposition (1). The
monopole part of the gauge field $U_{mon}(x)$ is responsible for the classical
part of the energy of the hadronic string, whereas the monopoleless part
$U_{mod}(x)$ is associated with the fluctuating part of its energy, i.e.
$U_{mod}(x)$ reproduces perturbative results at small distances and
contributes to the nonperturbative physics at large distances.

Acknowledgments. Computer simulations were performed on the Central Linux
Cluster of the NRC âKurchatov Instituteâ - IHEP (Protvino) and Linux Cluster
of the NRC âKurchatov Instituteâ - ITEP (Moscow). This work was supported by
the Russian Foundation for Basic Research, grant no.20-02-00737 A. The authors
are grateful to G. Schierholz, T. Suzuki, S. Syritsyn, V. Braguta, A. Nikolaev
for participation at the early stages of this work and for useful discussions.

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