11institutetext: Physics Department, University of Ioannina, Ioannina
GR-45110, Greece

# Relative field line helicity of a large eruptive solar active region

K. Moraitis1 S. Patsourakos1 A. Nindos1

(Received … / Accepted …)

###### Abstract

Context. Magnetic helicity is a physical quantity of great importance in the
study of astrophysical and natural plasmas. Although a density for helicity
cannot be defined, a good proxy for it is field line helicity. The appropriate
quantity for use in solar conditions is relative field line helicity (RFLH).

Aims. This work aims to study in detail the behaviour of RFLH, for the first
time, in a solar active region (AR).

Methods. The target active region is the large, eruptive AR 11158. In order to
compute RFLH and all other quantities of interest we use a non-linear force-
free reconstruction of the AR coronal magnetic field of excelent quality.

Results. We find that the photospheric morphology of RFLH is quite different
than that of the magnetic field or of the electrical current, and this is not
sensitive to the chosen gauge in the computation of RFLH. The value of
helicity experiences a large decrease, $\sim 25\%$ of its pre-flare value,
during an X-class flare of the AR, a change that is also depicted in the
photospheric morphology of RFLH. Moreover, the area of this change coincides
with the area that encompasses the flux rope, the magnetic structure that
later erupted.

Conclusions. The use of RFLH can provide important information about the value
and location of the magnetic helicity expelled from the solar atmosphere
during eruptive events.

###### Key Words.:

Sun: fundamental parameters – Sun: magnetic fields – Sun: flares –
Magnetohydrodynamics (MHD) – Methods: numerical

##  1 Introduction

Magnetic helicity is a quantity that represents an important restriction in
the evolution of astrophysical plasmas in the magnetohydrodynamics (MHD)
description, since it is conserved in ideal MHD (Woltjer 1958), and
approximately conserved in non-ideal conditions (Taylor 1974; Pariat et al.
2015). Magnetic helicity indicates the complexity of a magnetic field, as it
measures the twist, writhe, and interlinking of the magnetic field lines.
Mathematically, it is defined as the volume integral
$H=\int_{V}\mathbf{A}\cdot\mathbf{B}\,{\rm d}V$, where $\mathbf{B}$ is the
magnetic field in the volume $V$, and $\mathbf{A}$ the corresponding vector
potential such that $\mathbf{B}=\nabla\times\mathbf{A}$. Helicity thus depends
on the chosen gauge for $\mathbf{A}$ unless the volume of interest is bounded
by a magnetic flux surface. This condition is not met in the Sun where
magnetic flux is exchanged between the photosphere and the solar interior. The
appropriate quantity in this case is relative magnetic helicity (Berger &
Field 1984), which is defined with the help of a reference magnetic field.

An important application of the conservation property of helicity in the Sun
is to the initiation of coronal mass ejections (CMEs). These eruptions usually
involve the expulsion of highly-twisted, i.e., helical, magnetic structures,
the flux ropes (FRs, Green et al. 2018; Patsourakos et al. 2020). According to
a popular scenario (Rust 1994; Low 1994), CMEs are the means for the corona to
remove the constantly accumulating helicity from the solar interior. In cases
like this it would be desirable to be able to pinpoint the locations on the
Sun, or in the solar corona, where helicity is more important.

The classical and relative helicities however are non-local quantities, due to
their dependence on the vector potential, which means that the respective
densities lack a proper definition. The most meaningful way to define a
helicity density is with field line helicity (FLH, Antiochos 1987; Berger
1988). Field line helicity is given by the line integral
$h=\int_{C}\,\mathbf{A}\cdot{\rm d}\bm{l}$ along a field line $C$. FLH is a
function of the field lines, which expresses the flux of the magnetic field
through the surface bounded by the field line, when this is closed, otherwise,
it has no direct physical meaning.

The appropriate generalization of FLH for use at the solar environment, the
relative field line helicity (RFLH), was only recently developed (Yeates &
Page 2018; Moraitis et al. 2019). These works defined two distinct forms for
relative FLH which differ in the used gauge. Both forms however, accurately
reproduce relative helicity when all field lines that pierce the boundary of
the volume are considered.

Field line helicity, in its classical or relative form, has been employed in
various solar applications. In one application, FLH was used to trace the
connectivity changes caused by reconnection in a magnetic field setup which
simulated coronal loops (Russell et al. 2015). In another example, Yeates &
Hornig (2016) have applied FLH to the global magnetic field of the Sun in
order to study the distribution of helicity in the solar corona. An
interesting application of FLH was in the determination of the FRs location in
an approximation of the solar corona, through a threshold in FLH (Lowder &
Yeates 2017). The properties of FLH have also been examined theoretically in
Aly (2018). Most of the applications of FLH and RFLH so far are therefore
either in idealized situations, or to simulations of the actual solar
environment.

A first, more realistic application of RFLH to observed solar ARs was made in
Moraitis et al. (2019). Using a non-linear force-free (NLFF) reconstruction of
the coronal magnetic field of a solar active region at a specific instant, the
authors examined the relation of RFLH with the helicity flux density (e.g.,
Pariat et al. 2005). They showed that the two quantities exhibit overall
similar photospheric spatial distributions, while some minor differences were
also observed. This work highlighted the potential of using RFLH in actual
solar conditions.

In this work, we go one step further, as our purpose is to study the behaviour
of relative field line helicity in a large observed active region (AR) over an
extended time interval. In Sect. 2 we define all quantities of interest, and
we describe the observational data, as well as the implementation details,
that we use. In Sect. 3 we present the results of the application of RFLH in
the chosen AR, focusing on a specific large flare. Finally, in Sect. 4 we
summarize and discuss the results of the paper.

##  2 Methodology

###  2.1 Active region selection

The active region chosen for the first detailed study of the behaviour of RFLH
in solar conditions is AR 11158. The reasons for this choice are many. First,
there is an extensive literature about this AR, with many different of its
characteristics examined (e.g., Sun et al. 2012; Tziotziou et al. 2013).
Second, during the passage of the AR from the solar disk it exhibited a range
of behaviours, with most notable, the numerous flares (up to the X-class) and
the CMEs that accompanied many of them. Third, a high-quality reconstruction
of the AR’s coronal magnetic field, that is necessary in our computations, was
available to us (Thalmann et al. 2019b).

We employed two data products from the Solar Dynamic Observatory (SDO, Pesnell
et al. 2012) in this work. These were the sharp$\\_$cea$\\_$720s data series
from the Helioseismic and Magnetic Imager (HMI, Scherrer et al. 2012)
instrument and the lev1$\\_$uv$\\_$24s one at 1600 Å from the Atmospheric
Imaging Assembly (AIA) instrument (Lemen et al. 2012). The time interval that
we examined was between 12 February 2011, 00:00 UT and 16 February 2011, 00:00
UT. In this interval AR 11158 exhibited two M-class flares, an M6.6 on 13
February with peak time at 17:38 UT and an M2.2 on 14 February at 17:26 UT,
and the X2.2 flare on 15 February at 01:56 UT, all of which were eruptive.

###  2.2 Quantities of interest

####  2.2.1 Relative magnetic helicity

The appropriate magnetic helicity for the Sun is relative magnetic helicity
(Berger & Field 1984). For a three-dimensional (3D) magnetic field
$\mathbf{B}$ in a finite volume $V$, this is given by the Finn & Antonsen
(1985) formula

$H_{r}=\int_{V}(\mathbf{A}+\mathbf{A}_{\mathrm{p}})\cdot(\mathbf{B}-\mathbf{B}_{\mathrm{p}})\,{\rm
d}V.$ (1)

The magnetic field $\mathbf{B}_{\mathrm{p}}$ serves as a reference field, and
usually it is chosen to be potential, while $\mathbf{A}$,
$\mathbf{A}_{\mathrm{p}}$ are the respective vector potentials of the two
magnetic fields. Relative magnetic helicity is independent of the chosen
gauges of the two vector potentials, as long as the normal components of the
magnetic fields are the same on the boundary of the volume, $\partial V$. This
condition can be written as

$\left.\hat{n}\cdot\mathbf{B}\right|_{\partial
V}=\left.\hat{n}\cdot\mathbf{B}_{\mathrm{p}}\right|_{\partial V},$ (2)

with $\hat{n}$ denoting the outward-pointing unit normal on $\partial V$.

####  2.2.2 Energies

Although we are not interested in the evolution of the various energies in
this work, there are energy-related parameters that help us identify the
quality of a given magnetic field. We introduce these parameters in the
following.

The energy of a magnetic field $\mathbf{B}$ is given by the well-known
relation

$E=\frac{1}{2\mu_{0}}\int_{V}\mathbf{B}^{2}{\rm d}V,$ (3)

where $\mu_{0}$ is the magnetic permeability of the vacuum. A useful
decomposition of this energy is based on the unique splitting of the magnetic
field into potential and current-carrying components,
$\mathbf{B}=\mathbf{B}_{\mathrm{p}}+\mathbf{B}_{\mathrm{j}}$. Following Valori
et al. (2013), each component can be further divided into solenoidal and non-
solenoidal parts, as
$\mathbf{B}_{\mathrm{p}}=\mathbf{B}_{\mathrm{p,s}}+\mathbf{B}_{\mathrm{p,ns}}$
and
$\mathbf{B}_{\mathrm{j}}=\mathbf{B}_{\mathrm{j,s}}+\mathbf{B}_{\mathrm{j,ns}}$.
By defining the energy of each component from Eq. (3) we end up with the
following decomposition

$E=E_{\mathrm{p,s}}+E_{\mathrm{p,ns}}+E_{\mathrm{j,s}}+E_{\mathrm{j,ns}}+E_{\mathrm{mix}},$
(4)

where $E_{\mathrm{mix}}$ denotes the energy corresponding to all the cross
terms. For a perfectly solenoidal magnetic field, Eq. (4) reduces to the more
familiar form $E=E_{\mathrm{p,s}}+E_{\mathrm{j,s}}$, where $E_{\mathrm{j,s}}$
can be identified as the free energy.

The non-solenoidal parts of the magnetic fields and the respective terms in
Eq. (4) are non-physical and stem solely from numerical reasons. Collecting
these terms together, leads to the definition of the energy

$E_{\mathrm{div}}=E_{\mathrm{p,ns}}+E_{\mathrm{j,ns}}+|E_{\mathrm{mix}}|,$ (5)

which is a measure of how much a numerical model of a magnetic field deviates
from solenoidality; lower values of $E_{\mathrm{div}}$, and consequently of
the energy ratio $E_{\mathrm{div}}/E$, indicate a less solenoidal magnetic
field. Regarding helicity computations, Valori et al. (2016) have set a
threshold of $E_{\mathrm{div}}/E\lesssim 0.08$ for helicity values to be
reliable, which was later refined to $E_{\mathrm{div}}/E\lesssim 0.05$ by
Thalmann et al. (2019a), where the various energies for the same AR 11158 were
discussed.

Another energy ratio that is related to the quality of a magnetic field with
respect to its solenoidality is $|E_{\rm mix}|/E_{\rm j,s}$, which shows the
importance of the cross term in Eq. (4) relative to the free energy. Values of
$|E_{\rm mix}|/E_{\rm j,s}\lesssim 0.35$ are related with magnetic fields of
higher quality (Thalmann et al. 2020).

####  2.2.3 Relative field line helicity

#### a. Unconstrained gauge

The most general expression for the field line helicity that corresponds to
relative magnetic helicity was derived in Moraitis et al. (2019), and it is
reviewed here for completeness. Relative field line helicity (RFLH), $h_{r}$,
can be considered as the density of relative magnetic helicity per unit
magnetic flux. This follows from the relation

$H_{r}=\oint_{\partial V}h_{r}\,{\rm d}\Phi,$ (6)

where ${\rm d}\Phi=\left|\hat{n}\cdot\mathbf{B}\right|\,{\rm d}S$ is the
elementary magnetic flux on the boundary (${\rm d}S$: elementary area).
Equation (6) results directly from Eq. (1) after expanding the integrand of
the latter and splitting the volume element along the field lines of
$\mathbf{B}$ as ${\rm d}V={\rm d}\mathbf{S}\cdot{\rm d}\bm{l}$, or, of
$\mathbf{B}_{\mathrm{p}}$ as ${\rm d}V={\rm d}\mathbf{S}\cdot{\rm
d}\bm{l}_{\mathrm{p}}$ (${\rm d}\bm{l}$, ${\rm d}\bm{l}_{\mathrm{p}}$:
elementary lengths along respective field lines). The only assumption made in
deriving Eq. (6) is the reasonable smoothness of the magnetic field, and
especially the lack of null points where field lines are discontinuous.
Additionally, this relation is useful when most of the field lines of
$\mathbf{B}$ are connected to the boundary at both ends, i.e., when the number
of closed field lines inside the volume is limited.

As its name indicates, RFLH is a function of the field lines that assigns a
single value to each field line (pair of footpoints), and so, it can be viewed
as a 2D map that depends on the location on the boundary. Denoting the part of
the boundary where magnetic flux enters into (leaves) the volume as $\partial
V^{+}$ ($\partial V^{-}$), and the footpoints of a generic field line of
$\mathbf{B}$ as $\alpha_{+}\in\partial V^{+}$, $\alpha_{-}\in\partial V^{-}$
(and as $\alpha_{p+}\in\partial V^{+}$, $\alpha_{p-}\in\partial V^{-}$ for
$\mathbf{B}_{\mathrm{p}}$, respectively), then RFLH is given by any of the
following expressions

$h_{r}^{+}=\int_{\alpha_{+}}^{\alpha_{-}}\,(\mathbf{A}+\mathbf{A}_{\mathrm{p}})\cdot{\rm
d}\bm{l}-\int_{\alpha_{+}}^{\alpha_{p-}}\,(\mathbf{A}+\mathbf{A}_{\mathrm{p}})\cdot{\rm
d}\bm{l}_{\mathrm{p}},$ (7)

or

$h_{r}^{-}=\int_{\alpha_{+}}^{\alpha_{-}}\,(\mathbf{A}+\mathbf{A}_{\mathrm{p}})\cdot{\rm
d}\bm{l}-\int_{\alpha_{p+}}^{\alpha_{-}}\,(\mathbf{A}+\mathbf{A}_{\mathrm{p}})\cdot{\rm
d}\bm{l}_{\mathrm{p}},$ (8)

or, by their average, as

$h_{r}^{0}=\frac{1}{2}\left(h_{r}^{+}+h_{r}^{-}\right).$ (9)

The first expression involves the integration along field lines of
$\mathbf{B}$ and $\mathbf{B}_{\mathrm{p}}$ that have common their positive-
polarity footpoints, $\alpha_{+}=\alpha_{p+}$, as shown by the red and green
field lines in Fig. 1. When using $h_{r}^{+}$ in Eq. (6), only the positive-
polarity part of the boundary, $\partial V^{+}$, should be considered, so that
each field line is counted once.

Similarly, if the field lines of the two fields have in common the negative-
polarity footpoints, $\alpha_{-}=\alpha_{p-}$, then the RFLH expression given
by Eq. (8) should be used along the $\partial V^{-}$ boundary (red and blue
field lines in Fig. 1). When the whole boundary is considered, the respective
RFLH, Eq. (9), involves all field lines shown in Fig. 1. In all cases the RFLH
is expressed as a line integral along the field lines of $\mathbf{B}$ relative
to the same quantity along those of $\mathbf{B}_{\mathrm{p}}$, thus justifying
its name.

Figure 1: Sketch of the field lines involved in the definition of relative
field line helicity.

#### b. Berger & Field gauge

The expressions for RFLH given above make no assumption on the gauge of the
vector potentials. RFLH however is a gauge-dependent quantity, as it can
easily be seen from any of Eqs. (7)-(9), and this can be exploited in order to
derive simpler expressions for the RFLH. A commonly-used gauge for the two
vector potentials is given by the relation

$\left.\hat{n}\times\mathbf{A}_{\mathrm{p}}\right|_{\partial
V}=\left.\hat{n}\times\mathbf{A}\right|_{\partial V}.$ (10)

This gauge condition can be fulfilled with the tangential components of
$\mathbf{A}_{\mathrm{p}}$ determined from those of $\mathbf{A}$, while the
latter can be in any gauge. The condition given by Eq. (10) leads to the
elimination of the cross terms in the integrand of Eq. (1), which then reads

$H_{r}=\int_{V}\mathbf{A}\cdot\mathbf{B}\,{\rm
d}V-\int_{V}\mathbf{A}_{\mathrm{p}}\cdot\mathbf{B}_{\mathrm{p}}\,{\rm d}V.$
(11)

This gauge was used in the original definition of relative magnetic helicity
by Berger & Field (1984). Starting from Eq. (11), Yeates & Page (2018) defined
the following relative field line helicity

$\tilde{h}_{r}^{+}=\int_{\alpha_{+}}^{\alpha_{-}}\,\mathbf{A}\cdot{\rm
d}\bm{l}-\int_{\alpha_{+}}^{\alpha_{p-}}\,\mathbf{A}_{\mathrm{p}}\cdot{\rm
d}\bm{l}_{\mathrm{p}},$ (12)

written here for the positive-polarity boundary, with analogous to Eqs. (8)
and (9) relations holding for the other expressions, $\tilde{h}_{r}^{-}$,
$\tilde{h}_{r}^{0}$. These relative RFLHs are constructed similarly to the
RFLHs in the uncontsrained gauge, they employ however a specific gauge
condition.

###  2.3 Implementation

####  2.3.1 Magnetic field modelling

All physical quantities that interest this work require as input the 3D
magnetic field in the coronal volume. The 3D field at each instant is
reconstructed from the corresponding observed HMI magnetogram with a NLFF
method. This uses a weighted optimization approach (Wiegelmann & Inhester
2010), after pre-processing the horizontal components of the field on the
photosphere so that it becomes more compatible with the force-free assumption,
and additionally, smoothing of the original vector magnetogram. The full
details of the method can be found in Thalmann et al. (2019b) where this
reconstruction was first used.

We only mention here that the original HMI data were slightly cropped so that
to leave quiet-Sun regions outside, and then rebinned by a factor of four, to
the resolution of $2\arcsec$ per pixel. The resulting NLFF field occupies the
volume $215\,{\rm Mm}\times 130\,{\rm Mm}\times 185\,{\rm Mm}$, and it is
discretized by $148\times 92\times 128$ grid points. In total, there are 115
snapshots with the typical cadence of 1 hr, except around the M6.6 and the
X2.2 flares, when HMI’s highest cadence of 12 minutes was used.

The morphology of the extrapolated magnetic field for a snapshot $\sim$ 1 hr
before the X-class flare is shown in Fig. 2. A low-lying flux rope surrounded
by an arcade field can be identified there, as is already known by other works
(e.g., Sun et al. 2012; Nindos et al. 2012). Moreover, Fig. 2 shows that the
RFLH values of the flux rope are higher than in the arcade field.

Figure 2: Morphology of the 3D magnetic field of AR 11158 on 01:11 UT of 15
February 2011, for the NLFF extrapolation used in this work. Field lines are
coloured according to the value of RFLH, which is positive for all selected
field lines, and they are overplotted on the photospheric distribution of the
vertical magnetic field, with the red contours corresponding to the values
$|B_{z}|=500\,{\rm G}$.

One of the reasons for choosing the specific AR for this study is the
excellent quality of the reconstruction of the AR’s 3D magnetic field. The
quality of the field is intended as the accuracy in fulfilling the assumptions
of the NLFFF method, as quantified by the low values of divergence-, and
force-freeness reported in Thalmann et al. (2019b). The respective parameters
used by the authors are the average absolute fractional flux increase and the
average current-weighted angle between the current and the magnetic field
(Wheatland et al. 2000).

Besides these parameters the high level of divergence-freeness of the magnetic
field is further shown with the following three parameters. The first is the
recently-proposed modification of the fractional flux increase which, unlike
the original parameter, is independent of the mesh size (Gilchrist et al.
2020). This is quite low, with a mean value over all the 115 snapshots of
$\langle|f_{d}|\rangle=(1.50\pm 0.06)\times 10^{-9}\,\mathrm{m}^{-1}$.
Additionally, the energy ratio proposed by Valori et al. (2016) is well below
the threshold of 0.08, or even of 0.05, with a value of $E_{\rm div}/E=(6.1\pm
0.3)\times 10^{-3}$. The other energy ratio (Thalmann et al. 2020) is also
much lower than the respective limit of 0.35, with a value of $|E_{\rm
mix}|/E_{\rm j,s}=0.051\pm 0.006$. Such high quality of the reconstructed
magnetic field thus quarantees the reliability of the subsequent helicity
computations.

####  2.3.2 Numerical computations

The computation of all the physical quantities can be performed in a three-
step process, once the magnetic field is provided by the NLFF method. The
first step is the computation of the potential field that satisfies the
condition of Eq. (2). This is derived by numerically solving Laplace’s
equation. The second step involves the computation of the vector potentials
from the respective magnetic fields. This is done by adopting the DeVore
(2000) gauge as this was modified by Valori et al. (2012). The full details
involved in these two steps are described in Moraitis et al. (2014).

The third step is required only for the RFLH computations, and it involves two
different field line integrations, one for $\mathbf{B}$, and one for
$\mathbf{B}_{\mathrm{p}}$. These integrations are performed with the fast and
robust method that is described in Moraitis et al. (2019). A difference with
that work is that here we consider only the field lines with (at least one)
footpoint on the photospheric part of the boundary. We thus neglect the
lateral and top parts of the boundary, as our experience has shown that a very
small piece of information is lost. Once the field lines are obtained, we
compute the RFLH expression in the unconstrained gauge, $h_{r}^{0}$, from Eq.
(9).

For the computation of the RFLH in the Berger & Field gauge,
$\tilde{h}_{r}^{0}$, the three-step process is slightly different than the one
just described. The reason is that the computation is performed with the code
provided by Yeates & Page (2018), which uses a different calculation for the
potential magnetic field, and treats all vector quantities in a staggered grid
instead of being colocated.

##  3 Results

###  3.1 RFLH morphology

Following the methodology in Sect. 2 we calculate all quantities of interest
for all available snapshots of the NLFF magnetic field model. In Fig. 3 we
show four snapshots from the evolution of the morphology of RFLH on the
photospheric plane, $z=0$, in the unconstrained gauge, $h_{r}^{0}$, and also,
of the vertical magnetic field, $B_{z}$, and of the vertical electrical
current, $j_{z}=\left(\nabla\times\mathbf{B}\right)_{z}/\mu_{0}$, also on the
photospheric level. Each snapshot provides an example of the field
configuration in each of the four intervals in which the major flares divide
our study interval into: one before the M6.6 flare, one between the two
M-class flares, one between the M2.2 and the X2.2 flares, and one after the
X2.2 flare.

Figure 3: Photospheric maps of vertical magnetic field, $B_{z}$ (left),
vertical electrical current, $j_{z}$ (middle), and RFLH (right) in AR 11158
for four snapshots during the studied interval. The red contours correspond to
the vertical magnetic field values $|B_{z}|=500\,{\rm G}$ at each instant.

The morphology of the vertical magnetic field evolved from two initial nearby
bipoles that converged into a highly sheared bipolar flux distribution
internal to the outer polarities (e.g., Chintzoglou & Zhang 2013). The
vertical current distribution primarily develops around the polarity inversion
lines (PILs), and it follows to a large degree the evolution of the magnetic
field, as the comparison with the respective contours shows in Fig. 3. The
RFLH morphology is however different than the other distributions. It is more
extended, and it has a much less coincidence with the magnetic field contours.
The sign of the RFLH is mostly positive, in agreement with the AR helicity
sign. The highest (positive) RFLH values are located around the core
polarities of the AR. This central part of RFLH becomes more intense until
around the time of the X-class flare, and finally relaxes to lower values. The
behaviour of RFLH around the flare is examined in more detail in Sect. 3.3.
The lowest RFLH values are found to the east periphery of the AR (left in the
respective panels of Fig. 3), with only a small coincidence with the nearby
magnetic field polarity.

The corresponding evolution of the relative helicity budgets, as calculated by
the volume method of Eq. (1), and of the RFLH method of Eq. (6) on the full
magnetogram, is shown in Fig. 4 with the black and grey curves, respectively.
We note that from 13 February onwards, the helicities obtained with the two
methods agree to a large extent. We quantify this agreement in the bottom
panel of Fig. 4 where we notice that the relative difference between the two
curves is within $\lesssim 5\%$, with an average absolute value of 2.2%. In
calculating these numbers we omit the day of February 12 when the values of
helicity are very low and consequently the relative difference much higher
than in the following days. The closeness of the two helicity curves indicates
that the RFLH method works as expected, since it exhibits similar levels of
agreement as in Moraitis et al. (2019). It also shows that most of the field
lines contributing to helicity are indeed rooted in the phothosphere rather
than on the lateral boundaries. Both methods indicate a large drop in helicity
during the X-class flare, which we further discuss in the following sections.

Figure 4: Evolution of relative helicity in AR 11158 as calculated by the
volume (Eq. (1), black line), and the RFLH methods (Eq. (6), grey line) in the
top panel, and of their relative difference in the bottom, during the whole
studied interval. The vertical lines denote the times of the two M-, and of
the X-class flares.

We note here that the morphology of RFLH is not sensitive to the gauge used in
its computation, at least in areas of high RFLH values. This can be seen in
Fig. 5 where we compare the morphology obtained with the two gauges of Sect. 2
for the snapshot at 07:59 UT of 15 February 2011. We see that the two
distributions have similar morphologies, although they also exhibit some minor
differences in small patches. The good agreement of the two maps is depicted
in the high value of the pixel-to-pixel linear correlation coefficient between
them, which is 0.85. The main difference between the two distributions is in
the magnitude of RFLH which is by a factor of $\lesssim 2$ smaller in the
Berger & Field gauge. The morphology in this gauge is also more smooth because
of the Coulomb gauge used in the computation of the vector potentials. Similar
differences exhibit the other snapshots as well.

Figure 5: Photospheric morphology of the RFLH in the unconstrained gauge
(top), and in the Berger & Field gauge (bottom), for AR 11158 on 07:59 UT of
15 February 2011. The two images exhibit a pixel-to-pixel linear correlation
coefficient of 0.85.

###  3.2 Flare-related changes during the X2.2 flare

We now focus on the X2.2 flare of AR 11158 of 15 February 2011. The flare’s
onset time was 01:44 UT and its peak time was 01:56 UT. Using observations
from SDO’s AIA instrument at 1600 Å we are able to identify two regions
related to the occurence of the flare. The first is the location of the flare
ribbons which are shown in the top panel of Fig. 6, at the time of their
maximum extent (01:48 UT) before the saturation of the image from the flare
emission. According to the standard model for the 3D flares (Janvier et al.
2013), the flare ribbons correspond to the footprints of the quasi-separatrix
layer (QSL) that separates the flux rope from the surrounding, confining
field. This image resulted after correcting the original AIA image with
standard methods (solar software’s aia$\\_$prep routine), and transforming it
to the same projection and field of view (FOV) as the HMI images of Fig. 3,
following the recipes of Sun (2013). The latter procedure slightly deforms the
shape of the image, but this is minimal due to the proximity of the AR to the
solar disk centre. We approximate the region corresponding to the central,
highest-intensity contour with the red polygon shown in the bottom panel of
Fig. 6. The second region of interest includes the central polarities of the
AR where the flare originated during its peak (as identified by eye inspection
of AIA’s images), although nearly the whole AR participated in this flare.
This region is shown as a green box in the bottom panel of Fig. 6, and it
encompasses the area covered by the ribbons when they reach their maximum
extent.

Figure 6: AIA image at 1600 Å of the same AR region shown in Fig. 3, saturated
at the intensity of 1180, and overplotted with the contours at the same value,
at the beginning of the X2.2 flare of AR 11158 (top). Zoom of the vertical
magnetic field 2D map at the same time, with the two regions of interest
(bottom). Red contours are the same as in the top panel, only rebinned to the
resolution of the magnetic field image. Yellow contours correspond to the
magnetic field value $B_{z}=0\,{\rm G}$.

In the panels (a), (b), (d), and (e) of Fig. 7 we show the evolution of the
RFLH distribution in the corresponding areas around the X-class flare. We note
that the overall morphology has small variations in a two-hours interval
around the flare. The most important difference is the large decrease in the
RFLH magnitude between the snapshots at 01:47 UT and 01:59 UT, i.e., during
the maximum of the flare. This is more evident in panels (c) and (f) where the
difference of the images at 01:47 UT and at 01:59 UT with the image at 01:11
UT is shown, respectively. While the first difference image (panel c) shows
small variations of both signs in the whole FOV, the second image (panel f)
depicts a clear decrease in RFLH which is contained in the green box.

The use of RFLH thus enables us to identify the locations where helicity is
lost from the AR during the flare and the respective eruption, and to also
calculate its value. It is also interesting that the area of the RFLH decrease
includes the location of the flux rope, and so we can deduce that a part of
the flux rope was expelled during the flare. In other words, we are able to
verify that the CME source coincides with the wider flux rope region. This
conclusion agrees with many previous studies (e.g., Zhang et al. 2012;
Patsourakos et al. 2013; Nindos et al. 2015; Chintzoglou et al. 2015; Nindos
et al. 2020).

Figure 7: Selected snapshots of the zoomed-in photospheric morhology of RFLH
around the X2.2 flare of AR 11158 (panels a, b, d, e). Also shown are the two
boxes defined in Fig. 6, and the contours of the vertical magnetic field
values at $|B_{z}|=500\,{\rm G}$.
Difference images of the photospheric RFLH at the beginning of the flare (at
01:47 UT, panel c) and after the flare (at 01:59, panel f), with respect to
the corresponding pre-flare image at 01:11 UT. The boxes are the same as in
the rest panels.

In Fig. 8 we show the evolution of various helicity budgets around the X2.2
flare. The black and gray curves are a zoom-in around the time of the X2.2.
flare of the same helicity budgets as calculated by the volume and the RFLH
methods for the full magnetogram, shown in Fig. 4. We also show the errors
stemming from the difference of the helicity curves from their five-point
moving averages, which correspond to one-hour intervals. We use this type of
error as an indication for the actual errors, since the computationally-
deduced ones, by the method of Moraitis et al. (2014), are very low.

In addition, Fig. 8 depicts the helicity budgets calculated by the RFLH method
for the two boxes of Fig. 6, as the green and red lines. In calculating these,
we employ Eq. (6) but consider only those field lines that have footpoints
inside the respective areas.

Figure 8 reiterates the good agreement of $\lesssim 5\%$ between the
helicities obtained by the volume and the RFLH methods (black and grey curves
respectively). Both curves exhibit a 25% drop in helicity during the flare,
which corresponds to $1.48\,10^{42}\,\mathrm{Mx}^{2}$ for the volume method,
and to the slightly larger $1.56\,10^{42}\,\mathrm{Mx}^{2}$ for the RFLH one.
The drop in the volume helicity is larger than the computed errors and so it
is not an artifact but a real drop, a conslusion that holds for the other
helicity curves as well. Similar sharp changes in other parameters related to
the magnetic field have been reported previously for the same AR (Liu et al.
2012; Sun et al. 2017).

Another conclusion that can be drawn from Fig. 8 is that the helicities
obtained by the RFLH method for the full FOV and for the green box (grey and
green curves respectively) agree to a large extent. This can be explained with
the majority of the AR’s helicity being contained in the green box, a
conclusion also supported by the RFLH morphology in Fig. 7. The agreement is
more pronounced before the flare, when the two curves agree to within 0.5%,
while after the flare this falls to 5%, which should be attributed to the
rearrangement of the magnetic field during the flare, and to a lesser extent,
to the evolution of the arc-shaped RFLH structure just above the green box in
Fig. 7. Finally, the drop in helicity coming from the green box is 20%, a bit
less than in the full FOV, and it corresponds to a value of
$1.35\,10^{42}\,\mathrm{Mx}^{2}$.

Focusing next to the red box that approximates the flare’s ribbons we note
that the helicity contained in it is the half of the entire AR. The relative
drop in helicity during the flare is similar to the other curves, $\sim 22$ %
of its pre-flare value; its absolute value however is much smaller,
corresponding to $7.1\,10^{41}\,\mathrm{Mx}^{2}$. This small relative drop
indicates that the region stays non-potential even after the expulsion of the
flux rope. On the other hand, the higher (absolute) helicity drop within the
green box suggests that the effect of the CME is to change the entire
surrounding of the flux rope. An alternative explanation could be that the
eruption changes the mutual helicity between the systems of the flux rope and
the surrounding field, in which case, the RFLH would reflect the non-local
nature of helicity. This would also relate to the mutual-to-self conversion of
helicity (Tziotziou et al. 2013), which was suggested for the same active
region.

Figure 8: Evolution of relative helicities in AR 11158 during a two-hours
interval centered on the X2.2 flare. Shown are the relative helicities as
calculated by the volume method (Eq. (1), black line), and by the RFLH method
using Eq. (9) in the whole AR (grey line), and in the two boxes of Fig. 6 (red
and green lines). The error bars are computed from the difference of the
curves from their five-point moving averages. The vertical lines denote the
onset and peak times of the X-class flare.

###  3.3 Relation with the eruption of February 15

The X2.2 flare of AR 11158 on February 15 was an eruptive one, that is, it was
accompanied by a CME. This was observed with STEREO’s COR1 coronagraph at 2:05
UT of February 15, and it had a peak velocity of $\sim
1300\,\mathrm{km}\,\mathrm{s}^{-1}$ (Maričić et al. 2014). The ejected
structure took a part of the AR’s helicity which, according to the previous
section, should be positive (right-handed) and no more than the maximum
estimated helicity decrease of $\sim 1.5\,10^{42}\,\mathrm{Mx}^{2}$. The CME
was recorded three days later, starting at February 18 around 20:00 UT, as an
Interplanetary Coronal Mass Ejection (ICME), using in situ data at the
Lagrange point L1 from the Wind spacecraft (Maričić et al. 2014; Lepping et
al. 2015). This ICME was also listed in several other catalogs (Richardson &
Cane 2010; Chi et al. 2016; Hess & Zhang 2017) at a similar time interval.

From the fitting of the ICME with a linear force-free magnetic cloud (MC)
model, Lepping et al. (2015, Table 2 therein) find a value for the axial
magnetic field strength of $B_{0}=11.4\,\mathrm{nT}$, and for the radius of
the MC, $r_{0}=0.06\,\mathrm{AU}$. Further assuming the arbitrary, but
consistent however with energetic electron recordings in ICMEs (Kahler et al.
2011), length for the MC of $L\sim 2\,\mathrm{AU}$, one can estimate the
helicity of the ICME to $H_{r}\sim 2\,10^{41}\,\mathrm{Mx}^{2}$, following the
equations in Dasso et al. (2006). This value is significantly smaller than the
entire AR’s helicity, and also of the helicity contained in the two boxes. The
comparison with the corresponding helicity drops shows that the MC’s helicity
is a factor of $\sim 7$ smaller than the helicity lost from the whole AR, or
from the green box area. It is closer however to the helicity lost in the
flare ribbons’ area (red box), but still, a factor of $\sim 3$ smaller. A
problem with the helicity reported in Lepping et al. (2015) is that this is
left-handed, i.e., it has opposite sign compared to the AR. Moreover, as
discussed by Lepping et al. (2015), the quality of the MC fitting for this
ICME is poor, exhibiting convergence issues. Therefore, this ICME does not
represent the most optimal case for comparisons with solar observations.
Despite all the uncertainties in the estimation of the MC’s helicity, the
absolute value obtained is consistent with the upper limit derived from the
helicity drop in the whole AR.

##  4 Discussion

This work dealt with the first detailed study of relative field line helicity
in a solar active region. The target AR 11158 exhibited intense activity
during the four-days study interval. The computation of RFLH was based on the
relevant recent developments, and on a high-quality NLFF model for the AR’s
coronal magnetic field, thus ensuring reliable helicity estimations.

The examination of RFLH’s photospheric morphology showed that this is quite
different than the respective distributions of the magnetic field and of the
electrical current. Furthermore, the morphology of RFLH was shown not to be
sensitive on the employed gauge in its computation. The total AR helicity as
estimated by the RFLH agrees to a large extent with the more accurate, volume
helicity method, as was already demonstrated before.

Our results are consistent with the picture drawn by other works on AR 11158,
e.g., that of Jing et al. (2012). These authors find a similar helicity
evolution, both overall, and during the X-class flare. The helicity of the AR
exhibited a sharp decrease during the X2.2 flare, $\sim 25\%$ of its pre-flare
value. A similar sharp drop was reported in Jing et al. (2012), which,
moreover, resulted from the respective drop in the right-handed helicity. This
is exactly what the RFLH photospheric maps of Sect. 3.2 reveal.

The main advantage of using RFLH is the additional spatial information that it
can provide about the locations where helicity is more important, and its
respective values. In the case of the X2.2 flare of AR 11158, we found that
the decrease in helicity is coming mostly from the wider flux rope area.
Additionally, by examining two observationally-deduced, flare-related regions,
we were able to monitor the individual helicity evolution in each of them. The
helicity drop during the flare in these regions did not reveal a clear
relation with the helicity of the observed ICME, although there are many
uncertainties in the derivation of the latter.

As a conclusion, this work pointed out the usefulness of RFLH in solar
applications. The spatial information provided by RFLH can help in situations
such as the one examined in this paper, i.e., in the determination of the pre-
eruptive structures of coronal mass ejections. A possible next step in the
study of RFLH would be to perform similar analysis to a number of ARs with
different levels of activity, and different characteristics, and juxtapose
them with the helicity of the associated ICMEs at 1 AU or even much closer in
the corona or inner heliosphere with Parker Solar Probe (Fox et al. 2016) and
Solar Orbiter (Müller et al. 2020). This would enable the examination of the
behaviour of RFLH in different environments, and of the possibility of
discriminating solar activity with the help of relative field line helicity.

###### Acknowledgements.

The authors thank the referee for carefully reading the paper and providing
constructive comments. This research is co-financed by Greece and the European
Union (European Social Fund – ESF) through the Operational Programme “Human
Resources Development, Education and Lifelong Learning” in the context of the
project “Reinforcement of Postdoctoral Researchers – 2nd Cycle” (MIS-5033021),
implemented by the State Scholarships Foundation (IKY). It also benefited from
the discussions within the international team “Magnetic Helicity in
Astrophysical Plasmas” which was supported by the International Space Science
Institute. The authors also thank J. Thalmann for providing the NLFF
reconstruction of the AR’s magnetic field.

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$\displaystyle\omega^{-+}(k)=-\varepsilon_{1}(q)-\varepsilon_{1}(q^{\prime})\hskip
5.69046pt{\rm where}\hskip 5.69046ptk=\iota\pi-q-q^{\prime}\hskip
5.69046pt{\rm and}\hskip 5.69046pt\iota=\pm 1$ $\displaystyle{\rm for}\hskip
5.69046ptq\in[-k_{F\downarrow},k_{F\downarrow}]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq^{\prime}\in[-k_{F\downarrow},k_{F\downarrow}]\,,$ (58)

and

$\displaystyle\omega^{+-}(k)=\varepsilon_{1}(q)-\varepsilon_{1}(q^{\prime})\hskip
5.69046pt{\rm where}\hskip 5.69046ptk=\iota\pi+q-q^{\prime}\hskip
5.69046pt{\rm and}\hskip 5.69046pt\iota=\pm 1$ $\displaystyle{\rm for}\hskip
5.69046pt|q|\in[k_{F\downarrow},k_{F\uparrow}]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq^{\prime}\in[-k_{F\downarrow},k_{F\downarrow}]\,,$ (59)

respectively. Here $\varepsilon_{1}(q)$ is the $1$-band energy dispersion
given in Eq. (89) of Appendix B. The spectrum $\omega^{xx}(k)$ of the
transverse dynamical structure factor $S^{xx}(k,\omega)$ associated with the
lower continuum in Fig. 2 results from combination of the two spectra
$\omega^{-+}(k)$ and $\omega^{+-}(k)$ in Eqs. (58) and (59), respectively.

The spectrum $\omega^{zz}(k)$ associated with the lower continuum in Fig. 3 is
again within a $k$ extended zone scheme given by,

$\displaystyle\omega^{zz}(k)=\varepsilon_{1}(q)-\varepsilon_{1}(q^{\prime})\hskip
5.69046pt{\rm where}\hskip 5.69046ptk=q-q^{\prime}$ $\displaystyle{\rm
for}\hskip 5.69046pt|q|\in[k_{F\downarrow},k_{F\uparrow}]\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq^{\prime}\in[-k_{F\downarrow},k_{F\downarrow}]\,.$ (60)

The upper thresholds of the two-parametric spectra, Eqs. (58) and (59), have
the following one-parametric spectra for spin densities $m\in]0,1[$,

$\displaystyle\omega^{+-}_{ut}(k)$ $\displaystyle=$ $\displaystyle
2\mu_{B}\,h-\varepsilon_{1}(k_{F\downarrow}-k)\hskip 5.69046pt{\rm
where}\hskip 5.69046ptk=k_{F\downarrow}-q\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[0,k_{F\downarrow}]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq\in[0,k_{F\downarrow}]\,,$ (61) $\displaystyle=$
$\displaystyle\varepsilon_{1}(q)-\varepsilon_{1}(q^{\prime})\hskip
5.69046pt{\rm where}\hskip 5.69046ptk=\pi+q-q^{\prime}\hskip 5.69046pt{\rm
for}\hskip 5.69046ptk\in[k_{F\downarrow},\pi]\hskip 5.69046pt{\rm and}\hskip
5.69046ptv_{s}(q)=v_{s}(q^{\prime})\hskip 5.69046pt$ $\displaystyle{\rm
with}\hskip 5.69046ptq\in[-k_{F\uparrow},-k_{F\downarrow}]\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq^{\prime}\in[-k_{F\downarrow},0]\,,$

and

$\omega^{-+}_{ut}(k)=-2\varepsilon_{1}\left({\pi-k\over 2}\right)\hskip
5.69046pt{\rm where}\hskip 5.69046ptk=\pi-2q\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[(k_{F\uparrow}-k_{F\downarrow}),\pi]\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq\in[-k_{F\downarrow},0]\,,$ (62)

respectively. The function $v_{1}(q)$ is in Eq. (61) the $1$-band group
velocity defined in Eq. (91) of Appendix B.

The upper threshold spectrum $\omega^{xx}_{ut}(k)$ of the combined spectra,
Eqs. (58) and (59), is given by,

$\displaystyle\omega^{xx}_{ut}(k)$ $\displaystyle=$
$\displaystyle\omega^{+-}_{ut}(k)\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[0,k^{xx}_{ut}]$ (63) $\displaystyle=$
$\displaystyle\omega^{-+}_{ut}(k)\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[k^{xx}_{ut},\pi]\,,$

where the momentum $k^{xx}_{ut}$ is such that
$\omega^{+-}_{ut}(k^{xx}_{ut})=\omega^{-+}_{ut}(k^{xx}_{ut})$.

On the other hand, the one-parametric upper threshold spectrum associated with
the two-parametric longitudinal spectrum, Eq. (60), reads for $m\in]0,1[$,

$\displaystyle\omega^{zz}_{ut}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{1}(q)-\varepsilon_{1}(q^{\prime})\hskip
5.69046pt{\rm where}\hskip 5.69046ptk=q-q^{\prime}\hskip 5.69046pt{\rm
for}\hskip 5.69046ptv_{s}(q)=v_{s}(q^{\prime})\hskip 5.69046pt{\rm and}\hskip
5.69046ptk\in[0,k_{F\uparrow}]\hskip 5.69046pt{\rm with}$ (64) $\displaystyle
q\in[k_{F\downarrow},k_{F\uparrow}]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq^{\prime}\in[0,k_{F\downarrow}]\,,$ $\displaystyle=$ $\displaystyle
2\mu_{B}\,h-\varepsilon_{1}(k_{F\uparrow}-k)\hskip 5.69046pt{\rm where}\hskip
5.69046ptk=k_{F\uparrow}-q\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[k_{F\uparrow},\pi]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq\in[-k_{F\downarrow},0]\,.$

At $k=0,k_{F\downarrow},\pi$ and $k=0,k_{F\uparrow}-k_{F\downarrow},\pi$, the
upper threshold spectra, Eqs. (61) and (62), respectively, are given by,

$\displaystyle\omega^{+-}_{ut}(0)=W_{1}^{h}=2\mu_{B}\,h\,;\hskip
8.5359pt\omega^{+-}_{ut}(k_{F\downarrow})=W_{1}=2\mu_{B}\,h+W_{1}^{p}\,;\hskip
8.5359pt\omega^{+-}_{ut}(\pi)=0$
$\displaystyle\omega^{-+}_{ut}(k_{F\uparrow}-k_{F\downarrow})=0\,;\hskip
8.5359pt\omega^{-+}_{ut}(\pi)=2W_{1}^{p}\,.$ (65)

Here $W_{1}=W_{1}^{p}+W_{1}^{h}$, $W_{1}^{p}$, and $W_{1}^{h}$ are the energy
bandwidths of the full $1$-band, its occupied Fermi sea, and unoccupied sea,
respectively, in Eqs. (103) and (105) of Appendix B.

At $k=0,k_{F\uparrow},\pi$ the upper threshold spectrum $\omega^{zz}_{ut}(k)$
reads,

$\omega^{zz}_{ut}(0)=0\,;\hskip
8.5359pt\omega^{zz}_{ut}(k_{F\uparrow})=W_{1}=2\mu_{B}\,h+W_{1}^{p}\,;\hskip
8.5359pt\omega^{zz}_{ut}(\pi)=W_{1}^{h}=2\mu_{B}\,h\,.$ (66)

The spin dynamical structure factors’s line shape near the lower thresholds of
the spectra, Eqs. (58), (59), and (60), has the general form provided in Eq.
(12). In the case of (i) $S^{-+}(k,\omega)$ and (ii) $S^{+-}(k,\omega)$ and
$S^{zz}(k,\omega)$ such lower thresholds refer to (i) a single $1$-branch line
and (ii) two sections of a $1$-branch line, respectively. Their spectra can be
expressed in terms of the excitation momentum $k$ or of the $1$-band momentum
$q$ and are given by,

$\displaystyle\omega^{-+}_{lt}(k)$ $\displaystyle=$
$\displaystyle-\varepsilon_{1}(k_{F\uparrow}-k)\hskip 5.69046pt{\rm and}\hskip
5.69046ptk=k_{F\uparrow}-q\hskip 5.69046pt{\rm where}$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle[(k_{F\uparrow}-k_{F\downarrow}),\pi]\hskip
5.69046pt{\rm and}\hskip 5.69046ptq\in]-k_{F\downarrow},k_{F\downarrow}]\,,$
(67) $\displaystyle\omega^{+-}_{lt}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{1}(k-k_{F\uparrow})\hskip 5.69046pt{\rm and}\hskip
5.69046ptk=k_{F\uparrow}+q\hskip 5.69046pt{\rm where}$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle[0,(k_{F\uparrow}-k_{F\downarrow})]\hskip
5.69046pt{\rm and}\hskip 5.69046ptq\in[-k_{F\uparrow},-k_{F\downarrow}]\,,$
$\displaystyle=$ $\displaystyle-\varepsilon_{1}(k_{F\uparrow}-k)\hskip
5.69046pt{\rm and}\hskip 5.69046ptk=k_{F\uparrow}-q\hskip 5.69046pt{\rm
where}$ $\displaystyle k$ $\displaystyle\in$
$\displaystyle[(k_{F\uparrow}-k_{F\downarrow}),\pi]\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq\in[-k_{F\downarrow},k_{F\downarrow}]\,,$ (68)
$\displaystyle\omega^{zz}_{lt}(k)$ $\displaystyle=$
$\displaystyle-\varepsilon_{1}-k_{F\downarrow}[_{F\downarrow}-k)\hskip
5.69046pt{\rm and}\hskip 5.69046ptk=k_{F\downarrow}-q\hskip 5.69046pt{\rm
where}$ $\displaystyle k$ $\displaystyle\in$
$\displaystyle[0,2k_{F\downarrow}]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq\in[-k_{F\downarrow},k_{F\downarrow}]\,,$ $\displaystyle=$
$\displaystyle\varepsilon_{1}(k-k_{F\downarrow})\hskip 5.69046pt{\rm
and}\hskip 5.69046ptk=k_{F\downarrow}+q\hskip 5.69046pt{\rm where}$
$\displaystyle k$ $\displaystyle\in$
$\displaystyle[2k_{F\downarrow}),\pi]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq\in[k_{F\downarrow},k_{F\uparrow}]\,.$ (69)

The lower threshold’s spectrum $\omega^{xx}_{lt}(k)$ of $S^{xx}(k,\omega)$ has
the same expression as that of $S^{+-}(k,\omega)$,

$\displaystyle\omega^{xx}_{lt}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{1}(k-k_{F\uparrow})\hskip 5.69046pt{\rm and}\hskip
5.69046ptk=k_{F\uparrow}+q\hskip 5.69046pt{\rm where}$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle[0,(k_{F\uparrow}-k_{F\downarrow})]\hskip
5.69046pt{\rm and}\hskip 5.69046ptq\in[-k_{F\uparrow},-k_{F\downarrow}]\,,$
$\displaystyle=$ $\displaystyle-\varepsilon_{1}(k_{F\uparrow}-k)\hskip
5.69046pt{\rm and}\hskip 5.69046ptk=k_{F\uparrow}-q\hskip 5.69046pt{\rm
where}$ $\displaystyle k$ $\displaystyle\in$
$\displaystyle[(k_{F\uparrow}-k_{F\downarrow}),\pi]\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq\in[-k_{F\downarrow},k_{F\downarrow}]\,.$ (70)

Finally, the $k$ dependent expressions and the limiting values of the energy
gaps in Eqs. (36)-(38) are provided. The energy gap $\Delta_{\rm gap}^{+-}(k)$
is finite and positive for $0<m<1$. For spin densities $m\in]0,\tilde{m}]$
where $\tilde{m}\approx 0.317$ it reads,

$\displaystyle\Delta_{\rm gap}^{+-}(k)$ $\displaystyle=$
$\displaystyle-2\mu_{B}\,h+\varepsilon_{2}(k)+\varepsilon_{1}(k_{F\downarrow}-k)\hskip
5.69046pt{\rm for}\hskip 5.69046ptk\in[0,(k_{F\uparrow}-k_{F\downarrow})]$
$\displaystyle\Delta_{\rm gap}^{+-}(k)$ $\displaystyle=$ $\displaystyle
2\mu_{B}\,h-\varepsilon_{1}(k_{F\uparrow}-k)+\varepsilon_{1}(k_{F\downarrow}-k)\hskip
5.69046pt{\rm for}\hskip
5.69046pt[(k_{F\uparrow}-k_{F\downarrow}),{\tilde{k}}[$
$\displaystyle\Delta_{\rm gap}^{+-}(k)$ $\displaystyle=$ $\displaystyle
2\mu_{B}\,h-W_{2}\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in]{\tilde{k}},k_{F\downarrow}]$ $\displaystyle\Delta_{\rm
gap}^{+-}(k)$ $\displaystyle=$ $\displaystyle
4\mu_{B}\,h-W_{2}-\varepsilon_{1}(k_{F\downarrow}-k)+\varepsilon_{1}(q)-\varepsilon_{1}(k+q-\pi)\hskip
5.69046pt{\rm for}\hskip 5.69046ptk\in[k_{F\downarrow},2k_{F\downarrow}]\hskip
5.69046pt{\rm and}$ $\displaystyle
q\in[-(k_{\bullet}-k_{F\uparrow}+k_{F\downarrow}),0]$
$\displaystyle\Delta_{\rm gap}^{+-}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{2}(k-2k_{F\downarrow})+\varepsilon_{1}(q)-\varepsilon_{1}(k+q-\pi)\hskip
5.69046pt{\rm for}\hskip 5.69046ptk\in[2k_{F\downarrow},\pi]\hskip
5.69046pt{\rm and}$ (71) $\displaystyle{\rm for}\hskip 5.69046pt{\rm
spin}\hskip 5.69046pt{\rm densities}\hskip 5.69046ptm\in]0,\tilde{m}]\,.$

For spin densities $m\in[\tilde{m},1[$ its expression is,

$\displaystyle\Delta_{\rm gap}^{+-}(k)$ $\displaystyle=$
$\displaystyle-2\mu_{B}\,h+\varepsilon_{2}(k)+\varepsilon_{1}(k_{F\downarrow}-k)\hskip
5.69046pt{\rm for}\hskip 5.69046ptk\in[0,{\tilde{k}}[$
$\displaystyle\Delta_{\rm gap}^{+-}(k)$ $\displaystyle=$ $\displaystyle
2\mu_{B}\,h-W_{2}\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in]{\tilde{k}},k_{F\downarrow}]$ $\displaystyle\Delta_{\rm
gap}^{+-}(k)$ $\displaystyle=$ $\displaystyle
4\mu_{B}\,h-W_{2}-\varepsilon_{1}(k_{F\downarrow}-k)+\varepsilon_{1}(q)-\varepsilon_{1}(k+q-\pi)\hskip
5.69046pt{\rm for}\hskip 5.69046ptk\in[k_{F\downarrow},2k_{F\downarrow}]\hskip
5.69046pt{\rm and}$ $\displaystyle
q\in[-(k_{\bullet}-k_{F\uparrow}+k_{F\downarrow}),0]$
$\displaystyle\Delta_{\rm gap}^{+-}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{2}(k-2k_{F\downarrow})+\varepsilon_{1}(q)-\varepsilon_{1}(k+q-\pi)\hskip
5.69046pt{\rm for}\hskip 5.69046ptk\in[2k_{F\downarrow},\pi]\hskip
5.69046pt{\rm and}$ (72) $\displaystyle
q\in[-k_{F\downarrow},-(k_{\bullet}-k_{F\uparrow}+k_{F\downarrow})]$
$\displaystyle{\rm for}\hskip 5.69046pt{\rm spin}\hskip 5.69046pt{\rm
densities}\hskip 5.69046ptm\in[\tilde{m},1[\,.$

In the above equations, the momentum $\tilde{k}$ is defined by the relations,
Eq. (28), and the momentum $k_{\bullet}$ satisfies the following equation,

$v_{1}(k_{\bullet})=v_{1}(k_{\bullet}-k_{F\uparrow}+k_{F\downarrow})\hskip
5.69046pt{\rm where}\hskip 5.69046ptk_{\bullet}>k_{F\downarrow}\,.$ (73)

On the other hand, for spin densities $m\in]0,\tilde{m}]$, the energy gap
$\Delta_{\rm gap}^{-+}(k)=\Delta^{-+}(k)-\omega^{-+}_{ut}(k)$ where
$\Delta^{-+}(k)=\Delta^{+-}(k)$ reads,

$\displaystyle\Delta_{\rm gap}^{-+}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{2}(k)\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[0,(k_{F\uparrow}-k_{F\downarrow})]$ $\displaystyle\Delta_{\rm
gap}^{-+}(k)$ $\displaystyle=$ $\displaystyle
4\mu_{B}\,h-\varepsilon_{1}(k_{F\uparrow}-k)+2\varepsilon_{1}\left({\pi-k\over
2}\right)\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[(k_{F\uparrow}-k_{F\downarrow}),{\tilde{k}}[$
$\displaystyle\Delta_{\rm gap}^{-+}(k)$ $\displaystyle=$ $\displaystyle
4\mu_{B}\,h-W_{2}-\varepsilon_{1}(k_{F\downarrow}-k)+2\varepsilon_{1}\left({\pi-k\over
2}\right)\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in]{\tilde{k}},2k_{F\downarrow}]$ $\displaystyle\Delta_{\rm
gap}^{-+}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{2}(k-2k_{F\downarrow})+2\varepsilon_{1}\left({\pi-k\over
2}\right)\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[2k_{F\downarrow},\pi]\,,$ (74)

whereas for $m\in[\tilde{m},1[$ it is given by,

$\displaystyle\Delta_{\rm gap}^{-+}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{2}(k)\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[0,{\tilde{k}}[$ $\displaystyle\Delta_{\rm gap}^{-+}(k)$
$\displaystyle=$ $\displaystyle
4\mu_{B}\,h-W_{2}-\varepsilon_{1}(k_{F\downarrow}-k)\hskip 5.69046pt{\rm
for}\hskip 5.69046ptk\in]{\tilde{k}},(k_{F\uparrow}-k_{F\downarrow})]$
$\displaystyle\Delta_{\rm gap}^{-+}(k)$ $\displaystyle=$ $\displaystyle
4\mu_{B}\,h-W_{2}-\varepsilon_{1}(k_{F\downarrow}-k)+2\varepsilon_{1}\left({\pi-k\over
2}\right)\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[(k_{F\uparrow}-k_{F\downarrow}),2k_{F\downarrow}]$
$\displaystyle\Delta_{\rm gap}^{-+}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{2}(k-2k_{F\downarrow})+2\varepsilon_{1}\left({\pi-k\over
2}\right)\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[2k_{F\downarrow},\pi]\,.$ (75)

The energy gap $\Delta_{\rm gap}^{+-}(k)$ is given by the constant energy
scale $2\mu_{B}\,h-W_{2}$ where $W_{2}$ is the energy bandwidth of the
$2$-band in Eqs. (103) and (105) of Appendix B for the following $k$ and spin
density $m$ values and intervals,

$\displaystyle\Delta_{\rm gap}^{+-}(k)$ $\displaystyle=$ $\displaystyle
2\mu_{B}\,h-W_{2}$ $\displaystyle k$ $\displaystyle=$ $\displaystyle 0\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,1[$ $\displaystyle k$
$\displaystyle=$ $\displaystyle k_{F\uparrow}-k_{F\downarrow}\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,1/3]$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle]{\tilde{k}},k_{F\downarrow}]\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,1[\,.$ (76)

The energy scale $2\mu_{B}\,h-W_{2}\geq 0$ has the following limiting values,

$2\mu_{B}\,h-W_{2}=0\hskip 5.69046pt{\rm for}\hskip 5.69046ptm\rightarrow
0\hskip 14.22636pt{\rm and}\hskip 14.22636pt2\mu_{B}\,h-W_{2}=J\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\rightarrow 1\,.$ (77)

At $k=\pi$ the energy gap $\Delta_{\rm gap}^{+-}(k)$ reads,

$\Delta_{\rm gap}^{+-}(\pi)=4\mu_{B}\,h\hskip 5.69046pt{\rm for}\hskip
5.69046ptm\in]0,1[\,.$ (78)

It has the following limiting values,

$\Delta_{\rm gap}^{+-}(\pi)=0\hskip 5.69046pt{\rm for}\hskip
5.69046ptm\rightarrow 0\hskip 14.22636pt{\rm and}\hskip 14.22636pt\Delta_{\rm
gap}^{+-}(\pi)=4J\hskip 5.69046pt{\rm for}\hskip 5.69046ptm\rightarrow 1\,.$
(79)

The energy gap $\Delta_{\rm gap}^{-+}(k)$ has the following values at
$k=0,k_{F\uparrow}-k_{F\downarrow},\pi$,

$\displaystyle\Delta_{\rm gap}^{-+}(0)$ $\displaystyle=$ $\displaystyle
4\mu_{B}\,h-W_{2}\hskip 5.69046pt{\rm for}\hskip 5.69046ptm\in]0,1[$
$\displaystyle\Delta_{\rm gap}^{-+}(k_{F\uparrow}-k_{F\downarrow})$
$\displaystyle=$ $\displaystyle 4\mu_{B}\,h\hskip 5.69046pt{\rm for}\hskip
5.69046ptm\in]0,\tilde{m}]$ $\displaystyle\Delta_{\rm gap}^{-+}(\pi)$
$\displaystyle=$ $\displaystyle 4\mu_{B}\,h-2W_{1}^{p}\hskip 5.69046pt{\rm
for}\hskip 5.69046ptm\in]0,1[\,.$ (80)

For small spin densities $m\in]0,\bar{m}]$ where $\bar{m}\approx 0.276$ and
the momentum intervals given in Eq. (29) the inequality $\Delta_{\rm
gap}^{-+}(k)<0$ holds. In such a $k$ intervals, $\Delta_{\rm gap}^{-+}(k)$
equals the energy gap $\Delta_{\rm gap}^{xx}(k)=\Delta_{\rm gap}^{-+}(k)<0$.
At $k=\pi$ such gaps thus have negative and positive values for spin densities
$m<\bar{m}_{0}$ and $m>\bar{m}_{0}$, respectively, where $\bar{m}_{0}\approx
0.239$. The corresponding limiting values read,

$\displaystyle\Delta_{\rm gap}^{xx}(\pi)$ $\displaystyle=$
$\displaystyle\Delta_{\rm gap}^{-+}(\pi)=-\pi J\hskip 5.69046pt{\rm for}\hskip
5.69046ptm\rightarrow 0$ (81) $\displaystyle=$ $\displaystyle 0\hskip
5.69046pt{\rm for}\hskip 5.69046ptm=\bar{m}_{0}\approx 0.239$ $\displaystyle=$
$\displaystyle 4J\hskip 5.69046pt{\rm for}\hskip 5.69046ptm\rightarrow 1\,.$

##  Appendix B Some useful quantities

In this Appendix some quantities needed for our study are defined and
corresponding useful limiting behaviors are provided. The quantum problem
studied in this paper is described by the spin-$1/2$ $XXX$ chain in a
longitudinal magnetic field, Eq. (1), acting in the subspace considered in
Sec. III.1. It involves a subset of Bethe ansatz equations. That associated
with the $1$-band is given by,

$\displaystyle q_{j}$ $\displaystyle=$ $\displaystyle
2\arctan\left(\Lambda_{1}(q_{j})\right)-{2\over
L}\sum_{j^{\prime}=1}^{N_{\uparrow}}\,N_{1}(q_{j^{\prime}})\arctan\left({\Lambda_{1}(q_{j})-\Lambda_{1}(q_{j^{\prime}})\over
2}\right)$ (82) $\displaystyle-$ $\displaystyle{2\over
L}\sum_{j^{\prime}=1}^{N_{\uparrow}-N_{\downarrow}+N_{2}}\,N_{2}(q_{j^{\prime}})\left\\{\arctan\left(\Lambda_{1}(q_{j})-\Lambda_{2}(q_{j^{\prime}})\right)+\arctan\left({\Lambda_{1}(q_{j})-\Lambda_{2}(q_{j^{\prime}})\over
3}\right)\right\\}$ $\displaystyle{\rm where}\hskip
14.22636ptj=1,...,N_{\uparrow}\hskip 14.22636pt{\rm and}\hskip
14.22636ptN_{2}=0,1\,.$

That associated with the $2$-band reads,

$\displaystyle q_{j}$ $\displaystyle=$ $\displaystyle
2\arctan\left({\Lambda_{2}(q_{j})\over 2}\right)-{2\over
L}\sum_{j^{\prime}=1}^{N_{\uparrow}}\,N_{1}(q_{j^{\prime}})\left\\{\arctan\left(\Lambda_{2}(q_{j})-\Lambda_{1}(q_{j^{\prime}})\right)+\arctan\left({\Lambda_{2}(q_{j})-\Lambda_{1}(q_{j^{\prime}})\over
3}\right)\right\\}$ (83) $\displaystyle{\rm where}\hskip
14.22636ptj=1,...,N_{\uparrow}-N_{\downarrow}+N_{2}\hskip 14.22636pt{\rm
and}\hskip 14.22636ptN_{2}=0,1\,.$

In these equations, $N_{1}(q_{j^{\prime}})=1$ and $N_{2}(q_{j^{\prime}})=1$
for occupied momentum values $q_{j^{\prime}}$ and $N_{1}(q_{j^{\prime}})=0$
and $N_{2}(q_{j^{\prime}})=0$ for unoccupied momentum values $q_{j^{\prime}}$.

The subspace considered in Sec. III.1 is spanned by excited energy eigenstates
populated either by a number $N_{1}=N_{\downarrow}$ of $1$-particles or by a
number $N_{1}=N_{\downarrow}-2$ of $1$-particles and a single $2$-particle. In
the case of the latter class of states, the Bethe-ansatz equation, Eq. (83),
does not include a third term, that in the case of the full Hilbert space
involves the spin rapidity differences
$\Lambda_{2}(q_{j})-\Lambda_{2}(q_{j^{\prime}})$. Indeed, when $\delta
N_{2}=1$ one has that such a term only contributes for $q_{j}=q_{j^{\prime}}$
at which values it vanishes.

The $1$-band Bethe ansatz rapidity is real and associated with the rapidity
function $\Lambda_{1}(q_{j})$ in the above Bethe-ansatz equations. In the case
of general $n$-strings of length $n>1$, the corresponding complex non-real
Bethe ansatz rapidities have in the thermodynamic limit the following form
Takahashi_71 ,

$\displaystyle\Lambda_{n,l}(q_{j})$ $\displaystyle=$
$\displaystyle\Lambda_{n}(q_{j})+i(n+1-2l)\hskip 5.69046pt{\rm for}\hskip
5.69046ptl=1,...,n\hskip 5.69046pt{\rm and}\hskip 5.69046ptj=1,...,L_{n}\hskip
5.69046pt{\rm where}$ $\displaystyle L_{n}$ $\displaystyle=$ $\displaystyle
N_{n}+N^{h}_{n}\hskip 5.69046pt{\rm and}\hskip
5.69046ptN^{h}_{n}=2S+\sum_{n^{\prime}=n+1}^{\infty}2(n^{\prime}-n)N_{n^{\prime}}\,.$
(84)

This general expression also applies for $n=1$, $\Lambda_{1,1}(q_{j})$ being
real and equal to $\Lambda_{1}(q_{j})$. In the general $n=1,...,\infty$ case,
$\Lambda_{n}(q_{j})$ are real rapidity functions defined by the set of coupled
Bethe-ansatz equations associated with the full Hilbert space not given here,
$q_{j}$ such that $q_{j+1}-q_{j}=2\pi/L$ are the corresponding $n$-band
discrete momentum values whose number is $L_{n}$, $N_{n}$ is both the number
of $n$-band occupied momentum values and the number of $n$-particles and thus
of $n$-strings when $n>2$, $N_{n}^{h}$ is that of $n$-band unoccupied momentum
values, and $S$ is the energy eigenstate’s spin such that $2S$ gives the
number of unpaired physical spins $1/2$ that are not paired and thus are not
part of the $n$ particles’s internal degrees of freedom.

In the present case of the subspace considered in Sec. III.1, the problem
simplifies. The $2$-band rapidity function $\Lambda_{2}(q_{j})$ that appears
in Eqs. (82) and (83) is the real part of the following two Bethe ansatz
complex rapidities associated with a $n$-string of length $n=2$,

$\Lambda_{2,l}(q_{j})=\Lambda_{2}(q_{j})+i(3-2l)\hskip 5.69046pt{\rm
for}\hskip 5.69046ptl=1,2\hskip 5.69046pt{\rm and}\hskip
5.69046ptj=1,...,N_{\uparrow}-N_{\downarrow}+N_{2}\,.$ (85)

This expression refers to that given in Eq. (84) for $n=2$ and $l=1,2$.

The momentum values $q_{j}$ in Eqs. (82), (83), and (85) are given by,

$q_{j}={2\pi\over L}\,I^{n}_{j}\hskip 5.69046pt{\rm for}\hskip
5.69046ptn=1,2\,,$ (86)

where the quantum numbers $I^{n}_{j}$ are either integers or half-odd integers
according to the following boundary conditions Takahashi_71 ,

$\displaystyle I_{j}^{1}$ $\displaystyle=$ $\displaystyle 0,\pm 1,\pm
2,...\hskip 14.22636pt{\rm for}\hskip 4.26773ptN_{\uparrow}\hskip
4.26773pt{\rm odd}$ $\displaystyle=$ $\displaystyle\pm 1/2,\pm 3/2,\pm
5/2,...\hskip 14.22636pt{\rm for}\hskip 4.26773ptN_{\uparrow}\hskip
4.26773pt{\rm even}$ $\displaystyle I_{j}^{2}$ $\displaystyle=$ $\displaystyle
0,\pm 1,\pm 2,...\hskip 14.22636pt{\rm for}\hskip 4.26773ptN_{2}=1\,.$ (87)

In the thermodynamic limit, we often use continuous momentum variables $q$
that replace the discrete $1$\- and $2$-bands momenta $q_{j}$ such that
$q_{j+1}-q_{j}=2\pi/L$. They read $q\in[-k_{F\uparrow},k_{F\uparrow}]$ and
$q\in[-(k_{F\uparrow}-k_{F\downarrow}),(k_{F\uparrow}-k_{F\downarrow})]$,
respectively. In that limit, the momenta $k_{F\downarrow}$ and $k_{F\uparrow}$
are given by,

$k_{F\downarrow}={\pi\over 2}(1-m)\,;\hskip 5.69046ptk_{F\uparrow}={\pi\over
2}(1+m)\,;\hskip 5.69046ptk_{F}={\pi\over 2}\,,$ (88)

for the spin-density interval, $m\in]0,1[$ where $k_{F}=\lim_{m\rightarrow
0}k_{F\downarrow}=\lim_{m\rightarrow 0}k_{F\uparrow}$.

The energy dispersions $\varepsilon_{1}(q)$ and $\varepsilon_{2}(q)$ that
appear in the spectra of the excited energy eigenstates are defined as,

$\varepsilon_{1}(q)={\bar{\varepsilon}_{1}}(\Lambda_{1}(q))\hskip
5.69046pt{\rm for}\hskip 5.69046ptq\in[-k_{F\uparrow},k_{F\uparrow}]\hskip
5.69046pt{\rm where}\hskip
5.69046pt{\bar{\varepsilon}_{1}}(\Lambda)=\int_{B}^{\Lambda}d\Lambda^{\prime}\,2J\eta_{1}(\Lambda^{\prime})\,,$
(89)

where $B$ is defined below and,

$\displaystyle\varepsilon_{2}(q)$ $\displaystyle=$ $\displaystyle
4\mu_{B}\,h+\varepsilon_{2}^{0}(q)\hskip 5.69046pt{\rm for}\hskip
5.69046ptq\in[-(k_{F\uparrow}-k_{F\downarrow}),(k_{F\uparrow}-k_{F\downarrow})]\hskip
5.69046pt{\rm where}$ $\displaystyle\varepsilon_{2}^{0}(q)$ $\displaystyle=$
$\displaystyle{\bar{\varepsilon}}_{2}^{0}(\Lambda_{2}(q))\hskip 5.69046pt{\rm
and}\hskip
5.69046pt{\bar{\varepsilon}}_{2}^{0}(\Lambda)=\int_{\infty}^{\Lambda}d\Lambda^{\prime}\,2J\eta_{2}(\Lambda^{\prime})\,,$
(90)

respectively.

The corresponding $1$\- and $2$-bands group velocities are given by,

$v_{1}(q)={\partial\varepsilon_{1}(q)\over\partial q}\hskip 5.69046pt{\rm
and}\hskip 5.69046ptv_{2}(q)={\partial\varepsilon_{2}(q)\over\partial q}\,.$
(91)

The distribution $2J\eta_{1}(\Lambda)$ appearing in Eq. (89) is the solution
of the integral equation,

$2J\eta_{1}(\Lambda)={4J\,\Lambda\over(1+\Lambda^{2})^{2}}+\int_{-B}^{B}d\Lambda^{\prime}\,G(\Lambda,\Lambda^{\prime})\,2J\eta_{1}(\Lambda^{\prime})\,.$
(92)

The kernel $G(\Lambda,\Lambda^{\prime})$ appearing here is given by,

$G(\Lambda,\Lambda^{\prime})=-{1\over{2\pi}}\left({1\over{1+((\Lambda-\Lambda^{\prime})/2)^{2}}}\right)\,.$
(93)

The values of the distribution $2J\eta_{2}(\Lambda)$ in Eq. (90) are
determined by those of $2J\eta_{1}(\Lambda)$ as follows,

$2J\eta_{2}(\Lambda)={J\over 2}{\Lambda\over\left(1+\left({\Lambda\over
2}\right)^{2}\right)^{2}}-\frac{1}{\pi}\int_{-B}^{B}d\Lambda^{\prime}\,{2J\eta_{1}(\Lambda^{\prime})\over
1+\left(\Lambda-\Lambda^{\prime}\right)^{2}}-\frac{1}{3\pi}\int_{-B}^{B}d\Lambda^{\prime}\,{2J\eta_{1}(\Lambda^{\prime})\over
1+\left({\Lambda-\Lambda^{\prime}\over 3}\right)^{2}}\,,$ (94)

where the distribution $2J\eta_{1}(\Lambda)$ is the solution of Eq. (92).

The rapidity distribution function $\Lambda_{1}(q)$ where
$q\in[-k_{F\uparrow},k_{F\uparrow}]$ in the argument of the auxiliary
dispersion ${\bar{\varepsilon}_{1}}$ in Eq. (89) is defined in terms of its
$1$-band inverse function $q=q_{1}(\Lambda)$ where
$\Lambda\in[-\infty,\infty]$. The latter is defined by the equation,

$q=q_{1}(\Lambda)=2\arctan(\Lambda)-\frac{1}{\pi}\int_{-B}^{B}d\Lambda^{\prime}\,2\pi\sigma(\Lambda^{\prime})\,\arctan\left({\Lambda-\Lambda^{\prime}\over
2}\right)\hskip 5.69046pt{\rm for}\hskip
5.69046pt\Lambda\in[-\infty,\infty]\,.$ (95)

The rapidity distribution function $\Lambda_{2}(q)$ where
$q\in[-(k_{F\uparrow}-k_{F\downarrow}),(k_{F\uparrow}-k_{F\downarrow})]$ in
the argument of the auxiliary dispersion ${\bar{\varepsilon}}_{2}^{0}$ in Eq.
(90) is also defined in terms of its $2$-band inverse function
$q=q_{2}(\Lambda)$ where $\Lambda\in[-\infty,\infty]$ as follows,

$q_{2}(\Lambda)=2\arctan\left({\Lambda\over
2}\right)-\frac{1}{\pi}\int_{-B}^{B}d\Lambda^{\prime}\,2\pi\sigma(\Lambda^{\prime})\arctan\left(\Lambda-\Lambda^{\prime}\right)-\frac{1}{\pi}\int_{-B}^{B}d\Lambda^{\prime}\,2\pi\sigma(\Lambda^{\prime})\arctan\left({\Lambda-\Lambda^{\prime}\over
3}\right)\hskip 5.69046pt{\rm for}\hskip
5.69046pt\Lambda\in[-\infty,\infty]\,.$ (96)

The distribution $2\pi\sigma(\Lambda)$ in Eqs. (95) and (96) is the solution
of the integral equation,

$2\pi\sigma(\Lambda)={2\over
1+\Lambda^{2}}+\int_{-B}^{B}d\Lambda^{\prime}\,G(\Lambda,\Lambda^{\prime})\,2\pi\sigma(\Lambda^{\prime})\,,$
(97)

whose kernel is given in Eq. (93). Such a distribution obeys the sum rule,

$\frac{1}{\pi}\int_{-B}^{B}d\Lambda\,2\pi\sigma(\Lambda)=(1-m)\,.$ (98)

The parameter $B=\Lambda_{1}(k_{F\downarrow})$ appearing in the above
equations has the limiting behaviors,

$B=\Lambda_{1}(k_{F\downarrow})\hskip 5.69046pt{\rm with}\hskip
5.69046pt\lim_{m\rightarrow 0}B=\infty\hskip 5.69046pt{\rm and}\hskip
5.69046pt\lim_{m\rightarrow 1}B=0\,.$ (99)

Useful reference values of the rapidity functions $\Lambda_{1}(q)$ and
$\Lambda_{2}(q)$ are,

$\displaystyle\Lambda_{1}(0)$ $\displaystyle=$ $\displaystyle 0\hskip
5.69046pt{\rm and}\hskip 5.69046pt\Lambda_{1}(\pm k_{F\uparrow})=\pm\infty$
$\displaystyle\Lambda_{2}(0)$ $\displaystyle=$ $\displaystyle 0\hskip
5.69046pt{\rm and}\hskip
5.69046pt\Lambda_{2}(\pm(k_{F\uparrow}-k_{F\downarrow}))=\pm\infty\,.$ (100)

The $1$-band energy dispersion,

$\varepsilon_{1}^{0}(q)={\bar{\varepsilon}_{1}}^{0}(\Lambda_{1}(q))\hskip
5.69046pt{\rm where}\hskip
5.69046pt{\bar{\varepsilon}_{1}}^{0}(\Lambda)=\int_{\infty}^{\Lambda}d\Lambda^{\prime}\,2J\eta_{1}(\Lambda^{\prime})\,,$
(101)

whose zero-energy level is for $m>0$ shifted realtive to that of
$\varepsilon_{1}(q)$ defines the spin density curve as follows,

$h(m)=-{\varepsilon_{1}^{0}(k_{F\downarrow})\over
2\mu_{B}}|_{m=1-2k_{F\downarrow}/\pi}\in]0,h_{c}[\,.$ (102)

Here $h_{c}=J/\mu_{B}$ is the critical field for fully polarized
ferromagnetism achieved when $m\rightarrow 1$ and thus
$k_{F\downarrow}\rightarrow 0$.

The $1$\- and $2$-band energy dispersions $\varepsilon_{1}(q)$ and
$\varepsilon_{2}(q)$, Eqs. (89) and (90), respectively, have limiting values,

$\displaystyle\varepsilon_{1}(0)$ $\displaystyle=$ $\displaystyle-
W_{1}^{p}\,;\hskip 14.22636pt\varepsilon_{1}(\pm k_{F\downarrow})=0\,;\hskip
14.22636pt\varepsilon_{1}(\pm k_{F\uparrow})=W_{1}^{h}=2\mu_{B}\,h$
$\displaystyle\varepsilon_{2}(0)$ $\displaystyle=$ $\displaystyle
W_{1}^{h}-W_{2}=4\mu_{B}\,h-W_{2}\,;\hskip
14.22636pt\varepsilon_{2}(\pm(k_{F\uparrow}-k_{F\downarrow}))=4\mu_{B}\,h\,,$
(103)

where $W_{1}=W_{1}^{p}+W_{1}^{h}$, $W_{1}^{p}$, and $W_{1}^{h}$ are the
$1$-band energy bandwidth, occupied Fermi sea energy bandwidth, and unoccupied
sea energy bandwidth, respectively, and $W_{2}$ is the $2$-band energy
bandwidth. Such energy scales have the limiting behaviors,

$\lim_{m\rightarrow 0}W_{1}=W_{1}^{p}={\pi\over 2}\,J\hskip 5.69046pt{\rm
and}\hskip 5.69046pt\lim_{m\rightarrow 0}W_{2}=0\,,$ (104)

and

$\lim_{m\rightarrow 1}W_{1}=W_{1}^{h}=2\mu_{B}\,h_{c}=2J\hskip 5.69046pt{\rm
and}\hskip 5.69046pt\lim_{m\rightarrow 1}W_{2}=J\,.$ (105)

In the $m\rightarrow 0$ limit, the $2$-band does not exist in the ground
state. In that limit, it reduces to $q=0$ with $\varepsilon_{2}(0)=0$ for
energy eigenstates for which $N_{2}=1$. In the same limit, the $1$-band energy
dispersions and group velocity can be written as,

$\displaystyle\varepsilon_{1}(q)$ $\displaystyle=$
$\displaystyle\varepsilon_{1}^{0}(q)=-J{\pi\over 2}\cos q\hskip 14.22636pt{\rm
and}\hskip 14.22636ptv_{1}(q)=J{\pi\over 2}\sin q$ (106) $\displaystyle{\rm
for}\hskip 5.69046ptq\in[-\pi/2,\pi/2]\hskip 5.69046pt{\rm and}\hskip
5.69046ptm\rightarrow 0\,.$

For $(1-m)\ll 1$, the $1$-band energy dispersions and group velocity, Eq.
(91), behave as,

$\displaystyle\varepsilon_{1}(q)$ $\displaystyle=$ $\displaystyle-J\,(\cos
q-1)+J\,(1-m)\sin q\,\arctan\left({1\over 2}\tan\left({q\over
2}\right)\right)$ $\displaystyle\varepsilon_{1}^{0}(q)$ $\displaystyle=$
$\displaystyle-J\,(\cos q+1)+J\,(1-m)\sin q\,\arctan\left({1\over
2}\tan\left({q\over 2}\right)\right)$ $\displaystyle v_{1}(q)$
$\displaystyle=$ $\displaystyle J\sin q+J\,(1-m)\left\\{{\sin q\over
1+3\cos^{2}\left({q\over 2}\right)}+\cos q\,\arctan\left({1\over
2}\tan\left({q\over 2}\right)\right)\right\\}$ (107) $\displaystyle{\rm
for}\hskip 5.69046ptq\in\left[-{\pi\over 2}(1+m),{\pi\over
2}(1+m)\right]\hskip 5.69046pt{\rm and}\hskip 5.69046pt(1-m)\ll 1\,.$

Also for $(1-m)\ll 1$, the behaviors of the $2$-band energy dispersion and
group velocity are,

$\displaystyle\varepsilon_{2}(q)$ $\displaystyle=$ $\displaystyle 4J-{J\over
2}\,(1+\cos q)+{J\over 2}\,(1-m)\sin q\left\\{\arctan\left(2\tan\left({q\over
2}\right)\right)+\arctan\left({2\over 3}\tan\left({q\over
2}\right)\right)\right\\}$ $\displaystyle\varepsilon_{2}^{0}(q)$
$\displaystyle=$ $\displaystyle\varepsilon_{2}(q)-4J$ $\displaystyle v_{2}(q)$
$\displaystyle=$ $\displaystyle{J\over 2}\sin q+{J\over 2}\,(1-m)\sin
q\left\\{{1\over 1+3\sin^{2}\left({q\over 2}\right)}+{3\over
4+5\cos^{2}\left({q\over 2}\right)}\right\\}$ (108) $\displaystyle+{J\over
2}\,(1-m)\cos q\left\\{\arctan\left(2\tan\left({q\over
2}\right)\right)+\arctan\left({2\over 3}\tan\left({q\over
2}\right)\right)\right\\}$ $\displaystyle{\rm for}\hskip 5.69046ptq\in[-\pi
m,\pi m]\hskip 5.69046pt{\rm and}\hskip 5.69046pt(1-m)\ll 1\,.$

From analysis of the above expressions, one finds that for $(1-m)\ll 1$ the
following equality holds,

$v_{1}(k_{F\downarrow})=v_{2}(k_{F\uparrow}-k_{F\downarrow})=J{\pi\over
2}(1-m)\,.$ (109)

The $1$-particle phase shifts play an important role in the dynamical
properties. They are given by,

$2\pi\,\Phi_{1,n}(q,q^{\prime})=2\pi\,\bar{\Phi}_{1,n}\left(\Lambda,\Lambda^{\prime}\right)\hskip
5.69046pt{\rm where}\hskip 5.69046pt\Lambda=\Lambda_{1}(q)\hskip 5.69046pt{\rm
and}\hskip 5.69046pt\Lambda^{\prime}=\Lambda_{n}(q^{\prime})\,.$ (110)

The rapidity phase shifts
$2\pi\bar{\Phi}_{1,n}\left(\Lambda,\Lambda^{\prime}\right)$ on the right-hand
side of the above equality are functions of the rapidity-related variables
$\Lambda$ for the $n$-bands. In the case of the excited energy eigenstates
that span the subspace considered in Sec. III.1, the quantum number $n$ has
the values $1$ and $2$. In units of $2\pi$, the corresponding $n=1,2$ rapidity
phase shifts $2\pi\bar{\Phi}_{1,n}\left(\Lambda,\Lambda^{\prime}\right)$ are
defined by the following integral equations,

$\bar{\Phi}_{1,1}(\Lambda,\Lambda^{\prime})={1\over\pi}\arctan\left({\Lambda-\Lambda^{\prime}\over
2}\right)+\int_{-B}^{B}d\Lambda^{\prime\prime}\,G(\Lambda,\Lambda^{\prime\prime})\,{\bar{\Phi}}_{1,1}(\Lambda^{\prime\prime},\Lambda^{\prime})\,,$
(111)

and

$\bar{\Phi}_{1,2}\left(\Lambda,\Lambda^{\prime}\right)={1\over\pi}\arctan(\Lambda-\Lambda^{\prime})+{1\over\pi}\arctan\left({\Lambda-\Lambda^{\prime}\over
3}\right)+\int_{-B}^{B}d\Lambda^{\prime\prime}\,G(\Lambda,\Lambda^{\prime\prime})\,{\bar{\Phi}}_{1,2}(\Lambda^{\prime\prime},\Lambda^{\prime})\,.$
(112)

The kernel $G(r,r^{\prime})$ in these equations is given in Eq. (93).

The phase shifts in units of $2\pi$ that appear in the expressions of the
branch-line exponents, Eqs. (43), (48), (50), and (51), are given by,

$\Phi_{1,1}\left(\iota k_{F\downarrow},q\right)=\bar{\Phi}_{1,1}\left(\iota
B,\Lambda_{1}(q)\right)\hskip 5.69046pt{\rm and}\hskip
5.69046pt\Phi_{1,2}\left(\iota
k_{F\downarrow},q\right)=\bar{\Phi}_{1,2}\left(\iota
B,\Lambda_{2}(q)\right)\hskip 5.69046pt{\rm where}\hskip 5.69046pt\iota=\pm
1\,.$ (113)

In the $m\rightarrow 0$ limit, the rapidity phase shift
$\bar{\Phi}_{1,1}(\Lambda,\Lambda^{\prime})$ (in units of $2\pi$) is given by,

$\displaystyle\bar{\Phi}_{1,1}(\Lambda,\Lambda^{\prime})$ $\displaystyle=$
$\displaystyle{i\over 2\pi}\,\ln\left({\Gamma\left({1\over
2}+i{(\Lambda-\Lambda^{\prime})\over
4}\right)\Gamma\left(1-i{(\Lambda-\Lambda^{\prime})\over
4}\right)\over\Gamma\left({1\over 2}-i{(\Lambda-\Lambda^{\prime})\over
4}\right)\Gamma\left(1+i{(\Lambda-\Lambda^{\prime})\over
4}\right)}\right)\hskip 5.69046pt{\rm for}\hskip
5.69046pt\Lambda\neq\iota\infty$ (114) $\displaystyle=$
$\displaystyle{\iota\over 2\sqrt{2}}\hskip 5.69046pt{\rm for}\hskip
5.69046pt\Lambda=\iota\infty\hskip 5.69046pt{\rm and}\hskip
5.69046pt\Lambda^{\prime}\neq\iota\infty$ $\displaystyle=$
$\displaystyle\iota\left({3\over 2\sqrt{2}}-1\right)\hskip 5.69046pt{\rm
for}\hskip 5.69046pt\Lambda=\Lambda^{\prime}=\iota\infty\hskip 5.69046pt{\rm
where}\hskip 5.69046pt\iota=\pm 1\,,$

where $\Gamma(x)$ is the usual gamma function.

From the use of Eq. (114), one finds that in the $m\rightarrow 0$ limit for
which $k_{F}=\pi/2$ the phase shift
$\Phi_{1,1}\left(\iota\pi/2,q\right)=\lim_{m\rightarrow
0}\Phi_{1,1}\left(\iota k_{F\downarrow},q\right)$ reads (in units of $2\pi$),

$\displaystyle\Phi_{1,1}\left(\iota\pi/2,q\right)$ $\displaystyle=$
$\displaystyle{\iota\over 2\sqrt{2}}\hskip 5.69046pt{\rm for}\hskip
5.69046ptq\neq\iota k_{F}$ (115) $\displaystyle=$
$\displaystyle\iota\left({3\over 2\sqrt{2}}-1\right)\hskip 5.69046pt{\rm
for}\hskip 5.69046ptq=\iota k_{F}\hskip 5.69046pt{\rm where}\hskip
5.69046pt\iota=\pm 1\,.$

In the $m\rightarrow 0$ limit, the phase shift
$\Phi_{1,2}\left(\iota\pi/2,q\right)=\lim_{m\rightarrow
0}\Phi_{1,2}\left(\iota k_{F\downarrow},q\right)$ has in units of $2\pi$ the
following value,

$\lim_{m\rightarrow
0}\Phi_{1,2}\left(\iota\pi/2,q\right)=\Phi_{1,2}\left(\iota\pi/2,0\right)={\iota\over\sqrt{2}}\,.$
(116)

In the opposite $m\rightarrow 1$ limit in which $k_{F\downarrow}\rightarrow
0$, the phase shifts $\Phi_{1,1}\left(\iota k_{F\downarrow},q\right)$ and
$\Phi_{1,2}\left(\iota k_{F\downarrow},q\right)$ behave as,

$\displaystyle\lim_{m\rightarrow 1}\Phi_{1,1}(\iota k_{F\downarrow},q)$
$\displaystyle=$
$\displaystyle\Phi_{1,1}(0,q)=-{1\over\pi}\arctan\left({1\over
2}\tan\left({q\over 2}\right)\right)$ $\displaystyle\lim_{m\rightarrow
1}\Phi_{1,2}(\iota k_{F\downarrow},q)$ $\displaystyle=$
$\displaystyle\Phi_{1,2}(0,q)=-{1\over\pi}\arctan\left(2\tan\left({q\over
2}\right)\right)-{1\over\pi}\arctan\left({2\over 3}\tan\left({q\over
2}\right)\right)\,.$ (117)

The phase-shift related parameters $\xi_{1\,1}$ and $\xi_{1\,2}^{0}$ also
appear in the expressions of the branch-line exponents, Eqs. (43), (48), (50),
and (51). The $1$-band Fermi-points phase-shift parameter $\xi_{1\,1}$ is such
that Carmelo_94 ,

$\displaystyle(\xi_{1\,1})^{\pm 1}$ $\displaystyle=$ $\displaystyle
1+\sum_{\iota=\pm 1}(\iota)^{1\pm 1\over
2}\,\Phi_{1,1}\left(k_{F\downarrow},\iota k_{F\downarrow}\right)\,.$ (118)

It is also related to the phase shift $\Phi_{1,2}(k_{F\downarrow},q)$ in Eq.
(113) as follows,

$\displaystyle\xi_{1\,1}$ $\displaystyle=$ $\displaystyle-\Phi_{1,2}(\pm
k_{F\downarrow},(k_{F\uparrow}-k_{F\downarrow}))$ (119) $\displaystyle=$
$\displaystyle\Phi_{1,2}(\pm
k_{F\downarrow},-(k_{F\uparrow}-k_{F\downarrow}))\,.$

From manipulations of the phase-shift integral equation, Eq. (111), and of Eq.
(118) one finds that the parameter $\xi_{1\,1}$ given by,

$\displaystyle\xi_{1\,1}$ $\displaystyle=$
$\displaystyle\xi_{1\,1}\left(B\right)\,.$ (120)

The function $\xi_{1\,1}(r)$ on the right-hand side of this equation at
$\Lambda=B$ is the solution of the integral equation,

$\xi_{1\,1}(\Lambda)=1+\int_{-B}^{B}d\Lambda^{\prime}\,G(\Lambda,\Lambda^{\prime})\,\xi_{1\,1}(\Lambda^{\prime})\,,$
(121)

where the kernel $G(\Lambda,\Lambda^{\prime})$ is given in Eq. (93).

The parameter $\xi_{1\,1}$ continuously increases upon increasing the spin
density from $\xi_{1\,1}=1/\sqrt{2}$ as $m\rightarrow 0$ to $\xi_{1\,1}=1$ for
$m\rightarrow 1$, so that its limiting values are,

$\lim_{m\rightarrow 0}\xi_{1\,1}={1\over\sqrt{2}}\hskip 14.22636pt{\rm
and}\hskip 14.22636pt\lim_{m\rightarrow 1}\xi_{1\,1}=1\,.$ (122)

Finally, the parameter $\xi_{1\,2}^{0}$ that also appears in the momentum
dependent exponents is given by,

$\xi_{1\,2}^{0}=2\Phi_{1,2}(k_{F\downarrow},0)\,,$ (123)

where the phase shift $\Phi_{1,2}(k_{F\downarrow},q)$ is defined in Eq. (113).
At $q=0$, it is such that $\Phi_{1,2}(\iota
k_{F\downarrow},0)=\iota\,\Phi_{1,2}(k_{F\downarrow},0)$. This justifies why
$\iota\,\xi_{1\,2}^{0}=2\Phi_{1,2}(\iota
k_{F\downarrow},0)=\iota\,2\Phi_{1,2}(k_{F\downarrow},0)$ for $\iota=\pm 1$.

The parameter $\xi_{1\,2}^{0}$ continuously decreases upon increasing the spin
density from $\xi_{1\,2}^{0}=\sqrt{2}$ as $m\rightarrow 0$ to
$\xi_{1\,2}^{0}=0$ for $m\rightarrow 1$. Consitent, it follows from Eqs. (116)
and (117) that,

$\lim_{m\rightarrow 0}\xi_{1\,2}^{0}=\sqrt{2}\hskip 14.22636pt{\rm and}\hskip
14.22636pt\lim_{m\rightarrow 1}\xi_{1\,2}^{0}=0\,.$ (124)

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