# Bethe strings in the dynamical structure factor of the spin-$1/2$ Heisenberg
$XXX$ chain

José M. P. Carmelo  Center of Physics of University of Minho and University of
Porto, P-4169-007 Oporto, Portugal  Department of Physics, University of
Minho, Campus Gualtar, P-4710-057 Braga, Portugal  Boston University,
Department of Physics, 590 Commonwealth Ave, Boston, MA 02215, USA  Tilen
Čadež  Center for Theoretical Physics of Complex Systems, Institute for Basic
Science (IBS), Daejeon 34126, Republic of Korea  Pedro D. Sacramento  CeFEMA,
Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais,
P-1049-001 Lisboa, Portugal

(11 August 2020)

###### Abstract

Recently there has been a renewed interest in the spectra and role in
dynamical properties of excited states of the spin-$1/2$ Heisenberg
antiferromagnetic chain in longitudinal magnetic fields associated with Bethe
strings. The latter are bound states of elementary magnetic excitations
described by Bethe-ansatz complex non-real rapidities. Previous studies on
this problem referred to finite-size systems. Here we consider the
thermodynamic limit and study it for the isotropic spin-$1/2$ Heisenberg $XXX$
chain in a longitudinal magnetic field. We confirm that also in that limit the
most significant spectral weight contribution from Bethe strings leads to
$(k,\omega)$-plane gapped continua in the spectra of the spin dynamical
structure factors $S^{+-}(k,\omega)$ and $S^{xx}(k,\omega)=S^{yy}(k,\omega)$.
The contribution of Bethe strings to $S^{zz}(k,\omega)$ is found to be small
at low spin densities $m$ and to become negligible upon increasing that
density above $m\approx 0.317$. For $S^{-+}(k,\omega)$, that contribution is
found to be negligible at finite magnetic field. We derive analytical
expressions for the line shapes of $S^{+-}(k,\omega)$,
$S^{xx}(k,\omega)=S^{yy}(k,\omega)$, and $S^{zz}(k,\omega)$ valid in the
$(k,\omega)$-plane vicinity of singularities located at and just above the
gapped lower thresholds of the Bethe-string states’s spectra. As a side result
and in order to provide an overall physical picture that includes the relative
$(k,\omega)$-plane location of all spectra with a significant amount of
spectral weight, we revisit the general problem of the line-shape of the
transverse and longitudinal spin dynamical structure factors at finite
magnetic field and excitation energies in the $(k,\omega)$-plane vicinity of
other singularities. This includes those located at and just above the lower
thresholds of the spectra that stem from excited states described by only real
Bethe-ansatz rapidities.

##  I Introduction

Recently, there has been a renewed interest in bound states of elementary
magnetic excitations named Bethe strings known long ago Bethe_31 ;
Takahashi_71 ; Gaudin_71 ; Taka-AN ; Gaudin_14 . In spite of Bethe strings
being a rather theoretical issue, as they were first found and identified
within the Bethe ansatz solution of spin and electronic integrable models in
some classes of energy eigenstates described by complex non-real spin
rapidities Bethe_31 ; Takahashi_71 , this renewed interest is actually due, in
part, to their experimental identification and realization in spin-chain
compounds whose magnetic properties are described by the spin-$1/2$ Heisenberg
chain in longitudinal magnetic fields Bera_20 ; Wang_18 ; Kohno_09 ; Kohno_10
; Stone_03 . This applies to that model isotropic point in the case of
experimental studies of some classes of such compounds Kohno_09 ; Kohno_10 ;
Stone_03 ; Heilmann_78 .

The present paper addresses only theoretical issues of that interesting
physical problem. Most previous studies on the spin dynamical properties of
the spin-$1/2$ $XXX$ chain in a longitudinal magnetic field focused on the
contribution from energy eigenstates described by real Bethe-ansatz
rapidities, which are associated with most spectral weight of the spin
dynamical structure factors. Several such studies considered finite-size
systems and relied on different methods. This includes for instance numerical
diagonalizations Lefmann-96  and evaluation of matrix elements between Bethe-
ansatz states Muller ; Karbach_00 ; Karbach_02 . Previous studies that
considered the thermodynamic limit Carmelo_15A , were also limited to the
contribution to the spin dynamical structure factors from energy eigenstates
described by real Bethe-ansatz rapidities. Concerning the specific issue of
the contribution of Bethe strings to the spin dynamical properties of
spin-$1/2$ $XXX$ chain in a longitudinal magnetic field, the few previous
studies considered finite-size systems Kohno_09 ; Kohno_10 .

In the case of that spin-$1/2$ chain, Bethe strings Bethe_31 , which here we
call $n$-strings, have for $n>1$ and in the thermodynamic limit Takahashi_71
been shown to be bound states of $n=2,...,\infty$ singlet pairs of the model
physical spins $1/2$ Carmelo_15 ; Carmelo_17 ; Carmelo_18 . (The physical
meaning of the form of the spin-$1/2$ $XXX$ chain’s $n$-strings in that limit
is an issue shortly further discussed below in Sec. III.1.) Energy eigenstates
described by only real Bethe-ansatz rapidities lack such bound pairs and are
populated by unbound singlet pairs of such physical spins Carmelo_15 ;
Carmelo_17 ; Carmelo_18 . On the other hand, there are predictions according
to which for the large spin-$S$ Heisenberg $XXX$ chain in longitudinal
magnetic fields, Bethe strings could rather be bound states of spin-$1$
magnons Johnson_86 ; Dally_20 .

In this paper we address the problem of the contribution of $n$-strings to the
spin dynamical properties of spin-$1/2$ $XXX$ chain in a longitudinal magnetic
field, in the thermodynamic limit. Based on a relation between the level of
negativity of the momentum dependent exponents that control the
$(k,\omega)$-plane line shape of the spin dynamical structure factors near
singularities and the amount of spectral weight existing in their vicinity,
respectively, we confirm that in the thermodynamic limit, as in the case of
finite-size systems Kohno_09 ; Kohno_10 , the only contribution from excited
energy eigenstates populated by $n$-strings that leads to a $(k,\omega)$-plane
gapped continuum in the spectrum of the spin dynamical structure factors
refers to $S^{+-}(k,\omega)$ and thus also to
$S^{xx}(k,\omega)=S^{yy}(k,\omega)$. On the other hand, the contribution from
$n$-strings states to $S^{zz}(k,\omega)$ is found to be small at low spin
densities and to become negligible upon increasing it beyond a spin density,
$\tilde{m}\approx 0.317$. For the spin dynamical structure factor
$S^{-+}(k,\omega)$, that contribution is found to be negligible at any finite
magnetic field.

The main goal of this paper is thus the study of the line shape of the spin
dynamical structure factors $S^{+-}(k,\omega)$, $S^{xx}(k,\omega)$, and
$S^{zz}(k,\omega)$ at and just above singularities located at the
$(k,\omega)$-plane gapped lower thresholds of the spectra associated with
$n$-string states. To reach that goal, we extend the dynamical theory of Ref.
Carmelo_15A  to a larger subspace, which allows to account for the
contribution from the latter states to the spin dynamical structure factors.
We then derive analytical expressions valid in the thermodynamic limit for
line shape of these factors in the $(k,\omega)$-plane vicinity of the
singularities under consideration.

Complementarily and as a side result, in order to to provide an overall
physical picture that includes the relative $(k,\omega)$-plane location of all
features with a significant amount of spectral weight, we account for the
contributions from all types of states that lead to gapped and gapless lower
threshold singularities in the spin dynamical structure factors. This includes
both excited states with and without $n$-strings. (As mentioned above, the
contribution from the latter states, shortly revisited in this paper, is known
to lead to the largest amount of spin dynamical structure factors’s spectral
weight Lefmann-96 ; Muller ; Karbach_00 ; Karbach_02 ; Carmelo_15A .)

The paper is organized as follows. The model and the spin dynamical structure
factors are the subjects of Sec. II. In Sec. III the spectral functionals that
control the extended dynamical theory’s general expressions of the dynamical
structure factors are introduced. Such factors’s spectra are studied in Sec.
IV. The line shape near their singularities is the issue addressed in Sec. V.
The subject of Sec. VI is the limiting behaviors of the spin dynamical
structure factors. Finally, the discussion and concluding remarks are
presented in Sec. VII. Two Appendices provide useful side information needed
for the studies of this paper.

##  II The model and the spin dynamical structure factors

The spin-$1/2$ Heisenberg $XXX$ chain with exchange integral $J$ and length
$L\rightarrow\infty$ in a longitudinal magnetic field $h$ for spin densities
$m\in]0,1[$, which describes $N=\sum_{\sigma=\uparrow,\downarrow}N_{\sigma}$
physical spins $1/2$ of projection $\sigma$, is a paradigmatic example of an
integrable strongly correlated system Bethe_31 ; Takahashi_71 . Its
Hamiltonian is given by,

$\hat{H}=J\sum_{j=1}^{L}\sum_{a=x,y,z}{\hat{S}}_{j}^{a}{\hat{S}}_{j+1}^{a}+2\mu_{B}h\,{\hat{S}}^{z}\,.$
(1)

For simplicity, we have taken here $g=2$, $\hat{\vec{S}}_{j}$ is the
spin-$1/2$ operator at site $j=1,...,N$ with components $\hat{S}_{j}^{x,y,z}$,
$\mu_{B}$ is the Bohr magneton, and
${\hat{S}}^{z}=\sum_{j=1}^{N}\hat{S}^{z}_{j}$ is the diagonal generator of the
global spin $SU(2)$ symmetry algebra. We denote the energy eigenstate’s spin
projection by $S^{z}=-(N_{\uparrow}-N_{\downarrow})/2\in[-S,S]$ where
$S\in[0,N/2]$ is their spin. Units of lattice spacing and Planck constant one
are used in this paper.

Due to the rotational symmetry in spin space, off-diagonal components of the
spin dynamical structure factors vanish, $S^{aa^{\prime}}(k,\omega)=0$ for
$a\neq a^{\prime}$ where $a$ and $a^{\prime}$ are given by $x,y,z$. In
addition, the two transverse components are identical,
$S^{xx}(k,\omega)=S^{yy}(k,\omega)$. In the present case of finite magnetic
fields $0<h<h_{c}$, one has that $S^{zz}(k,\omega)\neq S^{xx}(k,\omega)$. Here
$h_{c}=J/\mu_{B}$ is the magnetic field above which there is fully polarized
ferromagnetism. The corresponding magnetic energy scale, $2\mu_{B}\,h_{c}=2J$,
is associated with the quantum phase transition to fully polarized
ferromagnetism. In the opposite limit of zero magnetic field, one has that
$S^{zz}(k,\omega)=S^{xx}(k,\omega)$.

The dynamical structure factors $S^{aa}(k,\omega)$ are given by,

$\displaystyle S^{aa}(k,\omega)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}e^{-ikj}\int_{-\infty}^{\infty}dt\,e^{-i\omega
t}\langle GS|\hat{S}^{aa}_{j}(t)\hat{S}^{a}_{j}(0)|GS\rangle$ (2)
$\displaystyle=$
$\displaystyle\sum_{\nu}|\langle\nu|\hat{S}^{a}_{k}|GS\rangle|^{2}\delta(\omega-\omega^{\tau}_{\nu}(k))\hskip
5.69046pt{\rm for}\hskip 5.69046pta=x,y,z\,.$

Here the spectra read $\omega^{aa}_{f}(k)=(E_{\nu}^{aa}-E_{GS})$,
$E_{\nu}^{aa}$ refers to the energies of the excited energy eigenstates that
contribute to the $aa=xx,yy,zz$ dynamical structure factors, $\sum_{\nu}$ is
the sum over such states, $E_{GS}$ is the initial ground state energy, and
$\hat{S}^{a}_{k}$ are for $a=x,y,z$ the Fourier transforms of the usual local
$a=x,y,z$ spin operators $\hat{S}^{a}_{j}$, respectively.

The spin dynamical structure factor $S^{xx}(k,\omega)$ can be expressed as,

$S^{xx}(k,\omega)={1\over 4}\left(S^{+-}(k,\omega)+S^{-+}(k,\omega)\right)\,.$
(3)

One can then address the dynamical properties of $S^{xx}(k,\omega)$ in terms
of those of $S^{+-}(k,\omega)$ and $S^{-+}(k,\omega)$.

Since $S^{aa}(k,\omega)=S^{aa}(-k,\omega)$ for $a=x,y,z$ and thus also
$S^{+-}(k,\omega)=S^{+-}(-k,\omega)$ and $S^{-+}(k,\omega)=S^{-+}(-k,\omega)$,
in this paper we consider excitation momentum values $k>0$ in the first
Brillouin zone, $k\in[0,\pi]$. Another useful symmetry relating the spin
density intervals $m\in]-1,0]$ and $m\in]0,1[$ is such that,

$\displaystyle S^{-+}(k,\omega)|_{m}$ $\displaystyle=$ $\displaystyle
S^{+-}(k,\omega)|_{-m}\hskip 5.69046pt{\rm and}$ $\displaystyle
S^{+-}(k,\omega)|_{m}$ $\displaystyle=$ $\displaystyle
S^{-+}(k,\omega)|_{-m}\hskip 5.69046pt{\rm for}\hskip 5.69046ptm\in]0,1[\,.$
(4)

Hence, as mentioned above, we only consider explicitly the spin density
interval $m=2S^{z}/N\in]0,1[$. The subspace defined below in Sec. III.1 of the
quantum problem studied in this paper is spanned by some classes of energy
eigenstates with spin $S\in]0,N/2[$ and magnetic fields $0<h<h_{c}$ for which
the spin density belongs to the interval $m\in]0,1[$. ($N$ is even and odd
when the states spin $S$ is an integer and half-odd integer number,
respectively. In the latter case, the minimal spin value is $1/2$, rather than
$0$.)

Some useful selection rules tell us which classes of energy eigenstates have
nonzero matrix elements with the ground state. Let $|S,\alpha\rangle$,
$|S^{z},\beta\rangle$, and $|S,S^{z},\gamma\rangle$ denote energy eigenstates
where $S\in[0,N/2]$ is their spin, $S^{z}$ their spin projection, and
$\alpha$, $\beta$ and $\gamma$ represent all other quantum numbers needed to
uniquely specify these states, respectively. The selection rules given in the
following are derived from the properties of the operators $\hat{S}^{z}_{k}$
and $\hat{S}^{\pm}_{k}$ by straightforward manipulations involving their
operator algebra Muller .

At vanishing magnetic field, $h=0$, the following selection rules hold in the
thermodynamic limit,

$\displaystyle\langle
S,\alpha|\hat{S}^{a}_{k}|S^{\prime}\alpha^{\prime}\rangle$ $\displaystyle=$
$\displaystyle 0\hskip 5.69046pt{\rm for}\hskip 5.69046ptS=S^{\prime}=0\hskip
5.69046pt{\rm and}\hskip 5.69046pta=z,\pm$ $\displaystyle\langle
S,\alpha|\hat{S}^{a}_{k}|S^{\prime}\alpha^{\prime}\rangle$ $\displaystyle=$
$\displaystyle 0\hskip 5.69046pt{\rm for}\hskip 5.69046pt|S-S^{\prime}|\neq
0,1\hskip 5.69046pt{\rm and}\hskip 5.69046pta=z,\pm$ $\displaystyle\langle
S^{z},\beta|\hat{S}^{\pm}_{k}|S^{z^{\prime}},\beta^{\prime}\rangle$
$\displaystyle=$ $\displaystyle 0\hskip 5.69046pt{\rm for}\hskip
5.69046ptS^{z^{\prime}}\neq S^{z}\pm 1$ $\displaystyle\langle
S^{z},\beta|\hat{S}^{z}_{k}|S^{z^{\prime}},\beta^{\prime}\rangle$
$\displaystyle=$ $\displaystyle 0\hskip 5.69046pt{\rm for}\hskip
5.69046ptS^{z^{\prime}}\neq S^{z}\,.$ (5)

On the other hand, for finite magnetic fields $0<h<h_{c}$ of most interest for
our study, the following selection rules are valid in that limit,

$\displaystyle\langle
S,S,\gamma|\hat{S}^{\pm}_{k}|S^{\prime},S^{z^{\prime}},\gamma^{\prime}\rangle$
$\displaystyle=$ $\displaystyle 0\hskip 5.69046pt{\rm for}\hskip
5.69046ptS^{\prime}\neq S\pm 1\hskip 5.69046pt{\rm and}\hskip
5.69046ptS^{z^{\prime}}\neq S\pm 1$ $\displaystyle\langle
S,S,\gamma|\hat{S}^{z}_{k}|S^{\prime},S^{z^{\prime}},\gamma^{\prime}\rangle$
$\displaystyle=$ $\displaystyle 0\hskip 5.69046pt{\rm for}\hskip
5.69046ptS^{\prime}\neq S\hskip 5.69046pt{\rm and}\hskip
5.69046ptS^{z^{\prime}}\neq S\,.$ (6)

The spin dynamical structure factors satisfy the following sum rules,

$\displaystyle{1\over
2\pi^{2}}\int_{-\pi}^{\pi}dk\int_{0}^{\infty}d\omega\,S^{+-}(k,\omega)$
$\displaystyle=$ $\displaystyle(1+m)$ $\displaystyle{1\over
2\pi^{2}}\int_{-\pi}^{\pi}dk\int_{0}^{\infty}d\omega\,S^{-+}(k,\omega)$
$\displaystyle=$ $\displaystyle(1-m)$ $\displaystyle{1\over
2\pi^{2}}\int_{-\pi}^{\pi}dk\int_{0}^{\infty}d\omega\,S^{zz}(k,\omega)$
$\displaystyle=$ $\displaystyle{1\over 2}(1-m^{2})\,.$ (7)

The selection rules in Eq. (5) reveal that at $h=0$ and thus $m=0$ when
$S^{xx}(k,\omega)=S^{yy}(k,\omega)=S^{zz}(k,\omega)$, the longitudinal
dynamical structure factor $S^{zz}(k,\omega)$ is fully controlled by
transitions from the ground state for which $S^{z}=S=0$ to excited states with
spin numbers $S^{z}=0$ and $S=1$. That according to such rules the transverse
dynamical structure factors are at $h=0$ controlled by transitions from that
ground state to excited states with spin numbers $S^{z}=\pm 1$ and $S=1$, does
not prevent the equality $S^{zz}(k,\omega)=S^{xx}(k,\omega)$ imposed by the
spin $SU(2)$ symmetry.

This is different from the case for magnetic fields $0<h<h_{c}$ considered in
this paper. According to the selection rules, Eq. (6), the factor
$S^{zz}(k,\omega)\neq S^{xx}(k,\omega)$ is then controlled by transitions from
the ground state with spin numbers $S^{z}=-S$ to excited states with the same
spin numbers $S^{z}=-S$. According to the same selection rules, the dynamical
structure factors $S^{+-}(k,\omega)$ and $S^{-+}(k,\omega)$ are controlled by
transitions from the ground state with spin numbers $S^{z}=-S$ to excited
states with spin numbers $S^{z}\pm 1=-S\pm 1$.

##  III The spectral functionals of the extended dynamical theory

As reported in Sec. I, the general goal of this paper is the study of the
contribution from $n$-string states to the spin dynamical structure factors
given in Eq. (2) within the spin-$1/2$ $XXX$ chain in a longitudinal magnetic
field, Eq. (1). The dynamical theory used in our studies refers to an
extension of that introduced for the present model in Ref. Carmelo_15A . In
that reference, only the contribution to the spin dynamical structure factors
from energy eigenstates described by real Bethe-ansatz rapidities was
considered.

The theory of that reference is directly related to that introduced for the
one-dimensional Hubbard model in Ref. Carmelo_05 . The related dynamical
theories of Refs. Carmelo_15A ; Carmelo_05 ; Carmelo_08  are equivalent to and
account for the same microscopic processes Carmelo_18  as the mobile quantum
impurity model scheme of Refs. Imambekov_09 ; Imambekov_12  in the case of
integrable models.

The main difference of such an extended theory to that considered in Ref.
Carmelo_15A , refers to the Hamiltonian, Eq. (1), acting onto an extended
subspace, including $n$-string states. This involves different new forms for
the spectral functionals that control the momentum dependent exponents in the
spin dynamical structure factors’s expressions obtained in this paper for
$(k,\omega)$-plane regions near specific types of spectral features. For
simplicity, we do not provide here the details of the extended dynamical
theory that are common to those already given in Ref. Carmelo_15A , and rather
focus on the differences associated with its extension to the contribution
from $n$-string states.

The use of the extended dynamical theory provides useful information on the
$(k,\omega)$-plane distribution of the excited energy eigenstates’s spectra
that contain in the thermodynamic limit most spectral weight of
$S^{+-}(k,\omega)$, $S^{xx}(k,\omega)$, and $S^{zz}(k,\omega)$. Such spectra
are schematically represented in Figs. 1, 2, and 3, respectively.

Figure 1: The two $(k,\omega)$-plane lower and upper continuum regions where
for spin densities (a) $m=0.15$, (b) $m=0.25$, (c) $m=0.50$, and (d) $m=0.75$
there is in the thermodynamic limit more spectral weight in
$S^{+-}(k,\omega)$. The sketch of the $(k,\omega)$-plane distributions
represented here and in Figs. 2 and 3 does not provide information on the
relative amount of spectral weight contained within each spectrum’s grey
continuum. The three reference vertical lines mark the momenta (a)
[$k=k_{F\uparrow}-k_{F\downarrow}=3\pi/20$, $k=k_{F\downarrow}=17\pi/40$,
$k=2k_{F\downarrow}=17\pi/20$]; (b) [$k=k_{F\uparrow}-k_{F\downarrow}=\pi/4$,
$k=k_{F\downarrow}=3\pi/8$, $k=2k_{F\downarrow}=3\pi/4$]; (c)
[$k=k_{F\downarrow}=\pi/4$,
$k=k_{F\uparrow}-k_{F\downarrow}=2k_{F\downarrow}=\pi/2$]; (d)
[$k=k_{F\downarrow}=\pi/8$, $k=2k_{F\downarrow}=\pi/4$,
$k=k_{F\uparrow}-k_{F\downarrow}=3\pi/4$]. The lower and upper continuum
spectra are associated with excited energy eigenstates without and with
$n$-strings, respectively. In the thermodynamic limit, the $(k,\omega)$-plane
region between the upper threshold of the lower continuum and the gapped lower
threshold of the upper $n$-string continuum has nearly no spectral weight. In
the case of the gapped lower threshold of the $n$-string continuum, the
analytical expressions given in this paper refer to near and just above that
threshold whose subintervals refer to branch lines parts represented in the
figure by solid and dashed lines. The latter refer to $k$ intervals where the
momentum dependent exponents plotted in Fig. 4 are negative and positive,
respectively. In the former intervals, $S^{+-}(k,\omega)$ displays singularity
peaks.

After introducing the quantum problem’s extended subspace, the general
expressions of the spectral functionals under consideration are introduced in
the following. Specific expressions of the needed spectral functionals
suitable to the line-shape near the four types of spectral features considered
in our study are obtained. Finally, the issue concerning the $k$ intervals
where the corresponding momentum dependent exponents are valid is also
addressed.

###  III.1 The present quantum problem extended subspace

The quantum problem considered in this paper refers to the Hamiltonian, Eq.
(1), in a subspace spanned by two classes of energy eigenstates, populated and
not populated by $n$-strings, respectively. Our corresponding study of the
spin dynamical structure factors relies on the representation of such energy
eigenstates in terms of $n$-particle occupancy configurations, which is that
suitable to the dynamical theory used in this paper. Here $n=1,...,\infty$ is
the number of singlet pairs of physical spins $1/2$ that refer to their
internal degrees of freedom. The studies of Ref. Carmelo_15A  only involved
$n=1$ particles that in such a reference were named “pseudoparticles”.

In the thermodynamic limit, the Bethe-ansatz rapidities have the general form
given in Eq. (84) of Appendix B Takahashi_71 . For $n=1$ such Bethe-ansatz
rapidities are real and otherwise their imaginary part is finite. In that
equation, the Bethe-ansatz rapidities are partitioned in a configuration of
strings, where a $n$-string is a group of $l=1,...,n$ rapidities with the same
real part $\Lambda^{n}(q_{j})$. The number $n$ is called in the literature the
string length and the real part of the number $n$ of rapidities,
$\Lambda^{n}(q_{j})$, is called the string center.

For $n>1$ the $n$-particle internal degrees of freedom refer to a $n$-string,
whereas the $n$-band momentum $q_{j}$ in the argument of the real part of the
set of $l=1,...,n$ rapidities, $\Lambda^{n}(q_{j})$ in Eq. (84) of Appendix A,
describes its translational degrees of freedom. Each $n$-string contains a
number $n=2,...,\infty$ of bound singlet pairs of physical spins $1/2$, whose
number thus equals the length of the $n$-string Carmelo_15 ; Carmelo_17 ;
Carmelo_18 . The $l=1,...,n$ imaginary parts, $i(n+1-2l)$, of the set of $n$
rapidities of a $n$-string describe the binding of the $l=1,...,n$ pairs of
physical spins $1/2$. Consistently, that imaginary part vanishes at $n=1$, the
internal degrees of freedom of the $1$-particles corresponding to a single
unbound singlet pair of physical spins Carmelo_15 ; Carmelo_17 ; Carmelo_18 .
Their translational degrees of freedom refer again the $1$-band momentum
$q_{j}$. Energy eigenstates that are not populated and are populated by
$n$-particles with $n>1$ pairs, are described by only real Bethe-ansatz
rapidities and both real and complex non-real such rapidities, respectively.

Figure 2: The two $(k,\omega)$-plane lower and upper continuum regions where
for the same spin densities as in Fig. 1 there is in the thermodynamic limit
more spectral weight in $S^{xx}(k,\omega)$. The notations and the momenta
associated with the reference vertical lines are the same as in Fig. 1. The
additional part of the lower continuum relative to that of $S^{+-}(k,\omega)$
in Fig. 1 stems from the contributions from $S^{-+}(k,\omega)$. As a result,
for some $k$ intervals the upper $n$-string continuum overlaps with the lower
continuum.

The ground states with spin densities $0<m<1$ and corresponding longitudinal
magnetic fields $0<h<h_{c}$ are not populated by $n$-strings. Concerning the
amount of spectral weight of the spin dynamical structure factors originated
from transitions from such ground states to $n$-string states, transitions to
excited energy eigenstates populated by a single $2$-particle are found to be
dominant by far, as justified below. This is consistent with results for large
finite-size systems Kohno_09 ; Kohno_10 .

The $1$\- and $2$-particles carry $1$-band and $2$-band discrete momentum
values $q_{j}$, respectively, Eq. (86) of Appendix B, whose spacing is
$q_{j+1}-q_{j}=2\pi/L$. Accounting for $1/L$ contributions, the ground state
at a given spin density $0<m<1$ and corresponding longitudinal magnetic field
$0<h<h_{c}$ is populated by a number $N_{1}=N_{\downarrow}$ of $1$-particles
that fill a $1$-band Fermi sea as follows,

$\displaystyle q_{j}$ $\displaystyle\in$
$\displaystyle[q^{-1}_{F},q^{+1}_{F}]\hskip 5.69046pt{\rm where}\hskip
5.69046ptq_{j+1}-q_{j}=2\pi/L\hskip 5.69046pt{\rm and}$ $\displaystyle
q^{\iota}_{F}$ $\displaystyle=$ $\displaystyle\iota
k_{F\downarrow}-\iota{\pi\over L}\hskip 5.69046pt{\rm for}\hskip
5.69046ptN\hskip 5.69046pt{\rm even}\hskip 5.69046pt{\rm and}\hskip
5.69046pt\iota=\pm 1$ $\displaystyle q^{\iota}_{F}$ $\displaystyle=$
$\displaystyle\iota k_{F\downarrow}-(\iota\pm 1){\pi\over L}\hskip
5.69046pt{\rm for}\hskip 5.69046ptN\hskip 5.69046pt{\rm odd}\hskip
5.69046pt{\rm and}\hskip 5.69046pt\iota=\pm 1\,.$ (8)

Here $\iota=+1$ and $\iota=-1$ refer to the $1$-band right and left Fermi
points, respectively, $k_{F\downarrow}={\pi\over 2}(1-m)$ (as given in Eq.
(88) of Appendix B), and in the case of $N$ odd, $\pm 1$ refers in $(\iota\pm
1){\pi\over L}$ to two alternative Fermi point’s values.

In the present thermodynamic limit, we often use continuous momentum variables
$q$ that replace the discrete $1$\- and $2$-bands momentum values $q_{j}$ such
that $q_{j+1}-q_{j}=2\pi/L$. We can then consider for the studies of some
properties that $q^{\iota}_{F}=\iota k_{F\downarrow}$ and thus a ground-state
$1$-band occupied Fermi sea, $q\in[-k_{F\downarrow},k_{F\downarrow}]$.

As reported in Sec. I, there is a direct relation between the values of the
momentum dependent exponents that within the dynamical theory used here
control the line shape in the $(k,\omega)$-plane vicinity of the spin
dynamical structure factors spectral features and the amount of spectral
weight located near them: Negative exponents implies the occurrence of
singularities associated with a significant amount of spectral weight in their
$(k,\omega)$-plane vicinity.

The use of this criterion, reveals that in the present thermodynamic limit and
for magnetic fields $0<h<h_{c}$, the only significant contribution to
$S^{+-}(k,\omega)$ from excited energy eigenstates populated by $n$-particles
refers to those populated by a number $N_{\downarrow}-2$ of $1$-particles and
a single $2$-particle. There is as well a much weaker contribution at small
spin densities from states populated by a number $N_{\downarrow}-3$ of
$1$-particles and a single $3$-particle.

The only significant yet weak contribution to $S^{zz}(k,\omega)$ from
$n$-string states, refers to energy eigenstates populated by a number
$N_{\downarrow}-2$ of $1$-particles and a single $2$-particle. On the other
hand, the contribution from such excited energy eigenstates to
$S^{-+}(k,\omega)$ is found to be negligible, since all relevant exponents are
both positive and large.

Figure 3: The $(k,\omega)$-plane continuum region where for the same spin
densities as in Figs. 1 and 2 there is in the thermodynamic limit more
spectral weight in $S^{zz}(k,\omega)$. The reference vertical lines mark the
same momenta as in these figures. Contributions from excited states containing
$n$-strings are much smaller than for $S^{+-}(k,\omega)$ and
$S^{xx}(k,\omega)$ and do not lead to an upper continuum. The gapped lower
threshold of such states is though shown in the figures. Only when that
threshold coincides with the $\bar{1}^{\prime}$-branch line, which only occurs
for spin densities $0<m<\tilde{m}$ where $\tilde{m}\approx 0.317$,
singularities occur near and just above the $\bar{1}^{\prime}$-branch line,
which is represented by a solid (green) line. In the remaining parts of the
gapped lower threshold, which for spin densities $\tilde{m}<m<1$ means all of
it, the momentum dependent exponents are positive and there are no
singularities. This is equivalent to a negligible amount of spectral weight
near such lines.

The contribution to $S^{+-}(k,\omega)$ from energy eigenstates populated by a
number $N_{\downarrow}-3$ of $1$-particles and a single $3$ particle that
occurs for small values of the spin density is very weak. It is actually
inexistent in the vicinity of the $(k,\omega)$-plane singularities to which
the analytical expressions obtained in our study refer to. Indeed, except for
very small spin densities, $m\rightarrow 0$, the latter very weak
contributions occur in $(k,\omega)$-plane regions of higher excitation energy
$\omega$, above the gapped lower threshold of the spectrum continuum
associated with energy eigenstates populated by a number $N_{\downarrow}-2$ of
$1$-particles and a single $2$-particle whose expression is given below in
Sec. III.2. That spectrum refers to the upper continuum shown in Fig. 1.

The above spectral-weight analysis refers to the thermodynamic limit. Its
results are fully consistent with corresponding results reached by a
completely different method in the case of large finite-size systems Kohno_09
.

The subspace of the quantum problem studied in this paper is thus spanned by
an initial ground state for a given spin density $0<m<1$ and a corresponding
longitudinal magnetic field $0<h<h_{c}$ and its following excited energy
eigenstates: States described by both real and complex non-real Bethe-ansatz
rapidities populated by a number $N_{\downarrow}-2$ of $1$-particles and a
single $2$-particle whose internal degrees of freedom refer to a $n$-string of
length $n=2$; States populated by a number $N_{\downarrow}$ of $1$-particles
that are described only by real Bethe-ansatz rapidities.

###  III.2 General expressions of the extended dynamical theory’s spectral
functionals

The following number and current number deviations under transitions from a
ground state with $1$-band momentum distribution given in Eq. (8) to the
excited energy eigenstates that span the present subspace play an important
role in the extended dynamical theory’s expressions,

$\displaystyle\delta N_{1,\iota}^{F}\hskip 5.69046pt{\rm for}\hskip
5.69046pt\iota=1,-1\hskip 5.69046pt{\rm(right,left)}\hskip 5.69046pt{\rm
1-particles}$ $\displaystyle\delta N_{1}^{F}=\sum_{\iota=\pm 1}\delta
N_{1,\iota}^{F}\hskip 5.69046pt{\rm and}\hskip 5.69046pt\delta
J_{1}^{F}={1\over 2}\sum_{\iota=\pm 1}\iota\,\delta N_{1,\iota}^{F}$
$\displaystyle\delta J_{2}={\iota\over 2}\delta
N_{2}(q)|_{q=\iota\,(k_{F\uparrow}-k_{F\downarrow})}\hskip 5.69046pt{\rm
where}\hskip 5.69046pt\iota=\pm 1\,.$ (9)

Under some of the transitions from the ground state to the excited energy
eigenstates of the present subspace, the number of $2$-particles and/or that
of $1$-particles changes. This leads to number deviations $\delta N_{2}$
and/or $\delta N_{1}$, respectively. The specific number deviations $\delta
N_{1,\iota}^{F}$ in Eq. (9) refer only to changes of the $1$-particles numbers
at the left $(\iota=-1)$ or right $(\iota=1)$ $1$-band Fermi points, Eq. (8).
Exactly the same information is contained in the two Fermi points number
deviations $\delta N_{1,\iota}^{F}$, on the one hand, and in the corresponding
Fermi points number deviations $\delta N_{1}^{F}=\sum_{\iota=\pm 1}\delta
N_{1,\iota}^{F}$ and current number deviations $\delta J_{1}^{F}={1\over
2}\sum_{\iota=\pm 1}\iota\,\delta N_{1,\iota}^{F}$, on the other hand. The
overall $1$-particles number deviation $\delta N_{1}$ can be expressed as,

$\delta N_{1}=\delta N_{1}^{F}+\delta N_{1}^{NF}\,.$ (10)

Here $\delta N_{1}^{NF}$ refers to changes in the number of $1$-particles at
$1$-band momenta other than those at the Fermi points, Eq. (8).

For the current subspace, the $2$-band number deviations may read $\delta
N_{2}=0$ or $\delta N_{2}=1$. The $2$-band is empty in the ground state. For
that state its unoccupied momentum values $q_{j}$ such that
$q_{j+1}-q_{j}=2\pi/L$ refer to the range
$q_{j}\in[-(k_{F\uparrow}-k_{F\downarrow}-1/L),(k_{F\uparrow}-k_{F\downarrow}-1/L)]$
where $k_{F\downarrow}={\pi\over 2}(1-m)$ and $k_{F\uparrow}={\pi\over
2}(1+m)$ (as given in Eq. (88) of Appendix B.) The $2$-band momentum range
that is of interest for our studies rather refers to excited energy
eigenstates populated by a single $2$-particle. For these states, the
available discrete momentum values belong to the interval
$q_{j}\in[-(k_{F\uparrow}-k_{F\downarrow}),(k_{F\uparrow}-k_{F\downarrow})]$
where $q_{j+1}-q_{j}=2\pi/L$. The $2$-particle can occupy any of such $2$-band
discrete momentum values whose number is $N_{\uparrow}-N_{\downarrow}+1$. Only
when the $2$-particle is created at one of the two $2$-band’s limiting values,
$q=-(k_{F\uparrow}-k_{F\downarrow})$ or $q=(k_{F\uparrow}-k_{F\downarrow})$,
that process leads to a current number deviation $\delta J_{2}=-1/2$ and
$\delta J_{2}=1/2$, respectively, Eq. (9).

Within the extended dynamical theory, the line shape at and just above the
gapped lower thresholds of the $n$-string states of $S^{+-}(k,\omega)$,
$S^{xx}(k,\omega)$, and $S^{zz}(k,\omega)$ is for spin densities $0<m<1$ and
momenta in the range $k\in]0,\pi[$ of the following general form,

$\displaystyle S^{ab}(k,\omega)$ $\displaystyle=$ $\displaystyle
C_{ab}^{\Delta}\Bigl{(}\omega-\Delta_{\bar{n}}^{ab}(k))\Bigr{)}^{\zeta_{\bar{n}}^{ab}(k)}\hskip
5.69046pt{\rm for}\hskip 5.69046pt(\omega-\Delta_{\bar{n}}^{ab}(k))\geq 0$
(11) $\displaystyle{\rm where}\hskip
5.69046pt\bar{n}=2,\bar{1},\bar{1}^{\prime},2^{\prime}\hskip 5.69046pt{\rm
and}\hskip 5.69046ptab=+-,xx,zz$ $\displaystyle({\rm valid}\hskip
5.69046pt{\rm when}\hskip 5.69046pt\Delta_{\rm gap}^{ab}>0)\,.$

Here $C_{ab}^{\Delta}$ is a constant that has a fixed value for the $k$ and
$\omega$ ranges associated with small values of the energy deviation
$(\omega-\Delta_{\bar{n}}^{ab}(k))\geq 0$ and $\Delta_{\bar{n}}^{ab}(k)$ are
the spectra that define the $(k,\omega)$-plane shape of the gapped lower
thresholds of the $n$-string states’s in Figs. 1, 2, and 3. The analytical
expressions of such spectra are given below in Sec. III.2 and the general
expression of the exponents $\zeta_{\bar{n}}^{ab}(k)$ also appearing in Eq.
(11) is provided below. The indices
$\bar{n}=2,\bar{1},\bar{1}^{\prime},2^{\prime}$ in such spectra and exponents
label the branch lines or branch line sections that are part of the
corresponding gapped lower thresholds in some specific $k$ intervals defined
below in Sec. IV.1. Branch lines are types of spectral features that are
defined as within the dynamical theory of Ref. Carmelo_15A .

The quantity $\Delta_{\rm gap}^{ab}$ in Eq. (11) is the gap between the upper
thresholds of the lower continua associated with excited states described only
by real Bethe-ansatz rapidities and the gapped lower thresholds of the
$n$-string states’s spectra displayed in Figs. 1, 2, and 3. Only for $ab=xx$
there is overlap for small spin densities and some $k$ intervals between the
lower continuum and the $n$-string states’s upper continuum, as shown in Fig.
2 for spin densities $m=0.15$ and $m=0.25$. In the corresponding $k$
intervals, one has that $\Delta_{\rm gap}^{xx}<0$ and the general line-shape
expression given in Eq. (11) does not apply.

Indeed, that expression is valid provided there is no spectral weight or
nearly no spectral weight below the gapped lower thresholds of the $n$-string
states. In the present thermodynamic limit, the amount of spectral weight just
below such thresholds either vanishes or is extremely small. In the latter
case, the very weak coupling to it leads to a higher order contribution to the
line shape expressions given in that equation that can be neglected in such a
limit. Hence, the general expression of the spin dynamical structure factors
given in Eq. (11) is an excellent approximation for small values the energy
deviation $(\omega-\Delta_{\bar{n}}^{ab}(k))\geq 0$.

On the other hand, the line shape of the spin dynamical structure factors
$S^{ab}(k,\omega)$ where $ab=+-,-+,xx,zz$ at and just above their lower
thresholds of the lower spectra that for $ab=+-,xx,zz$ are shown in Figs. 1,
2, and 3 and are associated with excited energy eigenstates described only by
real Bethe-ansatz rapidities has been derived within the dynamical theory of
Ref. Carmelo_15A . It has the following general form, similar to that given in
Eq. (11),

$\displaystyle S^{ab}(k,\omega)$ $\displaystyle=$ $\displaystyle
C_{ab}\Bigl{(}\omega-\omega^{ab}_{lt}(k)\Bigr{)}^{\zeta_{1}^{ab}(k)}\hskip
5.69046pt{\rm for}\hskip 5.69046pt(\omega-\omega^{ab}_{lt}(k))\geq 0$ (12)
$\displaystyle{\rm where}\hskip 5.69046ptab=+-,-+,xx,zz\,.$

Again, here $C_{ab}$ are constants that have a fixed value for the $k$ and
$\omega$ intervals for which the energy deviation
$(\omega-\omega^{ab}_{lt}(k))\geq 0$ is small. The lower thresholds under
consideration refer to a single $1$-branch line that except for
$S^{-+}(k,\omega)$ has two $k$ interval sections. The $ab=+-,-+,zz$ lower
threshold’s spectra $\omega^{+-}(k)$, $\omega^{-+}(k)$, and $\omega^{zz}(k)$
in that deviation are given in Eqs. (67), (68), and (69) of Appendix A,
respectively.

There is no spectral weight below the lower thresholds associated with the
line-shape expression, Eq. (12). The general expression of the spin dynamical
structure factors given in that expression is thus exact for small values of
the energy deviation $(\omega-\omega^{ab}_{lt}(k))\geq 0$.

The branch-line exponents that appear in both Eqs. (11) and (12) have the same
following general form,

$\zeta^{ab}_{\bar{n}}(k)=-1+\sum_{\iota=\pm 1}(\Phi_{\iota}(q))^{2}\hskip
5.69046pt{\rm for}\hskip
5.69046pt\bar{n}=2,\bar{1},\bar{1}^{\prime},2^{\prime},1\,.$ (13)

The differences relative to the dynamical theory of Ref. Carmelo_15A , refer
to the form of the spectral functionals $\Phi_{\iota}(q)$ in this general
exponent’s expression that is suitable to each type of branch line. In the
following, the forms of such functionals specific to the four types of branch
line involved in our study are introduced.

Consistent with the occurrence of an infinite number of conservation laws
associated with the present quantum problem integrability, there is a
representation of the $n$-particles for which they only undergo zero-momentum
forward-scattering events. The corresponding phase shifts fully control the
spectral and dynamical properties. The extended dynamical theory uses such a
$n$-particle representation. Within it, $1$-particles carry discrete canonical
momentum values ${\bar{q}}_{j}={\bar{q}}(q_{j})$ defined below such that
${\bar{q}}_{j+1}-{\bar{q}}_{j}=2\pi/L+{\rm h.o.}$, rather than $1$-band
momentum values $q_{j}$ directly related to Bethe-ansatz quantum numbers, Eq.
(86) of Appendix B. The higher order $(1/L)^{2}$ terms in the spacing
${\bar{q}}_{j+1}-{\bar{q}}_{j}$ have no physical meaning in the present
thermodynamic limit. The key property of the $1$-particles canonical momentum
representation is the lack of energy interaction terms. This is what ensures
the occurrence of only zero-momentum forward-scattering events.

The initial ground state is populated by a macroscopic number
$N_{1}=N_{\downarrow}$ of $1$-particles and contains no $n$-particles with
$n>1$. Therefore, $1$-particles contribute to the dynamical properties both as
scatterers and scattering centers, whereas the $2$-particle contributes to
them as a scattering center only. As justified below, in the case of
scattering centers the same results are obtained in the thermodynamic limit
within the representations for which such centers created under transitions to
excited states carry momentum $q_{j}$ and canonical momentum
${\bar{q}}_{j}={\bar{q}}(q_{j})$, respectively. While canonical momentum can
also be introduced for the $2$-particle, for simplicity we thus remain using
$2$-band momentum $q_{j}$ for it.

From straightforward yet lengthly manipulations of the Bethe ansatz equations,
Eqs. (82)-(83) of Appendix B, one finds that for the excited energy
eigenstates that span the present subspace, the $1$-band rapidity functional
$\Lambda_{1}(q_{j})$ can be written in terms of the corresponding ground-state
rapidity function $\Lambda_{1}^{0}(q_{j})$ as follows,

$\Lambda_{1}(q_{j})=\Lambda_{1}^{0}({\bar{q}}_{j})\hskip 5.69046pt{\rm
for}\hskip 5.69046ptj=1,...,N_{\uparrow}\,.$ (14)

Here ${\bar{q}}_{j}={\bar{q}}(q_{j})$ where $j=1,...,N_{\uparrow}$ are the
following discrete canonical momentum values,

${\bar{q}}_{j}={\bar{q}}(q_{j})=q_{j}+{2\pi\over L}\Phi_{1}(q_{j})={2\pi\over
L}\left(I^{1}_{j}+\Phi_{1}(q_{j})\right)\,,$ (15)

and $I_{j}^{1}$ are the Bethe-ansatz $1$-band quantum numbers given in Eq.
(87) of Appendix B. The lack of energy interactions follows from in terms of
canonical momentum values the $1$-band rapidity function having for the
excited energy eigenstates the same form,
$\Lambda_{1}(q_{j})=\Lambda_{1}^{0}({\bar{q}}_{j})$, as for the corresponding
initial ground state. (For the ground state, $1$-band momentum values and
canonical momentum values are actually the same.)

The general expression of the $1$-band functional $\Phi_{1}(q_{j})$ in Eq.
(15) is in the case of the present subspace given by,

$\Phi_{1}(q_{j})=\sum_{j^{\prime}=1}^{N_{\uparrow}}\,\Phi_{1,1}(q_{j},q_{j^{\prime}})\,\delta
N_{1}(q_{j^{\prime}})+\sum_{j^{\prime}=1}^{N_{\uparrow}-N_{\downarrow}+N_{2}}\,\Phi_{1,2}(q_{j},q_{j^{\prime}})\,\delta
N_{2}(q_{j^{\prime}})\,,$ (16)

where $\Phi_{1,1}(q_{j},q_{j^{\prime}})$ and
$\Phi_{1,2}(q_{j},q_{j^{\prime}})$ are as further discussed below phase shifts
in units of $2\pi$. They are defined by Eqs. (110)-(112) of Appendix B. The
deviations $\delta N_{1}(q_{j^{\prime}})$ and $\delta N_{2}(q_{j^{\prime}})$
also appearing in Eq. (16) read,

$\delta N_{n}(q_{j})=N_{n}(q_{j})-N^{0}_{n}(q_{j})\hskip 5.69046pt{\rm
for}\hskip 5.69046ptn=1,2\,.$ (17)

Here $N_{n}(q_{j})$ and $N^{0}_{n}(q_{j})$ are the $1$-band and $2$-band
momentum distributions of the excited energy eigenstate and ground state,
respectively. Such momentum distributions appear in the functional
representation of the Bethe-ansatz equations, Eqs. (82)-(83) of Appendix B.
The ground-state distribution $N^{0}_{1}(q_{j})$ is associated with the
$1$-band compact occupancy, Eq. (8), whereas $N^{0}_{2}(q_{j})=0$ for
$q_{j}\in[-(k_{F\uparrow}-k_{F\downarrow}-1/L),(k_{F\uparrow}-k_{F\downarrow}-1/L)]$.

In the $\Phi_{1}(q_{j})$’s general expression, Eq. (16), the momentum values
$q_{j}$ and $q_{j^{\prime}}$ are associated with scatterers and scattering
centers, respectively. As mentioned above, one could associate canonical
momentum values both with the scatterars and with the scattering centers
created under the transitions from the ground state to the excited states.
However, in the case of the scattering centers, the form of the expression on
the right-hand side of Eq. (16) reveals that this leads to contributions of
order $(1/L)^{2}$ that have no physical meaning in the thermodynamic limit.
(The validity of the corresponding dynamical theory refers to that limit.)

For $\delta N_{1}(q_{j^{\prime}})=\pm 1$, the quantity
$\pm\Phi_{1,1}(q_{j},q_{j^{\prime}})=\Phi_{1,1}(q_{j},q_{j^{\prime}})\,\delta
N_{1}(q_{j^{\prime}})$ in Eq. (16) is the scattering phase shift in units of
$2\pi$ acquired by a $1$-particle of canonical momentum
${\bar{q}}_{j}={\bar{q}}(q_{j})$ (scatterer) upon creation of one $1$-hole
($-\Phi_{1,1}(q_{j},q_{j^{\prime}})$) and one $1$-particle
($+\Phi_{1,1}(q_{j},q_{j^{\prime}})$) at a momentum $q_{j^{\prime}}$ in the
$1$-band unoccupied and occupied Fermi sea (scattering centers), respectively.
(Given the one-to-one relation between the canonical momentum ${\bar{q}}_{j}$
and the momentum $q_{j}$, Eq. (15), scatterers of canonical momentum
${\bar{q}}_{j}$ can also be labelled by momentum $q_{j}$, which refers to two
representations of the same $1$-particle.)

On the other hand, the quantity
$\Phi_{1,2}(q_{j},q_{j^{\prime}})=\Phi_{1,2}(q_{j},q_{j^{\prime}})\,\delta
N_{2}(q_{j^{\prime}})$ in Eq. (16) is for $\delta N_{2}(q_{j^{\prime}})=1$ the
scattering phase shift in units of $2\pi$ acquired by a $1$-particle of
canonical momentum ${\bar{q}}_{j}={\bar{q}}(q_{j})$ (scatterer) under creation
of one $2$-particle at a momentum $q_{j^{\prime}}$ in the $2$-band interval
$q_{j^{\prime}}\in[-(k_{F\uparrow}-k_{F\downarrow}),(k_{F\uparrow}-k_{F\downarrow})]$
(scattering center).

Hence $\Phi_{1}(q_{j})$ is in Eq. (16) the overall phase shift in units of
$2\pi$ acquired by a $1$-particle of canonical momentum
${\bar{q}}_{j}={\bar{q}}(q_{j})$ under a transition from the ground state to
an excited energy eigenstate belonging to the present subspace.

Important quantities for the dynamical properties, are the following
$\iota=\pm 1$ deviations $\delta q_{F}^{\iota}$ from the values of the
$1$-band ground-state Fermi momenta, Eq. (8), under transitions to excited
energy eigenstates,

$\delta q_{F}^{\iota}=\iota\,{2\pi\over L}\delta N^{F}_{1,\iota}={2\pi\over
L}\left(\iota\,\delta N^{0,F}_{1,\iota}+\Phi_{1}^{0}\right)\,.$ (18)

Here $\delta N^{0,F}_{1,\iota}$ are the deviations in the number of
$1$-particles at the Fermi points which either vanish or are positive or
negative integer numbers. On the other hand, the actual number deviations,
$\delta N^{F}_{1,\iota}=\delta N^{0,F}_{1,\iota}+\iota\,\Phi_{1}^{0}$, can as
well be half-odd integers. Their extra term, $\iota\,\Phi_{1}^{0}$ where
$\iota=\pm 1$, stems from contributions from a non-scattering phase shift
$\Phi_{1}^{0}$ regulated by the boundary conditions in Eq. (87) of Appendix B.
It shifts all $1$-band’s discrete momentum values as, $q_{j}\rightarrow
q_{j}+(2\pi/L)\,\Phi_{1}^{0}$, and is given by,

$\displaystyle\Phi_{1}^{0}$ $\displaystyle=$ $\displaystyle 0\hskip
5.69046pt{\rm for}\hskip 5.69046pt{\rm for}\hskip 5.69046pt\delta N_{1}\hskip
5.69046pt{\rm even}$ (19) $\displaystyle=$ $\displaystyle\pm{1\over 2}\hskip
5.69046pt{\rm for}\hskip 5.69046pt\delta N_{1}\hskip 5.69046pt{\rm odd}\,.$

It follows from the form of Eqs. (15) and (16) that for the initial ground
state the equality ${\bar{q}}_{j}=q_{j}$ holds. Canonical momentum values
${\bar{q}}_{j}$ are different from the corresponding $1$-band momentum values
$q_{j}$ only for excited energy eigenstates. Hence the initial $\iota=\pm 1$
ground-state $1$-band Fermi momenta $q_{F}^{\iota}$ have exactly the same
values, Eq. (8), for the $1$-particle momentum $q_{j}$ and canonical momentum
${\bar{q}}_{j}$ representations.

The corresponding $\iota=\pm 1$ Fermi canonical momentum values deviations
$\delta{\bar{q}}_{F}^{\iota}$ fully control the momentum and spin density
dependence of the exponents $\zeta^{ab}_{\bar{n}}(k)$, Eq. (13), in the spin
dynamical structure factors’s general expressions, Eqs. (11) and (12). The
deviations $\delta{\bar{q}}_{F}^{\iota}$ are obtained from the excited state’s
$1$-band Fermi momentum values $q_{F}^{\iota}+\delta q_{F}^{\iota}$ under the
momentum - canonical-momentum transformation, Eq. (15), as follows,

$q_{F}^{\iota}+\delta q_{F}^{\iota}\rightarrow\bar{q}(q_{F}^{\iota}+\delta
q_{F}^{\iota})=q_{F}^{\iota}+\delta q_{F}^{\iota}+{2\pi\over
L}\Phi_{1}(q_{F}^{\iota}+\delta q_{F}^{\iota})=q_{F}^{\iota}+\delta
q_{F}^{\iota}+{2\pi\over L}\Phi_{1}(\iota k_{F\downarrow})+{\rm h.o.}\,.$ (20)

Neglecting contributions of order $(1/L)^{2}$ and accounting for the values of
$1$-band Fermi momenta being in the case of the initial ground state the same
for the momentum and canonical-momentum representations, this gives,

$\delta{\bar{q}}_{F}^{\iota}=\delta q_{F}^{\iota}+{2\pi\over L}\Phi_{1}(\iota
k_{F\downarrow})={2\pi\over L}\left(\iota\,\delta
N^{F}_{1,\iota}+\Phi_{1}(\iota k_{F\downarrow})\right)={2\pi\over
L}\left(\iota\,\delta N^{0,F}_{1,\iota}+\Phi_{1}^{0}+\Phi_{1}(\iota
k_{F\downarrow})\right)\,.$ (21)

Indeed, expanding ${2\pi\over L}\Phi_{1}(q_{F}^{\iota}+\delta q_{F}^{\iota})$
in both the ${\cal{O}}(1/L)$ corrections in the ground-state expression,
$q_{F}^{\iota}=\iota k_{F\downarrow}+{\cal{O}}(1/L)$, Eq. (8), and in the
deviation $\delta q_{F}^{\iota}$, leads to ${2\pi\over L}\Phi_{1}(\iota
k_{F\downarrow})$ plus contributions of order $(1/L)^{2}$. Those have no
physical meaning in the thermodynamic limit to which the validity of the
dynamical theory refers. Here $k_{F\downarrow}={\pi\over 2}(1-m)$, Eq. (88) of
Appendix B.

That there are no $n$-particles with $n>1$ in the ground state dictates why
only the $\iota=\pm 1$ Fermi canonical momentum values fluctuations play an
active role in the dynamical properties. Within the present extended dynamical
theory, such fluctuations though account for the creation of the $2$-particle
through the phase-shift $\Phi_{1,2}(q_{j},q_{j^{\prime}})$ in units of $2\pi$
appearing on the right-hand side of Eq. (16).

That the $\iota=\pm 1$ Fermi canonical momentum values fluctuations associated
with the deviations $\delta{\bar{q}}_{F}^{\iota}$ fully control the excitation
momentum $k$ and spin density dependence of the exponents, Eq. (13), follows
from the $\iota=\pm 1$ spectral functionals $\Phi_{\iota}$ in that equation
being such deviations $\delta{\bar{q}}_{F}^{\iota}$, Eq. (21), in units of the
quantum momentum spacing $2\pi/L$,

$\displaystyle\Phi_{\iota}={\delta{\bar{q}}_{F}^{\iota}\over(2\pi/L)}$
$\displaystyle=$ $\displaystyle\iota\,\delta N^{F}_{1,\iota}+\Phi(\iota
k_{F\downarrow})=\iota\,\delta N^{0,F}_{1,\iota}+\Phi_{1}^{0}+\Phi(\iota
k_{F\downarrow})$ (22) $\displaystyle=$ $\displaystyle\iota\,\delta
N^{F}_{1,\iota}+\sum_{j^{\prime}=1}^{N_{\uparrow}}\,\Phi_{1,1}(\iota
k_{F\downarrow},q_{j^{\prime}})\,\delta
N_{1}(q_{j^{\prime}})+\sum_{j^{\prime}=1}^{N_{\uparrow}-N_{\downarrow}+N_{2}}\,\Phi_{1,2}(\iota
k_{F\downarrow},q_{j^{\prime}})\,\delta N_{2}(q_{j^{\prime}})\,.$

The specific phase shifts in units of $2\pi$, $\Phi_{1,1}\left(\iota
k_{F\downarrow},q\right)$ and $\Phi_{1,2}\left(\iota k_{F\downarrow},q\right)$
where $\iota=\pm 1$, of the $1$-particles with momentum and canonical momentum
values at the $1$-band Fermi points that appear in the last expression of this
equation, are defined by Eq. (113) of Appendix B. Limiting behaviors of such
phase shifts are provided in Eqs. (114)-(117) of that Appendix.

In the case of the four types of branch lines considered in our study, some of
the deviations $\delta N_{1}(q_{j^{\prime}})$ and $\delta
N_{2}(q_{j^{\prime}})$ in Eq. (22) refer to $1$-band momenta
$q_{j^{\prime}}=\iota k_{F\downarrow}$ and $2$-band momenta
$q_{j^{\prime}}=\iota(k_{F\uparrow}-k_{F\downarrow})$ and $q_{j^{\prime}}=0$
where $\iota=\pm 1$. As a result, the expressions of the functionals
$\Phi_{\iota}$, Eq. (22), specific to the corresponding branch-line exponents,
Eq. (13), involve the phase-shifts related parameters $\xi_{1\,1}$ and
$\xi_{1\,2}^{0}$. Those are defined by Eqs. (118)-(122) and (123)-(124),
respectively, of Appendix B. Corresponding number and current number
deviations $\delta N_{1}^{F}$, $\delta J_{1}^{F}$, $\delta N_{2}$, and $\delta
J_{2}$, Eq. (9), then emerge in such expressions of the spectral functionals
$\Phi_{\iota}(q)$ that control the momentum and spin density dependences of
the branch-line exponents, Eq. (13).

We start by providing the three specific forms of the $\iota=\pm 1$ general
spectral functionals $\Phi_{\iota}$, Eq. (22), suitable to the $2$\- and
$2^{\prime}$-branch lines, $\bar{1}$-branch lines, and
$\bar{1}^{\prime}$-branch lines, respectively, that are part of the gapped
lower thresholds of the upper $n$-string spectra in Figs. 1-3. (The spectra of
such branch lines are given below in Sec. V.1.)

In the case of the $2$\- and $2^{\prime}$-branch lines, the form of the
$\iota=\pm 1$ spectral functionals $\Phi_{\iota}(q)$ is,

$\Phi_{\iota}(q)={\iota\,\delta N^{F}_{1}\over
2\xi_{1\,1}^{1}}+\xi_{1\,1}^{1}\,\delta J^{F}_{1}+\Phi_{1,2}(\iota
k_{F\downarrow},q)\hskip 5.69046pt{\rm for}\hskip 5.69046pts2-\hskip
5.69046pt{\rm and}\hskip 5.69046pts2^{\prime}{\rm-branch}\hskip 5.69046pt{\rm
lines}\,,$ (23)

where $\xi_{1\,1}=1+\sum_{\iota=\pm
1}(\iota)\,\Phi_{1,1}\left(k_{F\downarrow},\iota k_{F\downarrow}\right)$ (see
Eq. (118) of Appendix B.) For the excited energy eigenstates that contribute
to the singularities at and above the $2$\- and $2^{\prime}$-branch lines, the
maximum interval of the $2$-band momentum $q$ in Eq. (23) is
$q\in[0,(k_{F\uparrow}-k_{F\downarrow})[$ or
$q\in]-(k_{F\uparrow}-k_{F\downarrow}),0]$.

For the $\bar{1}^{\prime}$-branch lines, the form of the spectral functionals
is,

$\Phi_{\iota}(q)={\iota\,\delta N^{F}_{1}\over
2\xi_{1\,1}^{1}}+\xi_{1\,1}^{1}\,(\delta J^{F}_{1}-2\delta
J_{2})-\Phi_{1,1}(\iota k_{F\downarrow},q)\hskip 5.69046pt{\rm for}\hskip
5.69046pt\bar{1}^{\prime}{\rm-branch}\hskip 5.69046pt{\rm lines}\,,$ (24)

where it was accounted for that the phase shift $\Phi_{1,2}(\iota
k_{F\downarrow},\pm(k_{F\uparrow}-k_{F\downarrow}))$ can be written as
$\mp\xi_{1\,1}^{1}$ (see Eq. (119) of Appendix B.)

In the case of the $\bar{1}$-branch lines, the spectral functionals
$\Phi_{\iota}$, Eq. (22), have the following form,

$\Phi_{\iota}(q)={\iota\,\delta N^{F}_{1}\over
2\xi_{1\,1}^{1}}+{\iota\,\xi_{1\,2}^{0}\over 2}+\xi_{1\,1}^{1}\,\delta
J^{F}_{1}-\Phi_{1,1}(\iota k_{F\downarrow},q)\hskip 5.69046pt{\rm for}\hskip
5.69046pt\bar{1}{\rm-branch}\hskip 5.69046pt{\rm lines}\,,$ (25)

where $\xi_{1\,2}^{0}=\Phi_{1,2}(k_{F\downarrow},0)$ and it was accounted for
that $\Phi_{1,2}(\iota
k_{F\downarrow},0)=\iota\,\Phi_{1,2}(k_{F\downarrow},0)$ (see Eq. (123) of
Appendix B and text below it.) The maximum interval of the $1$-band momentum
is $q\in]-k_{F\downarrow},k_{F\downarrow}[$ in both Eqs. (24) and (25).

Finally, the expressions of the spectral functionals already considered in
Ref. Carmelo_15A  that control the momentum and spin density dependence of the
exponents associated with the $1$-branch lines are provided. Those refer to
parts of the lower thresholds of the lower continua in Figs. 1-3. The spectra
of such thresholds are given in Eqs. (67)-(70) of Appendix A. The
corresponding spectral functionals are of general form,

$\displaystyle\Phi_{\iota}(q)$ $\displaystyle=$ $\displaystyle{\iota\,\delta
N^{F}_{1}\over 2\xi_{1\,1}^{1}}+\xi_{1\,1}^{1}\,\delta
J^{F}_{1}\mp\Phi_{1,1}(\iota k_{F\downarrow},q)\hskip 5.69046pt{\rm
where}\hskip 5.69046pt{\rm for}\hskip 5.69046pt1{\rm-branch}\hskip
5.69046pt{\rm lines}$ $\displaystyle-$ $\displaystyle\rightarrow$
$\displaystyle\hskip 5.69046pt{\rm maximum}\hskip 5.69046pt{\rm
interval}\hskip 5.69046ptq\in]-k_{F\downarrow},k_{F\downarrow}[$
$\displaystyle+$ $\displaystyle\rightarrow$ $\displaystyle\hskip 5.69046pt{\rm
maximum}\hskip 5.69046pt{\rm intervals}\hskip
5.69046ptq\in[-k_{F\uparrow},-k_{F\downarrow}[\hskip 5.69046pt{\rm and}\hskip
5.69046ptq\in]k_{F\downarrow},k_{F\uparrow}]\,.$ (26)

Here $-$ and $+$ is the phase-shift sign in $\mp\Phi_{1,1}(\iota
k_{F\downarrow},q)$ suitable to $1$-branch lines involving $1$-hole and
$1$-particle creation, respectively, at a $1$-band momentum $q$ belonging to
the maximum intervals given in the equation.

The values of the $1$\- and $2$-bands number and current number deviations
that in the case of the transverse and longitudinal excited states are for
excitation momentum $k>0$ used in Eqs. (23)-(25) to reach the specific
expressions of the branch-line exponents given in Sec. V.1 are provided below
in Tables 1 and 2, respectively. Those of the $1$-band number and current
number deviations that are used in Eq. (26) to derive the expressions of the
branch-line exponents provided in Sec. V.2 are given below in Table 3.

###  III.3 Constrains to the momentum dependent exponents’s $k$ intervals

The exponents of general form, Eq. (13), that control the spin dynamical
structure factors’s expressions, Eqs. (11) and (12), at and above the $2$-,
$\bar{1}$-, $\bar{1}^{\prime}$-, $2^{\prime}$-branch lines and the $1$-branch
lines, respectively, depend on the excitation momentum $k\in]0,\pi[$ through
the $1$\- or $2$-band momentum values in the arguments of the spectral
functionals, Eqs. (23)-(26).

The $\bar{1}$\- and $\bar{1}^{\prime}$-branch lines’s exponents and those of
the $1$-branch lines involving $1$-hole creation are valid in $k$ ranges
corresponding to a maximum $1$-band momentum interval
$q\in[-(k_{F\downarrow}-\delta q_{1}),(k_{F\downarrow}-\delta q_{1})]$. On the
other hand, the exponents of the $1$-branch lines involving $1$-particle
creation are valid in $k$ ranges corresponding to maximum $1$-band momentum
intervals such that $|q|\in[(k_{F\downarrow}+\delta q_{1}),k_{F\uparrow}]$. In
both cases, $\delta q_{1}$, such that $\delta q_{1}/k_{F\uparrow}\ll 1$ for
$0<m<1$, is for the different branch lines very small or vanishes in the
thermodynamic limit.

In the very small $k$ intervals corresponding to the $1$-band intervals
$q\in[-(k_{F\downarrow}+\delta q_{1}),-(k_{F\downarrow}-\delta q_{1})]$ and
$q\in[(k_{F\downarrow}-\delta q_{1}),(k_{F\downarrow}+\delta q_{1})]$, the
exponents that control the line shape of the spin dynamical structure factors
have a different form, as given in Ref. Carmelo_15A . (See Eqs. (77)-(82) of
that reference.)

Similarly, the $2$\- and $2^{\prime}$-branch lines’s exponents that control
the line shape at and above some parts of the gapped lower thresholds refer to
$k$ ranges corresponding to $2$-band momentum maximum intervals
$q\in[-(k_{F\uparrow}-k_{F\downarrow}-\delta q_{2}),0]$ or
$q\in[0,(k_{F\uparrow}-k_{F\downarrow}-\delta q_{2})]$. Here $\delta q_{2}$,
such that $\delta q_{2}/(k_{F\uparrow}-k_{F\downarrow})\ll 1$ for $0<m<1$, is
for the different branch lines again very small or vanishes in the
thermodynamic limit. (And again, the spin dynamical structure factors
expressions are different and known for
$q\in[-(k_{F\uparrow}-k_{F\downarrow}),-(k_{F\uparrow}-k_{F\downarrow}-\delta
q_{2})]$ and $q\in[(k_{F\uparrow}-k_{F\downarrow}-\delta
q_{2}),(k_{F\uparrow}-k_{F\downarrow})]$ yet are not of interest for our
study.)

In the present thermodynamic limit, the maximum $1$-band $q$ intervals
corresponding to $k$ intervals for which the exponents, Eq. (13), are valid,
are represented in the following as $q\in]-k_{F\downarrow},k_{F\downarrow}[$
and $|q|\in]k_{F\downarrow},k_{F\uparrow}]$. Similarly, the $2$-band $q$
intervals corresponding to such $k$ intervals are represented as
$q\in]-(k_{F\uparrow}-k_{F\downarrow}),0]$ and
$q\in[0,(k_{F\uparrow}-k_{F\downarrow})[$.

Around the specific excitation momentum $k$ values where along a gapped lower
threshold or a lower threshold two neighboring branch lines or branch line
sections cross, there are small momentum widths in which the corresponding
lower threshold refers to a boundary line that connects the two branch lines
or branch line sections under consideration.

In the thermodynamic limit, such momentum intervals are in general negligible
and the corresponding small spectra’s deviations are not visible in the
spectra plotted in Figs. 1-3. In the cases they are small yet more extended,
the two branch lines or branch line sections run very near and just above the
(gapped or gapless) lower threshold and there is very little spectral weight
between it and such lines. In this case, the singularities on the two branch
lines or branch line sections remain the dominant spectral feature.

We again account for such negligible effects by replacing parenthesis $[$ or
$]$ by $]$ or $[$, respectively, at the $k$’s limiting values that separate
(gapped or gapless) lower thresholds’s $k$ intervals associated with two
neighboring branch lines or branch line sections.

##  IV Dynamical structure factors’s spectra

Here the spectra associated with the $(k,\omega)$-plane regions that contain
most spectral weight of the spin dynamical structure factors are introduced,
with emphasis in those associated with $n$-string states. The
$(k,\omega)$-plane distribution of such spectra is represented for
$S^{+-}(k,\omega)$, $S^{xx}(k,\omega)$, and $S^{zz}(k,\omega)$ in Figs. 1, 2,
and 3, respectively, for spin densities (a) $m=0.15$, (b) $m=0.25$, (c)
$m=0.50$, and (d) $m=0.75$. Note that the sketch of the $(k,\omega)$-plane
distributions represented in these figures does not provide information on the
relative amount of spectral weight contained within each spectrum’s grey
continuum.

In the case of the spin dynamical structure factors $S^{+-}(k,\omega)$ and
$S^{xx}(k,\omega)$, the figures show both a lower continuum $(k,\omega)$-plane
region, whose spectral weight is associated with excited states without
$n$-strings, and an upper continuum whose spectral weight stems from excited
states populated by such $n$-strings. In the case of $S^{zz}(k,\omega)$, the
contribution to spectral weight from excited states containing $n$-strings is
much weaker than for $S^{+-}(k,\omega)$ and $S^{xx}(k,\omega)$ and does not
lead to an upper continuum. The gapped lower threshold of such states’s
spectrum is represented in Fig. 3 by a well defined $(k,\omega)$-plane line.

At finite magnetic fields $0<h<h_{c}$ the contribution to the spectral weight
from excited states containing $n$-strings is negligible in the case of
$S^{-+}(k,\omega)$. Therefore, its lower continuum spectrum is not plotted
here. (It was previously studied in Ref. Carmelo_15A .) The additional part of
the lower continuum in Fig. 2 for $S^{xx}(k,\omega)={1\over
4}\left(S^{+-}(k,\omega)+S^{-+}(k,\omega)\right)$ relative to that of
$S^{+-}(k,\omega)$ represented in Fig. 1 stems actually from contributions
from $S^{-+}(k,\omega)$. As a result of such contributions, for small spin
densities and some $k$ intervals the upper $n$-string continuum of
$S^{xx}(k,\omega)$ overlaps with its lower continuum.

In the case of both $S^{+-}(k,\omega)$ and $S^{zz}(k,\omega)$, there is in the
present thermodynamic limit for spin densities $0<m<1$ and thus finite
magnetic fields $0<h<h_{c}$ very little spectral weight between the upper
threshold of the lower continuum associated with excited states described only
by real Berthe-ansatz rapidities and the gapped lower threshold of the
$n$-string states’s spectra in Figs. 1 and 3, respectively. The same applies
to $S^{xx}(k,\omega)$ in the $k$ intervals of Fig. 2 for which there is a gap
between the upper continuum associated with $n$-string states and the lower
continuum.

Indeed, in the thermodynamic limit nearly all the small amount of spectral
weight associated with excited energy eigenstates described by only real
Bethe-ansatz rapidities that involve the emergence of four $1$-band holes
under transitions from the ground state (named in the literature four-spinon
states), is contained inside the lower continuum in such figures. This also
applies to large finite systems at zero magnetic field, as shown in Fig. 4 of
Ref. Caux_06 . In the present case of a finite magnetic field, it also applies
to the spin-$1/2$ Heisenberg antiferromagnetic with anisotropy $\Delta<1$, see
Fig. 1 of Ref. Caux_05 .

In the case of $S^{+-}(k,\omega)$ at finite magnetic fields $0<h<h_{c}$, due
to the behavior of spin operators matrix elements between energy eigenstates
in the selection rules, Eq. (6), the spectral weight stemming from $n$-string-
less states existing in finite systems for $k\in[2k_{F\downarrow},\pi]$ and
$\omega$ values above the upper threshold of the lower spectrum in Fig. 1, is
negligible for a macroscopic system. Such a spectral weight decreases upon
increasing the system size, as confirmed by analysis of the spectra in the
$S^{+-}(k,\omega)$’s first row frames of Figs. 3 (a) and (b) of Ref. Kohno_09
for two finite-size systems with $N=320$ and $N=2240$ spins, respectively.

The further discussion of this important issue is presented below in the final
section of this paper, Sec. VII.

###  IV.1 The spin densities $\tilde{m}$, $\bar{m}$, and $\bar{m}_{0}$ and
related momentum values

The spectra of the gapped lower thresholds of $S^{+-}(k,\omega)$,
$S^{xx}(k,\omega)$, and $S^{zz}(k,\omega)$ studied below have a different form
for two spin density intervals, $m\in]0,\tilde{m}]$ and $m\in[\tilde{m},1[$,
respectively. Here $\tilde{m}\approx 0.317$ is the spin density at which the
following equality holds,

$W_{2}|_{m=\tilde{m}\approx
0.317}=-\varepsilon_{1}(2k_{F\downarrow}-k_{F\uparrow})|_{m=\tilde{m}\approx
0.317}\,.$ (27)

From the use of the value of $2$-band energy dispersion $\varepsilon_{2}(q)$
at $q=0$ given in Eq. (103) of Appendix B, the $2$-band energy bandwidth
$W_{2}$ appearing here can be expressed as
$W_{2}=4\mu_{B}h-\varepsilon_{2}(0)$.

Some specific excitation momentum $k$’s values separate momentum intervals of
the gapped lower threshold spectra of $S^{+-}(k,\omega)$, $S^{xx}(k,\omega)$,
and $S^{zz}(k,\omega)$ that refer to different types of momentum dependences.
Such specific $k$ values either equal a momentum denoted here by $\tilde{k}$
or their expression involves $\tilde{k}$. The latter momentum is defined by
the following relations,

$\displaystyle W_{2}$ $\displaystyle=$
$\displaystyle\varepsilon_{1}(k_{F\uparrow}-\tilde{k})-\varepsilon_{1}(k_{F\downarrow}-\tilde{k})\hskip
5.69046pt{\rm for}\hskip
5.69046pt\tilde{k}\geq(k_{F\uparrow}-k_{F\downarrow})\hskip 5.69046pt{\rm
and}\hskip 5.69046ptm\in]0,\tilde{m}]$ $\displaystyle W_{2}$ $\displaystyle=$
$\displaystyle
4\mu_{B}\,h-\varepsilon_{2}(\tilde{k})-\varepsilon_{1}(k_{F\downarrow}-\tilde{k})\hskip
5.69046pt{\rm for}\hskip
5.69046pt\tilde{k}\leq(k_{F\uparrow}-k_{F\downarrow})\hskip 5.69046pt{\rm
and}\hskip 5.69046ptm\in[\tilde{m},1[\,.$ (28)

The momentum $\tilde{k}$ is given by
$\tilde{k}=(k_{F\uparrow}-k_{F\downarrow})$ at $m=\tilde{m}$.

In the relations provided in Eqs. (27) and (28), as well as in the expressions
of the spin dynamical structure factors’s spectra given below and in Appendix
A, the quantities $\varepsilon_{1}(q)$ and $\varepsilon_{2}(q)$ are the $1$\-
and $2$-band energy dispersions, respectively, defined by Eqs. (89), (90), and
(92)-(98) of Appendix B. Limiting behaviors of such dispersions and
corresponding $1$\- and $2$-band group velocities that provide useful
information on the momentum and spin density dependences of the corresponding
spin dynamical structure factors’s spectra studied below are given in Eqs.
(103)-(109) of that Appendix.

The $(k,\omega)$-plane spectrum that contains most of $S^{xx}(k,\omega)$’s
spectral weight has features whose form is different for specific spin
densities and momentum intervals. As shown in Fig. 2 (a) for $m=0.15$ and (b)
$m=0.25$, for small spin densities there are $k$ intervals for which the
$S^{xx}(k,\omega)$’s gapped lower threshold of the $2$-string continuum
overlaps with the lower continuum associated with energy eigenstates described
by real Bethe-ansatz rapidities. Such $k$ intervals are given by,

$\displaystyle k$ $\displaystyle\in$ $\displaystyle[\bar{k}_{0},\pi]\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,\bar{m}_{0}]$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle[\bar{k}_{0},\bar{k}_{1}]\hskip 5.69046pt{\rm
for}\hskip 5.69046ptm\in]\bar{m}_{0},\bar{m}]\,,$ (29)

where the spin densities $\bar{m}_{0}$ and $\bar{m}$ are given by
$\bar{m}_{0}\approx 0.239$ and $\bar{m}\approx 0.276$, respectively.

The momenta $\bar{k}_{0}$ and $\bar{k}_{1}$ appearing in Eq. (29) are such
$k^{xx}_{ut}\leq\bar{k}_{0}\leq\bar{k}_{1}$ and
$\bar{k}_{0}\leq\bar{k}_{1}\leq\pi$. The equality, $\bar{k}_{0}=\bar{k}_{1}$,
holds at $m=\bar{m}$. At that spin density, $\bar{k}_{0}=\bar{k}_{1}$ has a
value very near and just above $2k_{F\downarrow}$. The spectra plotted in Fig.
2 (a) for $m=0.15$ and (b) for $m=0.25$ refer to the two types of $k$
intervals reported in Eq. (29), respectively. For (a), spin density $m=0.15$,
one has that $\bar{k}_{0}\approx 0.60\,\pi$, whereas for (b), spin density
$m=0.25$, the two limiting momenta read $\bar{k}_{0}\approx 0.71\,\pi$ and
$\bar{k}_{1}\approx 0.92\,\pi$.

###  IV.2 Dynamical structure factors’s spectra

The information on the constrains to the $k$ intervals provided in Sec. III.3
applies to the branch-line spectra studied below in Sec. V. On the other hand,
in the particular case of the small $k$ intervals corresponding to the small
$1$-band intervals $q\in[-(k_{F\downarrow}+\delta
q_{1}),-(k_{F\downarrow}-\delta q_{1})]$ and $q\in[(k_{F\downarrow}-\delta
q_{1}),(k_{F\downarrow}+\delta q_{1})]$ and $2$-band intervals
$q\in[-(k_{F\uparrow}-k_{F\downarrow}),-(k_{F\uparrow}-k_{F\downarrow}-\delta
q_{2})]$ and $q\in[(k_{F\uparrow}-k_{F\downarrow}-\delta
q_{2}),(k_{F\uparrow}-k_{F\downarrow})]$, such constrains do not apply to the
two-parametric spectra of the $(k,\omega)$-plane regions where there is more
spectral weight. They also do not apply to the corresponding one-parametric
spectra of the gapped lower thresholds and lower thresholds themselves. Only
the general line-shape expressions, Eqs. (11) and (12), are not valid in such
small $k$ intervals.

The remaining constrains mentioned in Sec. III.3 associated with small $k$
intervals of the (gapped and gapless) lower thresholds in the vicinity of
crossing points of two branch lines or branch line sections though apply to
the expressions of the one-parametric spectra of the gapped lower thresholds
and lower thresholds given in the following and in Appendix A. This applies,
for instance, to the small excitation momentum intervals in the vicinity of
$k=\tilde{k}$ (in the transverse case) and other $k$ values whose expression
involves the momentum $\tilde{k}$ (in the logitudinal case) defined by the
relations given in Eq. (28)

The $(k,\omega)$-plane upper continuum shown in Fig. 1 is associated with the
gapped upper spectrum of $S^{+-}(k,\omega)$ that stems from transitions from
the ground state to excited energy eigenstates populated by a number
$N_{s}=N_{\downarrow}-2$ of $1$-particles and a single $2$-particle. It is
given by,

$\displaystyle\omega^{+-}_{\Delta}(k)$ $\displaystyle=$
$\displaystyle-\varepsilon_{1}(q)+\varepsilon_{2}(q^{\prime})\hskip
5.69046pt{\rm and}\hskip 5.69046ptk=\iota k_{F\downarrow}-q+q^{\prime}\hskip
5.69046pt{\rm for}\hskip 5.69046pt\iota=\pm 1\hskip 5.69046pt{\rm where}$ (30)
$\displaystyle q\in[-k_{F\downarrow},k_{F\downarrow}]\,,\hskip 5.69046pt\hskip
5.69046ptq^{\prime}\in[0,(k_{F\uparrow}-k_{F\downarrow})]\hskip 5.69046pt{\rm
for}\hskip 5.69046pt\iota=1\hskip 5.69046pt{\rm and}$ $\displaystyle
q\in[-k_{F\downarrow},k_{F\downarrow}]\,,\hskip 5.69046pt\hskip
5.69046ptq^{\prime}\in[-(k_{F\uparrow}-k_{F\downarrow}),0]\hskip 5.69046pt{\rm
for}\hskip 5.69046pt\iota=-1\,.$

This spectrum has two branches corresponding to $\iota=\pm 1$, such that,

$k=k_{F\downarrow}-q+q^{\prime}\in[0,\pi]\hskip 14.22636pt{\rm and}\hskip
14.22636ptk=-k_{F\downarrow}-q+q^{\prime}\in[-\pi,0]\,.$ (31)

As mentioned above, $\varepsilon_{1}(q)$ and $\varepsilon_{2}(q)$ are here the
$1$\- and $2$-band energy dispersions, respectively, defined by Eqs. (89),
(90), and (92)-(98) of Appendix B.

We denote by $\Delta^{ab}(k)$ where $ab=+-,xx,zz$ the spectra of the
$n$-string excited states’s gapped lower thresholds of the spin dynamical
structure factors $S^{ab}(k,\omega)$. These gapped lower thresholds play an
important role in our study. For some of their $k$ intervals there are
singularities of power-law form, Eq. (11), in the spin dynamical structure
factors at and just above them. (In that equation, the extra lower index
$\bar{n}$ in $\Delta^{ab}(k)$ refers to the different branch lines or branch
line sections that coincide with the gapped lower threshold for well-defined
$k$ intervals given below in Sec. V.1.)

The spectra of the transverse gapped lower thresholds are such that,

$\Delta^{xx}(k)=\Delta^{+-}(k)\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[0,\pi]\,.$ (32)

(The equality $\Delta^{-+}(k)=\Delta^{+-}(k)$ also holds, yet, as reported
previously, the amount of $S^{-+}(k,\omega)$’s spectral weight produced by
$n$-string states is negligible in the thermodynamic limit at finite magnetic
fields.) The spectrum of the longitudinal gapped lower threshold is also
related to $\Delta^{+-}(k)$ as follows,

$\Delta^{zz}(k)=\Delta^{+-}(\pi-k)\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[0,\pi]\,.$ (33)

For smaller spin densities in the range $m\in]0,\tilde{m}]$, the spectrum of
the lower threshold is of the form,

$\displaystyle\Delta^{+-}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{2}(k)\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[0,(k_{F\uparrow}-k_{F\downarrow})]$ (34) $\displaystyle=$
$\displaystyle 4\mu_{B}\,h-\varepsilon_{1}(k_{F\uparrow}-k)\hskip
5.69046pt{\rm for}\hskip
5.69046ptk\in[(k_{F\uparrow}-k_{F\downarrow}),{\tilde{k}}[$ $\displaystyle=$
$\displaystyle 4\mu_{B}\,h-W_{2}-\varepsilon_{1}(k_{F\downarrow}-k)\hskip
5.69046pt{\rm for}\hskip 5.69046ptk\in]{\tilde{k}},2k_{F\downarrow}]$
$\displaystyle=$ $\displaystyle\varepsilon_{2}(k-2k_{F\downarrow})\hskip
5.69046pt{\rm for}\hskip 5.69046ptk\in[2k_{F\downarrow},\pi]\,.$

For larger spin densities $m\in[\tilde{m},1[$, the spectrum of the lower
threshold is slightly different and reads,

$\displaystyle\Delta^{+-}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{2}(k)\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[0,{\tilde{k}}[$ (35) $\displaystyle=$ $\displaystyle
4\mu_{B}\,h-W_{2}-\varepsilon_{1}(k_{F\downarrow}-k)\hskip 5.69046pt{\rm
for}\hskip 5.69046ptk\in]{\tilde{k}},2k_{F\downarrow}]$ $\displaystyle=$
$\displaystyle\varepsilon_{2}(k-2k_{F\downarrow})\hskip 5.69046pt{\rm
for}\hskip 5.69046ptk\in[2k_{F\downarrow},\pi]\,.$

The expressions of two-parametric transverse gapless spectra $\omega^{-+}(k)$
and $\omega^{+-}(k)$ previously studied in Ref. Carmelo_15A , whose
superposition gives $\omega^{xx}(k)$, and that of the longitudinal gapless
spectrum $\omega^{zz}(k)$ that (except for $\omega^{-+}(k)$) refer to the
lower continua in Figs. 1-3, are given in Eqs. (58)-(60) of Appendix A. The
corresponding excited energy eigenstates are described by only real Bethe-
ansatz rapidities. The related expressions of the one-parametric spectra of
their upper thresholds $\omega^{-+}_{ut}(k)$, $\omega^{+-}_{ut}(k)$,
$\omega^{xx}_{ut}(k)$, and $\omega^{zz}_{ut}(k)$ and lower thresholds
$\omega^{-+}_{lt}(k)$, $\omega^{+-}_{lt}(k)$, $\omega^{xx}_{lt}(k)$, and
$\omega^{zz}_{lt}(k)$ are also provided in that Appendix.

We consider the following energy gaps that refer to $(k,\omega)$-plane regions
with a negligible amount of spectral weight in the thermodynamic limit,

$\displaystyle\Delta_{\rm gap}^{+-}(k)$ $\displaystyle=$
$\displaystyle\Delta^{+-}(k)-\omega^{+-}_{ut}(k)\geq 0$
$\displaystyle\Delta_{\rm gap}^{xx}(k)$ $\displaystyle=$
$\displaystyle\Delta^{xx}(k)-\omega^{xx}_{ut}(k)$ $\displaystyle\Delta_{\rm
gap}^{zz}(k)$ $\displaystyle=$
$\displaystyle\Delta^{zz}(k)-\omega^{zz}_{ut}(k)\geq 0\,,$ (36)

where,

$\displaystyle\Delta_{\rm gap}^{xx}(k)$ $\displaystyle=$
$\displaystyle\Delta^{+-}(k)-\omega^{+-}_{ut}(k)\hskip 5.69046pt{\rm
for}\hskip 5.69046ptk\in[0,k^{xx}_{ut}]$ $\displaystyle\Delta_{\rm
gap}^{xx}(k)$ $\displaystyle=$
$\displaystyle\Delta^{+-}(k)-\omega^{-+}_{ut}(k)\hskip 5.69046pt{\rm
for}\hskip 5.69046ptk\in[k^{xx}_{ut},\pi]\,,$ (37)

and

$\Delta_{\rm gap}^{zz}(k)=\Delta_{\rm gap}^{+-}(\pi-k)\hskip 5.69046pt{\rm
for}\hskip 5.69046ptk\in[0,\pi]\,.$ (38)

The momentum $k^{xx}_{ut}>k_{F\uparrow}-k_{F\downarrow}$ in Eq. (37) is that
at which the equality,
$\omega^{-+}_{ut}(k^{xx}_{ut})=\omega^{+-}_{ut}(k^{xx}_{ut})$, holds. As
confirmed from analysis of Figs. 1-3, one has that $\Delta_{\rm
gap}^{+-}(k)\geq 0$ and $\Delta_{\rm gap}^{zz}(k)\geq 0$ for $k\in]0,\pi[$,
whereas $\Delta_{\rm gap}^{xx}(k)$ is negative for small spin densities
$m\in]0,\bar{m}]$, where $\bar{m}\approx 0.276$, in the $k$ intervals given in
Eq. (29).

It follows from the $k$ intervals given in that equation that at $k=\pi$ the
energy gap $\Delta_{\rm gap}^{xx}(k)$ has negative and positive values for
spin densities $m<\bar{m}_{0}$ and $m>\bar{m}_{0}$, respectively, where
$\bar{m}_{0}\approx 0.239$. Its corresponding negative, vanishing, and
positive limiting values are provided in Eq. (81) of Appendix A.

The upper threshold spectra $\omega^{-+}_{ut}(k)$, $\omega^{+-}_{ut}(k)$,
$\omega^{xx}_{ut}(k)$, $\omega^{zz}_{ut}(k)$ in Eqs. (36)-(38) are given in
Eqs. (61)-(64) of Appendix A. The upper thresholds of the lower continua in
Figs. 1, 2, and 3 refer to the spectra $\omega^{+-}_{ut}(k)$,
$\omega^{xx}_{ut}(k)$, and $\omega^{zz}_{ut}(k)$, respectively.

In Appendix A, the $k$ dependent expressions and the limiting values of the
energy gaps considered here are provided.

##  V The line shape near the singularities of the dynamical structure factors

The power-law singularities, Eq. (11), in the spin dynamical structure factors
$S^{+-}(k,\omega)$, $S^{xx}(k,\omega)$, and $S^{zz}(k,\omega)$’s expressions
occur in the $k$ intervals for which the corresponding momentum dependent
exponents are negative. This applies to $(k,\omega)$-plane regions at and just
above the $\bar{n}$-branch lines that are part of the corresponding gapped
lower thresholds of the $n$-string states’s spectra. As discussed in Sec.
III.1, the main contributions to the line shape in these $(k,\omega)$-plane
regions stems from transitions to excited states populated by a number
$N_{\downarrow}-2$ of $1$-particles and a single $2$-particle. The momentum
interval, $k\in]0,\pi[$, of the gapped lower thresholds under consideration is
divided into several subintervals, which correspond to a set of $2$-,
$\bar{1}^{\prime}$-, $\bar{1}$-, and $2^{\prime}$-branch lines, respectively.
(In the case of $S^{zz}(k,\omega)$, the order of the $\bar{1}^{\prime}$\- and
$\bar{1}$-branch lines’s $k$ subintervals is the reverse, $\bar{1}$\- and
$\bar{1}^{\prime}$-.)

branch line |  $k$ in terms of $q$ | $\delta N_{1}^{F}$ | $\delta J_{1}^{F}$ | $\delta N_{1}^{NF}$ | $\delta J_{2}$ | $\delta N_{2}$
---|---|---|---|---|---|---
$2$ |  $k=q$ (for $2$-band $q$) | $-1$ | $0$ | $0$ | $0$ | $1$
$\bar{1}^{\prime}$ |  $k=k_{F\uparrow}-q$ (for $1$-band $q$) | $0$ | $1/2$ | $-1$ | $1/2$ | $1$
$\bar{1}$ |  $k=k_{F\downarrow}-q$ (for $1$-band $q$) | $0$ | $1/2$ | $-1$ | $0$ | $1$
$2^{\prime}$ |  $k=2k_{F\downarrow}+q$ (for $2$-band $q$) | $-1$ | $1$ | $0$ | $0$ | $1$
Table 1: The excitation momenta $k>0$ of $S^{+-}(k,\omega)$ and
$S^{xx}(k,\omega)$ expressed in terms of $1$\- and $2$-bands momenta $q$ and
corresponding number and current number deviations for excited states
populated by a number $N_{\downarrow}-2$ of $1$-particles and a single
$2$-particle and thus described by both real and complex non-real rapidities
in the case of the $2$-branch line, $\bar{1}^{\prime}$-branch line,
$\bar{1}$-branch line, and $2^{\prime}$-branch line. For the momentum
intervals given in Eqs. (39)-(42), such branch lines are part of the
corresponding gapped lower thresholds. (In the case of $S^{xx}(k,\omega)$, the
data given here do not apply to branch-line intervals overlapping with those
given in Eq. (29.)

On the other hand, the lower thresholds of the gapless spectra associated with
the lower continua in Figs. 1-3 that were already studied in Ref. Carmelo_15A
, which stem from transitions to excited states that are populated by a number
$N_{\downarrow}$ of $1$-particles, either correspond to a single $1$-branch
line or to two sections of such a line.

Following the information provided in Sec. III.3, the constrains to the $k$
intervals where the expressions of the exponents in the general line-shape
expressions, Eqs. (11) and (12), are valid imply replacements of parenthesis
$[$ and/or $]$ by $]$ and/or $[$, respectively, in the limits of such
intervals.

###  V.1 The line shape at and just above the gapped lower threshold of
$S^{+-}(k,\omega)$, $S^{xx}(k,\omega)$, and $S^{zz}(k,\omega)$

In the case of $S^{xx}(k,\omega)$, the present study of the line shape at and
just above the gapped lower thresholds of the spectra plotted in Fig. 2 refers
to $k$ intervals for which the corresponding energy gap $\Delta_{\rm
gap}^{xx}(k)$ obeys the inequality $\Delta_{\rm gap}^{xx}(k)>0$. At small spin
densities, this excludes the $k$ intervals given in Eq. (29).

Otherwise, the line shape near the gapped lower thresholds of
$S^{+-}(k,\omega)$, $S^{xx}(k,\omega)$, and $S^{zz}(k,\omega)$ is given by Eq.
(11). The exponents in the expression given in that equation have a general
form provided in Eq. (13). The $\iota=\pm 1$ spectral functionals
$\Phi_{\iota}(q)$ in that exponent’s expression are in the case of the (i)
$2$\- and $2^{\prime}$-branch lines, (ii) $\bar{1}^{\prime}$-branch lines, and
(iii) $\bar{1}$-branch lines given in Eqs. (23), (24), and (25), respectively.

The relation of the excitation momentum $k$ to the $1$-band momentum $q$ or
$2$-band momentum $q$ in the $\Phi_{\iota}(q)$’s argument is branch-line
dependent. Hence it is useful to revisit the expressions of the spectra of the
gapped lower thresholds, Eqs. (32)-(35) and (33), for each of their branch
lines or branch line sections, including information on the relation between
$k$ and the $1$\- or $2$-band momenta $q$. Additional $1$-band and $2$-band
constrains reported in Sec. III.3 associated with the validity of the
corresponding expressions of the momentum dependent exponents are also
accounted for. This ensures that the branch-line spectra’s expressions are
given in the following for the $k$ intervals for which the dynamical structure
factor’s expression is of the form, Eq. (11).

Figure 4: The momentum dependence of the exponents that in the $k$ intervals
for which they are negative control the $S^{+-}(k,\omega)$ line shape near and
just above the (a) $\bar{n}=2$, (b) $\bar{n}=\bar{1}^{\prime}$, (c)
$\bar{n}=\bar{1}$, and (d) $\bar{n}=2^{\prime}$ branch lines for several spin
densities. The present branch lines are part of the gapped lower threshold of
the $n$-strings continuum displayed in Fig. 1. The same exponents, in the $k$
intervals for which they are negative, also control the $S^{xx}(k,\omega)$
line shape near and just above the corresponding branch lines in the
$n$-strings continuum displayed in Fig. 2. In the case of the
$\bar{n}=\bar{1}$ and $\bar{n}=2^{\prime}$ branch lines, this only applies to
the $k$ intervals for which there is a gap between them and the upper
threshold of the lower continuum (See Fig. 2).

In the case of $S^{+-}(k,\omega)$, the gapped lower threshold spectrum
$\Delta^{+-}(k)$ is divided into the following branch-line intervals,

$\displaystyle\Delta_{2}^{+-}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{2}(k)\hskip 5.69046pt{\rm and}\hskip
5.69046ptk=q\hskip 5.69046pt{\rm where}$ $\displaystyle k$ $\displaystyle\in$
$\displaystyle[0,(k_{F\uparrow}-k_{F\downarrow})[\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq\in[0,(k_{F\uparrow}-k_{F\downarrow})[\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,\tilde{m}]$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle[0,{\tilde{k}}[\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq\in[0,{\tilde{k}}[\hskip 5.69046pt\hskip 5.69046pt{\rm
for}\hskip 5.69046ptm\in[\tilde{m},1[\,,$ (39)
$\displaystyle\Delta_{\bar{1}^{\prime}}^{+-}(k)$ $\displaystyle=$
$\displaystyle 4\mu_{B}\,h-\varepsilon_{1}(k_{F\uparrow}-k)\hskip
5.69046pt{\rm and}\hskip 5.69046ptk=k_{F\uparrow}-q\hskip 5.69046pt{\rm
where}$ $\displaystyle k$ $\displaystyle\in$
$\displaystyle](k_{F\uparrow}-k_{F\downarrow}),\tilde{k}[\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq\in](k_{F\uparrow}-\tilde{k}),k_{F\downarrow}[\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,\tilde{m}]\,,$ (40)
$\displaystyle\Delta_{\bar{1}}^{+-}(k)$ $\displaystyle=$ $\displaystyle
4\mu_{B}\,h-W_{2}-\varepsilon_{1}(k_{F\downarrow}-k)\hskip 5.69046pt{\rm
and}\hskip 5.69046ptk=k_{F\downarrow}-q\hskip 5.69046pt{\rm where}$
$\displaystyle k$ $\displaystyle\in$
$\displaystyle\in]{\tilde{k}},2k_{F\downarrow}[\hskip 5.69046pt{\rm and}\hskip
5.69046ptq\in]-k_{F\downarrow},(k_{F\downarrow}-{\tilde{k}})[\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,\tilde{m}]$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle]{\tilde{k}},2k_{F\downarrow}[\hskip
5.69046pt{\rm and}\hskip
5.69046ptq\in]-k_{F\downarrow},(k_{F\downarrow}-{\tilde{k}})[\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in[\tilde{m},1[\,,$ (41)

and

$\displaystyle\Delta_{2^{\prime}}^{+-}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{2}(k-2k_{F\downarrow})\hskip 5.69046pt{\rm
and}\hskip 5.69046ptk=2k_{F\downarrow}+q\hskip 5.69046pt{\rm where}$
$\displaystyle k$ $\displaystyle\in$
$\displaystyle]2k_{F\downarrow},\pi[\hskip 5.69046pt{\rm and}\hskip
5.69046ptq\in]0,(k_{F\uparrow}-k_{F\downarrow})[\hskip 5.69046pt{\rm
for}\hskip 5.69046ptm\in]0,1[\,.$ (42)

branch line |  $k$ in terms of $q$ | $\delta N_{1}^{F}$ | $\delta J_{1}^{F}$ | $\delta N_{1}^{NF}$ | $\delta J_{2}$ | $\delta N_{2}$
---|---|---|---|---|---|---
$2$ |  $k=(k_{F\uparrow}-k_{F\downarrow})+q$ (for $2$-band $q$) | $-2$ | $-1$ | $0$ | $0$ | $1$
$\bar{1}$ |  $k=k_{F\uparrow}-q$ (for $1$-band $q$) | $-2$ | $-1/2$ | $-1$ | $0$ | $1$
$\bar{1}^{\prime}$ |  $k=k_{F\uparrow}-q$ (for $1$-band $q$) | $1$ | $-1/2$ | $-1$ | $-1/2$ | $1$
$2^{\prime}$ |  $k=\pi+q$ (for $2$-band $q$) | $-2$ | $0$ | $0$ | $0$ | $1$
Table 2: The same information as in Table 1 for the $2$-branch line,
$\bar{1}$-branch line, $\bar{1}^{\prime}$-branch line, and $2^{\prime}$-branch
line that for the momentum intervals given in Eqs. (44)-(47) are part of the
gapped lower threshold of $S^{zz}(k,\omega)$.

The corresponding $k$ dependent exponents of general form, Eq. (13), that
appear in the expression,
$S^{+-}(k,\omega)=C_{+-}^{\Delta}(\omega-\Delta_{\bar{n}}^{+-}(k))^{\zeta_{\bar{n}}^{+-}(k)}$,
Eq. (11) for $ab=+-$ and $\beta=2,\bar{1}^{\prime},\bar{1},2^{\prime}$, are
given by,

$\displaystyle\zeta_{2}^{+-}(k)$ $\displaystyle=$
$\displaystyle-1+\sum_{\iota=\pm 1}\left(-{\iota\over
2\xi_{1\,1}}+\Phi_{1,2}(\iota k_{F\downarrow},q)\right)^{2}\hskip
5.69046pt{\rm for}\hskip 5.69046ptq=k\hskip 5.69046pt{\rm where}$
$\displaystyle k$ $\displaystyle\in$
$\displaystyle]0,(k_{F\uparrow}-k_{F\downarrow})[\hskip 5.69046pt{\rm
for}\hskip 5.69046ptm\in]0,\tilde{m}]\hskip 5.69046pt{\rm and}\hskip
5.69046ptk\in]0,{\tilde{k}}[\hskip 5.69046pt{\rm for}\hskip
5.69046ptm\in[\tilde{m},1[$ $\displaystyle\zeta_{\bar{1}^{\prime}}^{+-}(k)$
$\displaystyle=$ $\displaystyle-1+\sum_{\iota=\pm 1}\left(-{\xi_{1\,1}\over
2}-\Phi_{1,1}(\iota k_{F\downarrow},q)\right)^{2}\hskip 5.69046pt{\rm
for}\hskip 5.69046ptq=k_{F\uparrow}-k\hskip 5.69046pt{\rm where}$
$\displaystyle k$ $\displaystyle\in$
$\displaystyle](k_{F\uparrow}-k_{F\downarrow}),\tilde{k}[\hskip 5.69046pt{\rm
for}\hskip 5.69046ptm\in]0,\tilde{m}]$ $\displaystyle\zeta_{\bar{1}}^{+-}(k)$
$\displaystyle=$ $\displaystyle-1+\sum_{\iota=\pm
1}\left(\iota{\xi_{1\,2}^{0}\over 2}+{\xi_{1\,1}\over 2}-\Phi_{1,1}(\iota
k_{F\downarrow},q)\right)^{2}\hskip 5.69046pt{\rm for}\hskip
5.69046ptq=k_{F\downarrow}-k\hskip 5.69046pt{\rm where}$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle]{\tilde{k}},2k_{F\downarrow}[\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,\tilde{m}]\hskip 5.69046pt{\rm
and}\hskip 5.69046ptk\in]{\tilde{k}},2k_{F\downarrow}[\hskip 5.69046pt{\rm
for}\hskip 5.69046ptm\in]0,\tilde{m}]$
$\displaystyle\zeta_{2^{\prime}}^{+-}(k)$ $\displaystyle=$
$\displaystyle-1+\sum_{\iota=\pm 1}\left(-{\iota\over
2\xi_{1\,1}}+\xi_{1\,1}+\Phi_{1,2}(\iota k_{F\downarrow},q)\right)^{2}\hskip
5.69046pt{\rm for}\hskip 5.69046ptq=k-2k_{F\downarrow}\hskip 5.69046pt{\rm
where}\hskip 5.69046ptk\in]2k_{F\downarrow},\pi[\,.$ (43)

The three $\iota=\pm 1$ spectral functionals $\Phi_{\iota}(q)$ in the general
expression, Eq. (13), specific to the exponents given in Eq. (43) for the
$S^{+-}(k,\omega)$’s $2$\- and $2^{\prime}$-branch lines,
$\bar{1}^{\prime}$-branch line, and $\bar{1}$-branch line are provided in Eqs.
(23), (24), and (25), respectively. The corresponding suitable specific values
of the number and current number deviations, Eq. (9), used in such functionals
are for the present branch lines given in Table 1.

The $S^{+-}(k,\omega)$’s $2$-, $\bar{1}^{\prime}$-, $\bar{1}$-, and
$2^{\prime}$-branch line exponents whose expressions are given in Eq. (43) are
plotted as a function of $k$ in Fig. 4. In the $k$ intervals of the gapped
lower threshold of the $n$-string continuum in Fig. 1 for which they are
negative, which are represented by solid lines, there are singularities at and
just above the corresponding $\bar{n}=2,\bar{1}^{\prime},\bar{1},2^{\prime}$
branch lines in the expression
$S^{+-}(k,\omega)=C_{+-}^{\Delta}(\omega-\Delta_{\bar{n}}^{+-}(k))^{\zeta_{\bar{n}}^{+-}(k)}$,
Eq. (11) for $ab=+-$.

The related $S^{xx}(k,\omega)$’s expression, Eq. (11) for $ab=xx$, also valid
at and just above the gapped lower threshold of the $n$-string continuum in
Fig. 2, is similar to that of $S^{+-}(k,\omega)$ and involves exactly the same
exponents. This though applies provided that in the corresponding $k$
intervals there is no overlap between that continuum and the lower continuum
associated with excited states described by only real Bethe-ansatz rapidities.
For small spin densities, this thus excludes the $k$ intervals given in Eq.
(29).

In the case of $S^{zz}(k,\omega)$, the expressions of the gapped lower
threshold spectrum $\Delta^{zz}(k)$, Eqs. (36) and (38), are for the $k$
intervals of each corresponding branch line given by,

$\displaystyle\Delta_{2}^{zz}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{2}(k-(k_{F\uparrow}-k_{F\downarrow}))\hskip
5.69046pt{\rm and}\hskip 5.69046ptk=(k_{F\uparrow}-k_{F\downarrow})+q\hskip
5.69046pt{\rm where}$ $\displaystyle k$ $\displaystyle\in$
$\displaystyle]0,(k_{F\uparrow}-k_{F\downarrow})[\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq\in]-(k_{F\uparrow}-k_{F\downarrow}),0[\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,1[\,,$ (44)
$\displaystyle\Delta_{\bar{1}}^{zz}(k)$ $\displaystyle=$ $\displaystyle
4\mu_{B}\,h-W_{2}-\varepsilon_{1}\left(k_{F\uparrow}-k\right)\hskip
5.69046pt{\rm and}\hskip 5.69046ptk=k_{F\uparrow}-q\hskip 5.69046pt{\rm
where}$ $\displaystyle k$ $\displaystyle\in$
$\displaystyle](k_{F\uparrow}-k_{F\downarrow}),(\pi-{\tilde{k}})[\hskip
5.69046pt{\rm and}\hskip
5.69046ptq\in]-(k_{F\downarrow}-{\tilde{k}}),k_{F\downarrow}[\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,\tilde{m}]$ $\displaystyle k$
$\displaystyle\in$
$\displaystyle](k_{F\uparrow}-k_{F\downarrow}),(\pi-{\tilde{k}})[\hskip
5.69046pt{\rm and}\hskip
5.69046ptq\in]-(k_{F\downarrow}-{\tilde{k}}),k_{F\downarrow}[\hskip
5.69046pt{\rm and}\hskip 5.69046pt{\rm for}\hskip
5.69046ptm\in[\tilde{m},1[\,,$ (45)
$\displaystyle\Delta_{\bar{1}^{\prime}}^{zz}(k)$ $\displaystyle=$
$\displaystyle 4\mu_{B}\,h-\varepsilon_{1}(k_{F\downarrow}-k)\hskip
5.69046pt{\rm and}\hskip 5.69046ptk=k_{F\downarrow}-q\hskip 5.69046pt{\rm
where}$ $\displaystyle k$ $\displaystyle\in$
$\displaystyle](\pi-{\tilde{k}}),2k_{F\downarrow}[\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq\in]-k_{F\downarrow},-(k_{F\uparrow}-\tilde{k})[\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,\tilde{m}]\,,$ (46)

and

$\displaystyle\Delta_{2^{\prime}}^{zz}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{2}(k-\pi)\hskip 5.69046pt{\rm and}\hskip
5.69046ptk=\pi+q\hskip 5.69046pt{\rm where}$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle]2k_{F\downarrow},\pi[\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq\in]-(k_{F\uparrow}-k_{F\downarrow}),0[\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,\tilde{m}]$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle](\pi-{\tilde{k}}),\pi[\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq\in]-{\tilde{k}},0[\hskip 5.69046pt{\rm for}\hskip
5.69046ptm\in[\tilde{m},1[\,.$ (47)

Figure 5: The same as in Fig. 4 for the $\bar{1}^{\prime}$ branch line of
$S^{zz}(k,\omega)$. For that dynamical structure factor, this exponent is the
only one that is negative and refers to singularities near the corresponding
small momentum intervals of the gapped lower threshold of the $n$-string
continuum in Fig. 3. Such singularities only emerge in $S^{zz}(k,\omega)$ for
spin densities $0<m<\tilde{m}$ where $\tilde{m}\approx 0.317$.

The corresponding $k$ dependent exponents of general form, Eq. (13), that
appear in the expression,
$S^{zz}(k,\omega)=C_{zz}^{\Delta}(\omega-\Delta_{\bar{n}}^{zz}(k))^{\zeta_{\bar{n}}^{zz}(k)}$,
Eq. (11) for $ab=zz$ and $\bar{n}=2,\bar{1},\bar{1}^{\prime},2^{\prime}$,
read,

$\displaystyle\zeta_{2}^{zz}(k)$ $\displaystyle=$
$\displaystyle-1+\sum_{\iota=\pm
1}\left(-{\iota\over\xi_{1\,1}}-\xi_{1\,1}+\Phi_{1,2}(\iota
k_{F\downarrow},q)\right)^{2}\hskip 5.69046pt{\rm for}\hskip
5.69046ptq=k-k_{F\uparrow}+k_{F\downarrow}\hskip 5.69046pt{\rm where}$
$\displaystyle k$ $\displaystyle\in$
$\displaystyle]0,(k_{F\uparrow}-k_{F\downarrow})[$
$\displaystyle\zeta_{\bar{1}}^{zz}(k)$ $\displaystyle=$
$\displaystyle-1+\sum_{\iota=\pm
1}\left(-{\iota\over\xi_{1\,1}}+\iota{\xi_{1\,2}^{0}\over 2}-{\xi_{1\,1}\over
2}-\Phi_{1,1}(\iota k_{F\downarrow},q)\right)^{2}\hskip 5.69046pt{\rm
for}\hskip 5.69046ptq=k_{F\uparrow}-k\hskip 5.69046pt{\rm where}$
$\displaystyle k$ $\displaystyle\in$
$\displaystyle](k_{F\uparrow}-k_{F\downarrow}),(\pi-{\tilde{k}})[\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,\tilde{m}]\hskip 5.69046pt{\rm
and}\hskip
5.69046ptk\in](k_{F\uparrow}-k_{F\downarrow}),(\pi-{\tilde{k}})[\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,\tilde{m}]$
$\displaystyle\zeta_{\bar{1}^{\prime}}^{zz}(k)$ $\displaystyle=$
$\displaystyle-1+\sum_{\iota=\pm 1}\left({\iota\over
2\xi_{1\,1}}+{\xi_{1\,1}\over 2}-\Phi_{1,1}(\iota
k_{F\downarrow},q)\right)^{2}\hskip 5.69046pt{\rm for}\hskip
5.69046ptq=k_{F\downarrow}-k\hskip 5.69046pt{\rm where}$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle](\pi-{\tilde{k}}),2k_{F\downarrow}[\hskip
5.69046pt{\rm for}\hskip 5.69046ptm\in]0,\tilde{m}]$
$\displaystyle\zeta_{2^{\prime}}^{zz}(k)$ $\displaystyle=$
$\displaystyle-1+\sum_{\iota=\pm
1}\left(-{\iota\over\xi_{1\,1}}+\Phi_{1,2}(\iota
k_{F\downarrow},q)\right)^{2}\hskip 5.69046pt{\rm for}\hskip
5.69046ptq=k-\pi\hskip 5.69046pt{\rm where}$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle]2k_{F\downarrow},\pi[\hskip 5.69046pt{\rm
for}\hskip 5.69046ptm\in]0,\tilde{m}]\hskip 5.69046pt{\rm and}\hskip
5.69046ptk\in](\pi-{\tilde{k}}),\pi[\hskip 5.69046pt{\rm for}\hskip
5.69046ptm\in[\tilde{m},1[\,.$ (48)

Also in the present case of $S^{zz}(k,\omega)$, the three $\iota=\pm 1$
spectral functionals $\Phi_{\iota}(q)$ in the general expression, Eq. (13),
specific to the $2$\- and $2^{\prime}$-branch lines, $\bar{1}$-branch line,
and $\bar{1}^{\prime}$-branch line are provided in Eqs. (23)-(25). The
corresponding suitable values of the number and current number deviations, Eq.
(9), given in Table 2 and used in such functionals to reach the expressions in
Eq. (48) are though different relative to those used for $S^{+-}(k,\omega)$.

The corresponding behaviors of the spin dynamical structure factor
$S^{zz}(k,\omega)$ are also qualitatively different from those of
$S^{+-}(k,\omega)$. Except for $\zeta_{\bar{1}^{\prime}}^{zz}(k)$, the
exponents in Eq. (48) are positive for all their $k$ intervals. The
$\bar{1}^{\prime}$-branch line’s exponent is plotted as a function of $k$ in
Fig. 5. It is negative for its whole $k$ subinterval, which is part of the $k$
interval of the gapped lower threshold in Figs. 3 (a) and (b). The
$\bar{1}^{\prime}$-branch line’s $m$-dependent subinterval is though either
small or that line is not part of the $S^{zz}(k,\omega)$’s gapped lower
threshold at all. Its momentum width decreases upon increasing $m$ up to a
spin density $\tilde{m}\approx 0.317$. For $\tilde{m}<m<1$, the
$\bar{1}^{\prime}$-branch line is not part of the $S^{zz}(k,\omega)$’s gapped
lower threshold spectrum. This is why for $m=0.50>\tilde{m}$ and
$m=0.75>\tilde{m}$ that line does not appear in the gapped lower threshold in
Figs. 3 (c) and (d), respectively.

One then concludes that gapped lower threshold’s singularities only emerge in
$S^{zz}(k,\omega)$ for spin densities $0<m<\tilde{m}$ at and just above the
$\bar{1}^{\prime}$-branch line, the corresponding line shape reading,
$S^{zz}(k,\omega)=C_{zz}^{\Delta}(\omega-\Delta_{\bar{1}^{\prime}}^{zz}(k))^{\zeta_{\bar{1}^{\prime}}^{+-}(k)}$.
That branch line $k$ subinterval width though strongly decreases upon
increasing $m$ up to $\tilde{m}$.

$1$-branch line |  $k$ in terms of $q$ |  $k$ interval | $\delta N_{1}^{F}$ | $\delta J_{1}^{F}$ | $\delta N_{1}^{NF}$
---|---|---|---|---|---
$-+$ | $k=k_{F\uparrow}-q$ | $k\in](k_{F\uparrow}-k_{F\downarrow}),\pi[$ | $0$ | $-1/2$ | $-1$
$+-$ | $k=k_{F\uparrow}+q$ | $k\in[0,(k_{F\uparrow}-k_{F\downarrow})[$ | $0$ | $-1/2$ | $1$
$+-$ | $k=k_{F\uparrow}-q$ | $k\in](k_{F\uparrow}-k_{F\downarrow}),\pi[$ | $2$ | $-1/2$ | $-1$
$zz$ | $k=k_{F\downarrow}-q$ | $k\in]0,2k_{F\downarrow}[$ | $1$ | $1/2$ | $-1$
$zz$ | $k=k_{F\downarrow}+q$ | $k\in]2k_{F\downarrow},\pi]$ | $-1$ | $1/2$ | $1$
Table 3: The excitation momenta $k>0$ of (i) $S^{-+}(k,\omega)$ and (ii)
$S^{+-}(k,\omega)$ and $S^{zz}(k,\omega)$ expressed in terms of $1$-band
momenta $q$ and corresponding number and current number deviations for the (i)
$1$-branch line and (ii) two $1$-branch line sections that coincide with the
corresponding lower thresholds of the lower continua shown in Figs. 1-3.

These behaviors are consistent with the $S^{zz}(k,\omega)$’s spectral weight
stemming from $n$-string states being suppressed upon increasing the spin
density $m$ within the interval $m\in]0,\tilde{m}]$. Within it, that weight
decreases upon increasing the spin density, becoming negligible for
$\tilde{m}<m<1$.

###  V.2 The line shape near the lower thresholds of the spin dynamical
structure factors

In order to provide an overall physical picture that accounts for all gapped
lower threshold’s singularities and lower threshold’s singularities in the
spin dynamical structure factors, here we shortly revisit their line shape
behavior at and just above the lower thresholds of the lower continua in Figs.
1-3. The corresponding dominant contributions are from excited states
described by only real Bethe-ansatz rapidities Carmelo_15A . Such lower
continua contain most spectral weight of the corresponding spin dynamical
structure factors.

In the case of the transverse dynamical structure factor,
$S^{xx}(k,\omega)={1\over 4}\left(S^{+-}(k,\omega)+S^{-+}(k,\omega)\right)$,
we consider the transitions to excited states that determine the line shape in
the vicinity of the lower thresholds of both $S^{+-}(k,\omega)$ and
$S^{-+}(k,\omega)$, respectively. The lower threshold’s spectrum
$\omega_{lt}^{xx}(k)$ of $S^{xx}(k,\omega)$ is given in Eq. (70) of Appendix
A. The corresponding two-parametric spectrum $\omega^{xx}(k)$ results from the
superposition of the corresponding two-parametric spectra, $\omega^{+-}(k)$
and $\omega^{-+}(k)$, Eqs. (58) and (59) of that Appendix, respectively. It
refers to the $(k,\omega)$-plane lower continuum in Fig. 2.

For spin densities $0<m<1$ and momenta $k\in]0,\pi[$, the line shape of the
spin dynamical structure factors $S^{ab}(k,\omega)$ where $ab=+-,-+,xx,zz$ at
and just above their lower thresholds has the general form given in Eq. (12).
In the case of $S^{xx}(k,\omega)$, this expression can be expressed as,

$\displaystyle S^{xx}(k,\omega)=S^{+-}(k,\omega)\hskip 5.69046pt{\rm
for}\hskip 5.69046ptk\in[0,(k_{F\uparrow}-k_{F\downarrow})[$
$\displaystyle\hskip 35.56593pt=S^{-+}(k,\omega)\hskip 5.69046pt{\rm
for}\hskip 5.69046ptk\in](k_{F\uparrow}-k_{F\downarrow}),\pi[\,.$ (49)

Figure 6: The momentum dependence of the exponent that controls the
$S^{xx}(k,\omega)$’s line shape near and just above the lower threshold of the
lower continuum in Fig. 2 for several spin densities. For
$k\in]0,(k_{F\uparrow}-k_{F\downarrow})[$ and
$k\in](k_{F\uparrow}-k_{F\downarrow}),\pi[$ that exponent corresponds to that
of $S^{+-}(k,\omega)$ and $S^{-+}(k,\omega)$, respectively.

The $k$ dependent exponents appearing in the spin dynamical factors’s
expression, Eq. (12), are of general form, Eq. (13). Their specific
expressions for the different $1$-branch lines and $1$-branch line sections
under consideration read,

$\displaystyle\zeta_{1}^{-+}(k)$ $\displaystyle=$
$\displaystyle-1+\sum_{\iota=\pm 1}\left(-{\xi_{1\,1}\over 2}-\Phi_{1,1}(\iota
k_{F\downarrow},q)\right)^{2}$ $\displaystyle{\rm for}\hskip
5.69046ptq=k_{F\uparrow}-k\hskip 5.69046pt{\rm and}\hskip
5.69046ptk\in](k_{F\uparrow}-k_{F\downarrow}),\pi[$
$\displaystyle\zeta_{1}^{+-}(k)$ $\displaystyle=$
$\displaystyle-1+\sum_{\iota=\pm 1}\left(-{\xi_{1\,1}\over 2}+\Phi_{1,1}(\iota
k_{F\downarrow},q)\right)^{2}$ $\displaystyle{\rm for}\hskip
5.69046ptq=k-k_{F\uparrow}\hskip 5.69046pt{\rm and}\hskip
5.69046ptk\in]0,(k_{F\uparrow}-k_{F\downarrow})[$
$\displaystyle\zeta_{1}^{+-}(k)$ $\displaystyle=$
$\displaystyle-1+\sum_{\iota=\pm
1}\left({\iota\over\xi_{1\,1}}-{\xi_{1\,1}\over 2}-\Phi_{1,1}(\iota
k_{F\downarrow},q)\right)^{2}$ $\displaystyle{\rm for}\hskip
5.69046ptq=k_{F\uparrow}-k\hskip 5.69046pt{\rm and}\hskip
5.69046ptk\in](k_{F\uparrow}-k_{F\downarrow}),\pi[$
$\displaystyle\zeta_{1}^{zz}(k)$ $\displaystyle=$
$\displaystyle-1+\sum_{\iota=\pm 1}\left({\iota\over
2\xi_{1\,1}}+{\xi_{1\,1}\over 2}-\Phi_{1,1}(\iota
k_{F\downarrow},q)\right)^{2}$ $\displaystyle{\rm for}\hskip
5.69046ptq=k_{F\downarrow}-k\hskip 5.69046pt{\rm and}\hskip
5.69046ptk\in]0,2k_{F\downarrow}[$ $\displaystyle\zeta_{1}^{zz}(k)$
$\displaystyle=$ $\displaystyle-1+\sum_{\iota=\pm 1}\left(-{\iota\over
2\xi_{1\,1}}+{\xi_{1\,1}\over 2}+\Phi_{1,1}(\iota
k_{F\downarrow},q)\right)^{2}$ (50) $\displaystyle{\rm for}\hskip
5.69046ptq=k-k_{F\downarrow}\hskip 5.69046pt{\rm and}\hskip
5.69046ptk\in]2k_{F\downarrow},\pi[\,.$

The corresponding $\iota=\pm 1$ spectral functionals $\Phi_{\iota}(q)$ in the
exponent expression, Eq. (13), have for the present $1$-branch lines and
$1$-branch line sections the general form given in Eq. (26). The values of the
$1$-band number and current number deviations, Eq. (9), that are used in Eq.
(26) to reach the above specific exponents’s expressions are provided in Table
3.

Figure 7: The momentum dependence of the exponent that controls the
$S^{zz}(k,\omega)$ line shape near and just above the lower threshold of the
lower continuum in Fig. 3 for several spin densities.

As confirmed by the form of the expressions given in Eqs. (67) and (68) of
Appendix A, the lower threshold’ spectra equality
$\omega^{+-}_{lt}(k)=\omega^{-+}_{lt}(k)$ holds for
$k\in[(k_{F\uparrow}-k_{F\downarrow}),\pi]$. On the other hand,
$\omega^{+-}_{lt}(k)=\omega^{xx}_{lt}(k)$ for $k\in[0,\pi]$, as confirmed by
inspection of Eqs. (68) and (70) of that Appendix. In the
$k\in](k_{F\uparrow}-k_{F\downarrow}),\pi[$ interval, the line shape of
$S^{xx}(k,\omega)={1\over 4}\left(S^{+-}(k,\omega)+S^{-+}(k,\omega)\right)$ is
controlled by the smallest of the exponents $\zeta_{1}^{-+}(k)$ and
$\zeta_{1}^{+-}(k)$ in Eq. (50), which turns out to be $\zeta_{1}^{-+}(k)$. In
the $k\in]0,(k_{F\uparrow}-k_{F\downarrow})[$ interval it is rather controlled
by the exponent $\zeta_{1}^{+-}(k)$. The resulting exponent
$\zeta_{1}^{xx}(k)$ is thus given by,

$\displaystyle\zeta_{1}^{xx}(k)$ $\displaystyle=$
$\displaystyle-1+\sum_{\iota=\pm 1}\left(-{\xi_{1\,1}\over 2}+\Phi_{1,1}(\iota
k_{F\downarrow},q)\right)^{2}\hskip 5.69046pt{\rm for}$ (51) $\displaystyle
q=k-k_{F\uparrow}\hskip 5.69046pt{\rm and}\hskip
5.69046ptk\in]0,(k_{F\uparrow}-k_{F\downarrow})[$ $\displaystyle=$
$\displaystyle-1+\sum_{\iota=\pm 1}\left(-{\xi_{1\,1}\over 2}-\Phi_{1,1}(\iota
k_{F\downarrow},q)\right)^{2}\hskip 5.69046pt{\rm for}$ $\displaystyle
q=k_{F\uparrow}-k\hskip 5.69046pt{\rm and}\hskip
5.69046ptk\in](k_{F\uparrow}-k_{F\downarrow}),\pi[\,.$

This exponent is plotted as a function of $k$ in Fig. 6. On the other hand,
the $1$-branch line exponent $\zeta_{1}^{zz}(k)$ whose expression is given in
Eq. (50) is plotted as a function of that excitation momentum in Fig. 7.

Both such exponents are for spin densities $0<m<1$ negative in the whole
momentum interval $k\in]0,\pi[$. It follows that there are singularities at
and just above the corresponding lower thresholds. (Due to a sign error, the
minus sign in the quantity $-\xi_{1\,1}/2$ appearing in Eq. (51) was missed in
Ref. Carmelo_15A  where the exponent $\zeta_{1}^{xx}(k)$ was named $\xi^{t}$.
Its momentum dependence plotted in Fig. 6 corrects that plotted in Figs. 2-4
of that reference.)

##  VI Limiting behaviors of the spin dynamical structure factors

At zero magnetic field, $h=0$, and thus spin density $m=0$, the spin dynamical
structure factor $S^{-+}(k,\omega)$ equals that obtained in the $m\rightarrow
0$ limit from $m>0$ values. On the other hand, at $h=0$ and $m=0$ the spin
dynamical structure factor $S^{+-}(k,\omega)$ equals that obtained in the
$m\rightarrow 0$ limit from $m<0$ values. This is consistent with the
relation, Eq. (4). As required by the $SU(2)$ symmetry, this confirms as well
that $S^{-+}(k,\omega)=S^{+-}(k,\omega)$ at $h=0$ and $m=0$. That symmetry
also requires that at $h=0$ and $m=0$ the overall equality
$S^{xx}(k,\omega)=S^{yy}(k,\omega)=S^{zz}(k,\omega)$ holds. (For the ranges
$0<h<h_{c}$ and $0<m<1$ considered in this paper, the equality
$S^{xx}(k,\omega)=S^{yy}(k,\omega)$ remains valid, whereas one has that
$S^{zz}(k,\omega)\neq S^{xx}(k,\omega)$.)

The corresponding two parametric spectrum at $h=0$ and $m=0$ of the spin
dynamical structure factors equals that obtained in the $m\rightarrow 0$ limit
from the spectrum $\omega^{-+}(k)$ whose expression is given in Eq. (58) of
Appendix A for $0<h<h_{c}$ and $0<m<1$. On the other hand, in that limit both
the spectra $\omega^{+-}(k)$ and $\omega^{zz}(k)$ in Eqs. (59) and (60) of
that Appendix become one-parametric and coincide with the lower threshold of
the two-parametric spectrum of
$S^{xx}(k,\omega)=S^{yy}(k,\omega)=S^{zz}(k,\omega)$ and thus also of
$S^{-+}(k,\omega)=S^{+-}(k,\omega)$ at $h=0$ and $m=0$.

What is then the origin of the two-parametric spectrum $S^{zz}(k,\omega)$ at
$h=0$ and $m=0$? At vanishing magnetic field, $h=0$, the selection rules, Eq.
(5), reveal that $S^{zz}(k,\omega)$ is fully controlled by transitions from
the $S=S^{z}=0$ ground state to triplet excited states with spin numbers $S=1$
and $S^{z}=0$. This is different from the case considered in this paper, for
which the initial ground state refers to $0<h<h_{c}$ and $0<m<1$. According to
the corresponding selection rules, Eq. (6), $S^{zz}(k,\omega)\neq
S^{xx}(k,\omega)$ is then controlled by transitions from the ground state with
spin numbers $S^{z}=S$ or $S^{z}=-S$ to excited energy eigenstates with the
same spin numbers, $S^{z}=S$ or $S^{z}=-S$, respectively.

For the initial ground state at $h=0$ and $m=0$, one has more generally that
(i) $S^{zz}(k,\omega)$ and (ii) $S^{+-}(k,\omega)$ and $S^{-+}(k,\omega)$ are
fully controlled by transitions to spin triplet $S=1$ excited states with (i)
$S^{z}=0$ and (ii) $S^{z}=\pm 1$, respectively. Their $1$-band two-hole
spectrum is obtained in the $m\rightarrow 0$ limit from that of
$S^{+-}(k,\omega)$ for $m<0$ and from that of $S^{-+}(k,\omega)$ for $m>0$.
One then finds that,

$\displaystyle\omega^{xx}(k)$ $\displaystyle=$
$\displaystyle\omega^{zz}(k)=-\varepsilon_{1}(q)-\varepsilon_{1}(q^{\prime})=J{\pi\over
2}(\cos(q)+\cos(q^{\prime}))\hskip 5.69046pt{\rm where}\hskip
5.69046ptk=\iota\pi-q-q^{\prime}\hskip 5.69046pt{\rm and}\hskip
5.69046pt\iota=\pm 1$ (52) $\displaystyle{\rm for}\hskip
5.69046ptq\in[-\pi/2,\pi/2]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq^{\prime}\in[-\pi/2,\pi/2]\,.$

These results are consistent with spin $SU(2)$ symmetry implying that the
triplet $S=1$ and $S^{z}=0$ excited energy eigenstates that control
$S^{zz}(k,\omega)$ have exactly the same spectrum, Eq. (52), as the triplet
$S=1$ and $S^{z}=\pm 1$ excited states that control $S^{+-}(k,\omega)$ and
$S^{-+}(k,\omega)$.

On the other hand, the two parametric spectrum given in Eq. (30) refers to the
$S^{+-}(k,\omega)$’s gapped continuum of the $n$-string states shown in Fig.
1. It was obtained in this paper for for $0<h<h_{c}$ and $0<m<1$. In the
$m\rightarrow 0$ limit from such $m>0$ values, it becomes again a one-
parametric line that coincides both with its $\bar{1}$\- and
$\bar{1}^{\prime}$-branch lines, which extend to $k\in]0,\pi[$, and with the
lower threshold of $S^{xx}(k,\omega)=S^{yy}(k,\omega)=S^{zz}(k,\omega)$ and
$S^{-+}(k,\omega)=S^{+-}(k,\omega)$ at $h=0$ and $m=0$.

It is confirmed in the following that, in spite of the singular behavior
concerning the class of excited states that control the spin dynamical
structure factors for magnetic fields $h=0$ and $h\neq 0$, respectively, the
same line shape at and above the lower thresholds of such factors is obtained
at $m=0$ and in the $m\rightarrow 0$ limit, respectively.

###  VI.1 Behaviors of the spin dynamical structure factors in the
$m\rightarrow 0$ limit

Taking the $m\rightarrow 0$ limit from $m>0$ values, one confirms that the
lower threshold’s spectrum, Eq. (67) of Appendix A, of $S^{-+}(k,\omega)$
expands to $k\in[0,\pi]$. One then finds that within the present dynamical
theory the corresponding line shape at and just above the $1$-branch line
becomes valid for $k\in]0,\pi[$. An exactly equal spectrum is obtained for the
lower threshold of $S^{+-}(k,\omega)$ in the $m\rightarrow 0$ limit from $m<0$
values. In the $m\rightarrow 0$ limit, the lower threshold’s spectrum of
$S^{xx}(k,\omega)={1\over 4}\left(S^{+-}(k,\omega)+S^{-+}(k,\omega)\right)$ is
then found to read,

$\omega^{xx}_{lt}(k)=-\varepsilon_{1}(\pi/2-k)=J{\pi\over 2}\sin k\hskip
5.69046pt{\rm for}\hskip 5.69046ptk\in]0,\pi[\,.$ (53)

On the other hand and as reported above, the gapped continuum, Eq. (30), of
the $n$-string states becomes in the $m\rightarrow 0$ limit a
$(k,\omega)$-plane line whose spectrum is as well given by
$\Delta_{\bar{1}^{\prime}}^{+-}(k)=\Delta_{\bar{1}}^{+-}(k)=J{\pi\over 2}\sin
k$ for $k\in]0,\pi[$. By use of the limiting behaviors $\lim_{m\rightarrow
0}\Phi_{1,1}\left(\pm k_{F\downarrow},q\right)=\pm 1/(2\sqrt{2})$ for
$q\neq\pm k_{F\downarrow}$, $\lim_{m\rightarrow 0}\Phi_{1,2}\left(\pm
k_{F},0\right)=\pm 1/\sqrt{2}$, and $\lim_{m\rightarrow
0}\xi_{1\,1}=1/\sqrt{2}$ provided in Eqs. (115), (116), and (122) of Appendix
B, one finds that the corresponding exponents
$\zeta_{\bar{1}^{\prime}}^{+-}(k)$ and $\zeta_{\bar{1}}^{+-}(k)$, Eq. (43), as
well as the exponent $\zeta_{1}^{xx}(k)$, Eq. (51), become in the
$m\rightarrow 0$ limit equal and read,

$\displaystyle\zeta_{1}^{xx}(k)$ $\displaystyle=$
$\displaystyle\zeta_{\bar{1}^{\prime}}^{+-}(k)=-1+\sum_{\iota=\pm
1}\left(-{\xi_{1\,1}\over 2}-\Phi_{1,1}(\iota\pi/2,q)\right)^{2}=-{1\over 2}$
$\displaystyle\zeta_{\bar{1}}^{+-}(k)$ $\displaystyle=$
$\displaystyle-1+\sum_{\iota=\pm 1}\left(\iota{\xi_{1\,2}^{0}\over
2}+{\xi_{1\,1}\over 2}-\Phi_{1,1}(\iota k_{F\downarrow},q)\right)^{2}=-{1\over
2}\,.$ (54)

The exponent $\zeta_{1}^{xx}(k)$, Eq. (51), has for $0<m<1$ two expressions
associated with two $k$ intervals, respectively. The momentum width of one of
such intervals vanishes in the $m\rightarrow 0$ limit. This is why only one of
its two expressions contributes to the equality in this equation, which only
occurs in that limit.

Consistent with spin $SU(2)$ symmetry requirements, we confirm that in the
$m\rightarrow 0$ limit in which the two parametric spectrum $\omega^{zz}(k)$
in Eq. (60) of Appendix A becomes a one-parametric line that coincides with
the lower threshold’s spectrum of $S^{zz}(k,\omega)$, Eq. (69) of that
Appendix, and in addition becomes equal to the lower threshold’s spectrum
$\omega^{xx}_{lt}(k)$, Eq. (53), and thus reads
$\omega^{zz}_{lt}(k)=\omega_{1}^{zz}(k)=J{\pi\over 2}\sin k$ for
$k\in]0,\pi[$.

Similarly, the $S^{zz}(k,\omega)$’s gapped $\bar{1}$\- and
$\bar{1}^{\prime}$-branch line spectra in Eqs. (45) and (46), respectively,
are in the $m\rightarrow 0$ limit found to become equal and gapless and to
expand to the whole $k\in]0,\pi[$ interval. In that limit, they coincide with
the $S^{zz}(k,\omega)$’s lower threshold, their spectrum thus again reading,
$\Delta_{\bar{1}^{\prime}}^{zz}(k)=J{\pi\over 2}\sin k$ for $k\in]0,\pi[$.

One then finds that in the $m\rightarrow 0$ limit the corresponding exponents
$\zeta_{\bar{1}}^{zz}(k)$ and $\zeta_{\bar{1}^{\prime}}^{zz}(k)$, Eq. (48),
obey the inequality,
$\zeta_{\bar{1}^{\prime}}^{zz}(k)<\zeta_{\bar{1}}^{zz}(k)$. The smaller
exponent $\zeta_{\bar{1}^{\prime}}^{zz}(k)$ is associated with the spectrum in
Eq. (46). That inequality has a physical meaning, at it reveals that the line
shape is controlled by the exponents $\zeta_{\bar{1}^{\prime}}^{zz}(k)$ and
$\zeta_{1}^{zz}(k)$ such that
$\zeta_{\bar{1}^{\prime}}^{zz}(k)=\zeta_{1}^{zz}(k)$ in that limit. By the use
of the behaviors provided in Eqs. (115) and (122) of Appendix B, one then
finds that in the same limit the exponents $\zeta_{1}^{zz}(k)$, Eq. (50), and
$\zeta_{\bar{1}^{\prime}}^{zz}(k)$, Eq. (48), equal as well those given in Eq.
(54) and read,

$\zeta_{1}^{zz}(k)=\zeta_{\bar{1}^{\prime}}^{zz}(k)=-1+\sum_{\iota=\pm
1}\left({\iota\over 2\xi_{1\,1}}+{\xi_{1\,1}\over
2}-\Phi_{1,1}(\iota\pi/2,q)\right)^{2}=-{1\over 2}\,.$ (55)

Also in this case, the exponent $\zeta_{1}^{zz}(k)$ given in Eq. (50) has for
$0<m<1$ two expressions associated with two $k$ intervals, respectively. Only
one of such expressions contributes to the equality in this equation, which
only holds in the $m\rightarrow 0$ limit. This follows from the momentum width
of one of the two $0<m<1$ excitation momentum $k$’s intervals considered in
Eq. (50) vanishing in that limit.

The above results then confirm that in the $m\rightarrow 0$ limit the line
shape at and just above the lower threshold of the spin dynamical structure
factors is controlled by the exponent $-1/2$, which is a result known to hold
at zero magnetic field Imambekov_09 ; Bougourzi_97 ; Caux_11 . The
corresponding line-shape expression reads,

$S^{aa}(k,\omega)=C\,(\omega-\omega(k))^{-1/2}\hskip 14.22636pt{\rm
where}\hskip 14.22636pt\omega(k)=J{\pi\over 2}\sin k\hskip 5.69046pt{\rm
for}\hskip 5.69046pt]0,\pi[\,,$ (56)

where $aa=xx,yy,zz$ and $C$ is a constant that has a fixed value for the $k$
and $\omega$ ranges corresponding to small values of the energy deviation
$(\omega-\omega(k))$.

###  VI.2 Behaviors of the spin dynamical structure factors in the
$m\rightarrow 1$ limit

The sum rules, Eq. (7), reveal that both the spin dynamical structure factrs
$S^{-+}(k,\omega)$ and $S^{zz}(k,\omega)$ vanish in the $m\rightarrow 1$
limit. This implies that in that limit and thus in the $h\rightarrow h_{c}$
limit, only $S^{xx}(k,\omega)$ dominates the spin dynamical structure factor.
Here $h_{c}=J/\mu_{B}$ is the critical field at which fully polarized
ferromagnetism is achieved.

At $h=h_{c}$ the line-shape expressions of general form given in Eqs. (11) and
(12) of the extended dynamical theory are not valid. Indeed, at that magnetic
field the problem refers to a different quantum phase associated with fully
polarized ferromagnetism and the line shape rather becomes of
$\delta$-function like type, given by,

$S^{xx}(k,\omega)={\pi\over
2}\delta\left(\omega-\omega^{xx}_{lt}(k)\right)\hskip 14.22636pt{\rm
where}\hskip 14.22636pt\omega^{xx}_{lt}(k)=J\,(1+\cos k)\hskip 5.69046pt{\rm
for}\hskip 5.69046pt[0,\pi]\,.$ (57)

##  VII Discussion and concluding remarks

In this paper, the contribution to the spin dynamical structure factors
$S^{+-}(k,\omega)$, $S^{xx}(k,\omega)$, and $S^{zz}(k,\omega)$ from excited
energy eigenstates populated by $n$-strings has been studied for magnetic
fields $0<h<h_{c}$, in the thermodynamic limit. (The contribution to
$S^{-+}(k,\omega)$ from such states was found to be negligible.) In that
limit, there is nearly no spectral weight in the $(k,\omega)$-plane gap region
between the upper threshold of the lower continuum shown in Figs. 1 and 3 for
$S^{+-}(k,\omega)$ and $S^{zz}(k,\omega)$, respectively, and the gapped lower
threshold of the $n$-string states’s spectrum. The same applies to
$S^{xx}(k,\omega)$ in Fig. 2 for the spin densities and $k$ intervals for
which there is no overlap between the $n$-string states’s upper continuum and
the lower continuum.

Concerning the negligible amount of spectral weight in the $(k,\omega)$-plane
gap regions, let us consider for instance the more involved case of
$S^{+-}(k,\omega)$. Similar conclusions apply to the simpler problems of the
other spin dynamical structure factors. The behavior of spin operators matrix
elements between energy eigenstates in the selection rules valid for magnetic
fields $0<h<h_{c}$, Eq. (6), has important physical consequences. It implies
that the spectral weight stemming from excited energy eigenstates described by
only real Bethe-ansatz rapidities existing in finite systems in a
$(k,\omega)$-plane region corresponding to the momentum interval
$k\in[2k_{F\downarrow},\pi]$ and excitation energy values $\omega$ above the
upper threshold of the lower continuum in Fig. 1, whose spectrum’s expression
is given in Eq. (59) of Appendix A, becomes negligible in the present
thermodynamic limit for a macroscopic system.

Our thermodynamic limit’s study is complementary to results obtained by
completely different methods for finite-size systems Kohno_09 ; Kohno_10 ;
Muller . The spectral weight located in that $(k,\omega)$-plane region is
found to decrease upon increasing the system size Kohno_09 . This is confirmed
by comparing the spectra represented in the first row frames of Figs. 3 (a)
and (b) of Ref. Kohno_09  for two finite-size systems with $N=320$ and
$N=2240$ spins, respectively, in the case under consideration of the spin
dynamical structure factor $S^{+-}(k,\omega)$.

More generally, the selection rules in Eqs. (5) and (6) are behind in the
thermodynamic limit nearly all spectral weight generated by transitions to
excited energy eigenstates described only by real Bethe-ansatz rapidities
being contained in the $(k,\omega)$-plane lower continuum shown in Fig. 1,
whose spectrum is given in Eq. (59) of Appendix A. Let us consider for
instance the $(k,\omega)$-plane spectral weight distributions shown in Fig. 18
of Ref. Muller  for $S^{+-}(k,\omega)$. As reported in that reference, due to
the interplay of the selections rules given in Eqs. (5) and (6) for $h=0$ and
$0<h<h_{c}$, respectively, the spectral weight existing between the continuous
lower boundary $\epsilon_{4L}$ and the upper boundary $\epsilon_{4U}$ at $h=0$
becomes negligible for finite magnetic fields $0<h<h_{c}$. In addition, the
spectral weight existing between the continuous lower boundary $\epsilon_{5L}$
and the upper boundary $\epsilon_{5U}$ for small finite-size systems, becomes
negligible in the thermodynamic limit for a macroscopic system. This is indeed
due to the selection rules, Eq. (6), as discussed in that reference. As also
reported in it, only the spectral weight below the continuous lower boundary
$\epsilon_{5L}(q)$, located in the $(k,\omega)$-plane between the lower
boundary $\epsilon_{6L}$ and the upper boundary $\epsilon_{6U}$ has a
significant amount of spectral weight.

This refers to the $(k,\omega)$-plane region where, according to the analysis
of Ref. Muller , for magnetic fields $0<h<h_{c}$ a macroscopic system has
nearly the whole spectral weight stemming from transitions to excited energy
eigenstates described by only Bethe-ansatz rapidities. Consistent with the
spectral weight in the present gap region being negligible, the
$(k,\omega)$-plane between the continuous lower boundary $\epsilon_{6L}$ and
the upper boundary $\epsilon_{6U}$ in Fig. 18 of that reference corresponds
precisely to the lower continuum shown in Fig. 1, whose spectrum is provided
in Eq. (59) of Appendix A.

Our results have focused on the contribution from $n$-string states. This
refers to the line shape at and just above the $(k,\omega)$-plane gapped lower
threshold’s spectra $\Delta_{\bar{n}}^{ab}(k)$ where $ab=+-,xx,zz$ and
${\bar{n}}$ refers to different branch lines. In well-defined $m$-dependent
$k$ subintervals, Eqs. (39)-(42) and (44)-(47), such branch lines coincide
with the gapped lower thresholds under consideration. In these physically
important $(k,\omega)$-plane regions, the spin dynamical structure factors
$S^{ab}(k,\omega)$ have the general analytical expression provided in Eq.
(11). In the case of $S^{+-}(k,\omega)$ and $S^{xx}(k,\omega)$, such gapped
lower thresholds refer to the $n$-string states’s upper continua shown in the
$(k,\omega)$-plane in Figs. 1 and 2, respectively.

The above results concerning the spectral weight in the gap regions being
negligible in the present thermodynamic limit, are consistent with the amount
of that weight existing just below the $(k,\omega)$-plane gapped lower
thresholds of the $n$-string states’s spectra shown in Figs. 1-3 being
vanishingly small or negligible. This is actually behind the validity at
finite longitudinal magnetic fields $0<h<h_{c}$ and in the thermodynamic limit
of the analytical expressions of the spin dynamical structure factors of
general form, Eq. (11), obtained in this paper.

The momentum dependent exponents that control the spin dynamical structure
factors’s line-shape in such expressions are given in Eq. (43) for
$S^{+-}(k,\omega)$ and $S^{xx}(k,\omega)$ and in Eq. (48) for
$S^{zz}(k,\omega)$. In the former case, the exponents associated with the
$(k,\omega)$-plane vicinity of the $2-$, $\bar{1}^{\prime}-$, $\bar{1}-$, and
$2^{\prime}$-branch lines are plotted in Fig. 4. Such lines refer to different
$k$ intervals of the gapped lower threshold of the $n$-string states’s spectra
of $S^{+-}(k,\omega)$ and $S^{xx}(k,\omega)$. The solid lines in Figs. 1 and 2
that belong to that gapped lower threshold correspond to $k$ intervals for
which the exponents are negative. In them, singularities occur in the spin
dynamical structure factors’s expression, Eq. (11), at and above the gapped
lower thresholds.

In the case of $S^{xx}(k,\omega)$, the expression given in that equation does
not apply for small spin densities in the range $m\in[0,\bar{m}]$ where
$\bar{m}\approx 0.276$ to the $k$ intervals given in Eq. (29). For these
momentum ranges, there is overlap between the lower continuum and upper
$n$-string states’s continuum, as shown in Figs. 2 (a) and (b). The two $k$
intervals provided in Eq. (29), $k\in[\bar{k}_{0},\pi]$ for
$m\in]0,\bar{m}_{0}]$ and $k\in[\bar{k}_{0},\bar{k}_{1}]$ for
$]\bar{m}_{0},\bar{m}]$ where $\bar{m}_{0}\approx 0.239$, apply to the spectra
plotted in these two figures, respectively. For spin density (a) $m=0.15$, the
momentum $\bar{k}_{0}$ is given by $\bar{k}_{0}\approx 0.60\,\pi$ whereas for
spin density (b) $m=0.25$ the two limiting momenta read $\bar{k}_{0}\approx
0.71\,\pi$ and $\bar{k}_{1}\approx 0.92\,\pi$.

On the other hand, the contribution to $S^{zz}(k,\omega)$ from excited states
populated by $n$-strings is much weaker than for $S^{+-}(k,\omega)$ and
$S^{xx}(k,\omega)$. It does not lead to a $(k,\omega)$-plane continuum. The
gapped lower threshold of such states is shown in Fig. 3. There the $k$
subinterval associated with the $\bar{n}=\bar{1}^{\prime}$ branch line is the
only one at and above which there are singularities. Out of the four branch-
line’s exponents whose expressions are provided in Eq. (48), only that of the
$\bar{n}=\bar{1}^{\prime}$ branch line is indeed negative. That line is
represented in the gapped lower threshold of $S^{zz}(k,\omega)$ shown in Fig.
3 by a solid (green) line. The corresponding exponent is plotted in Fig. 5.
That line’s $k$ subinterval is though small. Its momentum width decreases upon
increasing the spin density within the range $0<m\leq\tilde{m}$ where
$\tilde{m}\approx 0.317$. For spin densities $\tilde{m}\leq m<1$, that line is
not part of the gapped lower threshold, so that the contribution to
$S^{zz}(k,\omega)$ from excited states populated by $n$-strings becomes
negligible. Consistent, in Fig. 3 (c) for $m=0.50$ and (d) for $m=0.75$ that
line is lacking.

In order to to provide an overall physical picture that includes the relative
$(k,\omega)$-plane location of all spectra with a significant amount of
spectral weight, we also accounted for the contributions from all types of
excited energy eigenstates that lead to gapped and gapless lower threshold
singularities in the spin dynamical structure factors. This includes excited
energy eigenstates described only by real Bethe-ansatz rapidities and thus
without $n$-strings, which are known to lead to most spectral weight of the
sum rules, Eq. (7) Lefmann-96 ; Muller ; Karbach_00 ; Karbach_02 ; Carmelo_15A
. Their contribution to $S^{+-}(k,\omega)$, $S^{xx}(k,\omega)$, and
$S^{zz}(k,\omega)$ leads to the $(k,\omega)$-plane lower continua shown in
Figs. 1, 2, and 3, respectively.

While the present work is purely theoretical, the singularities to which
$(k,\omega)$-plane vicinity the analytical line-shape expressions obtained and
studied in this paper refer to are observed in inelastic neutron scattering
experiments on spin-chain compounds Kohno_09 ; Kohno_10 ; Stone_03 ;
Heilmann_78 . All cusp singularities at and near both the gapped lower
thresholds and lower thresholds found in this paper to occur in the
thermodynamic limit, indeed correspond to peaks shown in Fig. 4 of Ref.
Kohno_09  for CuCl2$\cdot$2N(C5D5) and in Fig. 5 of that reference for
Cu(C4H4N2)(NO3)2 at the finite values of the magnetic field considered in
these figures and suitable values of the exchange integral $J$. Also in that
reference such a correspondence was found within the spin-$1/2$ $XXX$ chain,
for finite-size systems.

CRediT authorship contribution statement

The three authors contributed equally to the formulation of the research goals
and aims. J. M. P. C. extended a suitable dynamical theory to a subspace that
accounts for the contribution from $n$-string states to the spin dynamical
structure factors and prepared the original draft. T. Č. and P. D. S. have
designed the computer programs to solve the integral equations that define the
phase shifts and spectra and T. Č. has implemented them.

Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

###### Acknowledgements.

J. M. P. C. would like to thank the Boston University’s Condensed Matter
Theory Visitors Program for support and Boston University for hospitality
during the initial period of this research. He acknowledges the support from
FCT through the Grants PTDC/FIS-MAC/29291/2017, SFRH/BSAB/142925/2018, and
POCI-01-0145-FEDER-028887. J. M. P. C. and T. Č. acknowledge the support from
FCT through the Grant UID/FIS/04650/2013. T. Č. gratefully acknowledges the
support by the Institute for Basic Science in Korea (IBS-R024-D1). P. D. S.
acknowledges the support from FCT through the Grants UID/CTM/04540/2013 and
UID/CTM/04540/2019. J. M. P. C. and T. Č. contributed equally to this work.

##  Appendix A Gapless continuum spectra and energy gaps of the $n$-string
states

Within a $k$ extended zone scheme, the $S^{-+}(k,\omega)$ and
$S^{+-}(k,\omega)$’s spectra associated with lower $(k,\omega)$-plane
continua, which for $S^{+-}(k,\omega)$ is shown in Fig. 1, read,