# One-particle spectral function singularities in a one-dimensional gas of
spin-$1/2$ fermions with repulsive delta-function interaction

T. Čadež  Beijing Computational Science Research Center, Beijing 100193, China
Center of Physics of University of Minho and University of Porto, P-4169-007
Oporto, Portugal  S. Nemati  Beijing Computational Science Research Center,
Beijing 100193, China  Center of Physics of University of Minho and University
of Porto, P-4169-007 Oporto, Portugal  University of Aveiro, Department of
Physics, P-3810193 Aveiro, Portugal  J. M. P. Carmelo  Boston University,
Department of Physics, 590 Commonwealth Ave, Boston, MA 02215, USA
Massachusetts Institute of Technology, Department of Physics, Cambridge, MA
02139, USA  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

(25 March 2019)

###### Abstract

The momentum, fermionic density, spin density, and interaction dependencies of
the exponents that control the $(k,\omega)$-plane singular features of the
one-fermion spectral functions of a one-dimensional gas of spin-$1/2$ fermions
with repulsive delta-function interaction both at zero and finite magnetic
field are studied in detail. Our results refer to energy scales beyond the
reach of the low-energy Tomonaga-Luttinger liquid and rely on the
pseudofermion dynamical theory for integrable models. The one-fermion spectral
weight distributions associated with the spectral functions studied in this
paper may be observed in systems of spin-$1/2$ ultra-cold fermionic atoms in
optical lattices.

##  I Introduction

The one-dimensional (1D) continuous fermionic gas with repulsive delta-
function interaction, which in this paper we call 1D repulsive fermion model,
was one of the first quantum problems solved by the Bethe ansatz (BA) Bethe_31
. This was achieved by Yang Yang_67  and by Gaudin Gaudin_67 . Yang’s solution
of the 1D repulsive fermion model was actually the precursor of the BA
solution of the lattice 1D Hubbard model by Lieb and Wu Lieb ; Lieb-03 ;
Takahashi ; Martins . That the latter is the simplest condensed-matter toy
model for the description of the role of correlations in the exotic properties
of 1D and quasi-1D lattice condensed matter systems spectral0 ; Dionys-87
justifies why for several decades it had attracted more attention than its
continuous cousin, the fermionic gas with repulsive delta-function
interaction. This refers both to its metallic and Mott-Hubbard insulator
phases DSF-n1 , the latter not existing in the case of the continuous 1D
repulsive fermion model.

However, in the last years the interest in that Yang-Baxter integrable model
has been renewed by its new found impact on experiments in both condensed
matter physics and ultra-cold atomic gases Batchelor_16 . The latter have
provided new opportunities for studying 1D systems of spin-$1/2$ fermions with
repulsive interaction Guan_13 ; Zinner_16 . The present model can indeed be
implemented with ultra-cold atoms Batchelor_16 ; Guan_13 ; Zinner_16 ;
Dao_2007 ; Stewart_08 ; Clement_09 ; Febbri_12 . Ultra-cold Fermi gases
trapped inside a tight atomic waveguide offer for instance the opportunity to
measure the spin-drag relaxation rate that controls the broadening of a spin
packet. It has been found that while the propagation of long-wavelength charge
excitations is essentially ballistic, spin propagation is intrinsically damped
and diffusive Polini_07 ; Rainis_08 . A related interesting problem is the
force applied to a spin-flipped fermion in a gas, which may lead to Bloch
oscillations of the fermion’s position and velocity. The existence of such
oscillations has been found crucially to depend on the viscous friction force
exerted by the rest of the gas on the spin excitation Gangardt_09 .

The ground-state energy of the relative motion of a system of two fermions
with spin up and spin down interacting via a delta-function potential in a 1D
harmonic trap has been calculated by combining the BA with the variational
principle Rubeni_12 . Recently, related ground-state properties of a 1D
repulsive Fermi gas subjected to a commensurate periodic optical lattice of
arbitrary intensity have been investigated by the use of continuous-space
quantum Monte-Carlo simulations Pilati_17 . The thermodynamic properties of
the model have also been recently studied using a specific lattice embedding
and the quantum transfer matrix. That allowed the derivation of an exact
system of only two nonlinear integral equations for the thermodynamics of the
homogeneous model, which is valid for all temperatures and values of the
chemical potential, magnetic field, and repulsive interaction Patu_16 .

Another issue that has contributed to the renewed interest in the 1D repulsive
fermion model is the relation of integrable Yang-Baxter equation fermionic
models to topology and quantum computing Kauffman_18 . That equation can act
as a parametric two-body quantum gate Zhang_12 . An experimental realization
of the Yang-Baxter equation through a Nuclear Magnetic Resonance
interferometric setup has actually verified its validity Vind_16 .

The model dynamical properties is another problem of scientific interest. The
behavior of dynamic structure factors of fermionic models differs dramatically
for integrable and non-integrable models Tatche_11 . The mobile quantum
impurity model (MQIM) has been used to derive dynamic response functions of
interacting one-dimensional spin-1/2 fermions Schmidt_10 ; Imambekov_12 . An
approximation relying on the bosonization technique and diagonalizing the
model to two Tomonaga-Luttinger liquid (TLL) Hamiltonians, was used in Ref.
Orignac_11  to obtain some general expressions for the spectral function at
zero spin density, expressed in terms of the Gauss hypergeometric function.
The up-spin and down-spin one-fermion spectral functions of the present model
in zero magnetic field and in a finite field have not been detailed studied
though.

In this paper a systematic and detailed study of the momentum dependent
exponents and energy spectra that control the line shape near the high-energy
singularities of both (i) the one-fermion removal and addition spectral
functions at zero magnetic field and (ii) the up-spin and down-spin one-
fermion removal and addition spectral functions at finite magnetic field is
conducted. (Our designation high energy refers to energy scales beyond the
reach of the low-energy TLL Tomonaga-50 ; Luttinger-63 ; Solyom-79 ; Voit ;
Sutherland-04 .)

The 1D repulsive fermion model describes $N=N_{\uparrow}+N_{\downarrow}$
spin-$1/2$ fermions, $N_{\uparrow}$ with up-spin projection and
$N_{\downarrow}$ with down-spin projection, which in real space have a
repulsive delta-function interaction. The model Hamiltonian in a chemical
potential $\mu$ and magnetic field $h$ is in units of $\hbar=1$ and bare mass
$m=1/2$ given by,

$\hat{H}=-\sum_{j=1}^{N}{\partial^{2}\over\partial
x_{j}^{2}}+2\,{\cal{C}}\sum_{j^{\prime}>j}\delta(x_{j}-x_{j^{\prime}})-\mu\,N+2\mu_{B}h\,{\hat{S}}^{z}\hskip
5.69046pt{\rm where}\hskip 5.69046pt{\hat{S}}^{z}=-{1\over
2}({\hat{N}}_{\uparrow}-{\hat{N}}_{\downarrow})\,.$ (1)

Here $\delta(x)$ denotes the Dirac delta-function distribution, $x_{j}$ is the
position of the $j$-$th$ fermion, ${\cal{C}}>0$ gives the strength of the
repulsive interaction, $\mu_{B}$ is the Bohr magneton, and the fermion number
operator reads
${\hat{N}}=\sum_{\sigma=\uparrow,\downarrow}\,\hat{N}_{\sigma}$. Moreover,
${\hat{S}}^{z}$ is the diagonal generator of the Hamiltonian ${\hat{H}}$
global spin $SU(2)$ symmetry algebra. The lowest-weight states (LWSs) and
highest-weight states (HWSs) of that $SU(2)$ symmetry algebra have numbers
$S=-S^{z}$ and $S=S^{z}$, respectively, where $S$ is the states spin and
$S^{z}$ is the corresponding projection. The latter is an eigenvalue of the
spin operator given in Eq. (1).

On the one hand, at zero magnetic field, $h=0$, and thus zero spin density,
$m=0$, our study focuses on the one-fermion spectral function,

$B_{\gamma}(k,\omega)=\sum_{\sigma=\uparrow,\downarrow}B_{\sigma,\gamma}(k,\omega)\hskip
5.69046pt{\rm for}\hskip 5.69046pt\gamma\,\omega\geq 0\,,$ (2)

where,

$\gamma=-1\hskip 5.69046pt{\rm for}\hskip 5.69046pt{\rm fermion}\hskip
5.69046pt{\rm removal}\hskip 5.69046pt{\rm and}\hskip 5.69046pt\gamma=+1\hskip
5.69046pt{\rm for}\hskip 5.69046pt{\rm fermion}\hskip 5.69046pt{\rm
addition}\,.$ (3)

On the other hand, for $h\neq 0$ and $m>0$ it addresses the up-spin and down-
spin one-fermion removal and addition spectral functions
$B_{\sigma,\gamma}(k,\,\omega)$ on the right-hand side of Eq. (2), which read,

$\displaystyle B_{\sigma,-1}(k,\,\omega)$ $\displaystyle=$
$\displaystyle\sum_{\nu^{-}}|\langle\nu^{-}|\,c_{k,\sigma}|\,GS\rangle|^{2}\,\delta(\omega+(E_{\nu^{-}}^{N_{\sigma}-1}-E_{GS}^{N_{\sigma}}))\hskip
5.69046pt{\rm for}\hskip 5.69046pt\omega\leq 0$ $\displaystyle
B_{\sigma,+1}(k,\,\omega)$ $\displaystyle=$
$\displaystyle\sum_{\nu^{+}}|\langle\nu^{+}|\,c^{\dagger}_{k,\sigma}|\,GS\rangle|^{2}\,\delta(\omega-(E_{\nu^{+}}^{N_{\sigma}+1}-E_{GS}^{N_{\sigma}}))\hskip
5.69046pt{\rm for}\hskip 5.69046pt\omega\geq 0\,,$ (4)

Here $c_{k,\sigma}$ and $c^{\dagger}_{k,\sigma}$ are up-spin and down-spin
fermion annihilation and creation operators, respectively, of momentum $k$ and
$|GS\rangle$ denotes the initial $N_{\sigma}$-fermion ground state of energy
$E_{GS}^{N_{\sigma}}$. The $\nu^{-}$ and $\nu^{+}$ summations run over the
$N_{\sigma}-1$ and $N_{\sigma}+1$-fermion excited energy eigenstates,
respectively, and $E_{\nu^{-}}^{N_{\sigma}-1}$ and
$E_{\nu^{+}}^{N_{\sigma}+1}$ are the corresponding energies.

Our main goal is deriving the $(k,\omega)$-plane line shape near the
singularities of the spectral functions in Eq. (2) at zero spin density,
$m=0$, and in Eq. (4) for $m>0$. This includes the detailed study of the
dependence of the exponents that control that line shape on the excitation
momentum, repulsive interaction ${\cal{C}}$, fermionic density
$n\in[0,\infty[$, and spin-density $m\in[0,n]$. For such spin densities, the
model ground states are LWSs of the spin $SU(2)$ symmetry algebra. Hence we
use the LWS formulation of the model BA solution.

The high-energy dynamical correlation functions of some integrable models
Altshuler ; Konik ; Fuksa_17 ; Pakuliaka_15  can be studied by the form-factor
approach. Form factors of the 1D repulsive fermion model up-spin and down-spin
fermion creation and annihilation operators involved in the spectral functions
studied here remains though an unsolved problem.

The present study of the momentum dependent exponents that control the line
shape near the singularities of the one-fermion spectral functions, Eqs. (2)
and (4), relies on the pseudofermion dynamical theory (PDT) introduced in Ref.
Carmelo_05  for the related lattice 1D Hubbard model, which applies to other
integrable systems as well Carmelo_18 ; Carmelo_16 ; Carmelo_15 , including
the present 1D repulsive fermion model. For the latter we use in our study an
exact representation suitable to the PDT in terms of pseudofermions of that
model BA solution in the subspace spanned by the ground state and one-fermion
excited energy eigenstates. The pseudofermions are generated by a unitary
transformation from corresponding pseudoparticles Carmelo_18 ; Carmelo_17 .
For simplicity, in this article the pseudofermions are called charge or spin
particles, depending on the BA branch they refer to.

The MQIM Imambekov_12  applies both to integrable and non-integrable models.
The previously introduced PDT Carmelo_05  applies only to integrable models.
In the case of the latter models, the PDT and MQIM lead to exactly the same
momentum dependent exponents in the power-law expressions of the spectral
functions near their edges of support. Indeed, for integrable models the two
methods have been shown to describe exactly the same fractionalized particles
microscopic mechanisms Carmelo_18 ; Carmelo_16 .

The remainder of the paper is organized as follows. The related $c$ and $s$
pseudoparticle and $c$ and $s$ particle representations, respectively, and
corresponding BA and PDT basic quantities needed for the study of the up-spin
and down-spin one-fermion spectral weights is the topic addressed in Section
II. In Section III the general types of one-fermion spectral singularities
studied in this paper are reported. The line shape near specific
$(k,\omega)$-plane one-fermion removal and addition branch and boundary lines
singularities of the spectral functions, Eqs. (2) and (4), is then studied in
Section IV. The low-energy TLL limit of the PDT one-fermion spectral function
expressions near their singularities is the issue addressed in Section V.
Finally, the discussion of the relevance and consequences of the results and
the concluding remarks are presented in Section VI.

##  II The $c$ and $s$ pseudoparticle and $c$ and $s$ particle representations

###  II.1 The BA equations and quantum numbers

Let $\\{|l_{\rm r},l_{s},{\cal{C}}\rangle\\}$ be the complete set of energy
eigenstates of the Hamiltonian $\hat{H}$, Eq. (1), associated with the BA
solution for ${\cal{C}}>0$. We call a Bethe state an energy eigenstate that is
a LWS of the spin $SU(2)$ symmetry algebra, which is here denoted by $|l_{\rm
r},l_{s}^{0},{\cal{C}}\rangle$. The ${\cal{C}}$-independent label $l_{s}$ in
general energy eigenstates $\\{|l_{\rm r},l_{s},{\cal{C}}\rangle\\}$ is a
short notation for the set of quantum numbers,

$l_{s}=S,n_{s}\hskip 5.69046pt{\rm where}\hskip
5.69046ptn_{s}=S+S^{z}=0,1,...,2S\,.$ (5)

For a Bethe state one then has that $n_{s}=0$, so that $l_{s}^{0}$ stands for
$S,0$. Furthermore, the label $l_{\rm r}$ refers to the set of all remaining
${\cal{C}}$-independent quantum numbers needed to uniquely specify an energy
eigenstate $|l_{\rm r},l_{s},{\cal{C}}\rangle$. This refers to occupancy
configurations of BA momentum quantum numbers $q_{j}={2\pi\over
L}\,I^{\beta}_{j}$. Here $I^{\beta}_{j}$ are successive integers,
$I^{\beta}_{j}=0,\pm 1,\pm 2,...$, or half-odd integers, $I^{\beta}_{j}=\pm
1/2,\pm 3/2,\pm 5/2,...$, according to well-defined boundary conditions. Their
allowed occupancies are zero and one. The index $\beta$ denotes several BA
branches of quantum numbers.

In the case of the up-spin and down-spin one-fermion removal and addition
spectral functions, the line-shape near their singularities does not involve
at finite magnetic field excited states described by spin complex BA
rapidities. Fortunately, the finite-field quantities lead to correct zero-spin
density results in the limit of zero spin density. Hence for our study only
the charge $\beta=c$ band and spin $\beta=s$ band momentum $q_{j}={2\pi\over
L}\,I^{\beta}_{j}$ branches described by real BA rapidities are needed.

The general non-LWSs $|l_{\rm r},l_{s},{\cal{C}}\rangle$ for which $n_{s}>0$
can be generated from the corresponding Bethe states $|l_{\rm
r},l_{s}^{0},{\cal{C}}\rangle$ as,

$|l_{\rm
r},l_{s},{\cal{C}}\rangle=\left(\frac{1}{\sqrt{{\cal{C}}_{s}}}({\hat{S}}^{+})^{n_{s}}\right)|l_{\rm
r},l_{s}^{0},{\cal{C}}\rangle\hskip 5.69046pt{\rm where}\hskip
5.69046pt{\cal{C}}_{s}=(n_{s}!)\prod_{j=1}^{n_{s}}(\,2S+1-j\,)\hskip
5.69046pt{\rm and}\hskip 5.69046ptn_{s}=1,...,2S\,.$

Here ${\hat{S}}^{+}={\hat{S}}^{x}+i{\hat{S}}^{y}$, ${\hat{S}}^{x}$ and
${\hat{S}}^{y}$ are usual spin component operators, and ${\cal{C}}_{s}$ is a
normalization constant.

The BA equations of the 1D repulsive fermion model, Eq. (1), in the subspace
spanned by the ground state and the one-fermion excited energy eigenstates
that contribute to the spectral weight in the vicinity of the spectral
functions singularities are of the form Yang_67 ,

$k_{j}=q_{j}-{2\over
L}\sum_{j^{\prime}=1}^{N_{\downarrow}}\arctan\left({2k_{j}-2\Lambda_{j^{\prime}}\over{\cal{C}}}\right)\hskip
5.69046pt{\rm where}\hskip 5.69046ptq_{j}={2\pi\over L}\,I_{j}^{c}\hskip
5.69046pt{\rm and}\hskip 5.69046ptj=1,...,\infty\,,$ (6)

and

${2\over
L}\sum_{j^{\prime}=1}^{N}\arctan\left({2\Lambda_{j}-2k_{j^{\prime}}\over{\cal{C}}}\right)=q_{j}-{2\over
L}\sum_{j^{\prime}=1}^{N_{\downarrow}}\arctan\left({\Lambda_{j}-\Lambda_{j^{\prime}}\over{\cal{C}}}\right)\hskip
5.69046pt{\rm where}\hskip 5.69046ptq_{j}={2\pi\over L}\,I_{j}^{s}\hskip
5.69046pt{\rm and}\hskip 5.69046ptj=1,...,N_{\uparrow}\,.$ (7)

The $c$ and $s$ band discrete momentum values in those equations,

$q_{j}={2\pi\over L}\,I_{j}^{c}\hskip 5.69046pt{\rm where}\hskip
5.69046pt\hskip 5.69046ptj=1,...,\infty\hskip 5.69046pt{\rm and}\hskip
5.69046ptq_{j}={2\pi\over L}\,I_{j}^{s}\hskip 5.69046pt{\rm where}\hskip
5.69046ptj=1,...,N_{\uparrow}\,,$ (8)

respectively, are directly related to the BA solution quantum numbers
$I_{j}^{c}$ and $I_{j}^{s}$, respectively. Those are such discrete momentum
values in units of $2\pi/L$. They are integers or half-odd integers according
to the following boundary conditions,

$\displaystyle I_{j}^{c}$ $\displaystyle=$ $\displaystyle 0,\pm 1,\pm
2,...\hskip 5.69046pt{\rm for}\hskip 5.69046ptN_{s}=N_{\downarrow}\hskip
5.69046pt{\rm even}$ (9) $\displaystyle=$ $\displaystyle\pm 1/2,\pm
3/2,5/2,...\hskip 5.69046pt{\rm for}\hskip 5.69046ptN_{s}=N_{\downarrow}\hskip
5.69046pt{\rm odd}\,,$

and

$\displaystyle I_{j}^{s}$ $\displaystyle=$ $\displaystyle 0,\pm 1,\pm
2,...\hskip 5.69046pt{\rm for}\hskip 5.69046ptN_{c}-N_{s}=N_{\uparrow}\hskip
5.69046pt{\rm odd}$ (10) $\displaystyle=$ $\displaystyle\pm 1/2,\pm
3/2,5/2,...\hskip 5.69046pt{\rm for}\hskip
5.69046ptN_{c}-N_{s}=N_{\uparrow}\hskip 5.69046pt{\rm even}\,,$

respectively. Hence under transitions from the ground state to one-fermion
removal or addition excited energy eigenstates there may occur shakeup effects
involving overall $\beta=c,s$ band discrete momentum shifts, $q_{j}\rightarrow
q_{j}+2\pi\Phi_{\beta}^{0}/L$. It follows directly from the boundary
conditions, Eqs. (9) and (10), that the non-scattering phase shift
$2\pi\Phi_{\beta}^{0}$ is given by,

$\displaystyle 2\pi\Phi_{c}^{0}$ $\displaystyle=$ $\displaystyle 0\hskip
5.69046pt{\rm for}\hskip 5.69046pt\delta N_{s}=\delta N_{\downarrow}\hskip
5.69046pt{\rm even}\hskip 5.69046pt{\rm and}\hskip
5.69046pt2\pi\Phi_{c}^{0}=\pm\pi\hskip 5.69046pt{\rm for}\hskip
5.69046pt\delta N_{s}=\delta N_{\downarrow}\hskip 5.69046pt{\rm odd}$
$\displaystyle 2\pi\Phi_{s}^{0}$ $\displaystyle=$ $\displaystyle 0\hskip
5.69046pt{\rm for}\hskip 5.69046pt\delta N_{c}-\delta N_{s}=\delta
N_{\uparrow}\hskip 5.69046pt{\rm even}\hskip 5.69046pt{\rm and}\hskip
5.69046pt2\pi\Phi_{s}^{0}=\pm\pi\hskip 5.69046pt{\rm for}\hskip
5.69046pt\delta N_{c}-\delta N_{s}=\delta N_{\uparrow}\hskip 5.69046pt{\rm
odd}\,.$ (11)

The complete set of the model energy eigenstates involves those whose spin
rapidities $\Lambda_{j}$ in Eqs. (6) and (7) are complex numbers. Fortunately,
as mentioned in Sec. I, such states do not contribute to the expressions of
the up-spin and down-spin one-fermion spectral functions in the vicinity of
the singularities studied in this paper.

###  II.2 The $c$ and $s$ pseudoparticle representation

Within the pseudoparticle representation Carmelo_18 , the energy eigenstates
are generated by exclusion-principle occupancy configurations of $N_{c}=N$
charge $c$ pseudoparticles over $j=1,...,\infty$ discrete $c$ band momentum
values $q_{j}$ and $N_{s}=N_{\downarrow}$ spin $s$ pseudoparticles over
$j=1,...,N_{\uparrow}$ discrete $s$ band momentum values $q_{j}$ in Eq. (8).
Hence within that representation each occupied $\beta=c,s$ band discrete
momentum $q_{j}$ corresponds to one $\beta=c,s$ pseudoparticle.

The $\beta=c,s$ band momentum distribution function $N_{\beta}(q_{j})$ reads
$N_{\beta}(q_{j})=1$ and $N_{\beta}(q_{j})=0$ for occupied and unoccupied
discrete momentum values $q_{j}$, respectively. The BA equations, Eqs. (6) and
(7), can then be written in a corresponding functional form as,

$k(q_{j})=q_{j}-{2\over
L}\sum_{j^{\prime}=1}^{N_{\uparrow}}N_{s}(q_{j^{\prime}})\arctan\left({2k(q_{j})-2\Lambda(q_{j^{\prime}})\over{\cal{C}}}\right)\hskip
5.69046pt{\rm where}\hskip 5.69046ptq_{j}={2\pi\over L}\,I_{j}^{c}\hskip
5.69046pt{\rm and}\hskip 5.69046ptj=1,...,\infty\,.$ (12)

and

$\displaystyle{2\over
L}\sum_{j^{\prime}=1}^{\infty}N_{c}(q_{j^{\prime}})\arctan\left({2\Lambda(q_{j})-2k(q_{j^{\prime}})\over{\cal{C}}}\right)$
$\displaystyle=$ $\displaystyle q_{j}-{2\over
L}\sum_{j^{\prime}=1}^{N_{\uparrow}}N_{s}(q_{j^{\prime}})\arctan\left({\Lambda(q_{j})-\Lambda(q_{j^{\prime}})\over{\cal{C}}}\right)$
$\displaystyle{\rm where}$ $\displaystyle q_{j}={2\pi\over L}\,I_{j}^{s}\hskip
5.69046pt{\rm and}\hskip 5.69046ptj=1,...,N_{\uparrow}\,.$ (13)

respectively.

For the excited energy eigenstates that contribute to the one-fermion spectral
weights distributions near singularities studied below in Section III the
numbers $N_{c}$ and $N_{s}$ of $c$ and $s$ pseudoparticles, respectively, and
the number $N_{s}^{h}$ of $s$ band holes are related to those of the
spin-$1/2$ fermions $N=N_{\uparrow}+N_{\downarrow}$, $N_{\uparrow}$, and
$N_{\downarrow}$ as follows,

$N_{c}=N\,,\hskip 5.69046ptN_{s}=N_{\downarrow}\hskip 5.69046pt{\rm and}\hskip
5.69046ptN_{s}^{h}=N_{c}-2N_{s}=N_{\uparrow}-N_{\downarrow}\,,$

so that,

$N=N_{c}\,,\hskip 5.69046ptN_{\uparrow}-N_{\downarrow}=N_{s}^{h}\,,\hskip
5.69046ptN_{\uparrow}=N_{s}+N_{s}^{h}\hskip 5.69046pt{\rm and}\hskip
5.69046ptN_{\downarrow}=N_{s}\,.$

The general energy spectrum of the one-fermion excited energy eigenstates
generated by $c$ and $s$ particle occupancy configurations reads,

$\delta
E=\sum_{\beta=c,s}\sum_{j=1}^{L_{\beta}}\varepsilon_{\beta}(q_{j})\delta
N_{\beta}(q_{j})+{1\over
L}\sum_{\beta=c,s}\sum_{\beta^{\prime}=c,s}\sum_{j=1}^{L_{\beta}}\sum_{j^{\prime}=1}^{L_{\beta^{\prime}}}{1\over
2}\,f_{\beta\,\beta^{\prime}}(q_{j},q_{j^{\prime}})\,\delta
N_{\beta}(q_{j})\delta N_{\beta^{\prime}}(q_{j^{\prime}})\,.$ (14)

Here $L_{c}=L$ is the system length that is given by $L\rightarrow\infty$ in
the thermodynamic limit, $L_{s}=N_{\uparrow}$, and the $\beta=c,s$ momentum
band distribution function deviations read,

$\delta N_{\beta}(q_{j})=N_{\beta}(q_{j})-N^{0}_{\beta}(q_{j})\hskip
5.69046pt{\rm where}\hskip 5.69046ptj=1,...,L_{\beta}\hskip 5.69046pt{\rm
and}\hskip 5.69046pt\beta=c,s\,.$ (15)

$N_{\beta}(q_{j})$ is in this equation the $\beta=c,s$ band pseudoparticle
momentum distribution function for excited states for which the deviation
$\delta N_{\beta}(q_{j})$ is small and thus in the thermodynamic limit involve
a vanishing density of $\beta$ pseudoparticles. The ground-state $\beta=c,s$
band pseudoparticle momentum distribution functions $N^{0}_{\beta}(q_{j})$
also appearing in Eq. (15) are given by,

$N_{c}^{0}(q_{j})=\theta(q_{j}-q_{Fc}^{-})\,\theta(q_{Fc}^{+}-q_{j})\hskip
5.69046pt{\rm and}\hskip
5.69046ptN_{s}^{0}(q_{j})=\theta(q_{j}-q_{Fs}^{-})\,\theta(q_{Fs}^{+}-q_{j})\,,$
(16)

where the distribution $\theta(x)$ reads $\theta(x)=1$ for $x>0$ and
$\theta(x)=0$ for $x\leq 0$. The $\beta=c,s$ Fermi points of the compact and
symmetrical occupancy configurations, Eq. (16), are associated with the Fermi
momentum values $q_{F\beta}^{\pm}$. If within the thermodynamic limit we
ignore unimportant $1/L$ corrections, one may consider that
$q_{F\beta}^{\pm}=\pm q_{F\beta}$ and thus that
$N_{\beta}^{0}(q_{j})=\theta(q_{F\beta}-|q_{j}|)$ for $\beta=c,s$. For
densities $0<n<\infty$ and $0<m<n$ the $\beta=c,s$ Fermi momenta $q_{F\beta}$
are given by,

$q_{Fc}=2k_{F}={\pi\over L}(N-1)\approx\pi\,n\hskip 5.69046pt{\rm and}\hskip
5.69046ptq_{Fs}=k_{F\downarrow}={\pi\over
L}(N_{\downarrow}-1)\approx\pi\,n_{\downarrow}={\pi\over 2}(n-m)\,.$ (17)

Within the thermodynamic limit, the $c$ and $s$ band discrete momentum values,
Eq. (8), such that $q_{j+1}-q_{j}=2\pi/L$, may be replaced by $c$ and $s$ band
continuum momentum variables $q$ and $q^{\prime}$, respectively. (In some
cases the $s$ band momentum $q^{\prime}$ may be denoted by $q$ yet in general
is called $q^{\prime}$.) The ground-state rapidity functions
$k_{0}(q)\in[-\infty,\infty]$ and $\Lambda_{0}(q^{\prime})\in[-\infty,\infty]$
whose domains are $q\in[-\infty,\infty]$ and
$q^{\prime}\in[-k_{F\uparrow},k_{F\uparrow}]$, respectively, are defined in
Eqs. (101)-(108) of Appendix A. In the case of the ground state, the BA
equations, Eqs. (12) and (13), and corresponding BA distributions are given in
Eqs. (101)-(110) of that Appendix.

Moreover, the $c$ and $s$ pseudoparticle energy dispersions in Eq. (14) are
defined as follows,

$\displaystyle\varepsilon_{c}(q)$ $\displaystyle=$
$\displaystyle{\bar{\varepsilon}}_{c}(k_{0}(q))\hskip 5.69046pt{\rm and}\hskip
5.69046pt\varepsilon_{s}(q^{\prime})={\bar{\varepsilon}}_{s}(\Lambda_{0}(q^{\prime}))\hskip
5.69046pt{\rm where}$ $\displaystyle{\bar{\varepsilon}}_{c}(k)$
$\displaystyle=$
$\displaystyle\int_{Q}^{k}dk^{\prime}\,\eta_{c}(k^{\prime})\hskip
5.69046pt{\rm and}\hskip
5.69046pt{\bar{\varepsilon}}_{s}(\Lambda)=\int_{B}^{\Lambda}d\Lambda^{\prime}\,\eta_{s}(\Lambda^{\prime})\,.$
(18)

The distributions $\eta_{c}(\Lambda)$ and $\eta_{s}(\Lambda)$ appearing here
are solutions of the integral equations given in Eqs. (111)-(114) of Appendix
A. In the ${\cal{C}}\rightarrow 0$ limit the $c$ and $s$ pseudoparticle energy
dispersions read,

$\displaystyle\varepsilon_{c}(q)$ $\displaystyle=$ $\displaystyle{q^{2}\over
2}-k_{F\uparrow}^{2}-k_{F\downarrow}^{2}\hskip 5.69046pt{\rm for}\hskip
5.69046pt|q|\leq 2k_{F\downarrow}$ $\displaystyle=$
$\displaystyle(|q|-k_{F\downarrow})^{2}-k_{F\uparrow}^{2}\hskip 5.69046pt{\rm
for}\hskip 5.69046pt|q|\geq 2k_{F\downarrow}$
$\displaystyle\varepsilon_{s}(q^{\prime})$ $\displaystyle=$
$\displaystyle(q^{\prime})^{2}-k_{F\downarrow}^{2}\,,$ (19)

whereas in the ${\cal{C}}\rightarrow\infty$ limit they are given by,

$\displaystyle\varepsilon_{c}(q)$ $\displaystyle=$ $\displaystyle
q^{2}-(2k_{F})^{2}$ $\displaystyle\varepsilon_{s}(q^{\prime})$
$\displaystyle=$ $\displaystyle 0\,.$ (20)

The $c$ and $s$ pseudoparticles have energy residual interactions associated
with the $f$ functions in the second-order terms of the energy functional, Eq.
(14). Such $f$ functions expression given below involves the $c$ and $s$ bands
group velocities,

$v_{c}(q)={\partial\varepsilon_{c}(q)\over\partial q}\hskip 5.69046pt{\rm
and}\hskip
5.69046ptv_{s}(q^{\prime})={\partial\varepsilon_{s}(q^{\prime})\over\partial
q^{\prime}}\,,$ (21)

associated with the energy dispersions, Eq. (18), respectively. They can be
expressed in terms of BA distributions, as given in Eq. (115) of Appendix A.

The $c$ and $s$ band group velocities at the corresponding Fermi points,

$\pm v_{c}\equiv v(\pm q_{Fc})=v(\pm 2k_{F})\hskip 5.69046pt{\rm and}\hskip
5.69046pt\pm v_{s}\equiv v(\pm q_{Fs})=v_{s}(\pm k_{F\downarrow})\,,$ (22)

play an important role, as they are the velocities of the low-energy particle-
hole processes near the $c$ and $s$ bands Fermi points. Their expression in
terms of BA distributions is provided in Eq. (116) of Appendix A.

Moreover, the $f$ functions expression involves the functions
$2\pi\Phi_{\beta,\beta^{\prime}}(q_{j},q_{j^{\prime}})$ defined below in
Section II.3. The latter are related to the residual interactions of the
$\beta$ pseudoparticle or pseudohole of momentum $q_{j}$ with a
$\beta^{\prime}$ pseudoparticle or pseudohole created at momentum
$q_{j^{\prime}}$ under a transition from the ground state to an excited energy
eigenstate. Those processes are behind the momentum function deviations, Eq.
(15), in the energy functional, Eq. (14). Specifically, the $f$ functions
expression reads,

$\displaystyle f_{\beta\,\beta^{\prime}}(q_{j},q_{j^{\prime}})$
$\displaystyle=$ $\displaystyle
v_{\beta}(q_{j})\,2\pi\,\Phi_{\beta,\beta^{\prime}}(q_{j},q_{j^{\prime}})+v_{\beta^{\prime}}(q_{j^{\prime}})\,2\pi\,\Phi_{\beta^{\prime},\beta}(q_{j^{\prime}},q_{j})$
$\displaystyle+$ $\displaystyle{1\over
2\pi}\sum_{\beta^{\prime\prime}=c,s}\sum_{\iota=\pm
1}v_{\beta^{\prime\prime}}\,2\pi\Phi_{\beta^{\prime\prime},\beta}(\iota
q_{F\beta^{\prime\prime}},q_{j})\,2\pi\Phi_{\beta^{\prime\prime},\beta^{\prime}}(\iota
q_{F\beta^{\prime\prime}},q_{j^{\prime}})\,.$

The $c$ and $s$ pseudoparticle energy dispersions, Eq. (18), can be written
as,

$\displaystyle\varepsilon_{c}(q)$ $\displaystyle=$
$\displaystyle\varepsilon_{c}^{0}(q)-\varepsilon_{c}^{0}(2k_{F})\hskip
5.69046pt{\rm for}\hskip 5.69046ptq\in]-\infty,\infty[\hskip 5.69046pt{\rm
and}$ $\displaystyle\varepsilon_{s}(q^{\prime})$ $\displaystyle=$
$\displaystyle\varepsilon_{s}^{0}(q^{\prime})-\varepsilon_{s}^{0}(k_{F\downarrow})\hskip
5.69046pt{\rm for}\hskip 5.69046ptq\in[-k_{F\uparrow},k_{F\uparrow}]\,,$ (23)

respectively. The energy dispersions $\varepsilon_{c}^{0}(q)$ and
$\varepsilon_{s}^{0}(q)$ in this equation fully control the dependence of the
magnetic field $h$ on the spin density $m$ and chemical potential $\mu$ on the
fermionic density $n$. Such dependencies are contained in the following
expressions of the energy scales $2\mu$ and $2\mu_{B}\,h$,

$2\mu=2\varepsilon_{c}^{0}(2k_{F})-\varepsilon_{s}^{0}(k_{F\downarrow})\hskip
5.69046pt{\rm and}\hskip
5.69046pt2\mu_{B}\,h=-\varepsilon_{s}^{0}(k_{F\downarrow})=\varepsilon_{s}(k_{F\uparrow})\,,$
(24)

where $\mu_{B}$ is the Bohr magneton. (See also Eqs. (117) Appendix A.)

For the present spin density interval, $m\in[0,n]$, the magnetic field varies
in the domain $h\in[0,h_{c}]$ where $h_{c}$ is the critical field for fully
polarized ferromagnetism achieved in the $m\rightarrow n$ and
$k_{F\downarrow}\rightarrow 0$ limits. An analytical expression for the energy
scale $2\mu_{B}\,h_{c}$ associated with the critical field $h_{c}$ can be
derived from the use of the expressions of
${\bar{\varepsilon}}_{s}^{\,0}(\Lambda)$ and $\varepsilon_{s}^{\,0}(q)$ in the
$m\rightarrow n$ limit given in Eqs. (138) and (139) of Appendix B,
respectively. It reads,

$2\mu_{B}h_{c}=-\varepsilon_{s}^{0}(0)|_{k_{F\downarrow}=0}={1\over
2\pi}\left({\cal{C}}^{2}+(2\pi n)^{2}\right)\arctan\left({2\pi
n\over{\cal{C}}}\right)-{\cal{C}}\,n\,.$ (25)

Its limiting behaviors are given in Eqs. (140) and (141) of Appendix B. (In
that Appendix simplified expressions in the fully polarized ferromagnetism
limit of several physical quantities are provided.)

###  II.3 The related $c$ and $s$ particle representation and corresponding
phase shifts

For the 1D repulsive fermion model in the subspace populated only by $c$ and
$s$ pseudoparticles considered here, the BA $c$ and $s$ rapidity functions
$k(q_{j})$ and $\Lambda(q_{j})$ of the excited energy eigenstates, which are
solutions of the BA equations, Eqs. (12) and (13), can be expressed in terms
of those of the corresponding initial ground state, $k_{0}(q)$ and
$\Lambda_{0}(q)$, respectively, defined in Eqs. (101)-(108) of Appendix A.
Specifically, $k(q_{j})=k_{0}({\bar{q}}_{j})$ and
$\Lambda(q_{j})=\Lambda_{0}({\bar{q}}_{j})$.

The set of $j=1,...,L_{\beta}$ values ${\bar{q}}_{j}={\bar{q}}(q_{j})$ in such
excited energy eigenstates rapidity expressions
$k(q_{j})=k_{0}({\bar{q}}(q_{j}))$ and
$\Lambda(q_{j})=\Lambda_{0}({\bar{q}}(q_{j}))$ are the $\beta=c,s$ band
discrete canonical momentum values. They read,

${\bar{q}}_{j}={\bar{q}}(q_{j})=q_{j}+{2\pi\Phi_{\beta}(q_{j})\over
L}={2\pi\over L}\left(I^{\beta}_{j}+\Phi_{\beta}(q_{j})\right)\hskip
5.69046pt{\rm where}\hskip 5.69046ptj=1,...,L_{\beta}\hskip 5.69046pt{\rm
and}\hskip 5.69046pt\beta=c,s\,.$ (26)

Here ${\bar{q}}_{j+1}-{\bar{q}}_{j}=2\pi/L+{\rm h.o.}$ where h.o. stands for
contributions of second order in $1/L$. The function $2\pi\Phi_{\beta}(q_{j})$
in Eq. (26) is defined below.

We call a $\beta=c,s$ particle each of the $N_{\beta}$ occupied $\beta$-band
discrete canonical momentum values ${\bar{q}}_{j}$ Carmelo_05 ; LE . We call a
$\beta$ hole the remaining $N_{\beta}^{h}$ unoccupied $\beta$-band discrete
canonical momentum values ${\bar{q}}_{j}$ of an excited energy eigenstate. (In
the case of the related 1D Hubbard model PDT, such particles were rather
called pseudofermions Carmelo_05 ; Carmelo_18 ; Carmelo_17 .) There is a $c$
and $s$ particle representation for each initial ground state and its excited
states. This holds for all fermionic and spin densities.

The set of $L_{\beta}$ discrete $\beta=c,s$ bare momentum values
$\\{q_{j}\\}$, Eq. (8), and the corresponding set of $L_{\beta}$ discrete
$\beta=c,s$ canonical momentum values $\\{{\bar{q}}_{j}\\}$, Eq. (26), are
equally ordered. This is because $I^{\beta}_{j+1}-I^{\beta}_{j}=1$ and
$\Phi_{\beta}(q_{j+1})-\Phi_{\beta}(q_{j})={\cal{O}}(1/L)$ in
${\bar{q}}_{j}={2\pi\over L}(I^{\beta}_{j}+\Phi_{\beta}(q_{j}))$. For
simplicity, in the case of some $q_{j}$ dependent physical quantities one then
often associates in the thermodynamic limit the bare momentum $q_{j}$ to the
$\beta=c,s$ particle of canonical momentum ${\bar{q}}_{j}={2\pi\over
L}(I^{\beta}_{j}+\Phi_{\beta}(q_{j}))$. (This is in spite of $q_{j}$ being the
discrete momentum value of the $\beta=c,s$ pseudoparticle which is transformed
into the $\beta=c,s$ particle under the $\beta$ pseudoparticle - $\beta$
particle unitary transformation.) Moreover, if in the thermodynamic limit one
replaces the sets of discrete $c$ and $s$ bare momentum values $\\{q_{j}\\}$
and $\\{q_{j}^{\prime}\\}$ by continuous momentum variables
$q\in[-\infty,\infty]$ and $q^{\prime}\in[-k_{F\uparrow},k_{F\uparrow}]$,
respectively, the corresponding sets of discrete $c$ and $s$ canonical
momentum values $\\{{\bar{q}}_{j}\\}$ and $\\{{\bar{q}}_{j}^{\prime}\\}$ are
replaced by undistinguishable continuous momentum variables. For the excited
states considered here, the exception is at the $c$ and $s$ bands Fermi
points, which are slightly shifted under the $q_{j}\rightarrow{\bar{q}}_{j}$
and $q_{j}^{\prime}\rightarrow{\bar{q}}_{j}^{\prime}$ unitary transformation,
respectively. (See Eq. (29) below.)

An example of such $q_{j}$ dependent physical quantities is
$2\pi\Phi_{\beta}(q_{j})$ in Eq. (26). It is a functional that involves the
deviations $\delta N_{\beta^{\prime}}(q_{j^{\prime}})$ defined in Eq. (15) and
reads,

$2\pi\Phi_{\beta}(q_{j})=\sum_{\beta^{\prime}=c,s}\,\sum_{j^{\prime}=1}^{L_{\beta^{\prime}}}\,2\pi\Phi_{\beta,\beta^{\prime}}(q_{j},q_{j^{\prime}})\,\delta
N_{\beta^{\prime}}(q_{j^{\prime}})\hskip 5.69046pt{\rm where}\hskip
5.69046ptj=1,...,L_{\beta}\hskip 5.69046pt{\rm and}\hskip
5.69046pt\beta=c,s\,,$ (27)

and $2\pi\Phi_{\beta,\beta^{\prime}}(q_{j},q_{j^{\prime}})$ is for $\beta=c,s$
and $\beta^{\prime}=c,s$ given by,

$\displaystyle 2\pi\Phi_{s,s}\left(q,q^{\prime}\right)$ $\displaystyle=$
$\displaystyle
2\pi\bar{\Phi}_{s,s}\left({2\Lambda(q)\over{\cal{C}}},{2\Lambda(q^{\prime})\over{\cal{C}}}\right)\,;\hskip
11.38092pt2\pi\Phi_{s,c}\left(q,q^{\prime}\right)=2\pi\bar{\Phi}_{s,c}\left({2\Lambda(q)\over{\cal{C}}},{2k(q^{\prime})\over{\cal{C}}}\right)$
$\displaystyle 2\pi\Phi_{c,c}\left(q,q^{\prime}\right)$ $\displaystyle=$
$\displaystyle
2\pi\bar{\Phi}_{c,c}\left({2k(q)\over{\cal{C}}},{2k(q^{\prime})\over{\cal{C}}}\right)\,;\hskip
11.38092pt2\pi\Phi_{c,s}\left(q,q^{\prime}\right)=2\pi\bar{\Phi}_{c,s}\left({2k(q)\over{\cal{C}}},{2\Lambda(q^{\prime})\over{\cal{C}}}\right)\,.$
(28)

The quantities on the right-hand side of these equations are functions of the
rapidity-related variables $r=2k/{\cal{C}}$ for the $c$ band and
$r=2\Lambda/{\cal{C}}$ for the $s$ band. They are uniquely defined by the
integral equations given in Eqs. (126)-(131) of Appendix A. (In such equations
they appear in units of $2\pi$.)

In the $c$ and $s$ particle representation,
$2\pi\Phi_{\beta,\beta^{\prime}}(q_{j},q_{j^{\prime}})$, Eq. (28), has a
precise physical meaning:
$2\pi\Phi_{\beta,\beta^{\prime}}(q_{j},q_{j^{\prime}})$ (and
$-2\pi\Phi_{\beta,\beta^{\prime}}(q_{j},q_{j^{\prime}})$) is the phase shift
acquired by a $\beta=c,s$ particle or hole of canonical momentum
${\bar{q}}_{j}={\bar{q}}(q_{j})$ upon scattering off a $\beta^{\prime}=c,s$
particle (and $\beta^{\prime}=c,s$ hole) of canonical momentum value
${\bar{q}}_{j^{\prime}}={\bar{q}}(q_{j^{\prime}})$ created under a transition
from the ground state to an excited energy eigenstate. For simplicity, in the
thermodynamic limit one often says it to be the phase shift acquired by a
$\beta=c,s$ particle or hole of momentum $q_{j}$ upon scattering off a
$\beta^{\prime}=c,s$ particle (and $\beta^{\prime}=c,s$ hole) of momentum
$q_{j^{\prime}}$. Indeed,
$2\pi\Phi_{\beta,\beta^{\prime}}(q_{j},q_{j^{\prime}})$ is expressed in terms
of those bare momentum values.

Such a phase shift is thus imposed to the $\beta=c,s$ particle or hole
scatterer by the $\beta^{\prime}=c,s$ particle or hole created under such a
transition, which plays the role of mobile scattering center. Within the MQIM
the latter is called a mobile quantum impurity.

It then follows that the functional $2\pi\Phi_{\beta}(q_{j})$, Eq. (27), in
the $\beta=c,s$ canonical momentum expression ${\bar{q}}_{j}=q_{j}+{2\pi\over
L}\Phi_{\beta}(q_{j})$, Eq. (26), is the phase shift acquired by a $\beta$
particle or hole of canonical momentum value ${\bar{q}}_{j}={\bar{q}}(q_{j})$
(or momentum value $q_{j}$) upon scattering off the set of $\beta^{\prime}$
particles and $\beta^{\prime}$ holes created under such a transition. Hence
the $\beta$ particle phase shift $2\pi\Phi_{\beta}(q_{j})$ has a specific
value for each ground-state - excited-state transition.

The overall phase shift,

$2\pi\Phi_{\beta}^{T}(q_{j})=2\pi\Phi_{\beta}^{0}+2\pi\Phi_{\beta}(q_{j})\hskip
5.69046pt{\rm where}\hskip 5.69046ptj=1,...,L_{\beta}\hskip 5.69046pt{\rm
and}\hskip 5.69046pt\beta=c,s\,,$

involves both a non-scattering term $2\pi\Phi_{\beta}^{0}$, Eq. (11), and the
scattering term $2\pi\Phi_{\beta}(q_{j})$, Eq. (27).

The scattering functional $2\pi\Phi_{\beta}(q_{j})$, Eq. (27), and the overall
functional $2\pi\Phi_{\beta}^{T}(q_{j})$ fully determine the deviations of the
$\beta=c,s$ Fermi canonical momentum values under transitions from the ground
state to one-fermion excited states as follows,

$\delta{\bar{q}}_{F\beta}^{\iota}=\left(\iota\,\delta
N^{0,F}_{\beta,\iota}+\Phi_{\beta}^{T}(\iota q_{F\beta})\right){2\pi\over
L}=\left(\iota\,\delta N^{F}_{\beta,\iota}+\Phi_{\beta}(\iota
q_{F\beta})\right){2\pi\over L}\hskip 5.69046pt{\rm where}\hskip
5.69046pt\beta=c,s\hskip 5.69046pt{\rm and}\hskip 5.69046pt\iota=\pm 1\,.$
(29)

Here such functionals appear in units of $2\pi$, $q_{Fc}=2k_{F}$ and
$q_{Fs}=k_{F\downarrow}$ as given in Eq. (17), $\delta
N^{F}_{\beta,\iota}=\delta N^{0,F}_{\beta,\iota}+\iota\,\Phi_{\beta}^{0}$ is
the deviation in the number of right $(\iota=1)$ and left $(\iota=-1)$ $\beta$
particles at the corresponding $\beta,\iota$ Fermi point, and $\delta
N^{0,F}_{\beta,\iota}$ is such a deviation without accounting for the effects
of the non-scattering phase shift $2\pi\Phi_{\beta}^{0}$, Eq. (11). Hence
while $\delta N^{0,F}_{\beta,\iota}$ either vanishes or is a positive or
negative integer number, the deviation $\delta N^{F}_{\beta,\iota}$ may be a
positive or negative half-odd integer number. Indeed, $2\pi\Phi_{\beta}^{0}$
has in units of $2\pi$ the values $\Phi_{\beta}^{0}=0,\pm 1/2$.

The exponents in the one-fermion spectral functions power-law expressions
given below in Sections III and V have different expressions in the high-
energy regime and in the low-energy TLL regime, respectively. On the one hand,
the TLL and the crossover to TLL regimes involve processes in the $c$ and $s$
bands whose continuum momentum absolute values are in the intervals
$|q|\in[2k_{F}-k_{Fc}^{0},2k_{F}+k_{Fc}^{0}]$ and
$|q^{\prime}|\in[k_{F\downarrow}-k_{Fs}^{0},k_{F\downarrow}+k_{Fs}^{0}]$,
respectively. Here $k_{Fc}^{0}/2k_{F}\ll 1$ and $k_{Fs}^{0}/k_{F\downarrow}\ll
1$. On the other hand, the high-energy regime involves processes in the
complementary $c$ band momentum intervals
$q\in[-2k_{F}+k_{Fc}^{0},2k_{F}-k_{Fc}^{0}]$,
$q\in]-\infty,-2k_{F}-k_{Fc}^{0}]$, and $q\in[2k_{F}+k_{Fc}^{0},\infty[$ and
$s$ band momentum intervals
$q^{\prime}\in[-k_{F\downarrow}+k_{Fs}^{0},k_{F\downarrow}-k_{Fs}^{0}]$,
$q^{\prime}\in[-k_{F\uparrow},-k_{F\downarrow}-k_{Fs}^{0}]$, and
$q\in[k_{F\downarrow}+k_{Fs}^{0},k_{F\uparrow}]$.

The one-fermion spectral functions exponents expressions studied below in
Section III involve the following general functionals, which are merely the
square of the Fermi canonical momentum value deviations, Eq. (29), in units of
$2\pi/L$,

$2\Delta_{\beta}^{\iota}=\left({\delta{\bar{q}}_{F\beta}^{\iota}\over(2\pi/L)}\right)^{2}=\left(\iota\delta
N^{F}_{\beta,\iota}+\Phi_{\beta}(\iota q_{F\beta})\right)^{2}\hskip
5.69046pt{\rm where}\hskip 5.69046pt\beta=c,s\hskip 5.69046pt{\rm and}\hskip
5.69046pt\iota=\pm 1\,.$ (30)

Finally, expression of the energy functional, Eq. (14), in the $c$ and $s$
particle representation involves the $\beta=c,s$ bands discrete canonical
momentum values ${\bar{q}}_{j}={\bar{q}}(q_{j})$, Eq. (26). One finds after
some algebra that in such a representation it reads up to ${\cal{O}}(1/L)$
order,

$\delta
E=\sum_{\beta=c,s}\sum_{j=1}^{L_{\beta}}\varepsilon_{\beta}({\bar{q}}_{j})\,\delta{\cal{N}}_{\beta}({\bar{q}}_{j})\,.$
(31)

Here $\delta{\cal{N}}_{\beta}({\bar{q}}_{j})=N_{\beta}(q_{j})$ and the
$\beta=c,s$ particle energy dispersions $\varepsilon_{\beta}({\bar{q}}_{j})$
have exactly the same form as those given in Eq. (18) with the bare momentum,
$q_{j}$, replaced by the corresponding canonical momentum,
${\bar{q}}_{j}={\bar{q}}(q_{j})$.

In contrast to the equivalent pseudoparticle energy functional, Eq. (14), that
in Eq. (31) has no energy interaction terms of second-order in the deviations
$\delta{\cal{N}}_{\beta}({\bar{q}}_{j})$. This has a deep physical meaning:
The $\beta=c,s$ particles generated from corresponding $\beta=c,s$
pseudoparticles by a $q_{j}\rightarrow{\bar{q}}_{j}$ uniquely defined unitary
transformation have no such interactions up to ${\cal{O}}(1/L)$ order.

Within the present thermodynamic limit, only finite-size corrections up to
that order are relevant for the spectral functions expressions. The property
that the excitation energy spectrum, Eq. (31), has no $c$ and $s$ particle
energy interactions plays a key role in the derivation by the PDT of the
general one-fermion spectral functions used below in our studies of Section
III. Indeed it allows them to be expressed in terms of a sum of convolutions
of $c$ and $s$ particle spectral functions. Moreover, the spectral weights of
the latter spectral functions can be expressed as Slater determinants of $c$
and $s$ particles operators.

Such spectral weights involve the functionals, Eq. (30), determined by the
Fermi canonical momentum value deviations, Eq. (29). Since the derivation
within the PDT of the general one-fermion spectral functions used below in
Section III is similar to that of other integrable models Carmelo_18 ;
Carmelo_16 ; Carmelo_15 , it is not reported in this paper.

##  III General types of one-fermion spectral singularities

###  III.1 The two-dimensional $(k,\omega)$-plane spectra where the one-
fermion spectral singularities are contained

The two-parametric excitation processes that are behind $(k,\omega)$-plane
one-fermion spectral weight distribution near the singularity branch lines and
boundary lines defined below involve both creation of one charge or hole
particle and creation of one spin or hole particle.

In contrast to the related lattice 1D Hubbard model Carmelo_18 , all charge
excitations of the present continuous model only involve real BA rapidities.
At finite magnetic field the same applies to spin part of the one-fermion
excitations studied in this paper whereas at zero magnetic field they involve
as well complex BA spin rapidities. Fortunately, due to both the spin $SU(2)$
symmetry and the lack of a spin energy gap at zero spin density, the same
zero-spin-density spectral-function expressions in the vicinity of the
singularities are reached by taking the limit of zero magnetic field in the
corresponding suitable finite-field expressions or by their direct derivation
at zero magnetic field. The former method used in this paper has the advantage
of only involving real BA spin rapidities.

At zero spin density, $m=0$, the transitions from the ground state under one-
fermion removal lead to excited energy eigenstates associated with the two-
parametric processes whose number deviations relative to those of the initial
ground state read,

$\delta N_{c}=-1\,;\hskip 5.69046pt\delta J_{c}^{F}=\pm 1/2\,;\hskip
5.69046pt\delta N_{s}=-1\,;\hskip 5.69046pt\delta J_{s}^{F}=0\,.$ (32)

Both the $\beta=c,s$ Fermi points current number deviations $\delta
J^{F}_{\beta}$ in this equation and the $\beta=c,s$ Fermi points number
deviations $\delta N^{F}_{\beta}$ are defined in terms of $\beta=c,s$ left
($\iota=-1$) and right ($\iota=+1$) Fermi points number deviations $\delta
N^{F}_{\beta,\iota}$ as follows,

$\delta N^{F}_{\beta}=\sum_{\iota=\pm 1}\delta N^{F}_{\beta,\iota}\hskip
5.69046pt{\rm and}\hskip 5.69046pt\delta J^{F}_{\beta}={1\over
2}\sum_{\iota=\pm 1}(\iota)\delta N^{F}_{\beta,\iota}\hskip 5.69046pt{\rm
for}\hskip 5.69046pt\beta=c,s$ (33)

The number deviations $\delta N^{F}_{c}$ and $\delta N^{F}_{s}$ are particular
cases of those given in Eq. (32). The latter refer to $c$ and $s$ band
momentum values, respectively, at and away of the Fermi points. While the
$\beta=c,s$ Fermi points number deviations and current number deviations
$\delta N^{F}_{\beta}$ and $\delta J^{F}_{\beta}$, respectively, defined in
Eq. (33) correspond to number fluctuations at the $\beta=c,s$ Fermi points,
one denotes by $\delta N_{\beta}^{NF}$ the $\beta=c,s$ the number deviations
that refer to creation of $\beta=c,s$ particles or holes away from those Fermi
points.

The $\omega<0$ energy spectrum of such excitations is of the form
$-\omega=\omega_{R}(k)=-\varepsilon_{c}(q)-\varepsilon_{s}(q^{\prime})$. It
has the following two branches,

$\displaystyle\omega_{R}(k)$ $\displaystyle=$
$\displaystyle-\varepsilon_{c}(q)-\varepsilon_{s}(q^{\prime})\hskip
5.69046pt{\rm where}\hskip 5.69046ptk=\pm 2k_{F}-q-q^{\prime}\hskip
5.69046pt{\rm for}$ $\displaystyle q$ $\displaystyle\in$
$\displaystyle[-2k_{F},2k_{F}]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq^{\prime}\in[-k_{F},k_{F}]\hskip 5.69046pt{\rm with}\hskip
5.69046pt{\rm intervals}$ $\displaystyle k$ $\displaystyle=$ $\displaystyle
2k_{F}-q-q^{\prime}\in[-k_{F},5k_{F}]\hskip 5.69046pt{\rm for}\hskip
5.69046pt{\rm branch}\hskip 5.69046ptA\,,$ $\displaystyle k$ $\displaystyle=$
$\displaystyle-2k_{F}-q-q^{\prime}\in[-5k_{F},k_{F}]\hskip 5.69046pt{\rm
for}\hskip 5.69046pt{\rm branch}\hskip 5.69046ptB\,.$ (34)

Here the two alternative contributions $\pm 2k_{F}$ to the excitation momentum
$k=\pm 2k_{F}-q-q^{\prime}$ result from $c$ band momentum shifts
$2\pi\Phi_{c}^{0}=\pm\pi$ in Eq. (11), such that ${2\pi\over
L}\Phi_{c}^{0}\times N_{c}=\pm\pi n=\pm 2k_{F}$ where $N_{c}$ is the number of
$c$ band Fermi sea occupied discrete momentum values $q_{j}$, Eq. (8). In
contrast, $2\pi\Phi_{s}^{0}=0$ for the present one-fermion removal
excitations.

At zero spin density, $m=0$, the transitions under one-fermion addition lead
to excited energy eigenstates associated with the two-parametric processes
whose number deviations relative to those of the initial ground state are
given by,

$\delta N_{c}=1\,;\hskip 5.69046pt\delta J_{c}^{F}=0\,;\hskip 5.69046pt\delta
N_{s}=0\,;\hskip 5.69046pt\delta J_{s}^{F}=\pm 1/2\,.$ (35)

The $\omega>0$ energy spectrum of such excitations is given by
$\omega=\omega_{A}(k)=\varepsilon_{c}(q)-\varepsilon_{s}(q^{\prime})$. It has
again two branches,

$\displaystyle\omega_{A}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{c}(q)-\varepsilon_{s}(q^{\prime})\hskip
5.69046pt{\rm where}\hskip 5.69046ptk=q-q^{\prime}\hskip 5.69046pt{\rm and}$
$\displaystyle k$ $\displaystyle\in$ $\displaystyle[k_{F},\infty]\hskip
5.69046pt{\rm for}\hskip 5.69046ptq\in[2k_{F},\infty]\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq^{\prime}\in[-k_{F},k_{F}]\hskip 5.69046pt{\rm for}\hskip
5.69046pt{\rm branch}\hskip 5.69046ptA$ $\displaystyle k$ $\displaystyle\in$
$\displaystyle[-\infty,-k_{F}]\hskip 5.69046pt{\rm for}\hskip
5.69046ptq\in[-\infty,-2k_{F}]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq^{\prime}\in[-k_{F},k_{F}]\hskip 5.69046pt{\rm for}\hskip
5.69046pt{\rm branch}\hskip 5.69046ptB\,.$ (36)

The transitions under up-spin one-fermion removal at spin density $m>0$ lead
to excited energy eigenstates associated with the two-parametric processes
whose number deviations relative to those of the initial ground state read,

$\delta N_{c}=-1\,;\hskip 5.69046pt\delta J_{c}^{F}=0\,;\hskip 5.69046pt\delta
N_{s}=0\,;\hskip 5.69046pt\delta J_{s}^{F}=\pm 1/2\,.$ (37)

Moreover, in general $\delta N_{s}^{F}=-1$ and $\delta N_{s}^{NF}=1$ with a
limiting case being $\delta N_{s}^{F}=\delta N_{s}^{NF}=0$.

The corresponding $\omega<0$ energy spectrum of such excitations reads
$-\omega=\omega_{R}^{\uparrow}(k)=-\varepsilon_{c}(q)+\varepsilon_{s}(q^{\prime})$.
It has the following two branches,

$\displaystyle\omega_{R}^{\uparrow}(k)$ $\displaystyle=$
$\displaystyle-\varepsilon_{c}(q)+\varepsilon_{s}(q^{\prime})\hskip
5.69046pt{\rm where}\hskip 5.69046ptk=-q+q^{\prime}\hskip 5.69046pt{\rm and}$
$\displaystyle k$ $\displaystyle\in$
$\displaystyle[-k_{F\uparrow},(2k_{F}+k_{F\uparrow})]\hskip 5.69046pt{\rm
for}\hskip 5.69046ptq\in[-2k_{F},2k_{F}]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq^{\prime}\in[k_{F\downarrow},k_{F\uparrow}]\hskip 5.69046pt{\rm
for}\hskip 5.69046pt{\rm branch}\hskip 5.69046ptA$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle[-(2k_{F}+k_{F\uparrow}),k_{F\uparrow}]\hskip
5.69046pt{\rm for}\hskip 5.69046ptq\in[-2k_{F},2k_{F}]\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq^{\prime}\in[-k_{F\uparrow},-k_{F\downarrow}]\hskip
5.69046pt{\rm for}\hskip 5.69046pt{\rm branch}\hskip 5.69046ptB\,.$ (38)

The transitions under up-spin one-fermion addition at spin density $m>0$ give
rise to excited energy eigenstates associated with the two-parametric
processes whose number deviations relative to those of the initial ground
state are given in Eq. (35).

The $\omega>0$ energy spectrum of such excitations spectrum is given by
$\omega=\omega_{A}^{\uparrow}(k)=\varepsilon_{c}(q)-\varepsilon_{s}(q^{\prime})$.
It has again two branches,

$\displaystyle\omega_{A}^{\uparrow}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{c}(q)-\varepsilon_{s}(q^{\prime})\hskip
5.69046pt{\rm for}\hskip 5.69046ptk=q-q^{\prime}\hskip 5.69046pt{\rm and}$
$\displaystyle k$ $\displaystyle\in$
$\displaystyle[k_{F\uparrow},\infty]\hskip 5.69046pt{\rm for}\hskip
5.69046ptq\in[2k_{F},\infty]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq^{\prime}\in[-k_{F\downarrow},k_{F\downarrow}]\hskip 5.69046pt{\rm
for}\hskip 5.69046pt{\rm branch}\hskip 5.69046ptA$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle[-\infty,-k_{F\uparrow}]\hskip 5.69046pt{\rm
for}\hskip 5.69046ptq\in[-\infty,-2k_{F}]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq^{\prime}\in[-k_{F\downarrow},k_{F\downarrow}]\hskip 5.69046pt{\rm
for}\hskip 5.69046pt{\rm branch}\hskip 5.69046ptB\,.$ (39)

The transitions under down-spin one-fermion removal lead at spin density $m>0$
to excited energy eigenstates associated with the two-parametric processes
whose number deviations relative to those of the initial ground state are
given in Eq. (32).

The $\omega<0$ energy spectrum of such excitations is of the form
$-\omega=\omega_{R}^{\downarrow}(k)=-\varepsilon_{c}(q)-\varepsilon_{s}(q^{\prime})$.
It has two branches corresponding to $\iota=\pm 1$,

$\displaystyle\omega_{R}^{\downarrow}(k)$ $\displaystyle=$
$\displaystyle-\varepsilon_{c}(q)-\varepsilon_{s}(q^{\prime})\hskip
5.69046pt{\rm for}\hskip 5.69046ptk=\pm 2k_{F}-q-q^{\prime}\hskip
5.69046pt{\rm where}\hskip 5.69046ptq\in[-2k_{F},2k_{F}]\hskip 5.69046pt{\rm
and}\hskip 5.69046ptq^{\prime}\in[-k_{F\downarrow},k_{F\downarrow}]\hskip
5.69046pt{\rm with}\hskip 5.69046pt{\rm intervals}$ $\displaystyle k$
$\displaystyle=$ $\displaystyle
2k_{F}-q-q^{\prime}\in[-k_{F\downarrow},(4k_{F}+k_{F\uparrow})]\hskip
5.69046pt{\rm for}\hskip 5.69046pt{\rm branch}\hskip 5.69046ptA$
$\displaystyle k$ $\displaystyle=$
$\displaystyle-2k_{F}-q-q^{\prime}\in[-(4k_{F}+k_{F\uparrow}),k_{F\downarrow}]\hskip
5.69046pt{\rm for}\hskip 5.69046pt{\rm branch}\hskip 5.69046ptB\,.$ (40)

Finally, the transitions under down-spin one-fermion addition give rise at
spin density $m>0$ to excited energy eigenstates associated with the two-
parametric processes whose number deviations relative to those of the initial
ground state are given by,

$\delta N_{c}=1\,;\hskip 5.69046pt\delta J_{c}^{F}=\pm 1/2\,;\hskip
5.69046pt\delta N_{s}=1\,;\hskip 5.69046pt\delta J_{s}^{F}=0\,.$

The $\omega>0$ energy spectrum of such excitations reads
$\omega=\omega_{A}^{\downarrow}(k)=\varepsilon_{c}(q)+\varepsilon_{s}(q^{\prime})$.
It has four branches,

$\displaystyle\omega_{A}^{\downarrow}(k)$ $\displaystyle=$
$\displaystyle\varepsilon_{c}(q)+\varepsilon_{s}(q^{\prime})\hskip
5.69046pt{\rm for}\hskip 5.69046ptk=\pm 2k_{F}+q+q^{\prime}\hskip
5.69046pt{\rm and}\hskip 5.69046pt{\rm sgn}\\{q^{\prime}\\}=\pm\hskip
5.69046pt{\rm for}\hskip 5.69046ptq^{\prime}\neq 0\hskip 5.69046pt{\rm
with}\hskip 5.69046pt{\rm intervals}$ $\displaystyle k$ $\displaystyle=$
$\displaystyle 2k_{F}+q+q^{\prime}\in[(4k_{F}+k_{F\uparrow}),\infty]\hskip
5.69046pt{\rm for}\hskip 5.69046pt{\rm branch}\hskip 5.69046ptA$
$\displaystyle\hskip 5.69046pt{\rm where}\hskip
5.69046ptq\in[2k_{F},\infty]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq^{\prime}\in[k_{F\downarrow},k_{F\uparrow}]$ $\displaystyle k$
$\displaystyle=$ $\displaystyle
2k_{F}+q+q^{\prime}\in[-\infty,k_{F\uparrow}]\hskip 5.69046pt{\rm for}\hskip
5.69046pt{\rm branch}\hskip 5.69046ptB$ $\displaystyle\hskip 5.69046pt{\rm
where}\hskip 5.69046ptq\in[-\infty,-2k_{F}]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq^{\prime}\in[k_{F\downarrow},k_{F\uparrow}]$ $\displaystyle k$
$\displaystyle=$
$\displaystyle-2k_{F}+q+q^{\prime}\in[-\infty,-(4k_{F}+k_{F\uparrow})]\hskip
5.69046pt{\rm for}\hskip 5.69046pt{\rm branch}\hskip 5.69046ptA^{\prime}$
$\displaystyle\hskip 5.69046pt{\rm where}\hskip
5.69046ptq\in[-\infty,-2k_{F}]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq^{\prime}\in[-k_{F\uparrow},-k_{F\downarrow}]$ $\displaystyle k$
$\displaystyle=$
$\displaystyle-2k_{F}+q+q^{\prime}\in[-k_{F\uparrow},\infty]\hskip
5.69046pt{\rm for}\hskip 5.69046pt{\rm branch}\hskip 5.69046ptB^{\prime}$ (41)
$\displaystyle\hskip 5.69046pt{\rm where}\hskip
5.69046ptq\in[2k_{F},\infty]\hskip 5.69046pt{\rm and}\hskip
5.69046ptq^{\prime}\in[-k_{F\uparrow},-k_{F\downarrow}]\,.$

In the present case of the one-fermion spectral functions, Eq. (2), and of the
up-spin and down-spin one-fermion spectral functions, Eq. (4), the one-
parametric branch lines that for some momentum subdomains correspond to
singularities are contained in the two-parametric spectra, Eqs. (32)-(36),
and, Eqs. (37)-(41), respectively. Spectra that do not contain singularities
generated by higher-order $c$ and $s$ particle processes and/or complex spin
rapidities are not considered in our present study.

###  III.2 The $(k,\omega)$-plane one-fermion spectral singularities on the
$c$ and $s$ branch lines

A branch line results from transitions to a well-defined subclass of the
excited energy eigenstates associated with such spectra. At zero spin density,
$m=0$, the one-parametric $(k,\omega)$-plane $\beta=c,s$ branch line spectrum
has the general form,

$\omega_{\beta}(k)=\pm\varepsilon_{\beta}(q)\geq 0\hskip 5.69046pt{\rm
where}\hskip 5.69046ptk=k_{0}\pm q\hskip 5.69046pt{\rm for}\hskip
5.69046pt\delta N_{\beta}(q)=\pm 1\hskip 5.69046pt{\rm and}\hskip
5.69046pt\beta=c,s\,.$ (42)

Here $\varepsilon_{\beta}(q)$ is the $\beta=c,s$ band energy dispersion, Eq.
(18), the momentum distribution function deviation, Eq. (15), reads $\delta
N_{\beta}(q)=+1$ and $\delta N_{\beta}(q)=-1$ for a particle and hole $\beta$
branch line, respectively. The momentum $k_{0}$ in Eq. (42) is given by,

$k_{0}=4k_{F}\,\delta J_{c}^{F}+2k_{F}\,\delta J_{s}^{F}\,.$ (43)

At zero spin density the branch-line singularities are found below to occur at
some $k$ intervals of two $c$ branch lines called $c^{+}$ and $c^{-}$ branch
lines, respectively, which refer to different subdomains of $q$ in $k=k_{0}\pm
q$, and at some $k$ intervals of one $s$ branch line.

The one-parametric $(k,\omega)$-plane $\beta=c,s$ branch line spectrum has for
spin densities $m>0$ a similar general form,

$\omega_{\beta}^{\sigma}(k)=\pm\varepsilon_{\beta}(q)\geq 0\hskip
5.69046pt{\rm where}\hskip 5.69046ptk=k_{0}\pm q\hskip 5.69046pt{\rm
for}\hskip 5.69046pt\delta N_{\beta}(q)=\pm 1\hskip 5.69046pt{\rm and}\hskip
5.69046pt\beta=c,s\,,$ (44)

where now $\sigma=\uparrow$ and $\sigma=\downarrow$ refers to the up-spin and
down-spin one-fermion spectral function, respectively, and the momentum
$k_{0}$, Eq. (43), more generally reads,

$k_{0}=4k_{F}\,\delta J_{c}^{F}+2k_{F\downarrow}\,\delta J_{s}^{F}\,.$ (45)

At finite spin density the branch-line singularities of the up-spin and down-
spin one-fermion spectral functions are found to occur at some $k$ intervals
of two $c$ branch lines called again $c^{+}$ and $c^{-}$ branch lines,
respectively, which correspond to different subdomains of the $c$ band
momentum $q$ in the excitation momentum expression $k=k_{0}\pm q$, and at some
$k$ intervals of one $s$ branch line.

For one-fermion excitations at zero spin density, the use of the PDT leads in
the case of the present model to the following general high-energy behavior in
the vicinity of a $\beta=c,s$ branch line,

$\displaystyle B_{\gamma}(k,\omega)$ $\displaystyle=$ $\displaystyle
C_{\gamma,\beta}\,\Bigl{(}\gamma\,\omega-\omega_{\beta}(k)\Bigr{)}^{\xi_{\beta}(k)}\hskip
5.69046pt{\rm for}\hskip 5.69046pt(\gamma\,\omega-\omega_{\beta}(k))\geq
0\hskip 5.69046pt{\rm and}\hskip 5.69046pt\gamma=\pm 1\hskip 5.69046pt{\rm
where}$ $\displaystyle\xi_{\beta}(k)$ $\displaystyle=$
$\displaystyle-1+2\Delta_{c}^{+1}+2\Delta_{c}^{-1}+2\Delta_{s}^{+1}+2\Delta_{s}^{-1}=-1+\sum_{\iota=\pm
1}(2\Delta_{c}^{\iota}(q)|_{q=\pm(k-k_{0})}+2\Delta_{s}^{\iota})\,.$ (46)

Here the momentum $k_{0}$ is provided in Eq. (43) and $\gamma=-1$ for fermion
removal and $\gamma=+1$ for fermion addition, as given in Eq. (3). The
simplified expressions of the functionals $2\Delta_{c}^{\iota}(q)$ and
$2\Delta_{s}^{\iota}$ appearing in Eq. (46) that are specific to a $\beta=c,s$
branch line involve a summation $\sum_{j^{\prime}=1}$ in the general phase-
shift functional expression, Eq. (27), that refers to creation of a single
$\beta^{\prime}=c,s$ particle or hole. Moreover, at zero spin density, $m=0$,
such functionals read,

$\displaystyle 2\Delta^{\iota}_{c}(q)$ $\displaystyle=$
$\displaystyle\left({\iota\over\xi_{0}}{\delta N^{F}_{c}\over
2}+\xi_{0}\left[\delta J^{F}_{c}+{\delta J^{F}_{s}\over
2}\right]+c_{\beta^{\prime\prime}}\,\Phi_{c,\beta^{\prime\prime}}(\iota
2k_{F},q)\right)^{2}\hskip 5.69046pt{\rm and}$ $\displaystyle
2\Delta^{\iota}_{s}$ $\displaystyle=$ $\displaystyle{1\over
2}\left({\iota\over 2}\left[2\delta N^{F}_{s}-\delta
N^{F}_{c}+c_{\beta^{\prime\prime}}(-1)^{\delta_{\beta^{\prime\prime},c}}\right]+\delta
J^{F}_{s}\right)^{2}\,.$ (47)

In this equation $\xi_{0}$ is the parameter defined in Eq. (135) of Appendix A
whose limiting values are $\xi_{0}=\sqrt{2}$ for ${\cal{C}}\rightarrow 0$ and
$\xi_{0}=1$ for ${\cal{C}}\rightarrow\infty$.

On the one hand, the two $\iota=\pm 1$ functionals $2\Delta^{\iota}_{c}(q)$
depend on the excitation momentum $k=k_{0}\pm q$, Eq. (42), through the $c$
band momentum $q$ where the momentum $k_{0}$ is given in Eq. (43). On the
other hand, that the two $\iota=\pm 1$ functionals $2\Delta^{\iota}_{s}$ in
Eq. (47) do not depend on the $s$ band momentum of the $\beta^{\prime\prime}$
particle or hole created under the one-fermion excitation follows from the
spin SU(2) symmetry. Indeed, that symmetry is behind the very simple behavior
of the $s$ particle phase shifts in units of $2\pi$ given in Eq. (125) of
Appendix A whose use in the general expression of the $\iota=\pm 1$
functionals $2\Delta^{\iota}_{s}$, Eq. (30), leads to the present simple
expressions.

For up-spin and down-spin one-fermion excitations at spin densities $m>0$, the
use of the PDT leads to the following general high-energy behavior near a
$\beta=c,s$ branch line,

$\displaystyle B_{\sigma,\gamma}(k,\omega)$ $\displaystyle=$ $\displaystyle
C_{\sigma,\gamma,\beta}\,\Bigl{(}\gamma\,\omega-\omega_{\beta}^{\sigma}(k)\Bigr{)}^{\xi_{\beta}^{\sigma}(k)}\hskip
5.69046pt{\rm for}\hskip
5.69046pt(\gamma\,\omega-\omega_{\beta}^{\sigma}(k))\geq 0\hskip 5.69046pt{\rm
and}\hskip 5.69046pt\gamma=\pm 1\hskip 5.69046pt{\rm where}$
$\displaystyle\xi_{\beta}^{\sigma}(k)$ $\displaystyle=$
$\displaystyle-1+2\Delta_{c}^{+1}+2\Delta_{c}^{-1}+2\Delta_{s}^{+1}+2\Delta_{s}^{-1}=-1+\sum_{\beta^{\prime}=c,s}\sum_{\iota=\pm
1}2\Delta_{\beta^{\prime}}^{\iota}(q)|_{q=\pm(k-k_{0})}\,.$ (48)

At $m>0$ all four $\beta=c,s$ and $\iota=\pm 1$ functionals
$2\Delta_{\beta}^{\iota}(q)$ in the exponent expression depend on the
excitation momentum $k=k_{0}\pm q$, Eq. (44), through the $\beta=c,s$ band
momentum $q$ where $k_{0}$ is given in Eq. (45).

The four $\beta^{\prime}=c,s$ and $\iota=\pm 1$ functionals
$2\Delta_{\beta^{\prime}}^{\iota}(q)$ in the exponent
$\xi_{\beta}^{\sigma}(k)$ expression, Eq. (48), specific to a $\beta=c,s$
branch line are again such that the summation $\sum_{j^{\prime}=1}$ in Eq.
(27) corresponds to creation of a single $\beta^{\prime}=c,s$ particle or
hole. Hence from the use of their general expression in Eq. (30) one finds,

$\displaystyle 2\Delta^{\iota}_{c}(q)$ $\displaystyle=$
$\displaystyle\left(\sum_{\beta^{\prime}=c,s}\left(\iota\,\xi^{0}_{c\,\beta^{\prime}}\,{\delta
N^{F}_{\beta^{\prime}}\over 2}+\xi^{1}_{c\,\beta^{\prime}}\,\delta
J^{F}_{\beta^{\prime}}\right)+c_{\beta^{\prime\prime}}\,\Phi_{c,\beta^{\prime\prime}}(\iota
2k_{F},q)\right)^{2}\hskip 5.69046pt{\rm and}$ $\displaystyle
2\Delta^{\iota}_{s}(q^{\prime})$ $\displaystyle=$
$\displaystyle\left(\sum_{\beta^{\prime}=c,s}\left(\iota\,\xi^{0}_{s\,\beta^{\prime}}\,{\delta
N^{F}_{\beta^{\prime}}\over 2}+\xi^{1}_{s\,\beta^{\prime}}\,\delta
J^{F}_{\beta^{\prime}}\right)+c_{\beta^{\prime\prime}}\,\Phi_{s,\beta^{\prime\prime}}(\iota
k_{F\downarrow},q^{\prime})\right)^{2}\,.$ (49)

Here $c_{\beta^{\prime\prime}}=1$ and $c_{\beta^{\prime\prime}}=-1$ for
creation of one $\beta^{\prime\prime}$ particle and of one
$\beta^{\prime\prime}$ hole, respectively. The band momenta $q$ and
$q^{\prime}$ belong to the intervals
$q\in[-2k_{F}+k_{Fc}^{0},2k_{F}-k_{Fc}^{0}]$ for creation of one
$\beta^{\prime\prime}=c$ hole, $q\in]-\infty,-2k_{F}-k_{Fc}^{0}]$ and
$q\in[2k_{F}+k_{Fc}^{0},\infty[$ for creation of one $\beta^{\prime\prime}=c$
particle,
$q^{\prime}\in[-k_{F\downarrow}+k_{Fs}^{0},k_{F\downarrow}-k_{Fs}^{0}]$ for
creation of one $\beta^{\prime\prime}=s$ hole and
$q^{\prime}\in[-k_{F\uparrow},-k_{F\downarrow}-k_{Fs}^{0}]$ and
$q\in[k_{F\downarrow}+k_{Fs}^{0},k_{F\uparrow}]$ for creation of one
$\beta^{\prime\prime}=s$ particle.

The definition of the phase shifts in Eqs. (47) and (49) in units of $2\pi$
involves their expression provided in Eq. (124) of Appendix A in terms of the
corresponding phase-shift functions of rapidity variables. The latter are
uniquely defined by solution of the coupled integral equations, Eqs.
(126)-(131) of that Appendix.

Furthermore, the parameters $\xi^{j}_{\beta\,\beta^{\prime}}$ appearing in Eq.
(49) are the following phase-shift superpositions,

$\xi^{j}_{\beta\,\beta^{\prime}}=\delta_{\beta,\beta^{\prime}}+\sum_{\iota=\pm
1}(\iota)^{j}\,\Phi_{\beta,\beta^{\prime}}\left(q_{F\beta},\iota
q_{F\beta^{\prime}}\right)\hskip 5.69046pt{\rm where}\hskip
5.69046pt\beta=c,s\,,\hskip 5.69046pt\beta^{\prime}=c,s\,,\hskip 5.69046pt{\rm
and}\hskip 5.69046ptj=0,1\,.$ (50)

(When $\beta=\beta^{\prime}$ and $\iota=1$ the second momentum
$q_{F\beta^{\prime}}$ in $\Phi_{\beta,\beta^{\prime}}\left(q_{F\beta},\iota
q_{F\beta^{\prime}}\right)$ reads $q_{F\beta}-2\pi/L$.) The behaviors of these
parameters and their limiting values are given in Eqs. (132)-(137) of Appendix
A.

Importantly, the one-electron spectral functions expressions in the vicinity
of a $\beta=c,s$ branch line, Eqs. (46) and (48), are valid provided that
$\xi_{\beta}(k)>-1$ and $\xi_{\beta}^{\sigma}(k)>-1$, respectively. That for a
given $\beta=c,s$ branch line $k$ range that exponents read
$\xi_{\beta}(k)=-1$ and $\xi_{\beta}^{\sigma}(k)=-1$, respectively, means that
the exact expression of the spectral function is not that given in those
equations. For these $k$ ranges the four functionals $2\Delta^{\iota}_{\beta}$
in Eqs. (47) and (49) vanish. In this case the PDT also provides the
corresponding behavior of the one-electron spectral functions in Eqs. (2) and
(4), which is $\delta$-function-like and given by,

$B_{\gamma}(k,\omega)=\delta\Bigl{(}\gamma\,\omega-\omega_{\beta}(k)\Bigr{)}\hskip
5.69046pt{\rm and}\hskip
5.69046ptB_{\sigma,\gamma}(k,\omega)=\delta\Bigl{(}\gamma\,\omega-\omega_{\beta}^{\sigma}(k)\Bigr{)}\,,$
(51)

respectively.

On the one hand, the $s$ branch line studied below corresponds to edges of
support of the one-fermion spectral functions, i.e it separates two regions
with finite and vanishing spectral weight, respectively. The underlying
physical mechanism behind the line shape near it follows from the requirement
of energy and momentum conservation. The excitation leads to creation of the
$s$ hole or $s$ particle on its energy dispersion and thus mass shell, which
carries almost the entire energy. The remaining momentum is absorbed by a
dressing of low-energy particle-hole processes near the Fermi points. The
expression of the PDT $s$ branch line momentum dependent exponent is exact.

On the other hand, the $c^{\pm}$ branch lines also studied in the following
run within the spectral-weight distribution continuum. In non-integrable
models their power-law singularities are broadened or even progressively
washed by relaxation processes of the $c$ band particle or $c$ band hole
created away from its band Fermi points Imambekov_12 . However, in the present
solvable model, its integrability is associated with the occurrence of an
infinite number of conservation laws. They ensure that the $c$ band multi-
particle scattering factorizes into two-particle scattering processes. This
prevents relaxation processes, so that the line shape near the $c^{\pm}$
branch lines remains power-law like.

Figure 1: The $\beta=c^{+},c^{-},s$ singular branch lines $k$ intervals (solid
lines) and other branch lines $k$ intervals (dashed lines) in units of
$2k_{F}=\pi\,n$ for which the corresponding exponent $\xi_{\beta}(k)$, Eq.
(46), is negative and positive, respectively, and the singular boundary lines
(dashed-dotted lines) of the weight distribution associated with the one-
fermion spectral function are plotted in the $(k,\omega)$ plane. The curves
refer to fermionic density $n=0.7$ for interaction (a) ${\cal{C}}=0.1$, (b)
${\cal{C}}=1.0$, and (c) ${\cal{C}}=10.0$ and fermionic density $n=2.0$ for
interaction (d) ${\cal{C}}=0.1$, (e) ${\cal{C}}=1.0$, and (f)
${\cal{C}}=10.0$. The branch line spectra plotted here are defined in Section
III. (Online, the $c^{+}$, $c^{-}$, and $s$ branch lines appear blue, red, and
green, respectively.)

Furthermore, in the present model there is very little continuum spectral
weight just above for fermion removal and just below for fermion addition the
$c^{\pm}$ branch lines. For finite repulsive interaction ${\cal{C}}$ the
$c^{\pm}$ branch lines exponents in Eqs. (46) and (48) are the leading zero-
order term of an expansion whose very small parameter is the coupling to that
small spectral weight. Their expressions are exact for large values of
${\cal{C}}$ and their use leads to exact spectral-function expressions in the
${\cal{C}}\rightarrow 0$ limit. At intermediate ${\cal{C}}$ values the higher-
order terms are extremely small and vanish when the exponents vanish. This
means that the exact exponents are negative and positive when their leading
terms are negative and positive, respectively. Otherwise, the $c^{\pm}$ branch
lines exponents in Eqs. (46) and (48) are a very good approximation.

###  III.3 The $(k,\omega)$-plane one-fermion removal spectral singularities
on boundary lines

There is a second type of high-energy $(k,\omega)$-plane feature in the
vicinity of which the PDT provides an analytical expression of the one-fermion
spectral functions. It is called a boundary line. In the case of the present
repulsive fermion model, such boundary lines exist in the one-fermion removal
spectral function at zero magnetic field and in the up-spin and down-spin one-
fermion removal spectral functions at finite field. It is generated by a
subclass of the general processes behind the branch $A$ or B of the $m=0$ one-
fermion removal two-parametric spectrum, Eq. (34), and $m>0$ up-spin and down-
spin one-fermion removal two-parametric spectra, Eqs. (38) and (40),
respectively.

Specifically, the up-spin one-fermion removal boundary line is generated by
processes where one $c$ band hole is created at a momentum value $q$ and one
$s$ band particle is created at a momentum value $q^{\prime}$ such that their
group velocities, Eq. (21), obey the equality $v_{c}(q)=v_{s}(q^{\prime})$.
Similarly, both the $m=0$ one-fermion removal boundary line and the $m>0$
down-spin one-fermion removal boundary line are generated by processes where
one $c$ band hole is created at a momentum value $q$ and one $s$ band hole is
created at a momentum value $q^{\prime}$ such that again
$v_{c}(q)=v_{s}(q^{\prime})$.

Figure 2: The singular branch lines $k$ ranges (solid lines) and other branch
lines $k$ ranges (dashed lines) in units of $2k_{F}=\pi\,n$ for which the
corresponding exponent $\xi_{\beta}^{\sigma}(k)$, Eq. (48), is negative and
positive, respectively, and the singular boundary lines (dashed-dotted lines)
of the weight distribution associated with the up-spin and down-spin one-
fermion spectral function are plotted in the $(k,\omega)$ plane. The curves
refer to fermionic density $n=1.0$ and spin density $m=0.15$ for up spin and
interaction (a) ${\cal{C}}=0.1$, (b) ${\cal{C}}=1.0$, and (c) ${\cal{C}}=10.0$
and for down spin and interaction (d) ${\cal{C}}=0.1$, (e) ${\cal{C}}=1.0$,
and (f) ${\cal{C}}=10.0$. The branch line spectra plotted here are defined in
Section III. (Online, the $c^{+}$, $c^{-}$, and $s$ branch lines appear blue,
red, and green, respectively.) The lines represented by sets of diamond
symbols contribute to the ${\cal{C}}\rightarrow 0$ one-fermion spectrum yet
are not branch lines.

That such a feature is a $(k,\omega)$-plane line results from the $c$ band and
$s$ band momentum values $q$ and $q^{\prime}$, respectively, not being
independent of each other because of the boundary-line velocity equality
constraint, $v_{c}(q)=v_{s}(q^{\prime})$. For the momentum $k$ domains for
which those one-fermion removal boundary lines exist, they are part of the
limiting line of the corresponding two-dimensional $(k,\omega)$ plane
branches. However and as further discussed below in Sections IV.6 and IV.7,
most of such a limiting line is not a boundary line and thus does not
correspond to a singularity.

At zero spin density, $m=0$, the removal one-fermion boundary line
$(k,\omega)$-plane spectrum has the following general form,

$\omega_{BL}(k)=\left(-\varepsilon_{c}(q)-\varepsilon_{s}(q^{\prime})\right)\,\delta_{v_{c}(q),v_{s}(q^{\prime})}\hskip
5.69046pt{\rm where}\hskip 5.69046ptk=\pm 2k_{F}-q-q^{\prime}\,.$ (52)

Near such a line at small energy deviation $(\omega+\omega_{BL}(k))$ values,
the one-fermion removal spectral function has the following behavior,

$B_{\gamma}(k,\omega)\propto\Bigl{(}\omega+\omega_{BL}(k)\Bigr{)}^{-1/2}\,.$
(53)

This expression is determined by the density of the two-parametric states
generated upon varying $q$ and $q^{\prime}$ within the corresponding $c$ and
$s$ band values, respectively.

The up-spin $(\sigma=\uparrow)$ and down-spin $(\sigma=\downarrow)$ one-
fermion removal boundary line $(k,\omega)$-plane spectrum has at spin density
$m>0$ the following general form,

$\omega_{BL}^{\sigma}(k)=\left(-\varepsilon_{c}(q)+\gamma_{\sigma}\,\varepsilon_{s}(q^{\prime})\right)\,\delta_{v_{c}(q),v_{s}(q^{\prime})}\hskip
5.69046pt{\rm where}\hskip
5.69046ptk=\pm[1-\gamma_{\sigma}]\,k_{F}-q+\gamma_{\sigma}\,q^{\prime}\,,$
(54)

where,

$\gamma_{\uparrow}=+1\hskip 5.69046pt{\rm and}\hskip
5.69046pt\gamma_{\downarrow}=-1\,.$ (55)

In the vicinity of such lines at small energy deviation
$(\omega+\omega_{BL}^{\sigma}(k))$ values the up-spin and down-spin one-
fermion removal spectral functions have the following behavior,

$B_{\sigma,\gamma}(k,\omega)\propto\Bigl{(}\omega+\omega_{BL}^{\sigma}(k)\Bigr{)}^{-1/2}\,.$
(56)

Again, this expression is determined by the density of the two-parametric
states generated upon varying $q$ and $q^{\prime}$ within the corresponding
$c$ and $s$ band values, respectively.

Figure 3: The same as Fig. 2 for spin density $m=0.65$ and interaction
${\cal{C}}=1.0$ for up spin and fermionic densities (a) $n=0.7$, (b) $n=1.0$,
and (c) $n=2.0$ and for down spin and fermionic densities (d) $n=0.7$, (e)
$n=1.0$, and (f) $n=2.0$.

##  IV Specific one-fermion removal and addition spectral singularities on
branch and boundary lines

In the following, the line shape behavior of the one-fermion spectral
function, Eq. (2), near the branch lines and boundary lines at zero spin
density and line shape behavior of the up-spin and down-spin one-fermion
spectral functions, Eq. (4), in the vicinity of the branch lines and boundary
lines is studied. Such lines are plotted in the $(k,\omega)$-plane in Figs.
1-3. The curves refer to repulsive interactions ${\cal{C}}=0.1$,
${\cal{C}}=1.0$, ${\cal{C}}=10.0$, fermionic densities $n=0.7$, $n=1.0$,
$n=2.0$, and corresponding spin densities $m=0$, $m=0.15$, $m=0.65$ such that
$m<n$. The $c^{+},c^{-},s$ branch lines are the only branch lines whose
exponent is negative for at least some $k$ interval and ${\cal{C}}$, $n$, and
$m$ ranges. At those $k$ intervals there are singularity cusps in the
corresponding one-fermion spectral functions. Those branch lines are in Figs.
1-3 represented by solid lines and dashed lines for the $k$ ranges for which
the corresponding momentum dependent exponent is negative and positive,
respectively. The one-fermion removal boundary lines also refer to singularity
cusps and are represented by dashed-dotted lines.

At zero spin density, $m=0$, all ${\cal{C}}=0$ non-interacting
$\delta$-function like one-fermion spectrum $k$ ranges are recovered from
specific branch lines in the ${\cal{C}}\rightarrow 0$ limit. For $m>0$ this
applies to most of the ${\cal{C}}=0$ non-interacting $\delta$-function like
up-spin and down-spin one-fermion spectrum $k$ ranges. The exceptions refer to
the ${\cal{C}}=0$ non-interacting up-spin one-fermion removal spectrum for the
momentum interval $k\in[-k_{F\downarrow},k_{F\downarrow}]$ and to the
${\cal{C}}=0$ non-interacting down-spin one-fermion addition spectrum for the
momentum intervals $k\in[-\infty,-k_{F\uparrow}]$ and
$k\in[k_{F\uparrow},\infty]$. The corresponding ${\cal{C}}=0$ non-interacting
$\delta$-function like one-fermion spectra are in these $k$ intervals
recovered in the ${\cal{C}}\rightarrow 0$ limit from well-defined
${\cal{C}}>0$ spectral features that are here called non-branch lines. Those
are represented for $m>0$ in Figs. 2 and 3 by sets of diamond symbols.

###  IV.1 The one-fermion removal and addition $c^{\pm}$ branch lines at zero
magnetic field

At zero magnetic field and thus zero spin density, $m=0$, the one-fermion
removal and addition $c^{\pm}$ branch lines are generated by one-parametric
processes that correspond to particular cases of the two-parametric processes
that generate the spectra in Eqs. (34) and (36), respectively. These lines
one-parametric spectra are plotted in Fig. 1 where they are contained within
such two-parametric spectra. (Online, the $c^{+}$ and $c^{-}$ branch lines are
blue and red, respectively, in these figures.)

The one-parametric spectra $\omega_{c^{\pm}}(k)$ and the corresponding
exponents $\xi_{c^{\pm}}(k)$ associated with these branch lines are related by
the following symmetry,

$\omega_{c^{+}}(k)=\omega_{c^{-}}(-k)\hskip 5.69046pt{\rm and}\hskip
5.69046pt\xi_{c^{+}}(k)=\xi_{c^{-}}(-k)\,.$ (57)

Considering both the $c^{+}$ and $c^{-}$ branch lines for $k\in[0,\infty]$ or
only the $c^{+}$ branch line for $k\in[-\infty,\infty]$ contains exactly the
same information. Here we chose the latter option.

The one-fermion removal and addition $c^{+}$ branch line refers to excited
energy eigenstates with the following number deviations relative to those of
the initial ground state,

$\delta N_{c}^{F}=0\,;\hskip 5.69046pt\delta
J_{c}^{F}=\delta_{\gamma,-1}/2\,;\hskip 5.69046pt\delta
N_{c}^{NF}=\gamma\,;\hskip 5.69046pt\delta
N_{s}^{F}=-\delta_{\gamma,-1}\,;\hskip 5.69046pt\delta J_{s}^{F}=\gamma/2\,.$

Figure 4: The exponent $\xi_{c^{+}}(k)=\xi_{c^{-}}(-k)$, Eq. (60), that
controls the singularities in the vicinity of the $c^{+}$ branch line whose
$(k,\omega)$-plane one-parametric spectrum is defined in Eqs. (58) and (59) is
plotted for the one-fermion removal and addition spectral function, Eq. (61),
as a function of the momentum $k/2k_{F}$. The soft grey region refers to the
ground-state occupied Fermi sea. The curves refer to several
${\cal{C}}=0.10-30.00$ values, fermionic densities $n$ (a) $0.3$, (b) $0.7$,
(c) $1.0$, (d) $2.0$ and spin density $m=0$. The type of exponent line
associated with each ${\cal{C}}$ value is for all figures the same. Dashed
horizontal and vertical lines denote specific momentum values between
different subbranches and momentum values where the ${\cal{C}}\rightarrow 0$
limiting value of the exponent changes, respectively.

The spectrum of general form, Eq. (42), that defines the $(k,\omega)$-plane
shape of the one-fermion removal and addition $c^{+}$ branch line is given by,

$\displaystyle\omega_{c^{+}}(k)$ $\displaystyle=$
$\displaystyle\gamma\,\varepsilon_{c}(q)\hskip 5.69046pt{\rm for}\hskip
5.69046pt\gamma=\pm 1\hskip 5.69046pt{\rm where}$ $\displaystyle q$
$\displaystyle\in$ $\displaystyle[-2k_{F},2k_{F}]\hskip 5.69046pt{\rm
for}\hskip 5.69046pt\sigma\hskip 5.69046pt{\rm branch}\hskip 5.69046ptA\hskip
5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=-1\hskip 5.69046pt{\rm removal}$
$\displaystyle q$ $\displaystyle\in$ $\displaystyle[2k_{F},\infty]\hskip
5.69046pt{\rm for}\hskip 5.69046pt\sigma\hskip 5.69046pt{\rm branch}\hskip
5.69046ptA\hskip 5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=+1\hskip
5.69046pt{\rm addition}$ $\displaystyle q$ $\displaystyle\in$
$\displaystyle[-\infty,-2k_{F}]\hskip 5.69046pt{\rm for}\hskip
5.69046pt\sigma\hskip 5.69046pt{\rm branch}\hskip 5.69046ptB\hskip
5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=+1\hskip 5.69046pt{\rm
addition}\,.$ (58)

Here $\varepsilon_{c}(q)$ is the $c$ band energy dispersion, Eq. (18) for
$\beta=c$. The excitation momentum $k$ is expressed in terms of the $c$ band
momentum $q$ as follows,

$\displaystyle k$ $\displaystyle=$ $\displaystyle\gamma\,q+k_{F}\hskip
5.69046pt{\rm with}\hskip 5.69046pt{\rm intervals}$ $\displaystyle k$
$\displaystyle\in$ $\displaystyle[-k_{F},3k_{F}]\hskip 5.69046pt{\rm
for}\hskip 5.69046pt\hskip 5.69046pt{\rm branch}\hskip 5.69046ptA\hskip
5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=-1\hskip 5.69046pt{\rm removal}$
$\displaystyle k$ $\displaystyle\in$ $\displaystyle[3k_{F},\infty]\hskip
5.69046pt{\rm for}\hskip 5.69046pt{\rm branch}\hskip 5.69046ptA\hskip
5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=+1\hskip 5.69046pt{\rm addition}$
$\displaystyle k$ $\displaystyle\in$ $\displaystyle[-\infty,-k_{F}]\hskip
5.69046pt{\rm for}\hskip 5.69046pt{\rm branch}\hskip 5.69046ptB\hskip
5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=+1\hskip 5.69046pt{\rm
addition}\,.$ (59)

As given in Eq. (57), the corresponding one-fermion removal and addition
$c^{-}$ branch line spectrum reads $\omega_{c^{-}}(k)=\omega_{c^{+}}(-k)$.

At excitation momentum $k=k_{F}$ the removal spectrum is such that,

$\omega_{c^{+}}(k_{F})=-\varepsilon_{c}(0)\hskip 5.69046pt{\rm and}\hskip
5.69046pt{\partial\omega_{c^{+}}(k)\over\partial k}|_{k=k_{F}}=0\,,$

where $-\varepsilon_{c}(0)>0$ is the energy bandwidth of the $c$ band occupied
Fermi sea, Eq. (118) of Appendix A at $m=0$. The limiting behaviors for
${\cal{C}}\rightarrow 0$ and ${\cal{C}}\rightarrow\infty$ of the spectrum,
Eqs. (58) and (59), are given in Eqs. (142) and (143) of Appendix C,
respectively.

The use within the PDT of the values of the functional, Eq. (47), specific to
the excited energy eigenstates that determine spectral weight distribution
near the $c^{\pm}$ branch lines, allows accessing the momentum dependence of
the exponents of general form, Eq. (48), that control such a line shape. The
exponent $\xi_{c^{+}}(k)=\xi_{c^{-}}(-k)$ is found to read,

$\xi_{c^{+}}(k)=\xi_{c^{-}}(-k)=-{3\over 4}+\sum_{\iota=\pm
1}\left({\xi_{0}\over 4}+\gamma\,\Phi_{c,c}(\iota 2k_{F},q)\right)^{2}\,,$
(60)

where the parameter $\xi_{0}$ is defined in Eq. (135) of Appendix A. The phase
shift $\Phi_{c,c}(\pm 2k_{F},q)$ is defined in Eq. (124) for $m=0$. The
exponents, Eq. (60), are plotted in Fig. 4 as a function of the momentum $k$.
The curves correspond to several ${\cal{C}}=0.10-30.00$ values and fermionic
densities $n$ (a) $0.3$, (b) $0.7$, (c) $1.0$, (d) $2.0$.

The specific form of the general PDT expression, Eq. (48), of the one-fermion
spectral function $B_{\gamma}(k,\omega)$, Eq. (2), in the vicinity of the
present $c^{\pm}$ branch lines is,

$B_{\gamma}(k,\omega)=C_{\gamma,c^{\pm}}\Bigl{(}\gamma\omega-\omega_{c^{\pm}}(k)\Bigr{)}^{\xi_{c^{\pm}}(k)}\hskip
5.69046pt{\rm for}\hskip 5.69046pt(\gamma\,\omega-\omega_{c^{\pm}}(k))\geq
0\hskip 5.69046pt{\rm and}\hskip 5.69046pt\gamma=\pm 1\,.$ (61)

Here $C_{\gamma,c^{\pm}}$ are constants that have a fixed value for the $k$
and $\omega$ ranges corresponding to small values of the energy deviation
$(\gamma\omega-\omega_{c^{\pm}}(k))$ and the spectra
$\omega_{c^{+}}(k)=\omega_{c^{-}}(-k)$ in that energy deviation are given in
Eqs. (58) and (59). The exponent $\xi_{c^{+}}(k)=\xi_{c^{-}}(-k)$ is defined
in Eq. (60).

In the ${\cal{C}}\rightarrow 0$ limit the $c^{+}$ branch line exponent for
one-fermion removal ($\gamma=-1$) reads,

$\lim_{{\cal{C}}\rightarrow 0}\xi_{c^{+}}(k)=0\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[-k_{F},3k_{F}]\hskip 5.69046pt{\rm and}\hskip
5.69046pt\gamma=-1\,.$

For one-fermion addition ($\gamma=+1$), one finds in that limit,

$\displaystyle\lim_{{\cal{C}}\rightarrow 0}\xi_{c^{+}}(k)$ $\displaystyle=$
$\displaystyle-1\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[-\infty,-k_{F}]\hskip 5.69046pt{\rm and}\hskip
5.69046pt\gamma=+1$ $\displaystyle=1$ $\displaystyle\hskip 5.69046pt{\rm
for}\hskip 5.69046ptk\in[3k_{F},\infty]\hskip 5.69046pt{\rm and}\hskip
5.69046pt\gamma=+1\,.$

Similar values for the exponent $\xi_{c^{-}}(k)$ are obtained upon exchanging
$k$ by $-k$. The important $c^{-}$ branch line subbranch is that of one-
fermion addition for which,

$\lim_{{\cal{C}}\rightarrow 0}\xi_{c^{-}}(k)=-1\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[k_{F},\infty]\hskip 5.69046pt{\rm and}\hskip
5.69046pt\gamma=+1\,.$

For the $k$ ranges for which $\lim_{{\cal{C}}\rightarrow
0}\xi_{c^{\pm}}(k)=-1$ for one-fermion addition, the line shape has not the
form given in Eq. (69). It rather is $\delta$-function like, Eq. (51). In the
present case, this gives,

$\displaystyle\lim_{{\cal{C}}\rightarrow 0}B_{+1}(k,\omega)$ $\displaystyle=$
$\displaystyle\delta\Bigl{(}\omega-\omega_{c^{+}}(k)\Bigr{)}=\delta\Bigl{(}\omega-(k^{2}-k_{F}^{2})\Bigr{)}$
(62) $\displaystyle{\rm for}\hskip 5.69046ptk\in[-\infty,-k_{F}]\hskip
5.69046pt{\rm and}\hskip 5.69046ptk\in[k_{F},\infty]\,,$

where the expression of the $c$ band energy dispersion in the
${\cal{C}}\rightarrow 0$ limit, Eq. (19) for $m=0$, has been used.

For the $k$ ranges for which the exponents are for ${\cal{C}}\rightarrow 0$
given by $0$ and/or $1$, the one-fermion spectral weight at and near the
corresponding branch lines vanishes in the ${\cal{C}}\rightarrow 0$ limit.

In the ${\cal{C}}\rightarrow\infty$ limit the $c^{\pm}$ branch line exponent
in Eq. (60) has the following values for its whole $k$ range,

$\lim_{{\cal{C}}\rightarrow\infty}\xi_{c^{\pm}}(k)=\lim_{{\cal{C}}\rightarrow\infty}\lim_{m\rightarrow
0}\xi_{c^{\pm}}^{\sigma}(k)=-{3\over 8}\,.$ (63)

###  IV.2 The up-spin and down-spin one-fermion removal and addition $c^{\pm}$
branch lines

The up-spin and down-spin one-fermion removal and addition $c^{\pm}$ branch
lines are generated by one-parametric processes that correspond to particular
cases of the two-parametric processes that generate the spectra, Eqs.
(38)-(41). Hence these lines one-parametric spectra plotted in Figs. 2 and 3
are contained within such two-parametric spectra. Those occupy well defined
regions in the $(k,\omega)$ plane.

As at zero spin density, Eq. (57), the one-parametric spectra
$\omega_{c^{\pm}}^{\sigma}(k)$ and the corresponding exponents
$\xi_{c^{\pm}}^{\sigma}(k)$ associated with these branch lines are related by
the symmetry, $\omega_{c^{+}}^{\sigma}(k)=\omega_{c^{-}}^{\sigma}(-k)$ and
$\xi_{c^{+}}^{\sigma}(k)=\xi_{c^{-}}^{\sigma}(-k)$. And again, considering
both the $c^{+}$ and $c^{-}$ branch lines for $k\in[0,\infty]$ or only the
$c^{+}$ branch line for $k\in[-\infty,\infty]$ contains exactly the same
information. Here we chose the latter option.

The up-spin and down-spin one-fermion removal and addition $c^{+}$ branch line
refers to excited energy eigenstates with the following number deviations
relative to those of the initial ground state,

$\delta N_{c}^{F}=0\,;\hskip 5.69046pt\delta
J_{c}^{F}=\delta_{\sigma,\downarrow}/2\,;\hskip 5.69046pt\delta
N_{c}^{NF}=\gamma\,;\hskip 5.69046pt\delta
N_{s}^{F}=\delta_{\sigma,\downarrow}\,\gamma\,;\hskip 5.69046pt\delta
J_{s}^{F}=\gamma_{\sigma}/2\,.$

The spectrum of general form, Eq. (44), that defines the $(k,\omega)$-plane
shape of the up-spin and down-spin one-fermion removal and addition $c^{+}$
branch line reads,

$\displaystyle\omega_{c^{+}}^{\sigma}(k)$ $\displaystyle=$
$\displaystyle\gamma\,\varepsilon_{c}(q)\hskip 5.69046pt{\rm for}\hskip
5.69046pt\gamma=\pm 1\hskip 5.69046pt{\rm where}$ $\displaystyle q$
$\displaystyle\in$ $\displaystyle[-2k_{F},2k_{F}]\hskip 5.69046pt{\rm
for}\hskip 5.69046pt\sigma\hskip 5.69046pt{\rm branch}\hskip 5.69046ptA\hskip
5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=-1\hskip 5.69046pt{\rm removal}$
$\displaystyle q$ $\displaystyle\in$ $\displaystyle[2k_{F},\infty]\hskip
5.69046pt{\rm for}\hskip 5.69046pt\sigma\hskip 5.69046pt{\rm branch}\hskip
5.69046ptA\hskip 5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=+1\hskip
5.69046pt{\rm addition}$ $\displaystyle q$ $\displaystyle\in$
$\displaystyle[-\infty,-2k_{F}]\hskip 5.69046pt{\rm for}\hskip
5.69046pt\sigma\hskip 5.69046pt{\rm branch}\hskip 5.69046ptB\hskip
5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=+1\hskip 5.69046pt{\rm
addition}\,.$ (64)

Here $\varepsilon_{c}(q)$ is the $c$ band energy dispersion, Eq. (18) for
$\beta=c$. The expression of the excitation momentum $k$ in terms of the $c$
band momentum $q$ is given by,

$\displaystyle k$ $\displaystyle=$
$\displaystyle\gamma\,q+k_{F\bar{\sigma}}\hskip 5.69046pt{\rm where}$
$\displaystyle k$ $\displaystyle\in$
$\displaystyle[-k_{F\sigma},(2k_{F}+k_{F\bar{\sigma}})]\hskip 5.69046pt{\rm
for}\hskip 5.69046pt\sigma\hskip 5.69046pt{\rm branch}\hskip 5.69046ptA\hskip
5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=-1\hskip 5.69046pt{\rm removal}$
$\displaystyle k$ $\displaystyle\in$
$\displaystyle[(2k_{F}+k_{F\bar{\sigma}}),\infty]\hskip 5.69046pt{\rm
for}\hskip 5.69046pt\sigma\hskip 5.69046pt{\rm branch}\hskip 5.69046ptA\hskip
5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=+1\hskip 5.69046pt{\rm addition}$
$\displaystyle k$ $\displaystyle\in$
$\displaystyle[-\infty,-k_{F\sigma}]\hskip 5.69046pt{\rm for}\hskip
5.69046pt\sigma\hskip 5.69046pt{\rm branch}\hskip 5.69046ptB\hskip
5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=+1\hskip 5.69046pt{\rm
addition}\,.$ (65)

In this equation,

${\bar{\uparrow}}=\downarrow\hskip 5.69046pt{\rm and}\hskip
5.69046pt{\bar{\downarrow}}=\uparrow\,,$ (66)

so that $k_{F\bar{\uparrow}}=k_{F\downarrow}$ and
$k_{F\bar{\downarrow}}=k_{F\uparrow}$. The two-parametric spectra branches $A$
and $B$ where the $c^{+}$ branch line is contained are defined in Eqs.
(38)-(41). The corresponding $k$ intervals of the $c^{-}$ branch line
subbranches are obtained from those provided here upon exchanging $k$ by $-k$.

Figure 5: The exponent $\xi_{c^{+}}^{\uparrow}(k)=\xi_{c^{-}}^{\uparrow}(-k)$,
Eq. (67), that controls the singularities in the vicinity of the $c^{+}$
branch line whose $(k,\omega)$-plane shape is defined by Eqs. (64) and (65)
for $\sigma=\uparrow$ is plotted for the up-spin one-fermion removal and
addition spectral function, Eq. (69) for $\sigma=\uparrow$, as a function of
the momentum $k/2k_{F}$. The soft grey region refers to the ground-state
occupied Fermi sea. The curves refer to several ${\cal{C}}=0.03-30.00$ values
and spin density $m=0.15$ and fermionic densities $n$ (a) $0.7$ and (b) $2.0$
and spin density $m=0.65$ and fermionic densities (c) $0.7$ and (d) $2.0$.

Combined analysis of the momentum $k$ intervals in Eq. (65) with the relation
$\omega_{c^{+}}^{\sigma}(k)=\omega_{c^{-}}^{\sigma}(-k)$ reveals that the up-
spin and down-spin one-fermion addition $c^{\pm}$ branch lines are the natural
continuation of the up-spin and down-spin one-fermion removal $c^{\pm}$ branch
lines, respectively.

The limiting behaviors for ${\cal{C}}\rightarrow 0$ and
${\cal{C}}\rightarrow\infty$ of the up-spin and down-spin one-fermion $c^{+}$
branch-line spectra, Eqs. (64) and (65), are given in Eqs. (144) and (145) of
Appendix C, respectively.

We use the values of the functional, Eq. (49), specific to the excited energy
eigenstates that determine spectral weight distribution near the $c^{\pm}$
branch lines, to access the momentum dependence of the exponents of general
form, Eq. (48), that control such a line shape. One finds,

$\xi_{c^{+}}^{\uparrow}(k)=\xi_{c^{-}}^{\uparrow}(-k)=-1+\sum_{\iota=\pm
1}\left({\xi_{c\,s}^{1}\over 2}+\gamma\,\Phi_{c,c}(\iota
2k_{F},q)\right)^{2}+\sum_{\iota=\pm 1}\left({\xi_{s\,s}^{1}\over
2}+\gamma\,\Phi_{s,c}(\iota k_{F\downarrow},q)\right)^{2}\,,$ (67)

for the up-spin one-fermion $c^{\pm}$ branch lines and,

$\displaystyle\xi_{c^{+}}^{\downarrow}(k)$ $\displaystyle=$
$\displaystyle\xi_{c^{-}}^{\downarrow}(-k)=-1+\sum_{\iota=\pm
1}\left({\iota\,\gamma\,\xi_{c\,s}^{0}\over
2}+{(\xi_{c\,c}^{1}-\xi_{c\,s}^{1})\over 2}+\gamma\,\Phi_{c,c}(\iota
2k_{F},q)\right)^{2}$ (68) $\displaystyle+$ $\displaystyle\sum_{\iota=\pm
1}\left({\iota\,\gamma\,\xi_{s\,s}^{0}\over
2}+{(\xi_{s\,c}^{1}-\xi_{s\,s}^{1})\over 2}+\gamma\,\Phi_{s,c}(\iota
k_{F\downarrow},q)\right)^{2}\,,$

for the down-spin one-fermion $c^{\pm}$ branch lines. The phase shifts
$\Phi_{c,c}(\pm 2k_{F},q)$ and $\Phi_{s,c}(\pm k_{F\downarrow},q)$ in those
exponents expressions are defined in Eq. (124) and the $j=0,1$ parameters
$\xi^{j}_{\beta\,\beta^{\prime}}$ are defined in Eq. (50).

The up-spin and down-spin one-fermion exponents are plotted in Figs. 5 and 6,
respectively, as a function of the momentum $k$. The curves correspond to
several ${\cal{C}}=0.03-30.00$ values, fermionic densities $n=0.7$ and $2.0$,
and spin densities $m=0.15$ and $m=0.65$.

The specific form of the general expression, Eq. (48), of the up-spin and
down-spin one-fermion spectral function $B_{\sigma,\gamma}(k,\omega)$, Eq.
(4), in the vicinity of the present $c^{\pm}$ branch lines is,

$B_{\sigma,\gamma}(k,\omega)=C_{\sigma,\gamma,c^{\pm}}\Bigl{(}\gamma\omega-\omega_{c^{\pm}}^{\sigma}(k)\Bigr{)}^{\xi_{c^{\pm}}^{\sigma}(k)}\hskip
5.69046pt{\rm for}\hskip
5.69046pt(\gamma\,\omega-\omega_{c^{\pm}}^{\sigma}(k))\geq 0\hskip
5.69046pt{\rm where}\hskip 5.69046pt\gamma=\pm 1\,,$ (69)

and $C_{\sigma,\gamma,c^{\pm}}$ are constants that have a fixed value for the
$k$ and $\omega$ ranges corresponding to small values of the energy deviation
$(\gamma\omega-\omega_{c^{\pm}}^{\sigma}(k))$. The spectra
$\omega_{c^{+}}^{\sigma}(k)=\omega_{c^{-}}^{\sigma}(-k)$ in that energy
deviation are given in Eqs. (64) and (65) and the exponents
$\xi_{c^{+}}^{\sigma}(k)=\xi_{c^{-}}^{\sigma}(-k)$ are defined in Eqs. (67)
and (68) for $\sigma=\uparrow$ and $\sigma=\downarrow$, respectively.

In ${\cal{C}}\rightarrow 0$ limit the $c^{+}$ branch line exponents for up-
spin one-fermion removal ($\gamma=-1$) read,

$\displaystyle\lim_{{\cal{C}}\rightarrow 0}\xi_{c^{+}}^{\uparrow}(k)$
$\displaystyle=$ $\displaystyle-1\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[-k_{F\uparrow},-k_{F\downarrow}]\hskip 5.69046pt{\rm and}\hskip
5.69046pt\gamma=-1$ (70) $\displaystyle=$ $\displaystyle 0\hskip 5.69046pt{\rm
for}\hskip 5.69046ptk\in[-k_{F\downarrow},3k_{F\downarrow}]\hskip
5.69046pt{\rm and}\hskip 5.69046pt\gamma=-1$ $\displaystyle=$ $\displaystyle
1\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[3k_{F\downarrow},(2k_{F}+k_{F\downarrow})]\hskip 5.69046pt{\rm
and}\hskip 5.69046pt\gamma=-1\,.$

For one-fermion addition ($\gamma=+1$), one finds,

$\displaystyle\lim_{{\cal{C}}\rightarrow 0}\xi_{c^{+}}^{\uparrow}(k)$
$\displaystyle=$ $\displaystyle-1\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[-\infty,-k_{F\uparrow}]\hskip 5.69046pt{\rm and}\hskip
5.69046pt\gamma=+1$ (71) $\displaystyle=1$ $\displaystyle\hskip 5.69046pt{\rm
for}\hskip 5.69046ptk\in[(2k_{F}+k_{F\bar{\sigma}}),\infty]\hskip
5.69046pt{\rm and}\hskip 5.69046pt\gamma=+1\,.$

Similar values for the exponent $\xi_{c^{-}}^{\uparrow}(k)$ are obtained upon
exchanging $k$ by $-k$. Important $c^{-}$ branch line subbranches are those
for which $\lim_{{\cal{C}}\rightarrow 0}\xi_{c^{-}}^{\uparrow}(k)=-1$. They
refer to the $k$ ranges,

$\displaystyle\lim_{{\cal{C}}\rightarrow 0}\xi_{c^{-}}^{\uparrow}(k)$
$\displaystyle=$ $\displaystyle-1\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[k_{F\downarrow},k_{F\uparrow}]\hskip 5.69046pt{\rm and}\hskip
5.69046pt\gamma=-1$ $\displaystyle=$ $\displaystyle-1\hskip 5.69046pt{\rm
for}\hskip 5.69046ptk\in[k_{F\uparrow},\infty]\hskip 5.69046pt{\rm and}\hskip
5.69046pt\gamma=+1\,.$

Figure 6: The exponent
$\xi_{c^{+}}^{\downarrow}(k)=\xi_{c^{-}}^{\downarrow}(-k)$, Eq. (68), that
controls the singularities in the vicinity of the $c^{+}$ branch line whose
$(k,\omega)$-plane one-parametric spectrum is defined by Eqs. (64) and (65)
for $\sigma=\downarrow$ is plotted for the one-fermion removal and addition
spectral function, Eq. (69) for $\sigma=\downarrow$, as a function of the
momentum $k/2k_{F}$. The soft grey region refers to the ground-state occupied
Fermi sea. The curves refer to several ${\cal{C}}=0.03-30.00$ values and spin
density $m=0.15$ and fermionic densities $n$ (a) $0.7$ and (b) $2.0$ and spin
density $m=0.65$ and fermionic densities (c) $0.7$ and (d) $2.0$. The type of
exponent line associated with each ${\cal{C}}$ value is for all figures the
same. Dashed horizontal and vertical lines denote specific momentum values
between different subbranches and momentum values where the
${\cal{C}}\rightarrow 0$ limiting value of the exponent changes, respectively.

As discussed previously, for the $k$ ranges for which
$\lim_{{\cal{C}}\rightarrow 0}\xi_{c^{\pm}}^{\uparrow}(k)=-1$, the line shape
has not the form given in Eq. (69). It rather is $\delta$-function like, Eq.
(51). In the present case this gives,

$\displaystyle\lim_{{\cal{C}}\rightarrow 0}B_{\uparrow,-1}(k,\omega)$
$\displaystyle=$
$\displaystyle\delta\Bigl{(}\omega+\omega_{c^{+}}^{\uparrow}(k)\Bigr{)}=\delta\Bigl{(}\omega-(k^{2}-k_{F\uparrow}^{2})\Bigr{)}$
$\displaystyle{\rm for}\hskip 5.69046ptk\in[-k_{F\uparrow},-k_{F\downarrow}]$
$\displaystyle\lim_{{\cal{C}}\rightarrow 0}B_{\uparrow,-1}(k,\omega)$
$\displaystyle=$
$\displaystyle\delta\Bigl{(}\omega+\omega_{c^{-}}^{\uparrow}(k)\Bigr{)}=\delta\Bigl{(}\omega-(k^{2}-k_{F\uparrow}^{2})\Bigr{)}$
$\displaystyle{\rm for}\hskip 5.69046ptk\in[k_{F\downarrow},k_{F\uparrow}]$
$\displaystyle\lim_{{\cal{C}}\rightarrow 0}B_{\uparrow,+1}(k,\omega)$
$\displaystyle=$
$\displaystyle\delta\Bigl{(}\omega-\omega_{c^{+}}^{\uparrow}(k)\Bigr{)}=\delta\Bigl{(}\omega-(k^{2}-k_{F\uparrow}^{2})\Bigr{)}$
$\displaystyle{\rm for}\hskip 5.69046ptk\in[-\infty,-k_{F\uparrow}]$
$\displaystyle\lim_{{\cal{C}}\rightarrow 0}B_{\uparrow,+1}(k,\omega)$
$\displaystyle=$
$\displaystyle\delta\Bigl{(}\omega-\omega_{c^{-}}^{\uparrow}(k)\Bigr{)}=\delta\Bigl{(}\omega-(k^{2}-k_{F\uparrow}^{2})\Bigr{)}$
(72) $\displaystyle{\rm for}\hskip 5.69046ptk\in[k_{F\uparrow},\infty]\,,$

where the expression of the $c$ band energy dispersion in the
${\cal{C}}\rightarrow 0$ limit, Eq. (19), has been used. The spectra
$\omega_{c^{\pm}}^{\uparrow}(k)$ indeed become in the ${\cal{C}}\rightarrow 0$
limit the corresponding exact ${\cal{C}}=0$ non-interacting spectra. In the
case of the up-spin one-fermion spectra this applies to all its momentum
intervals except for $k\in[-k_{F\uparrow},k_{F\uparrow}]$.

For the excitation momentum $k$ intervals for which the exponents are for
${\cal{C}}\rightarrow 0$ given by $0$ and/or $1$, the up-spin one-fermion
spectral weight at and near the corresponding branch lines vanishes in the
${\cal{C}}\rightarrow 0$ limit. Specifically, one finds that in the
${\cal{C}}\rightarrow 0$ limit the down-spin one-fermion removal exponent, Eq.
(68), has the following behaviors,

$\displaystyle\lim_{{\cal{C}}\rightarrow 0}\xi_{c^{+}}^{\downarrow}(k)$
$\displaystyle=$ $\displaystyle 1\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[-k_{F\downarrow},(k_{F\uparrow}-2k_{F\downarrow})]\hskip
5.69046pt{\rm and}\hskip 5.69046pt\gamma=-1$
$\displaystyle\lim_{{\cal{C}}\rightarrow 0}\xi_{c^{+}}^{\downarrow}(k)$
$\displaystyle=$ $\displaystyle 0\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[(k_{F\uparrow}-2k_{F\downarrow}),(2k_{F}+k_{F\downarrow})]\hskip
5.69046pt{\rm and}\hskip 5.69046pt\gamma=-1$
$\displaystyle\lim_{{\cal{C}}\rightarrow 0}\xi_{c^{+}}^{\downarrow}(k)$
$\displaystyle=$ $\displaystyle 1\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[(2k_{F}+k_{F\downarrow}),(2k_{F}+k_{F\uparrow})]\hskip
5.69046pt{\rm and}\hskip 5.69046pt\gamma=-1\,.$ (73)

The down-spin one-fermion addition exponent is found to behave in that limit
as,

$\lim_{{\cal{C}}\rightarrow 0}\xi_{c^{+}}^{\downarrow}(k)=1\hskip
5.69046pt{\rm for}\hskip
5.69046ptk\in[-\infty,-k_{F\downarrow}]\,;k\in[(2k_{F}+k_{F\uparrow}),\infty]\hskip
5.69046pt{\rm and}\hskip 5.69046pt\gamma=+1\,.$ (74)

Hence the down-spin one-fermion spectral weight at and near these branch lines
vanishes in the ${\cal{C}}\rightarrow 0$ limit both for one-fermion removal
and addition. Similar values for the exponent $\xi_{c^{-}}^{\downarrow}(k)$
are obtained upon exchanging $k$ by $-k$.

In the ${\cal{C}}\rightarrow\infty$ limit the $c^{\pm}$ branch lines exponents
in Eqs. (67) and (68) have for $m\rightarrow n$ the following values for their
whole $k$ intervals,

$\lim_{{\cal{C}}\rightarrow\infty}\xi_{c^{\pm}}^{\sigma}(k)=-{\gamma_{\sigma}\over
2}\hskip 5.69046pt{\rm for}\hskip 5.69046ptm\rightarrow n\,.$ (75)

On the one hand and as shown in Fig. 5, the main effect on the $k$ dependence
of the up-spin one-fermion removal and addition exponent
$\xi_{c^{+}}^{\uparrow}(k)=\xi_{c^{-}}^{\uparrow}(-k)$, Eq. (67), of
increasing the interaction ${\cal{C}}$ from ${\cal{C}}\ll 1$ to ${\cal{C}}\gg
1$ is to continuously changing its ${\cal{C}}\rightarrow 0$ values $-1$, $0$,
and $1$ for the $k$ ranges given in Eqs. (70) and (71) to a $k$ independent
value for $k\in[-\infty,\infty]$ as ${\cal{C}}\rightarrow\infty$, Eq. (63) and
Eq. (75) for $\sigma=\uparrow$. The latter smoothly changes from $-3/8$ for
$m\rightarrow 0$ to $-1/2$ for $m\rightarrow n$. The general trend of such an
exponent ${\cal{C}}$ dependence is the following: For the momentum $k$ ranges
for which it reads $0$ and $1$ in the ${\cal{C}}\rightarrow 0$ limit, it
decreases upon increasing ${\cal{C}}$; For the $k$ intervals for which it is
given by $-1$ in that limit, it rather increases for increasing ${\cal{C}}$
values.

On the other hand, the exponent
$\xi_{c^{+}}^{\downarrow}(k)=\xi_{c^{-}}^{\downarrow}(k)$, Eq. (68), plotted
in Fig. 6 becomes negative only for large ${\cal{C}}$ and small spin density
values. For ${\cal{C}}\rightarrow 0$ it reads $0$ and $1$ for the $k$
intervals provided in Eqs. (73) and (74). As ${\cal{C}}\rightarrow\infty$ it
continuously evolves to a $k$ independent value for $k\in[-\infty,\infty]$.
Such a value smoothly changes from $-3/8$ for $m\rightarrow 0$ to $1/2$ for
$m\rightarrow n$. The general trend of that exponent ${\cal{C}}$ dependence is
different upon changing the densities. As shown in Fig. 6, for some densities
it always decreases upon increasing ${\cal{C}}$. For other densities it first
decreases upon increasing ${\cal{C}}$ until reaching some minimum at a finite
${\cal{C}}$ value above which it increases upon further increasing
${\cal{C}}$.

###  IV.3 The one-fermion removal and addition $s$ branch line at zero
magnetic field

The one-parametric spectrum of this branch line is an even function of $k$,
$\omega_{s}(k)=\omega_{s}(-k)$. The corresponding exponent given below is also
an even function of $k$, $\xi_{s}(k)=\xi_{s}(-k)$. Hence for simplicity we
restrict our following analysis to $k\geq 0$. For such a momentum range the
one-fermion removal and addition parts of the $s$ branch line refer to excited
energy eigenstates with the following number deviations relative to those of
the initial ground state,

$\delta N_{c}^{F}=\gamma\,;\hskip 5.69046pt\delta
J_{c}^{F}=\delta_{\gamma,+1}/2\,;\hskip 5.69046pt\delta
N_{s}^{F}=\delta_{\gamma,+1}\,;\hskip 5.69046pt\delta J_{s}^{F}=0\,;\hskip
5.69046pt\delta N_{s}^{NF}=-1\,.$

The spectrum $\omega_{s}(k)$ of general form, Eq. (42), is for the present
branch line at $k>0$ given by,

$\displaystyle\omega_{s}(k)$ $\displaystyle=$
$\displaystyle-\varepsilon_{s}(q^{\prime})\hskip 5.69046pt{\rm where}$
$\displaystyle q^{\prime}$ $\displaystyle\in$ $\displaystyle[-k_{F},0]\hskip
5.69046pt{\rm for}\hskip 5.69046pt{\rm branch}\hskip 5.69046ptA\hskip
5.69046pt{\rm fermion}\hskip 5.69046pt{\rm removal}\hskip 5.69046pt\gamma=-1$
$\displaystyle q^{\prime}$ $\displaystyle\in$
$\displaystyle[-k_{F},k_{F}]\hskip 5.69046pt{\rm for}\hskip 5.69046pt{\rm
branch}\hskip 5.69046ptA\hskip 5.69046pt{\rm fermion}\hskip 5.69046pt{\rm
addition}\hskip 5.69046pt\gamma=+1\,.$ (76)

The relation of the $s$ band momentum $q^{\prime}$ to the excitation momentum
$k$ is,

$k=\delta_{\gamma,+1}\,2k_{F}-q^{\prime}\geq 0\,.$ (77)

The corresponding intervals of the excitation momentum $k$ are,

$\displaystyle k$ $\displaystyle\in$ $\displaystyle[0,k_{F}]\hskip
5.69046pt{\rm for}\hskip 5.69046pt{\rm fermion}\hskip 5.69046pt{\rm
removal}\hskip 5.69046pt\gamma=-1$ $\displaystyle k$ $\displaystyle\in$
$\displaystyle[k_{F},3k_{F}]\hskip 5.69046pt{\rm for}\hskip 5.69046pt{\rm
fermion}\hskip 5.69046pt{\rm addition}\hskip 5.69046pt\gamma=+1\,.$ (78)

Figure 7: The exponent $\xi_{s}(k)$, Eq. (82) for $\gamma=+1$, that controls
the singularities in the vicinity of the $s$ branch line whose
$(k,\omega)$-plane shape is defined by Eqs. (76) and (78) for $\gamma=+1$ is
plotted for the one-fermion addition spectral function, Eq. (80) for
$\gamma=+1$, as a function of the momentum $k/2k_{F}$. The curves refer to
several ${\cal{C}}=0.10-30.00$ values, fermionic densities $n$ (a) $0.3$, (b)
$0.7$, (c) $1.0$, (d) $2.0$ and spin density $m=0$. The type of exponent line
associated with each ${\cal{C}}$ value is for all figures the same.

The limiting behavior for ${\cal{C}}\rightarrow 0$ of the spectrum, Eqs. (76)
and (77), is given in Eq. (146) of Appendix C. For
${\cal{C}}\rightarrow\infty$ it reads $\omega_{s}(k)=0$ for its whole $k$
intervals.

One finds from inspection of the momentum $k$ intervals in Eq. (78) that the
one-fermion addition $s$ branch line is the natural continuation of the one-
fermion removal $s$ branch line. The momentum dependent exponent of general
form, Eq. (46), that controls the line shape near the one-fermion removal and
addition $s$ branch line is given by,

$\xi_{s}(k)=-1+\sum_{\iota=\pm 1}\left({\iota\over
2\xi_{0}}+(1+\gamma){\xi_{0}\over 4}-\gamma\,\Phi_{c,s}(\iota
2k_{F},q^{\prime})\right)^{2}\,.$ (79)

As reported above, the parameter $\xi_{0}$ is defined by Eq. (135) of Appendix
A. The phase shift $\Phi_{c,s}(\pm 2k_{F},q^{\prime})$ is defined in Eq. (124)
for $m=0$.

The $s$ branch line one-fermion exponents are plotted as a function of the
momentum $k$ in Fig. 7 for one-fermion addition and in Fig. 8 for one-fermion
removal. The curves correspond to several ${\cal{C}}=0.10-30.00$ values and
fermionic densities $n$ (a) $0.3$, (b) $0.7$, (c) $1.0$, (d) $2.0$.

The general expression, Eq. (46), of the one-fermion spectral function
$B_{\gamma}(k,\omega)$, Eq. (2), near the present $s$ branch lines reads,

$B_{\gamma}(k,\omega)=C_{\gamma,s}\Bigl{(}\gamma\omega-\omega_{s}(k)\Bigr{)}^{\xi_{s}(k)}\hskip
5.69046pt{\rm for}\hskip 5.69046pt(\gamma\,\omega-\omega_{s}(k))\geq 0\hskip
5.69046pt{\rm where}\hskip 5.69046pt\gamma=\pm 1\,,$ (80)

and $C_{\gamma,s}$ is a constant that has a fixed value for the $k$ and
$\omega$ ranges corresponding to small values of the energy deviation
$(\gamma\omega-\omega_{s}(k))$. The spectrum $\omega_{s}(k)$ in such an energy
deviation is that in Eq. (76).

Figure 8: The exponent $\xi_{s}(k)$, Eq. (82) for $\gamma=-1$, that controls
the singularities in the vicinity of the $s$ branch line whose
$(k,\omega)$-plane shape is defined by Eqs. (76) and (78) for $\gamma=-1$ is
plotted for the one-fermion removal spectral function, Eq. (80) for
$\gamma=-1$, as a function of the momentum $k/2k_{F}$. The soft grey region
refers to the ground-state occupied Fermi sea. The curves refer to several
${\cal{C}}=0.10-30.00$ values, fermionic densities $n$ (a) $0.3$, (b) $0.7$,
(c) $1.0$, (d) $2.0$ and spin density $m=0$. The type of exponent line
associated with each ${\cal{C}}$ value is for all figures the same.

The exponent $\xi_{s}(k)$, Eq. (79), in the spectral function expression, Eq.
(80), has in the ${\cal{C}}\rightarrow 0$ limit the following behavior for its
whole momentum $k$ intervals,

$\lim_{{\cal{C}}\rightarrow 0}\xi_{s}(k)=\gamma\,.$

Hence the $\gamma=+1$ addition one-fermion spectral weight at and near these
$s$ branch lines vanishes in the ${\cal{C}}\rightarrow 0$ limit.

As given generally in Eq. (51), for one-fermion removal for which
$\lim_{{\cal{C}}\rightarrow 0}\xi_{s}(k)=-1$ the line shape near the $s$
branch line is not of the power-law form, Eq. (80), in the
${\cal{C}}\rightarrow 0$ limit. In that limit it rather corresponds to the
following $\delta$-function-like one-fermion removal spectral weight
distribution,

$\lim_{{\cal{C}}\rightarrow
0}B_{-1}(k,\omega)=\delta\Bigl{(}\omega+\omega_{s}(k)\Bigr{)}=\delta\Bigl{(}\omega-(k^{2}-k_{F}^{2})\Bigr{)}\hskip
5.69046pt{\rm for}\hskip 5.69046ptk\in[-k_{F},k_{F}]\,,$ (81)

where the expression of the $s$ band energy dispersion for
${\cal{C}}\rightarrow 0$, Eq. (19) for $m=0$, has been used.

At zero spin density, $m=0$, the importance of the $s$ and $c^{\pm}$ branch
lines is confirmed by in the ${\cal{C}}\rightarrow 0$ limit they leading to
the whole ${\cal{C}}=0$ non-interacting $\delta$-function-like one-fermion
removal and addition spectrum. Specifically, the $s$ branch line gives rise in
the ${\cal{C}}\rightarrow 0$ limit to the ${\cal{C}}=0$ non-interacting one-
fermion removal spectrum, Eq. (81), for its whole momentum interval
$k\in[-k_{F},k_{F}]$. Furthermore, the $c^{+}$ and $c^{-}$ branch lines lead
in the ${\cal{C}}\rightarrow 0$ limit to the ${\cal{C}}=0$ non-interacting
one-fermion addition spectrum for its whole momentum intervals
$k\in[-\infty,-k_{F}]$ and $k\in[k_{F},\infty]$, respectively, as given in Eq.
(62).

In the opposite ${\cal{C}}\rightarrow\infty$ limit the exponent, Eq. (79), in
the spectral function expression, Eq. (80), that controls the line shape near
both the one-fermion removal and addition $s$ branch lines reads,

$\displaystyle\xi_{s}(k)$ $\displaystyle=$ $\displaystyle-{1\over
2}\left(1-\left({k\over\pi n}\right)^{2}\right)$ (82) $\displaystyle{\rm
for}\hskip 5.69046pt{\rm fermion}\hskip 5.69046pt{\rm removal}\hskip
5.69046pt{\rm at}\hskip 5.69046ptk\in[0,k_{F}]$ $\displaystyle{\rm for}\hskip
5.69046pt{\rm fermion}\hskip 5.69046pt{\rm addition}\hskip 5.69046pt{\rm
at}\hskip 5.69046ptk\in[k_{F},3k_{F}]\,.$

This implies that,

$\lim_{k\rightarrow 0}\xi_{s}(k)=-{1\over 2}\,;\hskip
5.69046pt\lim_{k\rightarrow k_{F}}\xi_{s}(k)=-{3\over 8}\,;\hskip
5.69046pt\lim_{k\rightarrow 2k_{F}}\xi_{s}(k)=0\,;\hskip
5.69046pt\lim_{k\rightarrow 3k_{F}}\xi_{s}(3k_{F})={5\over 8}\,.$

For ${\cal{C}}\rightarrow\infty$ and $m=0$ the one-fermion addition exponent
$\xi_{s}(k)$, Eq. (79), continuously changes from $\xi_{s}(k)=-3/8$ for
$k\rightarrow k_{F}$ to $\xi_{s}(k)=0$ for $k\rightarrow 2k_{F}$. For its
other $k$ ranges it is positive. In the case of one-fermion removal it
continuously changes in that limit from $\xi_{s}(k)=-1/2$ for $k\rightarrow 0$
to $\xi_{s}(k)=-3/8$ for $k\rightarrow k_{F}$.

###  IV.4 The up-spin and down-spin one-fermion removal and addition spectral
functions near the $s$ branch line

The up-spin and down-spin fermion removal and addition $s$ branch line is
generated by processes that correspond to particular cases of the two-
parametric processes that generate the spectra, Eqs. (38)-(41). For the up-
spin and down-spin one-fermion spectral functions its one-parametric spectrum
plotted in Figs. 2 and 3 is thus contained within such two-parametric spectra.
(Online, the $s$ branch lines are green in these figures.)

As at zero magnetic field, the one-parametric spectrum of this branch line is
an even function of $k$, $\omega_{s}^{\sigma}(k)=\omega_{s}^{\sigma}(-k)$. The
corresponding exponent given below is also an even function of $k$,
$\xi_{s}^{\sigma}(k)=\xi_{s}^{\sigma}(-k)$. Hence for simplicity we restrict
again our following analysis to $k\geq 0$. For such a momentum range the up-
spin and down-spin fermion removal and addition parts of the $s$ branch line
refer to excited energy eigenstates with the following number deviations
relative to those of the initial ground state,

$\delta N_{c}^{F}=\gamma\,;\hskip 5.69046pt\delta
J_{c}^{F}=\delta_{\sigma,\uparrow}/2\,;\hskip 5.69046pt\delta
N_{s}^{F}=\delta_{\sigma,\uparrow}\,\gamma\,;\hskip 5.69046pt\delta
J_{s}^{F}=0\,;\hskip 5.69046pt\delta N_{s}^{NF}=-\gamma_{\sigma}\,\gamma\,.$

The spectrum $\omega_{s}^{\sigma}(k)$ of general form, Eq. (44), is for the
present branch line at $k>0$ given by,

$\displaystyle\omega_{s}^{\sigma}(k)$ $\displaystyle=$
$\displaystyle-\gamma_{\sigma}\,\gamma\,\varepsilon_{s}(q^{\prime})\hskip
5.69046pt{\rm where}$ $\displaystyle q^{\prime}$ $\displaystyle\in$
$\displaystyle[-k_{F\uparrow},-k_{F\downarrow}]\hskip 5.69046pt{\rm for}\hskip
5.69046pt\uparrow\hskip 5.69046pt{\rm branch}\hskip 5.69046ptB\hskip
5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=-1\hskip 5.69046pt{\rm removal}$
$\displaystyle q^{\prime}$ $\displaystyle\in$
$\displaystyle[-k_{F\downarrow},k_{F\downarrow}]\hskip 5.69046pt{\rm
for}\hskip 5.69046pt\uparrow\hskip 5.69046pt{\rm branch}\hskip
5.69046ptA\hskip 5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=+1\hskip
5.69046pt{\rm addition}$ $\displaystyle q^{\prime}$ $\displaystyle\in$
$\displaystyle[-k_{F\downarrow},0]\hskip 5.69046pt{\rm for}\hskip
5.69046pt\downarrow\hskip 5.69046pt{\rm branches}\hskip 5.69046ptA\hskip
5.69046pt{\rm and}\hskip 5.69046ptB\hskip 5.69046pt{\rm fermion}\hskip
5.69046pt\gamma=-1\hskip 5.69046pt{\rm removal}$ $\displaystyle q^{\prime}$
$\displaystyle\in$ $\displaystyle[-k_{F\downarrow},k_{F\downarrow}]\hskip
5.69046pt{\rm for}\hskip 5.69046pt\downarrow\hskip 5.69046pt{\rm branch}\hskip
5.69046ptA\hskip 5.69046pt{\rm fermion}\hskip 5.69046pt\gamma=+1\hskip
5.69046pt{\rm addition}\,,$ (83)

and $\varepsilon_{s}(q^{\prime})$ is the $s$ band energy dispersion, Eq. (18)
for $\beta=s$. In the case of down-spin one-fermion removal both the branches
$A$ and $B$ of the spectrum, Eq. (40), contain the $s$ branch line.

Figure 9: The exponent $\xi_{s}^{\uparrow}(k)$, Eq. (87), that controls the
singularities in the vicinity of the $s$ branch line whose $(k,\omega)$-plane
shape is defined by Eqs. (83) and (84) for $\sigma=\uparrow$ and Eq. (85) is
plotted for the one-fermion removal and addition spectral function, Eq. (89)
for $\sigma=\uparrow$, as a function of the momentum $k/2k_{F}$. The soft grey
region refers to the ground-state occupied Fermi sea. The curves refer to
several ${\cal{C}}=0.03-30.00$ values and spin density $m=0.15$ and fermionic
densities $n$ (a) $0.7$ and (b) $2.0$ and spin density $m=0.65$ and fermionic
densities (c) $0.7$ and (d) $2.0$. The type of exponent line associated with
each ${\cal{C}}$ value is for all figures the same.

The relation of the $s$ band momentum $q$ to the excitation momentum $k$ is,

$k=\delta_{\sigma,\uparrow}\,2k_{F}-\gamma_{\sigma}\,\gamma\,q^{\prime}\geq
0\,.$ (84)

This gives,

$\displaystyle k$ $\displaystyle\in$
$\displaystyle[k_{F\downarrow},k_{F\uparrow}]\hskip 5.69046pt{\rm for}\hskip
5.69046pt\uparrow\hskip 1.42271pt{\rm fermion}\hskip 5.69046pt{\rm
removal}\hskip 5.69046pt\gamma=-1\,,$ $\displaystyle k$ $\displaystyle\in$
$\displaystyle[k_{F\uparrow},(2k_{F}+k_{F\downarrow})]\hskip 5.69046pt{\rm
for}\hskip 5.69046pt\uparrow\hskip 1.42271pt{\rm fermion}\hskip 5.69046pt{\rm
addition}\hskip 5.69046pt\gamma=+1\,,$ (85)

and

$\displaystyle k$ $\displaystyle\in$ $\displaystyle[0,k_{F\downarrow}]\hskip
5.69046pt{\rm for}\hskip 5.69046pt\downarrow\hskip 1.42271pt{\rm
fermion}\hskip 5.69046pt{\rm removal}\hskip 5.69046pt\gamma=-1\,,$
$\displaystyle k$ $\displaystyle\in$
$\displaystyle[k_{F\downarrow},k_{F\uparrow}]\hskip 5.69046pt{\rm for}\hskip
5.69046pt\downarrow\hskip 1.42271pt{\rm fermion}\hskip 5.69046pt{\rm
addition}\hskip 5.69046pt\gamma=+1\,.$ (86)

The limiting behavior for ${\cal{C}}\rightarrow 0$ of the spectrum, Eqs. (83)
and (84), is given Eq. (147) of Appendix C where the subdomains of the $k$
intervals that correspond to one-fermion addition $(\gamma=+1)$ and removal
$(\gamma=-1)$ are provided in Eqs. (85) and (86). For
${\cal{C}}\rightarrow\infty$ this spectrum reads $\omega_{s}(k)=0$ for its
whole $k$ intervals.

As for zero spin density, the momentum $k$ intervals in Eq. (86) reveal that
the up-spin and down-spin one-fermion addition $s$ branch line is the natural
continuation of the up-spin and down-spin one-fermion removal $s$ branch line.
The momentum dependent exponent of general form, Eq. (48), that controls the
line shape near the up-spin one-fermion removal and addition $s$ branch lines
is given by,

$\displaystyle\xi_{s}^{\uparrow}(k)$ $\displaystyle=$
$\displaystyle-1+\sum_{\iota=\pm
1}\left({\iota\,\gamma(\xi_{c\,c}^{0}+\xi_{c\,s}^{0})\over
2}+{\xi_{c\,c}^{1}\over 2}-\gamma\,\Phi_{c,s}(\iota
2k_{F},q^{\prime})\right)^{2}$ (87) $\displaystyle+$
$\displaystyle\sum_{\iota=\pm
1}\left({\iota\,\gamma(\xi_{s\,c}^{0}+\xi_{s\,s}^{0})\over
2}+{\xi_{s\,c}^{1}\over 2}-\gamma\,\Phi_{s,s}(\iota
k_{F\downarrow},q^{\prime})\right)^{2}\,.$

The exponent that controls it in the vicinity of the down-spin one-fermion
removal and addition $s$ branch line reads,

$\xi_{s}^{\downarrow}(k)=-1+\sum_{\iota=\pm
1}\left({\iota\,\xi_{c\,c}^{0}\over 2}+\Phi_{c,s}(\iota
2k_{F},q^{\prime})\right)^{2}+\sum_{\iota=\pm
1}\left({\iota\,\xi_{s\,c}^{0}\over 2}+\Phi_{s,s}(\iota
k_{F\downarrow},q^{\prime})\right)^{2}\,.$ (88)

This latter exponent has the same formal expression for $\gamma=-1$ and
$\gamma=+1$, respectively. The corresponding $q^{\prime}$ ranges are though
different, as given in Eq. (83). The phase shifts $\Phi_{c,s}(\pm
2k_{F},q^{\prime})$ and $\Phi_{s,s}(\pm k_{F\downarrow},q^{\prime})$ in those
exponents expressions are defined in Eq. (124) and the $j=0,1$ parameters
$\xi^{j}_{\beta\,\beta^{\prime}}$ are defined in Eq. (50).

Figure 10: The exponent $\xi_{s}^{\downarrow}(k)$, Eq. (88), that controls the
singularities in the vicinity of the $s$ branch line whose $(k,\omega)$-plane
one-parametric spectrum is defined by Eqs. (83) and (84) for
$\sigma=\downarrow$ and Eq. (86) is plotted for the down-spin one-fermion
removal and addition spectral function, Eq. (89) for $\sigma=\downarrow$, as a
function of the momentum $k/2k_{F}$. The soft grey region refers to the
ground-state occupied Fermi sea. The curves refer to several
${\cal{C}}=0.03-30.00$ values and spin density $m=0.15$ and fermionic
densities $n$ (a) $0.7$ and (b) $2.0$ and spin density $m=0.65$ and fermionic
densities (c) $0.7$ and (d) $2.0$. The type of exponent line associated with
each ${\cal{C}}$ value is for all figures the same.

The $s$ branch line one-fermion exponents are plotted as a function of the
momentum $k$ in Fig. 9 for up-spin one-fermion removal and addition and in
Fig. 10 for down-spin one-fermion removal and addition. The curves correspond
to several ${\cal{C}}=0.03-30.00$ values, fermionic densities $n=0.7$ and
$2.0$, and spin densities $m=0.15$ and $m=0.65$.

The general expression, Eq. (48), of the up-spin and down-spin one-fermion
spectral function $B_{\sigma,\gamma}(k,\omega)$, Eq. (4), is near the present
$s$ branch line given by,

$B_{\sigma,\gamma}(k,\omega)=C_{\sigma,\gamma,s}\Bigl{(}\gamma\omega-\omega_{s}^{\sigma}(k)\Bigr{)}^{\xi_{s}^{\sigma}(k)}\hskip
5.69046pt{\rm for}\hskip 5.69046pt(\gamma\,\omega-\omega_{s}^{\sigma}(k))\geq
0\hskip 5.69046pt{\rm where}\hskip 5.69046pt\gamma=\pm 1\,,$ (89)

and $C_{\sigma,\gamma,s}$ is a constant that has a fixed value for the $k$ and
$\omega$ ranges corresponding to small values of the energy deviation
$(\gamma\omega-\omega_{s}^{\sigma}(k))$. The spectrum $\omega_{s}^{\sigma}(k)$
in such an energy deviation is that in Eq. (83). The exponent
$\xi_{s}^{\sigma}(k)$ is given in Eqs. (87) and (88).

The exponent $\xi_{s}^{\sigma}(k)$ has the following related behavior in the
${\cal{C}}\rightarrow 0$ limit for its whole $k$ intervals,

$\lim_{{\cal{C}}\rightarrow 0}\xi_{s}^{\sigma}(k)=\gamma_{\sigma}\,.$

Hence the up-spin one-fermion spectral weight at and near these $s$ branch
lines vanishes in the ${\cal{C}}\rightarrow 0$ limit both for one-fermion
removal and addition.

As given generally in Eq. (51), for the $n$, $m$, and $k$ ranges for which
$\lim_{{\cal{C}}\rightarrow 0}\xi_{s}^{\downarrow}(k)=-1$ the line shape near
the branch line is not of the power-law form, Eq. (89). As for zero spin
density, in that limit it rather corresponds to the following
$\delta$-function-like down-spin one-fermion spectral weight distribution,

$\displaystyle\lim_{{\cal{C}}\rightarrow 0}B_{\downarrow,-1}(k,\omega)$
$\displaystyle=$
$\displaystyle\delta\Bigl{(}\omega+\omega_{s}^{\downarrow}(k)\Bigr{)}=\delta\Bigl{(}\omega-(k^{2}-k_{F\downarrow}^{2})\Bigr{)}$
$\displaystyle{\rm for}\hskip
5.69046ptk\in[-k_{F\downarrow},k_{F\downarrow}]\,,$
$\displaystyle\lim_{{\cal{C}}\rightarrow 0}B_{\downarrow,+1}(k,\omega)$
$\displaystyle=$
$\displaystyle\delta\Bigl{(}\omega-\omega_{s}^{\downarrow}(k)\Bigr{)}=\delta\Bigl{(}\omega-(k^{2}-k_{F\downarrow}^{2})\Bigr{)}$
(90) $\displaystyle{\rm for}\hskip
5.69046ptk\in[-k_{F\uparrow},-k_{F\downarrow}]\hskip 5.69046pt{\rm and}\hskip
5.69046ptk\in[k_{F\downarrow},k_{F\uparrow}]\,,$

where the expression of the $s$ band energy dispersion for
${\cal{C}}\rightarrow 0$, Eq. (19), has been used. The ${\cal{C}}\rightarrow
0$ limiting behavior reported in the latter equation for that energy
dispersion appearing in the spectrum $\omega_{s}^{\downarrow}(k)$, Eq. (83),
confirms that the latter spectrum becomes in the ${\cal{C}}\rightarrow 0$
limit the corresponding ${\cal{C}}=0$ non-interacting spin-down fermionic
spectrum, as given in Eq. (90). This applies to its whole $k$ interval except
for down-spin fermion addition for $k\in[-\infty,-k_{F\uparrow}]$ and
$k\in[k_{F\uparrow},\infty]$, as further discussed in Section IV.5.

For the $k$ interval for which $\lim_{{\cal{C}}\rightarrow
0}\xi_{s}^{\downarrow}(k)=0$, the down-spin one-fermion addition spectral
weight at and near the present $s$ branch line vanishes in the
${\cal{C}}\rightarrow 0$ limit.

For ${\cal{C}}\rightarrow\infty$ and $m\rightarrow n$ we find the following
exponent expressions for the up-spin one-fermion removal and down-spin one-
fermion addition $s$ branch line,

$\displaystyle\xi_{s}^{\uparrow}(k)$ $\displaystyle=$ $\displaystyle{1\over
2}\left({k\over\pi n}\right)^{2}+{2\over\pi^{2}}\left\\{\arctan\left({1\over
2}\cot\left({k\over 2n}\right)\right)\right\\}^{2}$ $\displaystyle\uparrow{\rm
fermion}\hskip 5.69046pt{\rm removal}\hskip 5.69046pt{\rm for}\hskip
5.69046ptk\in[0,2k_{F}]\,,$ $\displaystyle\xi_{s}^{\downarrow}(k)$
$\displaystyle=$ $\displaystyle-{1\over 2}\left(1-\left({k\over\pi
n}\right)^{2}\right)+{2\over\pi^{2}}\left\\{\arctan\left({1\over
2}\tan\left({k\over 2n}\right)\right)\right\\}^{2}$
$\displaystyle\downarrow{\rm fermion}\hskip 5.69046pt{\rm addition}\hskip
5.69046pt{\rm for}\hskip 5.69046ptk\in[0,2k_{F}]\,.$

This then implies that,

$\displaystyle\lim_{k\rightarrow 0}\xi_{s}^{\uparrow}(k)$ $\displaystyle=$
$\displaystyle{1\over 2}\,;\hskip 5.69046pt\lim_{k\rightarrow
k_{F}}\xi_{s}^{\uparrow}(k)={1\over 8}+2\left({1\over\pi}\arctan\left({1\over
2}\right)\right)^{2}\approx 0.16856\,;\hskip 5.69046pt\lim_{k\rightarrow
2k_{F}}\xi_{s}^{\uparrow}(k)={1\over 2}$ $\displaystyle\lim_{k\rightarrow
0}\xi_{s}^{\downarrow}(k)$ $\displaystyle=$ $\displaystyle-{1\over 2}\,;\hskip
5.69046pt\lim_{k\rightarrow k_{F}}\xi_{s}^{\downarrow}(k)=-{3\over
8}+2\left({1\over\pi}\arctan\left({1\over
2}\right)\right)^{2}\approx-0.33144\,;\hskip 5.69046pt\lim_{k\rightarrow
2k_{F}}\xi_{s}^{\downarrow}(k)={1\over 2}\,.$

Analysis of these expressions and values reveals that in the
${\cal{C}}\rightarrow\infty$ and $m\rightarrow n$ limits the up-spin one-
fermion removal exponent $\xi_{s}^{\uparrow}(k)$ smoothly decreases from
$\xi_{s}^{\uparrow}(k)=1/2$ for $k\rightarrow 0$ until it reaches a minimum
value at $k=k_{F}$. For $k>k_{F}$ it continuously increases to
$\xi_{s}^{\uparrow}(k)=1/2$ as $k\rightarrow 2k_{F}$. In the same limits, the
down-spin one-fermion addition exponent $\xi_{s}^{\downarrow}(k)$ smoothly
varies from $\xi_{s}^{\downarrow}(k)=-1/2$ for $k\rightarrow 0$ to
$\xi_{s}^{\downarrow}(k)=1/2$ for $k\rightarrow 2k_{F}$.

Moreover, analysis of Fig. 9 for other spin densities shows that the exponent
$\xi_{s}^{\uparrow}(k)$ only becomes negative for a part of the $s$ branch
line $k$ interval. It starts at $k=k_{F\downarrow}$ and ends at a $k$ momentum
that for smaller and larger spin density values refers to one-fermion addition
and removal, respectively. The ${\cal{C}}$ values for which it is negative
depend on the densities.

For ${\cal{C}}>0$ the exponent $\xi_{s}^{\downarrow}(k)$, whose $k$ dependence
is plotted in Fig. 10, is in general negative. The exception refers to a small
$k$ region. It corresponds to the larger $k$ values of its range. For the $k$
ranges for which it reads $-1$ for ${\cal{C}}\rightarrow 0$, it remains being
an increasing function of ${\cal{C}}$ for the whole ${\cal{C}}$ interval.
However, for the $k$ domains for which it is given by $0$ in the
${\cal{C}}\rightarrow 0$ limit, upon increasing ${\cal{C}}$ it first
decreases, goes through a minimum value, and then becomes an increasing
function of ${\cal{C}}$, until reaching its ${\cal{C}}\rightarrow\infty$ limit
$k$ dependent values.

###  IV.5 The up-spin one-fermion removal and down-spin one-fermion addition
“non-branch lines”

On the one hand and as discussed in Section IV.3, at zero magnetic field the
$c^{\pm}$ and $s$ branch lines lead in the ${\cal{C}}\rightarrow 0$ limit to
the ${\cal{C}}=0$ non-interaction $\delta$-function-like one-fermion addition
and removal spectra for their whole $k$ intervals, as given in Eqs. (62) and
(81).

On the other hand, at finite magnetic field and finite spin density the
$c^{\pm}$ and $s$ branch lines lead in the ${\cal{C}}\rightarrow 0$ limit to
most $k$ intervals of the ${\cal{C}}=0$ non-interaction $\delta$-function-like
up-spin and down-spin one-fermion addition and removal spectra, respectively.
This is confirmed from analysis of the up-spin one-fermion spectral function
expressions obtained from the $c^{\pm}$ branch lines in Eq. (72) and of the
expressions of the down-spin one-fermion spectral function obtained from the
$s$ branch lines in Eq. (90).

However and as mentioned above, at finite magnetic field some $k$ subintervals
of the ${\cal{C}}=0$ non-interacting removal up-spin and addition down-spin
one-fermion spectra do not stem from branch lines. This refers to the momentum
interval $k\in[-k_{F\downarrow},k_{F\downarrow}]$ for up-spin one-fermion
removal and to the momentum intervals $k\in[-\infty,-k_{F\uparrow}]$ and
$k\in[k_{F\uparrow},\infty]$ for down-spin one-fermion addition.

The ${\cal{C}}=0$ non-interacting up-spin one-fermion removal spectral weight
missing for $k\in[-k_{F\downarrow},k_{F\downarrow}]$ stems in the
${\cal{C}}\rightarrow 0$ limit from a ${\cal{C}}>0$ $s^{\prime}$ spectral
feature that is generated by transitions to excited energy eigenstates whose
number deviations relative to those of the initial ground state are given by,

$\delta N_{c}^{F}=\delta J_{c}^{F}=0\,;\hskip 5.69046pt\delta
N_{c}^{NF}=-1\,;\hskip 5.69046pt\delta N_{s}^{F}=1\,;\hskip 5.69046pt\delta
J_{s}^{F}=\pm 1\,;\hskip 5.69046pt\delta N_{s}^{NF}=-1\,.$

The one-parametric spectrum of this line reads,

$\displaystyle\omega_{s^{\prime}}^{\uparrow}(k)$ $\displaystyle=$
$\displaystyle-\varepsilon_{s}(-k)-\varepsilon_{c}(\pm
2k_{F\downarrow})=-\varepsilon_{s}(q^{\prime})-\varepsilon_{c}(\pm
2k_{F\downarrow})\hskip 5.69046pt{\rm where}\hskip
5.69046ptq^{\prime}\in[-k_{F\downarrow},k_{F\downarrow}]\hskip 5.69046pt{\rm
and}$ $\displaystyle k$ $\displaystyle=$
$\displaystyle-q^{\prime}\in[-k_{F\downarrow},k_{F\downarrow}]\,.$

Here $\varepsilon_{c}(q)$ and $\varepsilon_{s}(q^{\prime})$ are the $c$ and
$s$ band energy dispersions, Eq. (18). While the line shape expression near
the present $s^{\prime}$ line involves within the PDT state summations
difficult to be performed analytically for ${\cal{C}}>0$, in the
${\cal{C}}\rightarrow 0$ limit its exact line shape becomes $\delta$-function-
like,

$\lim_{{\cal{C}}\rightarrow
0}B_{\uparrow,-1}(k,\omega)=\delta\Bigl{(}\omega+\omega_{s^{\prime}}^{\uparrow}(k)\Bigr{)}=\delta\Bigl{(}\omega-(k^{2}-k_{F\uparrow}^{2})\Bigr{)}\hskip
5.69046pt{\rm for}\hskip 5.69046ptk\in[-k_{F\downarrow},k_{F\downarrow}]\,.$

Here the expression of the $c$ and $s$ energy dispersions for
${\cal{C}}\rightarrow 0$, Eq. (19), have been used.

The ${\cal{C}}=0$ non-interacting down-spin one-fermion addition spectral
weight missing for $k\in[-\infty,-k_{F\downarrow}]$ and
$k\in[k_{F\downarrow},\infty]$ stems in the ${\cal{C}}\rightarrow 0$ limit
from a ${\cal{C}}>0$ $c^{\prime}$ spectral feature that is generated by
transitions to excited energy eigenstates whose number deviations relative to
those of the initial ground state read,

$\delta N_{c}^{F}=0\,;\hskip 5.69046pt\delta J_{c}^{F}=\pm 1/2\,;\hskip
5.69046pt\delta N_{c}^{NF}=1\,;\hskip 5.69046pt\delta N_{s}^{F}=\delta
J_{s}^{F}=0\,;\hskip 5.69046pt\delta N_{s}^{NF}=1\,.$

The one-parametric spectrum of this line is given by,