# General Shiba mapping for on-site four-point correlation functions

Herbert Eßl$\,{}^{a}$<EMAIL_ADDRESS>Matthias Reitner$\,{}^{a}$
Giorgio Sangiovanni$\,{}^{b}$ Alessandro Toschia aInstitute of Solid State
Physics, TU Wien, 1040 Vienna, Austria  bInstitut für Theoretische Physik und
Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität
Würzburg, 97074 Würzburg, Germany

###### Abstract

By applying the Shiba mapping on the two particle level, we derive the
relation between the local four-point correlation functions of bipartite
lattice models with on-site electronic repulsion and those of the
corresponding models with attractive interaction in the most general setting.
In particular, we extend the results of [Phys. Rev. B, 101, 155148 (2020)],
which were limited to the rather specific situation of the static limit in
strictly particle-hole symmetric models, (i) by explicitly including both
magnetic field and different values of the chemical potentials, and (ii) by
considering the full dependence of the generalized susceptibilities on the
transfer (bosonic) Matsubara frequency. The derived formalism is then applied,
as a relevant benchmark, to the Hubbard atom, by investigating the general
properties of the divergences of its irreducible vertex functions as a
function of chemical potential and applied magnetic field. The resulting
phase-diagrams provide an insightful compass for future studies of the
breakdown of the self-consistent perturbation expansion beyond high-symmetric
regimes.

##  I Introduction

Due to the impressive algorithmic and computational advancements of the last
decade, performing quantum-many body calculations of four point correlation
functions has become achievable also in non-trivial, strongly correlated
parameter regimes. Evidently, this progress is highly relevant for several
reasons; among which we just recall here the precise calculations of vertex
corrections of physical susceptibilities [1, 2, 3, 4, 5] and conductivities
[6, 7, 8], the identification of the predominant scattering mechanisms
underlying intriguing photoemission or self-energy features [9, 10, 11, 12,
13, 14], and the implementation of advanced diagrammatic expansions [15, 16,
17, 18, 19] for non-perturbative regimes.

On the theoretical side, this development has triggered a quest for improving
our fundamental understanding of the many-electron properties on the two-
particle level and their associated Feynman diagrammatic formalism. In this
context, a considerable effort has been recently devoted to the analysis of
generalized susceptibilities and two-particle vertex functions [20, 21, 22,
23] beyond the standard textbook discussions, with a particular focus on their
high-frequency asymptotics [24, 25, 26] and on their algorithmic treatments
[27]. Considering the specific case of the physical interpretation of the
local (on-site) two-particle formalism, interesting information was obtained
in Ref. [28]. The Shiba transformation, which maps the quantities of a model
with local electrostatic repulsion $U>0$ to those of a corresponding
Hamiltonian with on-site attraction $U<0$, was applied to derive the explicit
relations between the on-site generalized two-particle susceptibilities of the
repulsive and their counterparts in the attractive Hubbard model. In fact, the
analytical derivations presented in [28] unveiled intrinsic symmetry
properties of the four-point correlation functions (such as two-particle
generalized susceptibilities and vertex functions) and provided pivotal
information for the investigation of the breakdown of the self-consistent
perturbation expansion in many-electron systems.

However, the analytical derivations of Ref. [28] were limited to a very
specific case: the Shiba transformation of the static (i.e., for zero
frequency transfer, $\omega=0$) two-particle generalized susceptibilities in
the particle-hole symmetric Hubbard model, i.e. half filling $(\mu=U/2)$ and
$SU(2)$ symmetry $(h=0)$. In this paper, our goal is to address and overcome
these notable restrictions, by providing the general expressions which
systematically map all two-particle generalized on-site susceptibilities of a
model with a purely local (Hubbard-like) interaction onto their counterparts
in the corresponding model with sign-flipped interaction. Importantly, these
mappings are established for arbitrary values of the chemical potential,
applied magnetic field, and bosonic Matsubara frequency $\omega$.

Eventually, to demonstrate the correctness and usefulness of the derived
expressions, we validate them through a hands-on application. Specifically, we
exploit these expressions to analyze the divergences affecting the irreducible
two-particle vertex functions of the Hubbard atom (i.e., of an isolated
interacting site) across different scattering sectors, as a function of
arbitrary chemical potential, magnetic field, temperature and interaction. The
obtained general phase-diagrams will provide an intuitive and robust guidance
to all future studies aiming at unveiling the formal [29, 30, 31, 32] and the
physical [33, 34, 35] aspects associated to the breakdown [29, 30, 31] of the
self-consistent perturbation theory for many-electron systems, beyond the
(unrealistic) assumption of perfect particle-hole symmetry made so far.

The structure of the paper is the following: In section II, we concisely
introduce the one- and two-particle formalism necessary for our derivations,
and we recall the general definition of the Shiba transformation. In section
III, we report the explicit derivation for the mapping of generalized on-site
two-particle quantities under the action of the Shiba transformation in the
most general case (i.e. arbitrary filling, magnetic field, and finite transfer
frequencies). Thereafter, in section IV we apply the derived relations to the
problem of the divergences of the irreducible vertex functions occurring in
the different scattering channels of the Hubbard atom with repulsive and
attractive interaction, whose study was hitherto restricted to the case of
perfect particle-hole symmetry. Finally, in section V we draw our conclusions,
outlining possible consequences of the results presented in our work.

##  II Formalism

###  II.1 Generalized Susceptibilities

When examining the fermionic four-point correlation functions of systems that
maintain time translation invariance, the associated generalized
susceptibilities, which characterize the scattering events between two
particles, depend only on three independent Matsubara frequencies (instead of
four). This reduction is due to the conservation of energy [20, 36].
Additionally, for spin conservation, only $6$ out of the $2^{4}$ spin
combinations for these correlation functions remain independent [20].

Therefore, for our considerations all possible (local) generalized
susceptibilities $\chi$ can be defined as follows [15, 20]:

$\displaystyle\begin{split}\chi_{\sigma\sigma^{\prime}}(\nu_{1},\nu_{2},\nu_{3})&=\int_{0}^{\beta}d\tau_{1}d\tau_{2}d\tau_{3}\,e^{-i\nu_{1}\tau_{1}}e^{i\nu_{2}\tau_{2}}e^{-i\nu_{3}\tau_{3}}\\\
&\bigl{(}\langle
T_{\tau}c_{\sigma}^{\dagger}(\tau_{1})c_{\sigma}(\tau_{2})c_{\sigma^{\prime}}^{\dagger}(\tau_{3})c_{\sigma^{\prime}}(0)\rangle\\\
-\langle
T_{\tau}&c_{\sigma}^{\dagger}(\tau_{1})c_{\sigma}(\tau_{2})\rangle\langle
T_{\tau}c_{\sigma^{\prime}}^{\dagger}(\tau_{3})c_{\sigma^{\prime}}(0)\rangle\bigr{)},\end{split}$
(1)
$\displaystyle\begin{split}\chi_{\overline{\sigma\sigma^{\prime}}}(\nu_{1},\nu_{2},\nu_{3})&=\int_{0}^{\beta}d\tau_{1}d\tau_{2}d\tau_{3}\,e^{-i\nu_{1}\tau_{1}}e^{i\nu_{2}\tau_{2}}e^{-i\nu_{3}\tau_{3}}\\\
&\langle
T_{\tau}c_{\sigma}^{\dagger}(\tau_{1})c_{\sigma^{\prime}}(\tau_{2})c_{\sigma^{\prime}}^{\dagger}(\tau_{3})c_{\sigma}(0)\rangle,\end{split}$
(2)

where $\sigma,\sigma^{\prime}$ denote the spins, $\nu_{i}$ are fermionic
Matsubara frequencies (where the fourth frequency is calculated by
$\nu_{4}=\nu_{1}-\nu_{2}+\nu_{3}$), $c^{(\dagger)}_{\sigma}$ annihilate
(create) an electron with spin $\sigma$, $T_{\tau}$ denotes the imaginary time
ordering operator, and $\langle\dots\rangle=\frac{1}{Z}\Tr(e^{-\beta H}\dots)$
the thermal expectation value.

Note that for non-local $\chi$, $c^{(\dagger)}_{i,\sigma}$ acquire an
additional lattice index $i$, and for space translation invariant systems, one
of the four lattice indexes can be set to zero.

By using two fermionic $\nu$, $\nu^{\prime}$, and one bosonic Matsubara
frequency $\omega$ one can define two convenient frequency conventions: the
particle-hole (ph) $\chi^{\nu,\nu^{\prime}\\!,\,\omega}$ and the particle-
particle (pp) $\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\text{pp}}$ notation

$\displaystyle\textbf{ph: }\nu_{1}$ $\displaystyle=\nu$
$\displaystyle\textbf{pp: }\nu_{1}$ $\displaystyle=\nu$ (3a)
$\displaystyle\nu_{2}$ $\displaystyle=\nu+\omega$ $\displaystyle\nu_{2}$
$\displaystyle=\omega-\nu^{\prime}$ (3b) $\displaystyle\nu_{3}$
$\displaystyle=\nu^{\prime}+\omega$ $\displaystyle\nu_{3}$
$\displaystyle=\omega-\nu$ (3c) $\displaystyle(\nu_{4}$
$\displaystyle=\nu^{\prime})$ $\displaystyle(\nu_{4}$
$\displaystyle=\nu^{\prime})$ (3d)

One can go from ph to pp notation by shifting
$\omega\rightarrow\omega-\nu-\nu^{\prime}$.

Instead of using the definitions in eqs. 1 and 2 directly, it is often more
useful to consider specific spin combinations, which correspond to physical
expectation values.

The spin combinations (channels) that we consider are charge (c), longitudinal
spin (s), the coupling between charge and longitudinal spin (cs) and (sc),
transversal spin ($S_{x}=S_{y}$) and the pairing channel (pair):

$\displaystyle\chi_{\text{c}}^{\nu,\nu^{\prime}\\!,\,\omega}$
$\displaystyle=\chi_{\uparrow\uparrow}^{\nu,\nu^{\prime}\\!,\,\omega}+\chi_{\downarrow\downarrow}^{\nu,\nu^{\prime}\\!,\,\omega}+\chi_{\uparrow\downarrow}^{\nu,\nu^{\prime}\\!,\,\omega}+\chi_{\downarrow\uparrow,}^{\nu,\nu^{\prime}\\!,\,\omega},$
(4) $\displaystyle\chi_{\text{s}}^{\nu,\nu^{\prime}\\!,\,\omega}$
$\displaystyle=\chi_{\uparrow\uparrow}^{\nu,\nu^{\prime}\\!,\,\omega}+\chi_{\downarrow\downarrow}^{\nu,\nu^{\prime}\\!,\,\omega}-\chi_{\uparrow\downarrow}^{\nu,\nu^{\prime}\\!,\,\omega}-\chi_{\downarrow\uparrow}^{\nu,\nu^{\prime}\\!,\,\omega},$
(5) $\displaystyle\chi_{\text{cs}}^{\nu,\nu^{\prime}\\!,\,\omega}$
$\displaystyle=\chi_{\uparrow\uparrow}^{\nu,\nu^{\prime}\\!,\,\omega}-\chi_{\downarrow\downarrow}^{\nu,\nu^{\prime}\\!,\,\omega}-\chi_{\uparrow\downarrow}^{\nu,\nu^{\prime}\\!,\,\omega}+\chi_{\downarrow\uparrow}^{\nu,\nu^{\prime}\\!,\,\omega},$
(6) $\displaystyle\chi_{\text{sc}}^{\nu,\nu^{\prime}\\!,\,\omega}$
$\displaystyle=\chi_{\uparrow\uparrow}^{\nu,\nu^{\prime}\\!,\,\omega}-\chi_{\downarrow\downarrow}^{\nu,\nu^{\prime}\\!,\,\omega}+\chi_{\uparrow\downarrow}^{\nu,\nu^{\prime}\\!,\,\omega}-\chi_{\downarrow\uparrow}^{\nu,\nu^{\prime}\\!,\,\omega},$
(7) $\displaystyle\chi_{S_{x}}^{\nu,\nu^{\prime}\\!,\,\omega}$
$\displaystyle=\chi_{S_{y}}^{\nu,\nu^{\prime}\\!,\,\omega}=\chi_{\overline{\uparrow\downarrow}}^{\nu,\nu^{\prime}\\!,\,\omega}+\chi_{\overline{\downarrow\uparrow}}^{\nu,\nu^{\prime}\\!,\,\omega},$
(8) $\displaystyle\chi_{\text{pair}}^{\nu,\nu^{\prime}\\!,\,\omega}$
$\displaystyle=-\chi_{\overline{\uparrow\downarrow},\text{pp}}^{\nu,\nu^{\prime}\\!,\,\omega}-\Big{(}\chi_{\overline{\downarrow\uparrow},\text{pp}}^{\nu,\nu^{\prime}\\!,\,\omega}\Big{)}^{*}.$
(9)

For the subsequent discussion it is convenient to view the generalized
susceptibilities as (infinite dimensional) matrices in the frequencies $\nu$,
$\nu^{\prime}$ and treat $\omega$ as an additional parameter.

To lighten the notation we are going to omit the fermionic frequencies $\nu$
and $\nu^{\prime}$ in the following and state the bosonic frequency $\omega$
only explicitly, when it is not the same for all quantities in an expression.
Within this notation all operations (e.g. multiplication, transposition,
inversion, …) are meant as matrix operations in the fermionic Matsubara
frequency space.

Note that for broken SU(2)-symmetry the previously separate charge and the
longitudinal spin channels get coupled into the longitudinal (L) channel

$\displaystyle\chi_{\text{L}}=\begin{pmatrix}\chi_{\text{c}}&\chi_{\text{cs}}\\\
\chi_{\text{sc}}&\chi_{\text{s}}\end{pmatrix},$ (10)

where the space of the longitudinal channel is a tensor product space of the
charge and spin fermionic Matsubara frequency spaces.

The generalized susceptibility matrices show interesting properties that
restrict their eigenvalues and eigenvector structure and depend the symmetries
of the considered model (see appendix A for an in depth discussion).

For $\omega=0$, the matrices of the transversal spin channel $\chi_{S_{x}}$ as
well as the pairing channel $\chi_{\text{pair}}$ can be identified as centro-
hermitian matrices [37] (see eq. 58), and therefore only have eigenvalues that
are real or come in complex conjugate pairs [38]. Moreover, the matrix of the
coupled longitudinal channel $\chi_{\text{L}}$ is classified as a
$\kappa-$real matrix eq. 60, which shows the same eigenvalue properties [39].

In addition, the generalized susceptibilities for all considered channels are
symmetric matrices and, hence, can be diagonalized by an orthogonal
transformation [40]. This, together with the centro-hermitian respectively
$\kappa-$real property, leads to the fact that eigenvectors $v_{\alpha}$ of
real eigenvalues $\lambda_{\alpha}$ have a symmetric/antisymmetric real part
$\real v_{\alpha}$ (in $\nu$) and an antisymmetric/symmetric imaginary part
$\imaginary v_{\alpha}$ (see eq. 63) [37]. For the longitudinal channel
eigenvectors are symmetric/antisymmetric in the charge respectively spin
subspace (see eqs. 64 and 64). fig. 1 shows a schematic example of a
$\kappa-$real matrix that is built from blocks of centro-hermitian matrices in
the fermionic Matsubara frequencies, as it is the case in the coupled
longitudinal channel.

Figure 1:  Schematic illustration of a $\kappa-$real matrix that is built from
blocks of centro-hermitian matrices, which is relevant for the coupled
longitudinal channel when the $SU(2)_{S}$-symmetry is violated.

A concise summary of the generalized susceptibility matrix properties for
$\omega=0$ in the different channels and under additional symmetries can be
seen in table 1, where _No Symmetry_ refers to time translation symmetry, spin
conservation and that the Hamiltonian is a real function of $c$ and
$c^{\dagger}$. All these symmetries can be derived by applying eqs. 54, 53,
55, 56 and 57 on the respective definition of the different channels.

Symmetry | $\chi_{\text{L}}^{\omega=0}$ | $\chi_{Sx}^{\omega=0}$ | $\chi_{\text{pair}}^{\omega=0}$
---|---|---|---
No symmetry |  $\kappa$-real, symmetric | centro-hermitian, symmetric | centro-hermitian, symmetric
$SU(2)_{S}$ | block-diagonal, kappa-real, symmetric | centro-hermitian, symmetric | bi-symmetric, real
$SU(2)_{P}$ | kappa-real, symmetric | bi-symmetric, real | centro-hermitian, symmetric
PH ($SU(2)_{S}\bigotimes SU(2)_{P}$) | block-diagonal, real, symmetric | bi-symmetric, real | bi-symmetric, real
Table 1: Matrix properties of the generalized susceptibilities under
additional symmetries where _No Symmetry_ refers to time translation symmetry,
spin conservation and that the Hamiltonian is a real function of $c$ and
$c^{\dagger}$.
For the additional symmetries we use: $SU(2)_{S}$ for Spin SU(2) symmetry
$(h=0)$, $SU(2)_{P}$ for Pseudo-Spin $SU(2)$ symmetry $(\delta\mu=0)$, PH for
particle-hole symmetry ($SU(2)_{S}$ and $SU(2)_{P}$ symmetry).

###  II.2 Shiba Transformation

The mapping—we apply in this paper—systematically maps all quantities of a
model with on-site interaction onto a corresponding model with sign-flipped
interaction. It has been used for the study in Ref. [28] and is derived by
applying a partial (local) particle-hole transformation, or Shiba
transformation [41], acting on the creation and annihilation operators with

$\displaystyle c_{\uparrow}\rightarrow
c_{\uparrow},c^{\dagger}_{\uparrow}\rightarrow
c^{\dagger}_{\uparrow},c_{\downarrow}\rightarrow
c^{\dagger}_{\downarrow}\text{ and }c^{\dagger}_{\downarrow}\rightarrow
c_{\downarrow}.$ (11)

By considering a local single orbital Hubbard Hamiltonian (written in a
symmetric form with $\delta\mu=\mu-U/2$) one readily sees that applying the
Shiba transformation

$\displaystyle\begin{split}H(U,&\delta\mu,h)=\\\
-\delta\mu(n_{\uparrow}+n_{\downarrow})-h(n_{\uparrow}-n_{\downarrow})&+U(n_{\uparrow}-1/2)(n_{\downarrow}-1/2)\\\
&\Bigg{\updownarrow}\text{Shiba}\\\
-\delta\mu(n_{\uparrow}-n_{\downarrow})-h(n_{\uparrow}+n_{\downarrow})&-U(n_{\uparrow}-1/2)(n_{\downarrow}-1/2)\\\
=H(-&U,h,\delta\mu),\end{split}$ (12)

($n_{\sigma}=c^{\dagger}_{\sigma}c_{\sigma}$) corresponds to interchanging the
values of $\delta\mu\leftrightarrow h$ as well as a sign flip in the
interaction $U$ (any constant terms that appear from the mapping in the
Hamiltonian have been omitted).

The Hamiltonian in eq. 12 corresponds to the Hubbard atom (HA), which we
consider for our exemplary study in section IV for different fillings and
external magnetic field $h$. The model can be solved analytically [42]; the
corresponding expressions for the one and two particle Green’s functions can
be found in appendix D

The (non-local) Shiba mapping can be also applied for a bipartite lattice
model, i.e. a lattice that can be divided into two sub lattices, where the
hopping from each site only connects to the other sub lattice. The Hamiltonian
of such a model reads

$\displaystyle\begin{split}H&=-\sum_{i,j,\sigma}t_{i,j}c^{\dagger}_{i,\sigma}c_{j,\sigma}-\delta\mu\sum_{i}(n_{i,\uparrow}+n_{i,\downarrow})\\\
&-h\sum_{i}(n_{i,\uparrow}-n_{i,\downarrow})+U\sum_{i}(n_{i,\uparrow}-1/2)(n_{i,\downarrow}-1/2),\end{split}$
(13)

where $t_{ij}$ is zero if $i$ and $j$ belong to the same sub lattice.
Assigning one sub lattices to all even and the other one to all odd sites
respectively we readily see that the non-local Shiba transformation
$c_{i,\downarrow}\rightarrow c^{\dagger}_{i,\downarrow}(-1)^{i}$ leaves the
hopping term eq. 13 invariant and otherwise follows eq. 12. Therefore, the
effect of the Shiba transformation on bipartite systems (with local on-site
interaction) can be regarded to be analog to purely local systems.

The Shiba mapping is linked to two sets of $SU(2)$ spin algebras, of which the
generators are the (local) spin operator $\bm{\mathcal{S}}$ with the
components

$\displaystyle\mathcal{S}_{x}=c_{\uparrow}^{\dagger}c_{\downarrow}+c_{\downarrow}^{\dagger}c_{\uparrow},$
(14)
$\displaystyle\mathcal{S}_{y}=-i(c_{\uparrow}^{\dagger}c_{\downarrow}-c_{\downarrow}^{\dagger}c_{\uparrow}),$
(15)
$\displaystyle\mathcal{S}_{z}=c_{\uparrow}^{\dagger}c_{\uparrow}-c_{\downarrow}^{\dagger}c_{\downarrow}$
(16)

and the (local) pseudospin operator $\bm{\mathcal{S}}_{p}$ with the components

$\displaystyle\mathcal{S}_{p,x}=c_{\uparrow}^{\dagger}c_{\downarrow}^{\dagger}+c_{\downarrow}c_{\uparrow},$
(17)
$\displaystyle\mathcal{S}_{p,y}=-i(c_{\uparrow}^{\dagger}c_{\downarrow}^{\dagger}-c_{\downarrow}c_{\uparrow}),$
(18)
$\displaystyle\mathcal{S}_{p,z}=c_{\uparrow}^{\dagger}c_{\uparrow}+c_{\downarrow}^{\dagger}c_{\downarrow}-1.$
(19)

Evidently, the Shiba transformation maps the spin $\bm{\mathcal{S}}$ to the
pseudospin $\bm{\mathcal{S}}_{p}$. Since the components of $\bm{\mathcal{S}}$
and $\bm{\mathcal{S}}_{p}$ commute, spin and pseudospin can be regarded as
independent set of operators[36].

The pseudospin can be given the following physical interpretation:
$\mathcal{S}_{p,x}$ and $\mathcal{S}_{p,y}$ correspond to the local Cooper
pair operators, which describe the real and imaginary part of the order
parameter of a strong-coupling (Bose-Einstein-like) superconductor, and
$\mathcal{S}_{p,z}$ describes the deviation of the density from half filling.

The Hamiltonian in the upper line of in eq. 12 commutes with all components of
$\bm{\mathcal{S}}$ if $h=0$, which we will denote as $SU(2)_{S}$ symmetry, and
with all components of $\bm{\mathcal{S}}_{p}$ for $\delta\mu=0$, referred to
as $SU(2)_{P}$ symmetry.

Systems that exhibit a symmetry under rotations generated by spin _and_
pseudospin, are symmetric under $SO(4)\simeq(SU(2)_{S}\otimes
SU(2)_{P})/\mathbb{Z}_{2}$ [36]. These $SO(4)$ symmetric systems are particle-
hole symmetric111Note that, for the scope of this paper, we adopt the standard
definition of particle-hole transformation, i.e. $c_{\sigma}\rightarrow
c^{\dagger}_{\sigma}$. Of course, in principle, one could also choose the
alternative definition of $c_{\sigma}\rightarrow c^{\dagger}_{-\sigma}$. The
latter definition would be consistent with the fact that the particle-hole
transformation changes the charge of the particle and therefore also its gyro-
magnetic moment. This would then be accounted for by flipping the spin in the
transformation. Using this alternative definition, the $SU(2)_{P}$ symmetry
would become equivalent to the particle-hole symmetry and the $SO(4)$ symmetry
would be achieved in the presence of simultaneous particle-hole and
$SU(2)_{S}$ symmetry..

The physical susceptibilities, describing the (linear) response of the system
on observable $A$ when observable $B$ is coupled to an external perturbation,

$\displaystyle\chi_{\text{AB}}^{\text{phys}}(\tau)=\langle
A(\tau)B(0)\rangle-\langle A\rangle\langle B\rangle$ (20)

are related to the generalized susceptibilities

$\displaystyle\chi_{r}^{\text{phys}}(\omega)=\frac{1}{\beta^{2}}\sum_{\nu\nu^{\prime}}\chi_{r}^{\nu\nu^{\prime}\omega}.$
(21)

Using eq. 21 and the definitions in eqs. 4, 5, 6, 7, 8 and 9 we find

$\displaystyle\chi_{\text{c}}^{\text{phys}}$
$\displaystyle:A=B=\mathcal{S}_{p,z},$ (22)
$\displaystyle\chi_{\text{s}}^{\text{phys}}$
$\displaystyle:A=B=\mathcal{S}_{z},$ (23)
$\displaystyle\chi_{\text{cs}}^{\text{phys}}$
$\displaystyle:A=\mathcal{S}_{p,z},B=\mathcal{S}_{z},$ (24)
$\displaystyle\chi_{\text{sc}}^{\text{phys}}$
$\displaystyle:A=\mathcal{S}_{z},B=\mathcal{S}_{p,z},$ (25)
$\displaystyle\chi_{S_{x}}^{\text{phys}}$ $\displaystyle:A=B=\mathcal{S}_{x},$
(26) $\displaystyle\chi_{\text{pair}}^{\text{phys}}$
$\displaystyle=A=B=\mathcal{S}_{p,x}.$ (27)

The mapping of the physical susceptibilities under the Shiba transformation is
thereby clearly identified from the corresponding mapping of the spin and
pseudospin components.

The more complicated case of the Shiba mapping for the local generalized
susceptibilities is derived in the next section III.

##  III Derivation of the mapping

In the following, we derive how the Shiba transformation in eq. 11 is acting
on the local222Note that all derivations in this section also hold for non-
local generalized susceptibilities of bipartite lattice models if the
following criterion is fulfilled: the algebraic sum of the lattice indices of
the considered generalized susceptibility is even. generalized
susceptibilities of the charge, longitudinal spin, transversal spin, and
pairing channel (see eqs. 4, 5, 6, 7, 8 and 9) for the most general cases—i.e.
arbitrary filling, external magnetic field, and arbitrary bosonic transfer
frequency $\omega$. Thereby, the coupling of the charge and longitudinal spin
channel for broken $SU(2)_{S}$ symmetry –i.e. finite magnetic field $h$– into
the longitudinal channel (c.f. eq. 10) has to be explicitly considered.

By applying the Shiba transformation on the general definition in eq. 1 we
find the relations:

$\displaystyle\begin{split}\chi_{\uparrow\uparrow,\text{U}}^{\nu,\nu^{\prime}\\!,\,\omega}=\chi_{\uparrow\uparrow,\text{-U}}^{\nu,\nu^{\prime}\\!,\,\omega},\;\;\chi_{\downarrow\downarrow,\text{U}}^{\nu,\nu^{\prime}\\!,\,\omega}=\chi_{\downarrow\downarrow,\text{-U}}^{-\nu-\omega,-\nu^{\prime}-\omega,\,\omega},\\\
\chi_{\uparrow\downarrow,\text{U}}^{\nu,\nu^{\prime}\\!,\,\omega}=-\chi_{\uparrow\downarrow,\text{-U}}^{\nu,-\nu^{\prime}-\omega,\,\omega},\;\;\text{and}\;\;\chi_{\downarrow\uparrow,\text{U}}^{\nu\nu^{\prime}}=-\chi_{\downarrow\uparrow,\text{-U}}^{-\nu-\omega,\nu^{\prime}\\!,\,\omega},\end{split}$
(28)

where the subscript $U$ refers to the model parameters $(U,\delta\mu,h)$ and
the subscript $-U$ to $(-U,h,\delta\mu$), i.e. the two models related by the
Shiba transformation in eq. 12.

With these building blocks we can now apply the Shiba mapping to the coupled
longitudinal channel.

We start with the case of $\omega=0$ and then generalize to finite bosonic
frequencies.

We will identify the similarity transformation $S$ that captures the action of
the Shiba transformation onto the generalized susceptibilities
$\chi_{\text{L,U}}$ and $\chi_{\text{L,-U}}$. Thereby, we employ the following
matrix $Q$ in Matsubara space:

$\displaystyle Q=\frac{1}{\sqrt{2}}\begin{pmatrix}\mathbb{1}&-J\\\
\mathbb{1}&J\end{pmatrix}\quad\text{with}\quad J=\matrixquantity(&&1\\\
&\udots&\\\ 1&&),$ (29)

where we divided the Matsubara frequencies into quadrants of positive $+$ and
negative $-$ frequencies

$\left(\begin{array}[]{c|c}\chi^{--}&\chi^{-+}\\\
\hline\cr\chi^{+-}&\chi^{++}\\\ \end{array}\right).$ (30)

The similarity transformation $Q^{T}\chi_{CS}Q$ separates the symmetric and
anti-symmetric part of a centro-symmetric matrix $(JAJ=A)$ into two blocks
along the diagonal and the symmetric and anti-symmetric part of a centro-skew-
symmetric matrix $(JAJ=-A)$ into two blocks along the skew diagonal.

Hence, a similarity transformation with $Q$ separates the real $\chi^{\prime}$
and imaginary part $\chi^{\prime\prime}$ of a centro-hermitian matrix
$\chi_{CH}$ $(J\chi_{CH}J=\chi_{CH}^{*})$ into the symmetric and anti-
symmetric components respectively

$\displaystyle
Q\chi_{CH}Q^{T}=\begin{pmatrix}\chi_{\mathcal{A}}^{\prime}&i\chi_{\mathcal{S}}^{\prime\prime}\\\
i\chi_{\mathcal{A}}^{\prime\prime}&\chi_{\mathcal{S}}^{\prime}\end{pmatrix}.$
(31)

A detailed definition of the different blocks can be found in appendix C.

By defining the transformations $\mathcal{Q}$ and $T$ in the tensor product
space of the coupled L channel

$\displaystyle\mathcal{Q}=\begin{pmatrix}Q&\mathbb{0}\\\
\mathbb{0}&Q\end{pmatrix}\quad\text{and}\quad
T=\left(\begin{array}[]{cc}\begin{array}[]{c|c}\mathbb{1}&\mathbb{0}\\\
\hline\cr\mathbb{0}&\mathbb{0}\end{array}&\begin{array}[]{c|c}\mathbb{0}&\mathbb{0}\\\
\hline\cr\mathbb{0}&\mathbb{1}\end{array}\\\
\hline\cr\begin{array}[]{c|c}\mathbb{0}&\mathbb{0}\\\
\hline\cr\mathbb{0}&\mathbb{1}\end{array}&\begin{array}[]{c|c}\mathbb{1}&\mathbb{0}\\\
\hline\cr\mathbb{0}&\mathbb{0}\end{array}\end{array}\right)$ (42)

(c.f. eq. 10) we can determine (see appendix C) the transformation $S$, which
relates the two sides of the Shiba transformation:

$\displaystyle\chi_{\text{L,U}}=S\chi_{\text{L,-U}}S\quad\text{with}\quad
S=\mathcal{Q}^{T}T\mathcal{Q}.$ (43)

In appendix C we show that this relation holds for arbitrary bosonic transfer
frequencies $\omega$, when adapting the $Q$ and $T$ matrices by shifting the
boundaries of the Matsubara frequency quadrants by $(\omega/2,\omega/2)$ in
analog to eq. 30.

Hence, for finite $\omega$ the susceptibility matrices can be viewed as
centro-hermitian in regard to the shifted Matsubara frequency basis (see
appendix A).

From eq. 43 (better visible in eq. 80) it follows that the Shiba
transformation leaves the real anti-symmetric part
$\chi^{\prime}_{\mathcal{A}}$ of each charge-spin components
($\chi_{c},\chi_{s},\chi_{cs},\chi_{sc}$) in $\chi_{L}$ invariant and
exchanges the real symmetric part $\chi^{\prime}_{\mathcal{S}}$ between
$\chi_{\text{c}}\leftrightarrow\chi_{\text{s}}$, and between
$\chi_{\text{cs}}\leftrightarrow\chi_{\text{sc}}$. For the imaginary parts
neither the symmetric nor the anti-symmetric part is invariant under the
mapping and they exchange according to the rules below:

$\displaystyle\begin{split}\chi_{\text{c,-U},\mathcal{A}}^{\prime}&\longleftrightarrow\chi_{\text{c,U},\mathcal{A}}^{\prime},\\\
\chi_{\text{s,-U},\mathcal{A}}^{\prime}&\longleftrightarrow\chi_{\text{s,U},\mathcal{A}}^{\prime},\\\
\chi_{\text{c,-U},\mathcal{S}}^{\prime}&\longleftrightarrow\chi_{\text{s,U},\mathcal{S}}^{\prime},\\\
\chi_{\text{cs,-U},\mathcal{A}}^{\prime}&\longleftrightarrow\chi_{\text{cs,U},\mathcal{A}}^{\prime},\\\
\chi_{\text{sc,-U},\mathcal{A}}^{\prime}&\longleftrightarrow\chi_{\text{sc,U},\mathcal{A}}^{\prime},\\\
\chi_{\text{cs,-U},\mathcal{S}}^{\prime}&\longleftrightarrow\chi_{\text{sc,U},\mathcal{S}}^{\prime},\\\
\chi_{\text{cs,-U},\mathcal{A}}^{\prime\prime}&\longleftrightarrow\chi_{\text{s,U},\mathcal{A}}^{\prime\prime},\\\
\chi_{\text{sc,-U},\mathcal{A}}^{\prime\prime}&\longleftrightarrow\chi_{\text{c,U},\mathcal{A}}^{\prime\prime},\\\
\chi_{\text{c,-U},\mathcal{S}}^{\prime\prime}&\longleftrightarrow\chi_{\text{cs,U},\mathcal{S}}^{\prime\prime},\\\
\chi_{\text{sc,-U},\mathcal{S}}^{\prime\prime}&\longleftrightarrow\chi_{\text{s,U},\mathcal{S}}^{\prime\prime}.\end{split}$
(44)

When we examine the eigenvectors of the generalized susceptibility in the
longitudinal channel, we show that the symmetric part in one subspace is
mapped from the charge to the spin subspace, and vice versa. However, the
antisymmetric part remains unaffected by this transformation (refer to eqs. 84
and 85).

Finally, we apply the Shiba transformation on the generalized susceptibilities
of the pairing and the transversal spin channel. From the transformation of
eq. 2 we get

$\displaystyle\chi_{S_{x},U}^{\omega}=\chi_{\text{pair,-U}}^{-\omega}$ (45)

(details in appendix C).

##  IV Application of the mapping

The mapping derived in section III can be useful in several practical
contexts, being applicable also beyond the purely local Hubbard atom
Hamiltonian in the first line of eq. 12. In fact, the expressions derived in
the previous section may also be applied to the local two-particle quantities
of bipartite lattice models with on-site electrostatic interaction, where
their mapping would work in a completely analogous fashion (see eq. 13). For
instance, our equations could be directly exploited to map the on-site
generalized susceptibilities of the Hubbard model on a generic $d-$dimensional
bipartite lattice with nearest neighbor333We recall that the hopping $t_{ij}$
transforms under the Shiba transformation for lattice models as:
$t_{ij}\rightarrow t_{ij}$ if one index belongs to an odd site and the other
on an even site and $t_{ij}\rightarrow\sigma t_{ij}$ otherwise. Hence, if the
hopping connects odd to odd or even to even sides (e.g. via a n.n.n hopping
$t^{\prime}$), this hopping term would change under the Shiba mapping from
$t^{\prime}$ to $-t^{\prime}$ for the down electrons only, limiting the
potential usefulness of the mapped expressions. hopping $t$ and
$(U,\delta\mu,h)$ to the corresponding lattice model with the same hopping and
$(-U,h,\delta\mu)$. Evidently, this feature renders the derived mapping also
immediately applicable to the local two-particle quantities of the widely used
Dynamical Mean-Field Theory (DMFT) [46]. On an even more general perspective,
it is worth mentioning that eqs. 43 and 45 would hold for the on-site
generalized susceptibilities of arbitrary lattice models, albeit with the
important caveat that the specific meaning of subindices $U$ and $-U$ need to
be properly reinterpreted444In practice, this can be done by determining the
precise effect of the Shiba transformation on the Hamiltonian of the
considered model and then defining the meaning of $U$ and $-U$ according to
this result.

Independent of the specific models considered, an evident advantage of the
mapping is to potentially reduce the effort of two-particle calculations, if
these need to be systematically performed in different regions of the phase-
diagram. Further, one could exploit the mapping to investigate the two-
particle properties of regimes (e.g., with $SU(2)_{S}$-broken symmetry) that
might otherwise not be directly and/or easily accessible with the numerical
algorithms at disposal.

Finally, the derived mapping gives also some fundamental insights on how two-
particle quantities of different scattering channels are connected to each
other and how they are linked to the symmetries of the system.

As an example to illustrate how this aspect of the mapping works, we look at
the problem of the two-particle irreducible vertex divergences in the Hubbard
atom, which is of considerable interest in the context for the investigation
of breakdown of the self-consistent perturbation expansion for many-electron
systems.

###  IV.1 Vertex divergences and their generalization

In this section, we extend the existing studies[48, 31, 22, 28] on the
divergences in the two-particle irreducible vertex of the Hubbard atom, which
have been hitherto restricted to the rather specific, highly symmetric case of
$\delta\mu=h=0$ (i.e., of perfect particle-hole symmetry, so $SU(2)_{S}$ and
$SU(2)_{P}$ symmetry). In particular, we aim here at individuating the
divergences of the two-particle irreducible vertices at transfer frequency
$\omega=0$ in all scattering channels for the whole parameter space of the
Hubbard atom with positive (repulsive) as well as negative (attractive) on-
site interaction, arbitrary filling, and in the presence of an external
magnetic field $h$.

We briefly recall here that the two-particle irreducible vertex function
$\Gamma_{r}^{\nu,\nu^{\prime}}(\omega)$ in a given scattering channel $r$
(e.g., charge, spin, pairing, etc.) is formally defined[49, 50, 20, 36] as the
kernel of the (self-consistent) Bethe-Salpeter equation (BSE) in the
corresponding sector $r$, and, as such, can be computed by inverting the BSE,
i.e.

$\Gamma_{r}^{\nu\nu^{\prime}}(\omega)\\!:=\\!\left[\chi_{r}^{-1}\right]^{\nu,\nu^{\prime}\\!,\,\omega}-\left[\chi_{r,0}^{-1}\right]^{\nu,\nu^{\prime}\\!,\,\omega},$

where the transversal channels has to be treated with care if
$H\notin\mathbb{R}$ and $\omega\neq 0$ (see appendix B)

Hence, the divergences of the $\Gamma_{r}$ are directly determined by the
vanishing of an eigenvalue of the corresponding generalized susceptibility
matrix $\chi_{r}^{\nu,\nu^{\prime}\\!,\,\omega}$ in the fermionic Matsubara
frequencies $\nu,\nu^{\prime}$ 555None of the different “bubble” terms
$\chi_{r,0}^{\nu,\nu^{\prime}\\!,\,\omega}$ can trigger a divergences, as they
are just diagonal in frequency spaces and defined by the product of two
Green’s functions..

As mentioned in section II.1 by perfect $SU(2)_{S}$ and $SU(2)_{P}$ symmetry,
all local two-particle quantities for zero-transfer frequency ($\omega=0$) are
real and symmetric matrices. Hence, the _sign change_ of an eigenvalue can
only happen if this eigenvalue vanishes. This eigenvalue is then responsible
for the divergence of $\Gamma_{r}^{\nu\nu^{\prime}}$. These sign changes, as
discussed in the recent literature [31, 28, 52, 34, 35], are crucial to drive
the suppression/enhancements of the corresponding physical static response of
the systems, and, in more complicated lattice systems, even to trigger
thermodynamic phase-instabilities [53, 52, 34, 37, 54, 55]. However, if either
$SU(2)_{S}$ ($h\neq 0$) or $SU(2)_{P}$ ($\delta\mu\neq 0$) symmetries are
broken, $\chi_{r}^{\nu\nu^{\prime}}$ has no longer to be a real and symmetric
matrix for all channels, but can be either a centro-hermitian or a
$\kappa$-real matrix (see table 1). Consequently, its eigenvalues are no
longer required to be real: They can be either real or appear in complex
conjugate pairs, consistent with the fundamental theorem of algebra [38, 39].

As a consequence, the discussed sign-changes of a eigenvalue of the
generalized susceptibilities, or at least of their real part, may now occur
continuously in the parameter space without the necessity that the eigenvalue
vanishes, which would trigger a vertex divergence. In fact, complex conjugate
pairs of eigenvalues can switch the sign of their real parts without vanishing
by keeping their imaginary parts finite. In this context, we define the
occurrence of a complex conjugate pair of eigenvalues with vanishing real
part, but finite imaginary part as _pseudo-divergence_. In this way, we can
extend the concept of vertex divergences to those cases, where sign-changes of
the real part of an eigenvalue pair of $\chi_{r}^{\nu\nu^{\prime}}$ are not
directly associated to an actual divergence of $\Gamma_{r}^{\nu\nu^{\prime}}$.
Loosely speaking, this generalization might recall the case of a branch cut of
the square root function which is not crossing through the origin, but rather
the negative real at a finite value by changing the sign of the imaginary
part. Eventually, the pseudo-divergences defined here in the physical
parameter space may be linked to the analytic properties of the generalized
susceptibilities[56, 57] and/or of the perturbative expansion of the electron
self-energy[10, 58, 59, 60] in abstract complex planes obtained, e.g. through
the complexification of the electronic interaction $U$[10, 56, 57].

###  IV.2 Dimensionless representation

Before starting the analysis of the vertex-divergences and pseudo-divergences
in the whole phase space of the Hubbard atom, it is convenient to make a
general consideration based on the formal Lehmann-representation of the on-
site666Note, however, that the presented arguments also apply for non-local
susceptibilities with the only difference that the $\sigma$ index must be
interpreted as combined $\sigma,k$ index generalized susceptibilities, whose
expression are reported, e.g., appendix D and in Ref. [62]. Let us consider a
model Hamiltonian, which is defined through $N$ real parameters
$\evaluated{\\{\varepsilon_{i}\\}}_{i=1}^{N}$ (e.g., the hopping $t$, the on-
site interaction $U$, etc.), where all $\varepsilon_{i}$ have the dimension of
an energy. Evidently, all eigenenergies of the many-electron Hamiltonian
considered will only depend on these parameters (or combination thereof),
i.e., $\\{E_{j}\\}=\\{E_{j}(\\{\varepsilon_{i}\\})\\}$. As a consequence, all
corresponding dimensionless quantities $\beta E_{j}$ must only be dependent on
the set $\\{\beta\epsilon_{i}\\}$ for all $j$. Then, by looking at the
Lehmann-representation of the generalized susceptibility, one readily realizes
that, apart from an overall scaling of $\beta^{3}$, the generalized
susceptibility can be always expressed in terms of the parameter set $\\{\beta
E_{j}\\}$ which can be fully expressed via the parameter set
$\\{\beta\varepsilon_{i}\\}$. Therefore, one can rewrite the Lehmann-
representation as:

$\displaystyle\chi^{\nu\nu^{\prime}\omega}_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}(\beta,\\{\varepsilon_{i}\\})=\beta^{3}f^{nn^{\prime}m}_{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}}(\\{\beta\varepsilon_{i}\\}),$
(46)

where $n,n^{\prime}$ and $m$ are the indices for $\nu,\nu^{\prime}$ and
$\omega$ and $\alpha_{i}$ is a generic set of quantum numbers. On the basis of
this observation, presenting our results for the location and the properties
of the vertex (pseudo-)divergences of the HA in the _dimensionless_ parameter
space $\\{\beta\delta\mu,\beta h,\beta U\\}$ turns out to be a particularly
convenient choice, as it allows for a “universal” representation of our
findings, directly applicable to any finite value of the temperature.

Before discussing the results, let us clarify the semantics. The nature of the
vertex (pseudo-)divergence is made clear by the color and the line-style in
the plot: Solid lines mark divergences and dashed lines mark pseudo-
divergences. For identifying the different channels, where the
(pseudo-)divergence occur, we adopt the following color coding, extending the
one introduced in existing literature: We use _red_ for a vertex
(pseudo-)divergence in the charge channel; _yellow_ for the pairing channel;
_green_ for the longitudinal spin channel; _blue_ for the transversal spin
channel; _bluish green_ for degenerate spin channel. Eventually, for broken
$SU(2)_{S}$ symmetry ($h\neq 0)$, where the coupled longitudinal channel needs
to be considered on a whole, a weighted mixing of green and red is used. The
proportion of red/green is determined by the percentage of the norm that the
vector has in the charge/spin subspace for more information see appendix E.
Further, we classify a divergence as “symmetric/antisymmetric” if the
corresponding eigenvector of the generalized susceptibility, which has a
vanishing eigenvalue, has a symmetric/antisymmetric real part and an
antisymmetric/symmetric imaginary part, provided that we use the norm eq. 62
to normalize the eigenvectors. Eventually, we recall that in the case of
perfect particle-hole/$SO(4)$ symmetry ($\delta\mu$ = $h=0$), it has been
analytically demonstrated in the literature[48, 22, 28] that the HA displays a
discrete set of vertex divergences, marked by colored dots in fig. 2, with
alternating properties for increasing (repulsive or attractive) interaction.
In particular, for $U>0$, the red dots on the $U$ axis of fig. 2 mark the
location of the irreducible vertex divergences in the charge channel
associated to an anti-symmetric eigenvector, while the orange dots mark the
simultaneous occurrence of vertex divergences in the charge and in the pairing
sector, associated to a symmetric eigenvector. Consistent with the mapping in
the perfect particle-hole regime [28], for $U<0$ one finds, exactly in
mirrored positions, alternating antisymmetric vertex divergences in the charge
sector (red dots) and symmetric ones in the spin-sector (bluish green dots).

###  IV.3 Vertex divergences of the HA: Limiting cases

We start then to illustrate the results of our investigation by considering
two special planes in the dimensionless phase space introduced above. These
correspond to the two special cases beyond perfect particle-hole/$SO(4)$
symmetry, where either the $SU(2)_{S}$ $(h=0$ left in fig. 2) or the
$SU(2)_{P}$ $(\delta\mu=0$ right in fig. 2) symmetry is preserved.

Figure 2:  Dimensionless phase space diagram of the Hubbard atom with
repulsive and attractive interaction, showing the location and the nature of
the (pseudo-)divergences of the irreducible vertex functions in the different
channels at the two special cases of (i) no magnetic field but out of half-
filling (left panel) and (ii) half filling in the presence of a magnetic field
(right panel). Solid lines mark divergences and dashed lines mark pseudo-
divergences, while, in order to identify the different channels, the following
color coding is adopted: _red_ for a vertex (pseudo-)divergence in the charge
channel; _yellow_ for the pairing channel; _green_ for the longitudinal spin
channel; _blue_ for the transversal spin channel; _bluish green_ for
degenerated spin channel. Further, when $SU(2)_{S}$ symmetry is broken ($h\neq
0)$, the coupled longitudinal channel needs to be considered on a whole (see
text) : In this case, a weighted mixing of green and red has been used.

Starting with the left panel of fig. 2 it should be noted that, as soon as the
$SU(2)_{P}$-symmetry is lifted for any $\delta\mu\neq 0$, the symmetric
divergences in the charge and pairing channel do no longer occur
simultaneously, yielding separate divergence lines (marked in yellow and red,
respectively, for the pairing and the charge sector). However, this is not the
only qualitative difference emerging between these two kind of vertex
divergences for $\delta\mu\neq 0$: As reported in table 1 the generalized
susceptibility in the pairing channel is a real and bi-symmetric matrix as
long as the $SU(2)_{S}$ symmetry is preserved. Therefore, differently from its
charge counterpart, its eigenvalues remain purely real numbers and the
associated vertex divergences actual divergences, without any possible
occurrence of exceptional points (EPs) and pseudo-divergences. For the same
reason, the symmetric nature of its singular eigenvectors is maintained for
arbitrary values of $\delta\mu$.

On a more geometrical viewpoint, one can easily observe in the left panel of
fig. 2 that the divergences lines of the charge sectors form elliptic shaped
loops in the phase space, terminating on both sides in EPs for positive and
negative values of $\delta\mu$. Henceforth, for the sake of conciseness, we
will refer to these specific divergence structures, including the associated
pseudo-divergences lines, with the short-hand term of _loops_. These loops
each consist of one symmetric (upper loop part) and one antisymmetric (lower
loop part) divergence line which meet at an EP. From this EP a pseudo-
divergence line emerges. Quantitatively, we note that the EPs of the loop
found at the lowest $\beta U$ values roughly corresponds to a hole/electron
doping of about $6$%. It is also important to stress, that the three loops
shown in fig. 2, as all the other vertex-divergence structures discussed in
the following, represent only the first three loop structures encountered by
increasing $\beta U$. In fact, consistent with Refs. [48, 22, 63], where the
occurrence of infinitely many vertex divergences at half filling has been
demonstrated, we also find infinitely many loops for increasing $\beta U$
values.

Turing now to the study of the attractive sector, we find that the divergence
lines show a completely different shape: They are disposed in a series of
parabolas one inside the other, as it can be seen on the bottom part of the
left panel in fig. 2. More specifically, considering first the charge sector,
the geometrical maximum of the parabola is represented by the (antisymmetric)
divergences located at particle-hole symmetry. Upon doping, an antisymmetric
divergence line meets with a symmetric divergence line out of half filling at
an EP, forming a pseudo-divergence line. The qualitatively different
geometrical structure compared to the elliptical shaped divergence line
structures in the repulsive sector corresponds to a specific difference: Each
symmetric divergence line extends down to $\beta U\rightarrow-\infty$. The
reason why this happens is the absence of symmetric divergences at half
filling in the charge sector,which prevents the formation of closed loops of
symmetric/antisymmetric divergences at difference with the repulsive sector.

In the spin sector, we observe an analogous situation777Note that, also in
this case, we only show the first three parabolas for each channel., only with
interchanged roles of the eigenvector symmetries: The symmetric divergence
lines, originating from the corresponding purely symmetric divergences at
half-filling, meet with an antisymmetric divergence line, extended down to
$\beta U\rightarrow-\infty$, at an EP, from which a pseudo-divergence emerges.

A second important difference w.r.t. the repulsive case is represented by the
orientation of the pseudo-divergences of both channels. These, after emerging
from the corresponding EPs, are directed towards smaller absolute values of
$\beta U$, in contrast to the pseudo-divergences in the repulsive sector. Even
more remarkably, they do cross the $U=0$ axis, continuing afterwards in the
repulsive sector. More specifically, one can classify the (pseudo-)divergences
with an integer number $N$ identifying the positive Matsubara frequency at
which their associated eigenvector displays its largest component. In this
way, we find that the $N^{th}$ pseudo-divergence of both channels crosses the
$U=0$ axis at $\beta\delta\mu=\beta\mu=\pm(2N-1)\pi$ (see appendix G). In
fact, this has to be the same point, because the generalized susceptibilities
of two-channels coincide for $U=0$, due to the absence of vertex corrections.
For more detailed information see [65].

In the case of preserved $SU(2)_{P}$ symmetry (i.e.,
$\delta\mu\\!=\\!0,h\\!\neq\\!0$, see fig. 2 right), the mapping relations
determine a perfectly “mirrored” disposition of vertex divergences and pseudo-
divergences, with the parabola structures now located in the repulsive sector
and the loop structure in the attractive one.

At the same time, as $SU(2)_{S}$ symmetry is broken for $h\neq 0$, the spin
channel splits in the spin transverse (blue lines) channel and in the spin
longitudinal channel (green lines), whereas the latter becomes coupled to the
charge one (red) in what we have defined as longitudinal sector. In
particular, consistent to the mapping, all pairing vertex divergences in the
repulsive sector of the $SU(2)_{S}$ case (yellow lines in the left panel fig.
2) are transformed in the corresponding divergences of the transversal channel
in the attractive sector with $SU(2)_{P}$ symmetry (blue lines in the right
panel of fig. 2). Indeed, for the same reasons explains above, the generalized
susceptibility of the spin transversal channels remains real and bi-symmetric
for $\delta\mu\\!=\\!0$ when the $SU(2)_{P}$ symmetry is preserved.
Analogously, all the loop and the parabola structures define now divergences
in the combined longitudinal sector, whereas the specific ratio between their
components in the charge and spin longitudinal subspaces is highlighted by the
red/green color scale888More quantitatively, one could observe the following:
Along a parabola towards the exceptional point the percentage of the
longitudinal spin subspace in the divergences increases. Remarkably, the
maximal mixing happens at the exceptional point. Far away from the exceptional
point at high $\beta U$ the divergences are almost completely in the
longitudinal spin subspace. Note that the pseudo-divergences in the
longitudinal are always maximally mixed (i.e. having weight 0.5 in the charge
and the spin subspace) if $SU(2)_{S}$ symmetry is present, which is shown in
eq. 98. , as detailed in appendix E. Moreover, a parabola in the pairing
channel (marked in a thicker yellow line in the right panel of fig. 2) is
superimposed to every second parabola of divergences in the longitudinal
sector.

###  IV.4 The T=0 limit

Before coming to the study of the most general case, where both $SU(2)_{S}$
and $SU(2)_{P}$ are violated, is worth to exploit our results to infer how the
vertex (pseudo-)divergences evolve in the $T\rightarrow 0$ limit. In fact, in
spite of its intrinsic interest (e.g. to discuss the relation between the
breakdown of self-consistent perturbation theory and the possible violation of
the Luttinger theorem), most of the analysis of the non-perturbative effects
on the two-particle level have been restricted to the finite $T$ case. The
choice of working in the dimensionless phase space offers, a valuable
opportunity to fill this gap, at least for the HA. Indeed, although our
temperature independent representation of the vertex divergences can be used,
strictly speaking, only to analyze finite $T$ cases, the possibility of
systematically extracting information for arbitrarily small temperatures and
evaluating the asymptotic behavior of vertex (pseudo-)divergence lines allow
for a description of the $T=0$ limit.

The results obtained are summarized in the schematic representation of the
$T\\!=\\!=0$ phase space fig. 3, while more details about the procedure used
can be found in appendix F.

Figure 3:  Schematic phase space diagram of the vertex (pseudo-)divergences
(bottom panels) in the Hubbard atom at $T=0$ for the two special cases shown
in fig. 2 (left: $h=0$ and $\delta\mu\neq 0$; right: $h\neq 0$ and
$\delta\mu=0$) compared to the corresponding filling
$n=n_{\uparrow}+n_{\downarrow}$ (upper left) and magnetization
$m=n_{\uparrow}-n_{\downarrow}$ (upper right) of the system. This schematic
phase space diagram should be understood as a continuous distribution of
(pseudo-)divergences of different nature in the whole phase space of the HA at
$T=0$: A divergence or pseudo-divergence of the vertex is found at every point
in the phase space, whereas the channel in which it occurs is encoded by the
color of the dense set of lines shown in the bottom panels. The plotting
conventions are as in fig. 2.

In particular, in the lower panels of fig. 3 we observe a continuous
distribution of vertex divergences at particle-hole symmetry
$(\delta\mu=h=0)$, which occur in the charge and pairing(spin) channels for
$U>0$ ($U<0$). The continuity of the vertex divergences distribution at $T=0$
evidently reflects the continuity of the Matsubara frequencies in that limit,
and appears consistent with the increasingly dense structure of the finite $T$
divergences reported (but not explicitly discussed) in previous works at
particle-hole symmetry[48, 22, 28].

Of greater interest are the results out of particle-hole symmetry. In
particular, for $h=0$ (lower left in fig. 3), we observe a V-shaped area in
the $U>0$ sector (marked by black lines in fig. 3) located between
$\delta\mu=-U/2$ and $\delta\mu=U/2$, which corresponds to the regime of the
HA where the ground state is half-filled (see upper left fig. 3). Remarkably,
for every parameter set within this V-shaped area a vertex divergence in the
paring channel and pseudo-divergence in the charge channel are found
simultaneously. Indeed, the latter ones stem from the “loop-like” structures
in the upper left panel of fig. 2, that collapse on the $\delta\mu=0$ axis in
the $T=0$ limit. Outside the V-shaped area, i.e. in the region which
corresponds to a totally empty or totally full ground-state for all $U$ (see
upper left fig. 3), we find the occurrence of simultaneous pseudo-divergences
in the spin and the charge channel everywhere (see fig. 2 lower left). The
pseudo-divergences originate form the parabola structures in the lower left
panel of fig. 2 as they progressively move towards the $\delta\mu=0$-axis in
the $U<0$ sector.

In the opposite situation of $\delta\mu=0$ (lower right panel in fig. 3) we
find, as expected, a “mirrored” distribution of divergences and pseudo-
divergences: the V-shaped region is now located in the $U<0$ sector, between
$\delta\mu=-U/2$ and $\delta\mu=U/2$. At the same time, due to the violation
of the $SU(2)_{S}$ symmetry and in accordance with the mapping, the V-shaped
region entails maximally mixed pseudo-divergences in the longitudinal sector
and divergences in the transversal spin channel, while pseudo-divergences in
the longitudinal and pairing channel are present outside.

###  IV.5 Vertex divergences of the HA: The general case

Figure 4:  Upper central panel: Sketch of the new coordinate system
$\\{U,r,\alpha\\}$ introduced to systematically interpolate the illustration
of the vertex (pseudo-)divergences of the HA for arbitrary values of $h$ and
$\delta\mu$ between the limiting cases shown in fig. 2, which correspond to
$\alpha=0^{\circ}$ and $90^{\circ}$. The corresponding (pseudo-)divergences in
the coupled longitudital sector already shown in fig. 2 are reproduced here,
for reference, in left/right upper panel. Lower panels: Dimensionless phase
space diagrams of the vertex (pseudo-)divergences in the coupled longitudinal
channel in the most general situation of finite magnetic field and arbitrary
filling. Note that only the first divergence line structure is displayed for
the different constant $\alpha$ planes chosen (see text). The plotting
conventions are as in fig. 2.  Figure 5:  Dimensionless phase space diagrams
of the vertex (pseudo-)divergences in the transversal spin channel (blue) and
in the pairing channel (yellow) for the Hubbard atom. Note that only the first
divergence line structure is displayed for different constant $\alpha$-planes.
The vertex (pseudo-)divergences of the coupled longitudinal channel, already
shown in fig. 4, are reproduced here in gray for reference. Figure 6: Left:
The phase space diagram of the vertex (pseudo-)divergences in longitudinal
channel are shown for $\alpha=35^{\circ}$ in the repulsive sector $U>0$ and
for $\alpha=55^{\circ}$ in the attractive sector $U<0$. These are the two
cases that map into each other according the the mapping derived in section
III. The plotting conventions are as in fig. 2. Middle: The eigenvectors that
corresponds to the (pseudo-)divergence at specific points, which are marked by
black stars. The eigenvectors are split into symmetric $(v_{\mathcal{S}})$ and
antisymmetric part ($v_{\mathcal{A}}$) to make the mapping more visible. Note
that (a) always corresponds to $\alpha=35^{\circ}$ while (b) corresponds to
$\alpha=55^{\circ}$. Right: The projected weights of the eigenvectors again
split into symmetric and antisymmetric part.

We turn eventually to the discussion of the vertex divergences of the HA in
the general case, i.e. for arbitrary values of $h$ and $\delta\mu$. For the
sake of compactness, we introduce the parameters $\alpha$ and $r$, as sketched
in central upper panel of fig. 4, in order to facilitate the variation of the
$\delta\mu,h$ values between the limiting cases of $h=0$ ($\alpha=0$) and
$\delta\mu=0$ ($\alpha=90^{\circ}$) and the visualization of the corresponding
results. Specifically, the parameter $\alpha$ and $r$ are defined as the
following set of cylinder coordinates:

$\displaystyle\delta\mu=r\cos{\alpha}\quad\text{and}\quad h=r\sin{\alpha}$
(47)

It is then easy to see that the sets of $\delta\mu,h$ values connected by the
mapping linking the attractive and the repulsive model, according to eq. 12,
corresponds to this simple relation for the polar coordinate $\alpha$:

$\displaystyle\alpha\leftrightarrow\alpha^{\prime}\quad\text{with}\quad\alpha+\alpha^{\prime}=90^{\circ},$
(48)

whereas $r$ remains unchanged.

Further, due to eqs. 50, 51 and 52, it is enough to consider
$\alpha\in[0^{\circ},90^{\circ}]$, as all other regions of the phase space can
be reconstructed from this interval.

In fig. 4, we show the evolution of the vertex divergence structures in the
longitudinal sector for representative intermediate parameter sets between the
limiting situations of fig. 2. The latter correspond to the case of
$\alpha=0^{\circ}$ and $\alpha=90^{\circ}$ and are reproduced for reference in
the two left/right upper panels of fig. 4 respectively. Note that, for the
sake of clarity, we only show here the lowest999In the coupled longitudinal
sector for arbitrary $\delta\mu$ and $h$, the labeling of a vertex divergence
line with $N$ indicate that the corresponding eigenvector has a maximum on the
$N_{th}$ Matsubara frequency in the charge or the spin subspace. Since in fig.
4 only the first divergence line structure is displayed, we show in fig. 8
also the second divergence line structure is shown for an exemplary case. loop
and parabola structures and their associated pseudo-divergences.

We consider first the situation where the variation of chemical potential is
larger than the magnetic field, i.e. $0^{\circ}<\alpha<45^{\circ}$, which we
illustrate by showing, as an example, the case of $\alpha=35^{\circ}$ in the
lower left panel of fig. 4. Interestingly, for $U>0$, in addition to the loop-
structures and their associated pseudo-divergences already discussed for
$\alpha=0$, a parabola structure appears at large values of $\beta U$. We can
also observe that the pseudo-divergence associated to this parabola at $U>0$
are connected with the pseudo-divergence emerging from the second parabola for
$U<0$.

In the reversed situation, where the magnetic field dominates over $\delta\mu$
(i.e.,for $45^{\circ}<\alpha<90^{\circ}$), in accordance with the mapping, the
phase diagrams are geometrically identical (the projected weights differ as it
is predicted by the mapping) as those for $\alpha^{\prime}=90^{\circ}-\alpha$,
with $0^{\circ}<\alpha^{\prime}<45^{\circ}$, but for the flipping of the $U$
and $-U$ sectors.

Remarkably, for exactly $\alpha=45^{\circ}$, where $h$ equals $\delta\mu$, the
loop structure touches its corresponding parabola at $U>0$, while the same
happens to the pair parabolas for $U<0$. In both cases, at the point where the
two structures touch, a perfectly horizontal (i.e., parallel to the $r$-axis)
pseudo-divergence line is emerging.

Our results thus unveil the specific evolution of the vertex
(pseudo-)-divergences between the two qualitatively different situations of
$SU(2)_{S}$ and $SU(2)_{P}$ symmetries, and, in particular, how the loop-
shaped divergence structures gets transformed in the parabola ones and vice-
versa. This also impacts the orientation of the pseudo-divergence lines: One
readily notes that those emerging from a parabola always cross the the $U=0$
axis at $r=\frac{\nu}{\sqrt{1\pm 2\cos{\alpha}\sin{\alpha}}}$, which is
explicitly demonstrated in eqs. 109 and 110, while the ones associated to the
loops never cross the non-interacting axis. Evidently, the condition
$\alpha=45^{\circ}$ marks the change between these two behaviors, which is
featured by the horizontal pseudo-divergence lines.

Finally, in fig. 5 we show for the same three exemplary cases as above (i.e.
$\alpha=35^{\circ},45^{\circ},55^{\circ}$) the vertex (pseudo-)divergences in
the transversal spin and in the pairing channel, restricting again the plot to
the first structures for each channel. The (pseudo-)divergences lines of the
longitudinal channel shown in fig. 6 are replotted in gray for reference.

All the divergence lines in these channels display a parabolic shape,
consistent to the fact that, at particle-hole symmetry, only symmetric
divergences for both transversal channels exist. However, different from the
longitudinal sector, the pseudo-divergences associated to the transversal spin
channel cross the $U=0$ axis at
$r=\frac{\nu}{\sqrt{(\cos{\alpha})^{2}-(\sin{\alpha})^{2}}}$, i.e. only for
$0^{\circ}\leq\alpha<45^{\circ}$, while those of the pairing channel cross the
$U=0$ axis at $r=\frac{\nu}{\sqrt{(\sin{\alpha})^{2}-(\cos{\alpha})^{2}}}$ and
therefore only for $45^{\circ}<\alpha\leq 90^{\circ}$.

###  IV.6 The mapping of the eigenvectors

Crucial information [31, 28, 52, 34, 37, 54] on the possible physical effects
associated to divergences is encoded in the frequency structure of the
associated eigenvectors, and, especially, in their overall symmetry
properties. While a detailed discussion of these properties is reported in
appendix A, which generalizes the corresponding particle-hole symmetric
results of [28], here we illustrate some essential aspects of the eigenvector
behavior, by hand of a pertinent example.

To this aim, we consider the following two cases, directly related by the
mapping, i.e., $\alpha=35^{\circ}$ for $U>0$ and $\alpha=55^{\circ}$ for $U<0$
(see left panel of fig. 6). The eigenvectors corresponding to the
(pseudo-)divergences of three selected parameter sets (marked by black stars
in the left panel) are then shown in the central panel of the figure, where
each eigenvector is split into its symmetric ($v_{\mathcal{S}}$) and and
antisymmetric $(v_{\mathcal{A}})$ part, while their projected weight in the
charge and spin subspace, calculated for the symmetric and antisymmetric part
of each eigenvector, is shown in the rightmost panel.

According to the relations in section III, the eigenvector marked by 1a is
mapped into 1b, 2a is mapped int 2b and so on. When considering 1a and 1b,
which are associated to an antisymmetric divergence of the vertex, we can
observe the characteristic symmetry properties of the eigenvectors
corresponding to a real eigenvalue in a $\kappa$-real matrix $\chi_{\text{L}}$
(see eq. 64). In particular, these eigenvectors split into purely real anti-
symmetric part and imaginary symmetric part. Analogously, 2a and 2b,
associated to a symmetric divergence display purely real symmetric/purely
imaginary anti-symmetric parts. On the other hand, 3a, 3b, 4a and 4b are
eigenvectors associated to a pseudo-divergence, and hence, to a pair of
complex conjugate eigenvalues. Therefore, the symmetric and anti-symmetric
parts of these eigenvectors are no longer purely real or imaginary.

Eventually, by comparing the a and b figures of each case, one can easily
recognize the behavior predicted by the mapping relations derived in the
previous section. The symmetric part of the eigenvector $v_{\mathcal{S}}$
changes from charge to spin subspace and vice versa, while the anti-symmetric
part $v_{\mathcal{A}}$ remains invariant under the mapping. This also reflects
into the relations of the corresponding weights, as it is readily seen when
projecting the portion of the symmetric/antisymmetric part of the eigenvectors
onto the charge or the spin subspace.

##  V Conclusions

In this work we have derived the explicit relations mapping the on-site two-
particle generalized susceptibilities of bipartite lattice models with
repulsive local interaction into those of corresponding models with attractive
interaction and vice versa. The presented derivations, which exploit the Shiba
transformation on the two-particle level (i) at arbitrary filling and/or in
the presence of a finite magnetic field, as well as (ii) for finite transfer
(bosonic) Matsubara frequency, extend the results of previous studies [28],
which were restricted to the case of static correlation functions in the
specific, and less realistic situation of perfect particle-hole symmetry of
the system.

Considering the increasing importance played by the calculation and the
manipulation of the two-particle quantities [20, 21, 23, 27] for different
cutting-edge algorithms [22, 25, 27] designed for treating strong correlations
in many-electron systems, the explicit derivation of the mapping relations on
the two-particle level might be useful in several contexts. For instance, it
may allow to reduce the computational effort of numerically heavy two-particle
calculations in challenging parameter regimes and provide rigorous benchmark
testbeds for computational schemes based on generalized on-site
susceptibilities and vertex functions beyond the special case of particle
hole-symmetry. Further, on a more fundamental level, our findings might
represent an useful guide for an improved understanding of the information
encoded in the correlation functions on the two-particle level.

In the latter respect, as a pertinent example for the applicability of the
mapping relations, we have systematically analyzed location and nature of the
divergences of the irreducible vertex functions in the Hubbard atom with
repulsive and attractive on-site interaction for arbitrary filling and finite
magnetic field, overcoming the parameter restrictions of preceding studies
[48, 22]. The evident symmetries characterizing our dimensionless phase space
diagrams for the divergences in the HA represent a direct consequence of the
derived mapping relations. This consideration allows, in turn, for an easier
comprehension of the intrinsic links existing between the various divergence
structures in different parameter regimes of the HA.

Further, among the properties characterizing the evolution of the
singularities in the different scattering channels as a function of filling
and magnetic field, it may be worth to stress here the appearance of the
exceptional points (EPs) at which the vertex divergences (associated to a
vanishing eigenvalue of the corresponding generalized susceptibility)
transform into pseudo-divergences (associated to a complex conjugate
eigenvalue pair with zero real part). In fact, the relevance of these EPs has
been recently discussed in the specific context of a possible topological
protection [37] of the thermodynamic phase-separation instabilities emerging
[68, 69, 52, 34, 54] in the proximity of a Mott-Hubbard metal-to-insulator
transitions of correlated lattice systems. Further, the EPs might play,
together with the pseudo-divergences discussed in our work as well as with the
pole structures [57] possibly appearing in the complex interaction plane[10],
an important role in controlling the convergence properties of the self-
consistent perturbation expansions for the many-electron problem [30, 70, 31,
22], beyond [32] the rather specific condition of perfect particle-hole
symmetry condition mostly investigated so far.

###### Acknowledgements.

We want to thank Sabine Andergassen, Massimo Capone, Sergio Ciuchi, Lorenzo
Crippa, Lorenzo Del Re, Dominik Fus, Anna Kauch, Erik van Loon, Georg
Rohringer, Stefan Rohshap and Thomas Schäfer for insightful discussions. We
acknowledge financial support from the Austrian Science Fund (FWF) via the
projects I 5487 (A.T.) and I 5868, project P01 of the research unit FOR 5249
of the German research foundation (DFG), (H.E.); M. R. acknowledges support as
a recipient of a DOC fellowship of the Austrian Academy of Sciences. G.S.
acknowledges support from the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) through QUAST FOR 5249 (Project No. 449872909, project
P5). Calculations have been partly performed on the Vienna Scientific Cluster.

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  * Note [11] Note that for $\mathcal{N}_{\text{c/s}}$ other definitions would be also possible, e.g.: a) using $\norm{v}_{T}=1$ and defining $\mathcal{N}_{\text{c/s}}=\norm{v_{\text{c/s}}}_{T}$. However, since $\mathcal{N}_{\text{c/s}}\in\mathbb{C}$, this definition cannot be used to define a percentage. b)using $\norm{v}_{\dagger}=1$ and defining $\mathcal{N}_{\text{c/s}}=\norm{v_{\text{c/s}}}_{\dagger}$. This definition is in fact equivalent to the definition in eq. 96. However, vectors normalized with $\norm{.}_{T}$ have the convenient property of either symmetric or antisymmetric real and imaginary parts in the charge and spin subspaces (if their eigenvalue is real). Hence, it is more practical to work with these vectors. The equivalence can be shown by comparing the two different normalizations of the same vector $v$: $\tilde{v}=\frac{v^{\prime}}{\norm{v^{\prime}}_{\dagger}}e^{i\varphi}$, where $\norm{\tilde{v}}_{\dagger}=1$, $\norm{v^{\prime}}_{T}=1$, and $\varphi$ is the phase of $\norm{\tilde{v}}_{T}$ The additional factor $1/\norm{v}^{2}_{\dagger}$ in the definition of the projected weight assures $\mathcal{N}_{\text{c}}+\mathcal{N}_{\text{s}}=1$.
  * Note [12] $v_{1}^{T}\cdot v_{2}=0$, because the vectors are orthogonal regarding the scalar product eq. 61. Further, $\norm{v_{2}}_{\dagger}=\norm{v_{1}}_{\dagger}$ holds because of the relation $v_{2}=Jv_{1}^{*}$.
  * Note [13] Indeed the results obtained with the simple scaling argument of the dimensionless phase space agree well with the results from the more rigorous treatment of the $T\rightarrow 0$ limit of the analytical formulas, where only special care has to be taken for the discontinuous parts of the phase space. This shows the usefulness of the dimensionless phase space.

##  Appendix A Symmetries of generalized susceptibilities

In this section, we study the properties of the local generalized
susceptibilities for the symmetries considered in the main text, with a
particular focus on $SU(2)_{S}$, $SU(2)_{P}$, and $SO(4)$ symmetry. For a
detailed derivation of these symmetry properties and an in-depth discussion,
the reader is revered to Ref. [36].

Doing a full particle-hole transformation ($c_{\sigma}\to
c^{\dagger}_{\sigma}$, $c^{\dagger}_{\sigma}\to c_{\sigma}$) on the
Hamiltonian in eq. 12 gives

$\displaystyle H(U,\delta\mu,h)\xleftrightarrow{\text{ph-
trafo}}H(U,-\delta\mu,-h)$

Therefore, the system is particle-hole symmetric (i.e., the Hamiltonian is
invariant under the particle-hole transformation) when
$\delta\mu\\!=\\!h\\!=\\!0$, which corresponds to $SO(4)$ symmetry.

Further, applying a full particle-hole transformation in the definitions of
eqs. 1 and 2 and then using crossing symmetry and complex conjugation (eqs. 54
and 53) lead to

$\displaystyle\chi\xleftrightarrow{\text{ph-trafo}}\chi^{*},$ (49)

and a sign change in the magnetic field just flips all spins of the
generalized susceptibility $(\sigma\rightarrow-\sigma,\;\forall\sigma)$.

We find the following relations for the different channels:

In the coupled longitudinal channel

$\displaystyle\begin{split}\chi_{\text{L}}\xleftrightarrow{\delta\mu,h\leftrightarrow-\delta\mu,-h}\chi_{\text{L}}^{*},\\\
\chi_{\text{L}}\xleftrightarrow{h\leftrightarrow-h}M\chi_{\text{L}}M,\\\
\chi_{\text{L}}\xleftrightarrow{\delta\mu\leftrightarrow-\delta\mu}M\chi_{\text{L}}^{*}M,\end{split}$
(50)

where $M=\begin{pmatrix}\mathbb{1}&\mathbb{0}\\\
\mathbb{0}&-\mathbb{1}\end{pmatrix}$, and for the transversal spin and the
pairing channel we find

$\displaystyle\begin{split}\chi_{S_{x}}\xleftrightarrow{\delta\mu,h\leftrightarrow-\delta\mu,-h}\chi_{S_{x}}^{*},\\\
\chi_{S_{x}}\xleftrightarrow{h\leftrightarrow-h}\chi_{S_{x}},\\\
\chi_{S_{x}}\xleftrightarrow{\delta\mu\leftrightarrow-\delta\mu}\chi_{S_{x}}^{*},\end{split}$
(51)

$\displaystyle\begin{split}\chi_{\text{pair}}\xleftrightarrow{\delta\mu,h\leftrightarrow-\delta\mu,-h}\chi_{\text{pair}}^{*},\\\
\chi_{\text{pair}}\xleftrightarrow{h\leftrightarrow-h}\chi_{\text{pair}}^{*},\\\
\chi_{\text{pair}}\xleftrightarrow{\delta\mu\leftrightarrow-\delta\mu}\chi_{\text{pair}}.\end{split}$
(52)

Since the transformations in eqs. 50, 51 and 52 change neither the eigenvalues
nor the projected weights (see eq. 96) of the generalized susceptibility
matrices, we confirm that the whole phase space can be generated by mapping
the calculated quantities from the sub region where $\delta\mu>0$ and $h>0$.
This corresponds to the interval $\alpha\in[0^{\circ},90^{\circ}]$ in the
cylinder coordinates of eq. 47.

In eqs. 53, 54, 55, 56 and 57 the properties of the generalized
susceptibilities under the specific symmetry relations are explicitly written
following Ref. [36].

Crossing symmetry and complex conjugation are fundamental “symmetries”, i.e.
they are always present, $H\in\mathbb{R}$ refers to the Hamiltonian being a
real function of $c$ and $c^{\dagger}$ and $SU(2)_{S}$/$SU(2)_{P}$ means
symmetric with respect to spin/pseudo-spin.

$\displaystyle\begin{split}\text{Crossing Symmetry:
}\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\sigma\sigma^{\prime},\text{ph}}&=\chi^{\omega+\nu^{\prime}\\!,\,\omega+\nu,\,-\omega}_{\sigma^{\prime}\sigma,\text{ph}},\\\
\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\overline{\sigma\sigma^{\prime}},\text{ph}}&=\chi^{\omega+\nu^{\prime}\\!,\,\omega+\nu,\,-\omega}_{\overline{\sigma^{\prime}\sigma},\text{ph}},\\\
\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\overline{\sigma\sigma^{\prime}},\text{pp}}&=\chi^{\omega-\nu\\!,\,\omega-\nu^{\prime}\\!,\,\omega}_{\overline{\sigma^{\prime}\sigma},\text{pp}},\end{split}$
(53)

$\displaystyle\begin{split}\text{Complex Conjugation:
}\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\sigma\sigma^{\prime},\text{ph}}&=\Big{(}\chi^{-\nu^{\prime}\\!,\,-\nu-,\,-\omega}_{\sigma^{\prime}\sigma,\text{ph}}\Big{)}^{*},\\\
\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\overline{\sigma\sigma^{\prime}},\text{ph}}&=\Big{(}\chi^{-\nu^{\prime}\\!,\,-\nu-,\,-\omega}_{\overline{\sigma\sigma^{\prime}},\text{ph}}\Big{)}^{*},\\\
\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\overline{\sigma\sigma^{\prime}},\text{pp}}&=\Big{(}\chi^{-\nu^{\prime}\\!,\,-\nu-,\,-\omega}_{\overline{\sigma\sigma^{\prime}},\text{pp}}\Big{)}^{*},\end{split}$
(54)

$\displaystyle\begin{split}\hat{H}\in\mathbb{R}\text{:
}\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\sigma\sigma^{\prime},\text{ph}}&=\chi^{\nu^{\prime}\\!,\,\nu,\,\omega}_{\sigma^{\prime}\sigma,\text{ph}},\\\
\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\overline{\sigma\sigma^{\prime}},\text{ph}}&=\chi^{\nu^{\prime}\\!,\,\nu,\,\omega}_{\overline{\sigma\sigma^{\prime}},\text{ph}},\\\
\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\overline{\sigma\sigma^{\prime}},\text{pp}}&=\chi^{\nu^{\prime}\\!,\,\nu,\,\omega}_{\overline{\sigma\sigma^{\prime}},\text{pp}},\end{split}$
(55)

$\displaystyle\begin{split}\text{$SU(2)_{S}$ symmetry:
}\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\sigma\sigma^{\prime},\text{ph}}&=\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{-\sigma-\sigma^{\prime},\text{ph}},\\\
\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\overline{\sigma\sigma^{\prime}},\text{ph}}&=\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\overline{-\sigma-\sigma^{\prime}},\text{ph}},\\\
\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\overline{\sigma\sigma^{\prime}},\text{pp}}&=\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\overline{-\sigma-\sigma^{\prime}},\text{pp}},\end{split}$
(56)

$\displaystyle\begin{split}\text{$SU(2)_{P}$ symmetry:
}\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\sigma\sigma^{\prime},\text{ph}}&=\Big{(}\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{-\sigma-\sigma^{\prime},\text{ph}}\Big{)}^{*},\\\
\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\overline{\sigma\sigma^{\prime}},\text{ph}}&=\Big{(}\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\overline{-\sigma-\sigma^{\prime}},\text{ph}}\Big{)}^{*},\\\
\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\overline{\sigma\sigma^{\prime}},\text{pp}}&=\Big{(}\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\overline{-\sigma-\sigma^{\prime}},\text{pp}}\Big{)}^{*}.\end{split}$
(57)

From eqs. 53, 54, 55, 56 and 57 the relations written in table 1 of the main
text can be derived.

In the following, we introduce the class of centro-hermitian matrices $C$
which fulfill

$\displaystyle JC^{*}J=C,\quad\text{with }J=\matrixquantity(&&1\\\ &\udots&\\\
1&&).$ (58)

Centro-hermitian matrices have eigenvalues that are either real or come in
complex conjugate pairs [38].

By using the symmetry eqs. 53, 54 and 55 we find that all generalized
susceptibilities are centro-hermitian for $\omega=0$. Note that for
$\chi_{\sigma\sigma^{\prime},\text{ph}}^{\omega=0}$ the centro-hermitian
property is fundamental, since only complex conjugation and crossing symmetry
(which are always present) are needed, whereas for
$\chi_{\overline{\sigma\sigma^{\prime}}\text{ph/pp}}^{\omega=0}$ also
$H\in\mathbb{R}$ is required (which could be violated in principal). Without
this restriction
$\chi_{\overline{\sigma\sigma^{\prime}}\text{ph/pp}}^{\omega=0}$ are per-
hermitian matrices [37].

Moreover, $\chi_{\sigma\sigma^{\prime},\text{ph}}$ can be considered as a
centro-hermitian matrix for finite $\omega$ when shifting the fermionic
Matsubara frequencies by $\nu^{(\prime)}\rightarrow\nu^{(\prime)}+\lfloor
m/2\rfloor 2\pi/\beta$ with $\omega=2\pi m/\beta$ ($\lfloor x\rfloor$ refers
to $\operatorname{floor}(x)$). This leads to

$\displaystyle\chi^{\nu,\nu^{\prime}\\!,\,\omega}_{\sigma\sigma^{\prime},\text{ph}}$
$\displaystyle=\Big{(}\chi^{-\nu-\omega\\!,\,-\nu^{\prime}-\omega,\,\omega}_{\sigma\sigma^{\prime},\text{ph}}\Big{)}^{*}$
$\displaystyle\Bigg{\downarrow}\text{frequency shift}$ (59)
$\displaystyle\chi^{\nu-\lfloor\omega/2\rfloor,\nu^{\prime}-\lfloor\omega/2\rfloor,\,\omega}_{\sigma\sigma^{\prime},\text{ph}}$
$\displaystyle=\Big{(}\chi^{-\nu-\lfloor\omega/2\rfloor,\,-\nu^{\prime}-\lfloor\omega/2\rfloor,\,\omega}_{\sigma\sigma^{\prime},\text{ph}}\Big{)}^{*},$

where $\lfloor\omega/2\rfloor$ is a short hand notation for $\lfloor
m/2\rfloor 2\pi/\beta$ such that $\nu-\omega/2$ is still a fermionic Matsubara
frequency. Thus, for any bosonic frequency $\omega$
$\chi_{\sigma\sigma^{\prime},\text{ph}}$ is a centro-hermitian matrix in the
frequency shifted space.

Moreover, we introduce the class of $\kappa$-real matrices [39], which also
have only real or complex conjugate pairs as eigenvalues. The generalized
susceptibility of the coupled longitudinal channel $\chi_{\mathrm{L}}$ is a
$\kappa$-real matrix which fulfills the relation

$\displaystyle\Pi
K^{*}\Pi=K\quad\text{with}\quad\Pi=\begin{pmatrix}J&\mathbb{0}\\\
\mathbb{0}&J\end{pmatrix}.$ (60)

Further, all considered channels are symmetric matrices if $H\in\mathbb{R}$
and can be diagonalized by a complex orthogonal transformation, if there is no
exceptional point (i.e. a point in parameter space where two or more
eigenvalues _and_ eigenvectors coalesce) [40, 37]. The corresponding inner
product of the orthogonal Euclidean quasi-norm is defined as

$\displaystyle\langle u,v\rangle:=u^{T}\cdot v=\sum_{i}^{n}u_{i}v_{i}$ (61)

and

$\displaystyle\norm{u}^{2}_{T}:=u^{T}\cdot u.$ (62)

Eigenvectors of symmetric and centro-hermitian matrices that are normalized
with respect to eq. 62 have the following properties [37]:

$\displaystyle\begin{split}&\text{if }\lambda_{\alpha}\in\mathbb{R}\text{ (and
not degenerate):}\;v_{\alpha}=\pm Jv_{\alpha}^{*},\\\ &\text{if
}\lambda_{\alpha}\in\mathbb{C}:\;\exists
v_{\alpha^{\prime}}=Jv_{\alpha}^{*},\\\ \end{split}$ (63)

where the relation for real eigenvalues leads to the fact that their
eigenvectors have a symmetric/antisymmetric real part with an
antisymmetric/symmetric imaginary part (with respect to the shifted Matsubara
frequency space $\nu^{(\prime)}\rightarrow\nu^{(\prime)}+\lfloor m/2\rfloor
2\pi/\beta$).

For the symmetric $\kappa$-real matrix $\chi_{\text{L}}$ we can make a similar
argument as in Ref. [37] to obtain:

$\displaystyle\begin{split}&\text{if }\lambda_{\alpha}\in\mathbb{R}\text{ (and
not degenerate):}\;v_{\alpha}=\pm\Pi v_{\alpha}^{*},\\\ &\text{if
}\lambda_{\alpha}\in\mathbb{C}:\;\exists v_{\alpha^{\prime}}=\Pi
v_{\alpha}^{*},\\\ \end{split}$ (64)

where the relation for real eigenvalues leads to the fact that their
eigenvectors have a symmetric/antisymmetric real part with an
antisymmetric/symmetric imaginary part in the charge and spin subspaces
respectively (with respect to the shifted Matsubara frequency space).

##  Appendix B BSE of the transversal channel

In general, $\chi_{S_{x}}$ and $\chi_{\text{pair}}$ do not represent decoupled
channels in the BSE. Even though $\chi_{S_{x}}=\chi_{S_{y}}$, they still
couple to each other for $\langle S_{z}\rangle\neq 0$ [19]. The same
considerations apply to the real and imaginary part of the pairing field
$\Delta$, which are given by $S_{p,x}$ and $S_{p,y}$, and couple for $\langle
S_{p,z}\rangle\neq 0$. Therefore the $S_{x}$/pairing channel generally (for
arbitrary $\omega$) only decouples within the transversal subspace when
$SU(2)_{S/P}$ symmetry is fulfilled. This can be shown from the symmetry
relations in appendix A and the expressions of the coupled channels in eqs. 65
and 66. This is an example of how the mapping “preserves” the decoupling of
the channels regarding the BSE. Instead the transverse components
$\chi_{\overline{\sigma\sigma^{\prime}}}$ and
$\chi_{\overline{\sigma\sigma^{\prime}},\text{pp}}$ remain independent in the
BSE. Hence, for the respective generalized susceptibility of the coupled BSE
we have to consider the transversal ph-channel (or
$\chi_{\overline{\sigma\sigma^{\prime}}}$ directly)

$\displaystyle\chi_{\text{T}}=\begin{pmatrix}\chi_{\overline{\uparrow\downarrow}}+\chi_{\overline{\downarrow\uparrow}}&\chi_{\overline{\uparrow\downarrow}}-\chi_{\overline{\downarrow\uparrow}}\\\
\chi_{\overline{\uparrow\downarrow}}-\chi_{\overline{\downarrow\uparrow}}&\chi_{\overline{\uparrow\downarrow}}+\chi_{\overline{\downarrow\uparrow}}\end{pmatrix}$
(65)

instead of the spin channel and the transversal pp-channel (or
$\chi_{\overline{\sigma\sigma^{\prime}},\text{pp}}$ directly)

$\displaystyle\chi_{\text{T,pp}}=\begin{pmatrix}-\chi_{\overline{\uparrow\downarrow},\text{pp}}-\chi_{\overline{\downarrow\uparrow},\text{pp}}^{*}&-\chi_{\overline{\uparrow\downarrow},\text{pp}}+\chi_{\overline{\downarrow\uparrow},\text{pp}}^{*}\\\
-\chi_{\overline{\uparrow\downarrow}}+\chi_{\overline{\downarrow\uparrow},\text{pp}}^{*}&-\chi_{\overline{\uparrow\downarrow},\text{pp}}-\chi_{\overline{\downarrow\uparrow},\text{pp}}^{*}\end{pmatrix}$
(66)

instead of the pairing channel. If the Hamiltonian is a real function of $c$
and $c^{\dagger}$ then
$\chi_{\overline{\uparrow\downarrow}}^{\nu,\nu^{\prime}\\!,\,\omega}=\chi_{\overline{\downarrow\uparrow}}^{\omega+\nu,\omega+\nu^{\prime}\\!,\,-\omega}$
and
$\chi_{\overline{\uparrow\downarrow},\text{pp}}^{\nu,\nu^{\prime}\\!,\,\omega}=(\chi_{\overline{\downarrow\uparrow},\text{pp}}^{\nu-\omega,\nu^{\prime}-\omega,\,\omega})^{*}$
(see appendix A), and both the BSE of the pairing channel and the transversal
spin channel decouple for $\omega=0$. Hence, to investigate the divergences of
the two-particle irreducible vertices for $\omega=0$, $\chi_{S_{x}}$ and
$\chi_{\text{pair}}$ can be used for the scope of this paper as it is done in
section IV.

##  Appendix C Calculations for the Shiba mapping of the generalized
susceptibilities

In this section, we derive the action of the Shiba transformation applied to
the local generalized susceptibilities. The results of this transformation are
shown in section III. For the derivation we start by showing how the
orthogonal transformation

$Q=\frac{1}{\sqrt{2}}\begin{pmatrix}\mathbb{1}&-J\\\
\mathbb{1}&J\end{pmatrix}$ (67)

separates the symmetric and antisymmetric parts of a centro-hermitian matrix.
Every centro-hermitian matrix $\chi_{CH}$ can be represented by the sum of a
centro-symmetric real matrix $\chi^{\prime}$ ($J\chi^{\prime}J=\chi^{\prime}$)
and a skew-centro-symmetric imaginary matrix $\chi^{\prime\prime}$
($J\chi^{\prime\prime}J=-\chi^{\prime\prime}$).

For even dimensions a centro-hermitian matrix can be represented by

$\displaystyle\begin{split}\chi_{CH}=\chi^{\prime}+i\chi^{\prime\prime}\quad\text{with}\quad\chi^{\prime}&=\begin{pmatrix}A^{\prime}&JB^{\prime}J\\\
B^{\prime}&JA^{\prime}J\end{pmatrix},\\\
\chi^{\prime\prime}&=\begin{pmatrix}A^{\prime\prime}&-JB^{\prime\prime}J\\\
B^{\prime\prime}&-JA^{\prime\prime}J\end{pmatrix},\end{split}$ (68)

where $\chi^{\prime(\prime)}$, $A^{\prime(\prime)}$ and $B^{\prime(\prime)}$
are real square matrices.

The transformation $Q$ acts on (skew-)centrosymmetric matrices as follows

$\displaystyle
Q\chi^{\prime}Q^{T}=\begin{pmatrix}A^{\prime}-JB^{\prime}&\mathbb{0}\\\
\mathbb{0}&A^{\prime}+JB^{\prime}\end{pmatrix}$ (69)

and

$\displaystyle
Q\chi^{\prime\prime}Q^{T}=\begin{pmatrix}\mathbb{0}&A^{\prime\prime}-JB^{\prime\prime}\\\
A^{\prime\prime}+JB^{\prime\prime}&\mathbb{0}\end{pmatrix}.$ (70)

$A^{\prime}-JB^{\prime}$ and $A^{\prime\prime}+JB^{\prime\prime}$ are referred
to as antisymmetric blocks and $A^{\prime}+JB^{\prime}$ and
$A^{\prime\prime}-JB^{\prime\prime}$ are termed symmetric blocks. To
understand why, we let $\chi^{\prime}$ and $\chi^{\prime\prime}$ act on a
generic vector $v$ which is decomposed into symmetric and antisymmetric parts:

$\displaystyle\begin{split}\chi^{\prime}v&=\begin{pmatrix}A^{\prime}&JB^{\prime}J\\\
B^{\prime}&JA^{\prime}J\end{pmatrix}\left[\begin{pmatrix}v_{\mathcal{S}}\\\
Jv_{\mathcal{S}}\end{pmatrix}+\begin{pmatrix}v_{\mathcal{A}}\\\
-Jv_{\mathcal{A}}\end{pmatrix}\right]\\\
&=\begin{pmatrix}(A^{\prime}+JB^{\prime})v_{\mathcal{S}}+(A^{\prime}-JB^{\prime})v_{\mathcal{A}}\\\
(C^{\prime}+JB^{\prime})v_{\mathcal{S}}+(C^{\prime}-JB^{\prime})v_{\mathcal{A}}\end{pmatrix}\end{split}$
(71)

and

$\displaystyle\begin{split}\chi^{\prime\prime}v&=\begin{pmatrix}A^{\prime\prime}&-JB^{\prime\prime}J\\\
B^{\prime\prime}&-JA^{\prime\prime}J\end{pmatrix}\left[\begin{pmatrix}v_{\mathcal{S}}\\\
Jv_{\mathcal{S}}\end{pmatrix}+\begin{pmatrix}v_{\mathcal{A}}\\\
-Jv_{\mathcal{A}}\end{pmatrix}\right]\\\
&=\begin{pmatrix}(A^{\prime\prime}-JB^{\prime\prime})v_{\mathcal{S}}+(A^{\prime\prime}+JB^{\prime\prime})v_{\mathcal{A}}\\\
(B^{\prime\prime}-JA^{\prime\prime})v_{\mathcal{S}}+(B^{\prime\prime}+JA^{\prime\prime})v_{\mathcal{A}}\end{pmatrix}.\end{split}$
(72)

As it can be seen $A^{\prime}-JB^{\prime}$ and
$A^{\prime\prime}+JB^{\prime\prime}$ couple to the antisymmetric part of the
vector and $A^{\prime}+JB^{\prime}$ and $A^{\prime\prime}-JB^{\prime\prime}$
couple to the symmetric part of the vector.

Therefore we identify the blocks in eq. 31 by

$\displaystyle\begin{split}\chi_{\mathcal{A}}^{\prime}=A^{\prime}-JB^{\prime},\\\
\chi_{\mathcal{S}}^{\prime}=A^{\prime}+JB^{\prime},\\\
\chi_{\mathcal{A}}^{\prime\prime}=A^{\prime\prime}+JB^{\prime\prime},\\\
\chi_{\mathcal{S}}^{\prime\prime}=A^{\prime\prime}-JB^{\prime\prime}.\end{split}$
(73)

As described in appendix A the generalized susceptibility
$\chi_{\sigma\sigma^{\prime},\text{ph}}$ is still a centro-hermitian matrix
for $\omega\neq 0$ if we consider the shifted space
$\nu^{(\prime)}\rightarrow\nu^{(\prime)}+\lfloor m/2\rfloor 2\pi/\beta$ with
$\omega=2m\pi/\beta$. As already mentioned in the main text the derivation for
$m=0$ (mod 2) is completely analogous to the $\omega=0$ case if the shifted
space is considered.

However, for $m\neq 0$ (mod 2) slight adaptations are needed since the matrix
acquires an odd frequency structure with a new central frequency
$\nu^{*}=\omega/2$. In other words, if we consider a finite frequency box
(e.g., in numerical calculations) the frequency shift leads to a
$(2N)\times(2N)$ matrix for $m=0$ (mod 2) and to a $(2N+1)\times(2N+1)$ matrix
for $m\neq 0$ (mod 2).

A centro-hermitian matrix of odd dimension can be depicted by

$\displaystyle\begin{split}\chi_{CH}=\chi^{\prime}+i\chi^{\prime\prime},\quad\text{with}\quad\chi^{\prime}&=\begin{pmatrix}A^{\prime}&a^{\prime}&JB^{\prime}J\\\
b^{\prime}&c&b^{\prime}J\\\
B^{\prime}&Ja^{\prime}&JA^{\prime}J\end{pmatrix},\\\
\quad\chi^{\prime\prime}&=\begin{pmatrix}A^{\prime\prime}&a^{\prime\prime}&-JB^{\prime\prime}J\\\
b^{\prime\prime}&0&-b^{\prime\prime}J\\\
B^{\prime\prime}&-Ja^{\prime\prime}&-JA^{\prime\prime}J\end{pmatrix},\end{split}$
(74)

where $A,B\in\mathbb{R}^{N\times N}$, $a\in\mathbb{R}^{N\times 1}$,
$b\in\mathbb{R}^{1\times N}$ and $c\in\mathbb{R}$. With

$\displaystyle Q=\frac{1}{\sqrt{2}}\begin{pmatrix}\mathbb{1}&0&-J\\\
0&\sqrt{2}&0\\\ \mathbb{1}&0&J\end{pmatrix}$ (75)

we can apply the transformation $Q\chi_{CH}Q^{T}$, in the same way as for the
even-dimensional case to identify the symmetric and anti-symmetric blocks of
the matrix in eq. 31:

$\displaystyle\begin{split}\chi_{\mathcal{A}}^{\prime}=A^{\prime}-JB^{\prime},\\\
\chi_{\mathcal{S}}^{\prime}=\begin{pmatrix}c&\sqrt{2}b^{\prime}\\\
\sqrt{2}a^{\prime}&A^{\prime}+JB^{\prime}\\\ \end{pmatrix},\\\
\chi_{\mathcal{A}}^{\prime\prime}=\begin{pmatrix}\sqrt{2}b^{\prime\prime}\\\
A^{\prime\prime}+JB^{\prime\prime}\end{pmatrix},\\\
\chi_{\mathcal{S}}^{\prime\prime}=\begin{pmatrix}\sqrt{2}a^{\prime\prime},&A^{\prime\prime}-JB^{\prime\prime}\end{pmatrix}.\end{split}$
(76)

Now we can start with the derivation of the mapping of the generalized
susceptibilities by writing the action of the Shiba transformation in eq. 28
as matrix products in the frequency shifted space:

$\displaystyle\begin{split}\chi_{\uparrow\uparrow,\text{U}}=\chi_{\uparrow\uparrow,\text{-U}},\;\;\chi_{\downarrow\downarrow,\text{U}}=J\chi_{\downarrow\downarrow,\text{-U}}J,\\\
\chi_{\uparrow\downarrow,\text{U}}=-\chi_{\uparrow\downarrow,\text{-U}}J,\;\;\text{and}\;\;\chi_{\downarrow\uparrow,\text{U}}=-J\chi_{\downarrow\uparrow,\text{-U}}.\end{split}$
(77)

With these building blocks, we can construct the coupled longitudinal matrix

$\chi_{\text{L}}=\begin{pmatrix}\chi_{\text{c}}&\chi_{\text{cs}}\\\
\chi_{\text{sc}}&\chi_{\text{s}}\end{pmatrix}$ (78)

and then apply the transformation

$\mathcal{Q}=\begin{pmatrix}Q&\mathbb{0}\\\ \mathbb{0}&Q\end{pmatrix}$ (79)

to it.

With simple matrix multiplication, we obtain

$\displaystyle\begin{split}\mathcal{Q}\chi_{\text{L,-U}}&\mathcal{Q}^{T}=\begin{pmatrix}Q\chi_{\text{c,-U}}Q^{T}&Q\chi_{\text{cs,-U}}Q^{T}\\\
Q\chi_{\text{sc,-U}}Q^{T}&Q\chi_{\text{s,-U}}Q^{T}\end{pmatrix}\\\
&=\begin{pmatrix}\chi_{\text{c,-U},\mathcal{A}}^{\prime}&i\chi_{\text{c,-U},\mathcal{S}}^{\prime\prime}&\chi_{\text{cs,-U},\mathcal{A}}^{\prime}&i\chi_{\text{cs,-U},\mathcal{S}}^{\prime\prime}\\\
i\chi_{\text{c,-U},\mathcal{A}}^{\prime\prime}&\chi_{\text{c,-U},\mathcal{S}}^{\prime}&i\chi_{\text{cs,-U},\mathcal{A}}^{\prime\prime}&\chi_{\text{cs,-U},\mathcal{S}}^{\prime}\\\
\chi_{\text{sc,-U},\mathcal{A}}^{\prime}&i\chi_{\text{sc,-U},\mathcal{S}}^{\prime\prime}&\chi_{\text{s,-U},\mathcal{A}}^{\prime}&i\chi_{\text{s,-U},\mathcal{S}}^{\prime\prime}\\\
i\chi_{\text{sc,-U},\mathcal{A}}^{\prime\prime}&\chi_{\text{sc,-U},\mathcal{S}}^{\prime}&i\chi_{\text{s,-U},\mathcal{A}}^{\prime\prime}&\chi_{\text{s,-U},\mathcal{S}}^{\prime}\end{pmatrix}\\\
&=\begin{pmatrix}\chi_{\text{c,U},\mathcal{A}}^{\prime}&i\chi_{\text{cs,U},\mathcal{S}}^{\prime\prime}&\chi_{\text{cs,U},\mathcal{A}}^{\prime}&i\chi_{\text{c,U},\mathcal{S}}^{\prime\prime}\\\
i\chi_{\text{sc,U},\mathcal{A}}^{\prime\prime}&\chi_{\text{s,U},\mathcal{S}}^{\prime}&i\chi_{\text{s,U},\mathcal{A}}^{\prime\prime}&\chi_{\text{sc,U},\mathcal{S}}^{\prime}\\\
\chi_{\text{sc,U},\mathcal{A}}^{\prime}&i\chi_{\text{s,U},\mathcal{S}}^{\prime\prime}&\chi_{\text{s,U},\mathcal{A}}^{\prime}&i\chi_{\text{sc,U},\mathcal{S}}^{\prime\prime}\\\
i\chi_{\text{c,U},\mathcal{A}}^{\prime\prime}&\chi_{\text{cs,U},\mathcal{S}}^{\prime}&i\chi_{\text{cs,U},\mathcal{A}}^{\prime\prime}&\chi_{\text{c,U},\mathcal{S}}^{\prime}\end{pmatrix}\\\
&=T\mathcal{Q}\chi_{\text{L,U}}\mathcal{Q}^{T}T,\end{split}$ (80)

where an appropriate $Q$ matrix has to be used depending on whether the
generalized susceptibility is an even or an odd matrix. From eq. 80 and the
fact that $\mathcal{Q}$ is an orthogonal transformation eq. 43 in the main
text is obtained.

In eq. 80 one can see that after the $\mathcal{Q}$ transformation
$\chi_{\text{L,U}}$ and $\chi_{\text{L,-U}}$ only differ by a permutation of
the different sub matrices. This reshuffling is done by the matrix $T$ in eq.
42 which needs to be adapted to eq. 81 for the odd-dimensional case

$\displaystyle
T=\begin{pmatrix}\mathbb{1}&0&\mathbb{0}&\mathbb{0}&0&\mathbb{0}\\\
0&0&0&0&1&0&\\\ \mathbb{0}&0&\mathbb{0}&\mathbb{0}&0&\mathbb{1}\\\
\mathbb{0}&0&\mathbb{0}&\mathbb{1}&0&\mathbb{0}\\\ 0&1&0&0&0&0&\\\
\mathbb{0}&0&\mathbb{1}&\mathbb{0}&0&\mathbb{0}\\\ \end{pmatrix}.$ (81)

We find that (i), a symmetric/anti-symmetric block of $\chi_{\text{L,U}}$
always maps to a symmetric/anti-symmetric block of $\chi_{\text{L,-U}}$ and
(ii), all $\chi_{\mathcal{A}}^{\prime}$ blocks are invariant under the
mapping. All other blocks get redistributed according to eq. 80, which is
summarized in eq. 44 in the main text.101010The $SO(4)$ symmetric case of Ref.
[28] can be obtained by noticing that for $SU(2)_{P}$ symmetry the matrices
have no imaginary part and for $SU(2)_{S}$ symmetry $\chi_{\text{cs}}$ and
$\chi_{\text{sc}}$ vanish. Note that for $\chi_{\mathrm{L}}$ the subscript
$\mathcal{A/S}$ refers to the coupling of the respective anti-/symmetric
components in the charge and spin subspace of a generic vector $v$. For the
even-dimensional case

$\displaystyle v=\begin{pmatrix}v_{\text{c},\mathcal{S}}\\\
Jv_{\text{c},\mathcal{S}}\\\ v_{\text{s},\mathcal{S}}\\\
Jv_{\text{s},\mathcal{S}}\end{pmatrix}+\begin{pmatrix}v_{\text{c},\mathcal{A}}\\\
-Jv_{\text{c},\mathcal{A}}\\\ v_{\text{s},\mathcal{A}}\\\
-Jv_{\text{s},\mathcal{A}}\end{pmatrix}$ (82)

and for the odd-dimensional case

$\displaystyle v=\begin{pmatrix}v_{\text{c},\mathcal{S}}\\\ m_{\text{c}}\\\
Jv_{\text{c},\mathcal{S}}\\\ v_{\text{s},\mathcal{S}}\\\ m_{\text{s}}\\\
Jv_{\text{s},\mathcal{S}}\end{pmatrix}+\begin{pmatrix}v_{\text{c},\mathcal{A}}\\\
0\\\ -Jv_{\text{c},\mathcal{A}}\\\ v_{\text{s},\mathcal{A}}\\\ 0\\\
-Jv_{\text{s},\mathcal{A}}\end{pmatrix}.$ (83)

Eventually, we consider the action of the Shiba transformation
$S=\mathcal{Q}^{T}T\mathcal{Q}$, which is orthogonal and symmetric, on the
eigenvectors and eigenvalues of $\chi_{\mathrm{L}}$.

Since $S$ is a orthogonal transformation $\chi_{\text{L,U}}$ and
$\chi_{\text{L,-U}}$ have the same eigenvalues. Further, if $v$ is a
eigenvector of $\chi_{\text{L,U}}$ then $Sv$ is an eigenvector of
$\chi_{\text{L,-U}}$ with the same eigenvalue. Transforming the eigenvector
$v$ we find

$\displaystyle\begin{split}Sv=S&\left[\begin{pmatrix}v_{\text{c},\mathcal{S}}\\\
Jv_{\text{c},\mathcal{S}}\\\ v_{\text{s},\mathcal{S}}\\\
Jv_{\text{s},\mathcal{S}}\end{pmatrix}+\begin{pmatrix}v_{\text{c},\mathcal{A}}\\\
-Jv_{\text{c},\mathcal{A}}\\\ v_{\text{s},\mathcal{A}}\\\
-Jv_{\text{s},\mathcal{A}}\end{pmatrix}\right]\\\
&=\begin{pmatrix}v_{\text{s},\mathcal{S}}\\\ Jv_{\text{s},\mathcal{S}}\\\
v_{\text{c},\mathcal{S}}\\\
Jv_{\text{c},\mathcal{S}}\end{pmatrix}+\begin{pmatrix}v_{\text{c},\mathcal{A}}\\\
-Jv_{\text{c},\mathcal{A}}\\\ v_{\text{s},\mathcal{A}}\\\
-Jv_{\text{s},\mathcal{A}}\end{pmatrix}\end{split}$ (84)

for the even dimensional case and

$\displaystyle\begin{split}Sv=S&\left[\begin{pmatrix}v_{\text{c},\mathcal{S}}\\\
m_{\text{c}}\\\ Jv_{\text{c},\mathcal{S}}\\\ v_{\text{s},\mathcal{S}}\\\
m_{\text{s}}\\\
Jv_{\text{s},\mathcal{S}}\end{pmatrix}+\begin{pmatrix}v_{\text{c},\mathcal{A}}\\\
0\\\ -Jv_{\text{c},\mathcal{A}}\\\ v_{\text{s},\mathcal{A}}\\\ 0\\\
-Jv_{\text{s},\mathcal{A}}\end{pmatrix}\right]\\\
&=\begin{pmatrix}v_{\text{s},\mathcal{S}}\\\ m_{\text{s}}\\\
Jv_{\text{s},\mathcal{S}}\\\ v_{\text{c},\mathcal{S}}\\\ m_{\text{c}}\\\
Jv_{\text{c},\mathcal{S}}\end{pmatrix}+\begin{pmatrix}v_{\text{c},\mathcal{A}}\\\
0\\\ -Jv_{\text{c},\mathcal{A}}\\\ v_{\text{s},\mathcal{A}}\\\ 0\\\
-Jv_{\text{s},\mathcal{A}}\end{pmatrix}\end{split}$ (85)

for the odd-dimensional case. Evidently, the symmetric part of the eigenvector
changes the subspace between charge and spin, while the anti-symmetric part of
the eigenvector is invariant under the transformation.

To obtain the Shiba transformation for the generalized susceptibility in the
transverse spin channel $\chi_{S_{x}}$ we apply the Shiba transformation on
eq. 2 and get

$\displaystyle\begin{split}\chi_{\overline{\uparrow\downarrow},\text{U}}^{\nu,\nu^{\prime}\\!,\,\omega}&=-\chi_{\overline{\uparrow\downarrow},\text{-U}}^{\nu,\nu^{\prime}\\!,\,-\nu-\nu^{\prime}-\omega}=-\chi_{\overline{\uparrow\downarrow},\text{pp,-U}}^{\nu,\nu^{\prime}\\!,\,-\omega},\\\
\chi_{\overline{\downarrow\uparrow},\text{U}}^{\nu,\nu^{\prime}\\!,\,\omega}&=-\chi_{\overline{\downarrow\uparrow},\text{-U}}^{-\nu,-\nu^{\prime}\\!,\,\nu+\nu^{\prime}+\omega}=-\chi_{\overline{\downarrow\uparrow},\text{pp,-U}}^{-\nu,-\nu^{\prime}\\!,\,\omega}=-\Big{(}\chi_{\overline{\downarrow\uparrow},\text{pp,-U}}^{\nu,\nu^{\prime}\\!,\,-\omega}\Big{)}^{*},\end{split}$
(86)

where we use the fact that shifting $\omega\rightarrow\omega-\nu-\nu^{\prime}$
is changing the ph-convention to the pp-convention and, in the second line, we
applied the property for complex conjugation eq. 54.

Together with the definition of $\chi_{S_{x}}^{\nu,\nu^{\prime}\\!,\,\omega}$
and $\chi_{\text{pair}}^{\nu,\nu^{\prime}\\!,\,\omega}$ in eqs. 8 and 9 we get

$\displaystyle\chi_{S_{x},\text{U}}^{\nu,\nu^{\prime}\\!,\,\omega}=\chi_{\text{pair,-U}}^{\nu,\nu^{\prime}\\!,\,-\omega},$
(87)

which is, in our notation, equivalent to eq. 45 in the main text.

##  Appendix D Analytical expression for the generalized susceptibility in the
Hubbard atom

Following Ref. [42], the one-particle Green’s function and the connected two-
particle Green’s functions for the HA in ph-notation for different spin
combinations are as follows:

$\displaystyle G_{1,\sigma}(\nu)=\frac{1-n_{-\sigma}}{i\nu+\mu+\sigma
h}+\frac{n_{-\sigma}}{i\nu+\mu+\sigma h-U},$ (88)

$\displaystyle
G_{2,\uparrow\uparrow}^{\text{con}}(\nu,\nu^{\prime},\omega)=\frac{\beta
U^{2}n_{\downarrow}(1-n_{\downarrow})(\delta_{\omega
0}-\delta_{\nu\nu^{\prime}})}{(i\nu+i\omega+\mu+h)(i\nu+i\omega+\mu+h-U)(i\nu^{\prime}+\mu+h)(i\nu^{\prime}+\mu+h-U)},$
(89)

$\displaystyle\begin{aligned}
&G_{2,\uparrow\downarrow}^{\text{con}}\left(\nu,\nu^{\prime},\omega\right)\\\
&=\frac{n_{\uparrow}+n_{\downarrow}-1}{i\nu+i\nu^{\prime}+i\omega+2\mu-U}\left(\frac{1}{i\nu+i\omega+\mu+h-U}+\frac{1}{i\nu^{\prime}+\mu-
h-U}\right)\left(\frac{1}{i\nu^{\prime}+i\omega+\mu-
h-U}+\frac{1}{i\nu+\mu+h-U}\right)\\\
&+\frac{n_{\uparrow}-n_{\downarrow}}{i\nu^{\prime}-i\nu-2h}\left(\frac{1}{i\nu^{\prime}+\mu-h}-\frac{1}{i\nu+\mu+h-U}\right)\left(\frac{1}{i\nu^{\prime}+i\omega+\mu-h}-\frac{1}{i\nu+i\omega+\mu+h-U}\right)\\\
&+\frac{\beta U^{2}\delta_{\omega
0}\left(e^{(2\mu-U)\beta}-e^{2\mu\beta}\right)}{\left(1+e^{(\mu+h)\beta}+e^{(\mu-h)\beta}+e^{(2\mu-U)\beta}\right)^{2}}\frac{1}{(i\nu+\mu+h)(i\nu+\mu+h-U)\left(i\nu^{\prime}+\mu-h\right)\left(i\nu^{\prime}+\mu-
h-U\right)}\\\
&+\frac{n_{\uparrow}-1}{(i\nu^{\prime}+i\omega+\mu-h)\left(i\nu^{\prime}+\mu-h\right)(i\nu+\mu+h-U)}+\frac{1-n_{\uparrow}}{(i\nu+i\omega+\mu+h-U)\left(i\nu+\mu+h-U\right)\left(i\nu^{\prime}+\mu-h\right)}\\\
&+\frac{1-n_{\downarrow}}{(i\nu+i\omega+\mu+h)\left(i\nu^{\prime}+\mu-
h-U\right)\left(i\nu+\mu+h-U\right)}+\frac{n_{\downarrow}-1}{\left(i\nu+i\omega+\mu+h\right)\left(i\nu+\mu+h-U\right)(i\nu^{\prime}+i\omega+\mu-h)}\\\
&+\frac{1-n_{\downarrow}}{(i\nu^{\prime}+\mu-h)\left(i\nu+i\omega+\mu+h\right)(i\nu^{\prime}+i\omega+\mu-h)}+\frac{1-n_{\downarrow}}{(i\nu^{\prime}+\mu-h)\left(i\nu+i\omega+\mu+h\right)\left(i\nu+\mu+h\right)}\\\
&+\frac{1-n_{\downarrow}}{(i\nu^{\prime}+\mu-
h-U)\left(i\nu+\mu+h\right)(i\nu^{\prime}+i\omega+\mu-
h-U)}+\frac{n_{\downarrow}-1}{(i\nu^{\prime}+\mu-
h-U)\left(i\nu+i\omega+\mu+h\right)\left(i\nu+\mu+h\right)}\\\
&+\frac{1-n_{\downarrow}}{\left(i\nu+i\omega+\mu+h-U\right)\left(i\nu+\mu+h\right)(i\nu^{\prime}+i\omega+\mu-
h-U)}+\frac{n_{\downarrow}-1}{(i\nu^{\prime}+\mu-h)\left(i\nu+i\omega+\mu+h-U\right)\left(i\nu+\mu+h\right)}\\\
&+\frac{-n_{\uparrow}}{(i\nu^{\prime}+\mu-
h-U)\left(i\nu+i\omega+\mu+h-U\right)(i\nu^{\prime}+i\omega+\mu-
h-U)}+\frac{-n_{\uparrow}}{(i\nu^{\prime}+\mu-
h-U)\left(i\nu+i\omega+\mu+h-U\right)\left(i\nu+\mu+h-U\right)},\end{aligned}$
(90)

where

$\displaystyle n_{\sigma}=\frac{e^{(\mu+\sigma
h)\beta}+e^{(2\mu-U)\beta}}{1+e^{(\mu+h)\beta}+e^{(\mu-h)\beta}+e^{(2\mu-U)\beta}}$
(91)

and $G_{2,\downarrow\downarrow}^{\text{con}}$ and
$G_{2,\downarrow\uparrow}^{\text{con}}$ can be obtained from eq. 89 and eq. 90
by changing $h$ to $-h$. Note that the terms in the first two lines in eq. 90
can become singular (i.e. proportional to $\delta_{\nu,\nu^{\prime}+\omega}$
and $\delta_{\nu,\nu^{\prime}}$) for $\mu=U/2$ respectively $h=0$; in these
cases the limit has to be taken carefully.

With these building blocks we can calculate all generalized susceptibilities
of the HA: The $\sigma\sigma^{\prime}$ spin indices can be calculated directly
with eqs. 92 and 93

$\displaystyle\chi_{\sigma\sigma}^{\nu,\nu^{\prime}\\!,\,\omega}=G_{2,\sigma\sigma}^{\text{con}}\left(\nu,\nu^{\prime},\omega\right)+\chi_{0,\sigma\sigma}^{\nu,\nu^{\prime}\\!,\,\omega},$
(92)
$\displaystyle\chi_{\sigma\sigma^{\prime}}^{\nu,\nu^{\prime}\\!,\,\omega}=G_{2,\sigma\sigma^{\prime}}^{\text{con}}\left(\nu,\nu^{\prime},\omega\right)\quad\text{with}\quad\sigma\neq\sigma^{\prime}.$
(93)

The spin indices $\overline{\sigma\sigma^{\prime}}$ can be calculated with

$\displaystyle\chi_{\overline{\sigma\sigma^{\prime}}}^{\nu,\nu^{\prime}\\!,\,\omega}=-\chi_{\sigma\sigma^{\prime}}^{\nu,\nu+\omega,\,\nu-\nu^{\prime}}+\chi_{0,\overline{\sigma\sigma^{\prime}}}^{\nu,\nu^{\prime}\\!,\,\omega},$
(94)

where $\sigma\neq\sigma^{\prime}$ [36] and the bubble terms $\chi_{0}$ are
defined in eqs. 102 and 103.

Finally, the generalized susceptibilities in the particle-particle channel can
be calculated by performing the frequency shift
$\omega\rightarrow\omega-\nu-\nu^{\prime}$.

##  Appendix E Projected weight in the coupled longitudinal channel

To quantify how much an eigenvalue of $\chi_{\mathrm{L}}$ is attributed to
either the charge or the spin subspace respectively, we split each eigenvector
$v$ (normalized by $\norm{.}_{T}$ in eq. 62) into the charge and spin
subspaces

$\displaystyle v=\begin{pmatrix}v_{\text{c}}\\\ v_{\text{s}}\end{pmatrix}.$
(95)

In doing so we can assign each eigenvector a projected weight which gives the
percentage contribution of the eigenvector in the charge/spin subspace
$\mathcal{N}_{\text{c}}/\mathcal{N}_{\text{s}}$ with

$\displaystyle\mathcal{N}_{\text{c/s}}=\frac{\norm{v_{\text{c/s}}}_{\dagger}^{2}}{\norm{v}_{\dagger}^{2}},$
(96)

where $\mathcal{N}_{\text{c}}+\mathcal{N}_{\text{s}}=1$ and
$\norm{u}_{\dagger}^{2}=u^{\dagger}\cdot u$111111Note that for
$\mathcal{N}_{\text{c/s}}$ other definitions would be also possible, e.g.: a)
using $\norm{v}_{T}=1$ and defining
$\mathcal{N}_{\text{c/s}}=\norm{v_{\text{c/s}}}_{T}$. However, since
$\mathcal{N}_{\text{c/s}}\in\mathbb{C}$, this definition cannot be used to
define a percentage. b)using $\norm{v}_{\dagger}=1$ and defining
$\mathcal{N}_{\text{c/s}}=\norm{v_{\text{c/s}}}_{\dagger}$. This definition is
in fact equivalent to the definition in eq. 96. However, vectors normalized
with $\norm{.}_{T}$ have the convenient property of either symmetric or
antisymmetric real and imaginary parts in the charge and spin subspaces (if
their eigenvalue is real). Hence, it is more practical to work with these
vectors. The equivalence can be shown by comparing the two different
normalizations of the same vector $v$:
$\tilde{v}=\frac{v^{\prime}}{\norm{v^{\prime}}_{\dagger}}e^{i\varphi}$, where
$\norm{\tilde{v}}_{\dagger}=1$, $\norm{v^{\prime}}_{T}=1$, and $\varphi$ is
the phase of $\norm{\tilde{v}}_{T}$ The additional factor
$1/\norm{v}^{2}_{\dagger}$ in the definition of the projected weight assures
$\mathcal{N}_{\text{c}}+\mathcal{N}_{\text{s}}=1$..

To prove that pseudo-divergences with $SU(2)_{P}$ symmetry ($\delta\mu=0$) are
always maximally “mixed” $(\mathcal{N}_{\text{c}}=\mathcal{N}_{\text{s}}=0.5)$
we first note that the Shiba mapping connects $SU(2)_{P}$ symmetric case with
the $SU(2)_{S}$ symmetric case ($h=0$), where the charge and the spin channel
are decoupled centro-hermitian matrices.

We consider the eigenvectors $v_{1}$ and $v_{2}$ of a complex conjugate pair
of eigenvalues, of the generalized susceptibility in the charge or the spin
channel. Because of eq. 63 the symmetric and antisymmetric parts of $v_{1}$
can be written as

$\displaystyle
v_{1}=\frac{1}{2}(v_{1}+v_{2}^{*})+\frac{1}{2}(v_{1}-v_{2}^{*})\text{ with
}v_{\mathcal{S/A}}=\frac{1}{2}(v_{1}\pm v_{2}^{*}).$ (97)

As we know from eq. 84 only the symmetric part of the eigenvector gets mapped
from charge/spin subspace to the spin/charge subspace.

Therefore, we must calculate the norm only for the symmetric/antisymmetric
part of the eigenvector

$\displaystyle\frac{\norm{v_{\mathcal{S/A}}}_{\dagger}^{2}}{\norm{v_{1}}_{\dagger}^{2}}=\frac{1}{4\norm{v_{1}}_{\dagger}^{2}}(v_{1}^{\dagger}\cdot
v_{1}\pm v_{2}^{T}\cdot v_{1}\pm v_{1}^{\dagger}\cdot v_{2}^{*}+v_{2}^{T}\cdot
v_{2}^{*})=\frac{1}{2},$ (98)

where $v_{1}^{T}\cdot v_{2}=0$ and
$\norm{v_{2}}_{\dagger}=\norm{v_{1}}_{\dagger}$121212$v_{1}^{T}\cdot v_{2}=0$,
because the vectors are orthogonal regarding the scalar product eq. 61.
Further, $\norm{v_{2}}_{\dagger}=\norm{v_{1}}_{\dagger}$ holds because of the
relation $v_{2}=Jv_{1}^{*}$.

Hence, we find that complex conjugate pairs, and therefore pseudo-divergences,
have the maximal mixing $(\mathcal{N}_{\text{c}}=\mathcal{N}_{\text{s}}=0.5)$
between the charge and longitudinal spin subspace when $SU(2)_{P}$ symmetry is
fulfilled.

##  Appendix F T=0 limit of the Hubbard atom

The advantage of considering a _dimensionless_ parameter space of
$\\{\beta\delta\mu,\beta h,\beta U\\}$ in the main text allows also for an
intuitive understanding of the $T\to 0$ limit.

In fig. 7 the first loop and the first parabola divergence line structure in
the charge channel for the $SU(2)_{S}$ symmetric case are shown for different
inverse temperatures $\beta=1/T$.

Figure 7:  The phase space diagram of the first loop and parabola divergence
line structure in the charge channel for different temperatures for the
$SU(2)_{S}$ symmetric case ($h=0$).

Here, we observe that, for $T\rightarrow 0$ ($\beta\to\infty$), the loop
structure contracts to a single point and both exceptional points of the
parabola approach $\delta\mu=0$.

The scaling suggested by the $T\rightarrow 0$ evolution of divergence lines
shown in fig. 7 outlines an intuitive procedure to get a first understanding
of the $T=0$ vertex divergence properties. This procedure generally works
satisfactory, except for the discontinuous parts of the phase space, located
at $\delta\mu=0$ for $U<0$ and at $\delta\mu=\pm U/2$ for $U>0$. In
particular, since the generalized susceptibility matrix becomes Hermitian for
$\delta\mu=0$ the exceptional points must transform either continuously into
two degenerate or discontinuously into two separate eigenvalues of two linear-
independent eigenvectors (one symmetric, one antisymmetric) for each value of
$U<0$ at $\delta\mu=0$, where a discontinuity appears. By carefully evaluating
the analytical expressions at $\delta\mu=0$ and $T=0$ (see Refs. [22, 65]) we
find a single vanishing eigenvalue with an antisymmetric eigenvector for every
$U<0$ and $\delta\mu=0$ at $T=0$. On the other hand, for $U>0$ where no
discontinuity appears one finds two degenerate vanishing eigenvalues (one with
symmetric one with antisymmetric eigenvector) for every $U<0$ and
$\delta\mu=0$ at $T=0$. This behavior matches the $T\rightarrow 0$ evolution
the loop structures formed by an symmetric and an antisymmetric divergence
that contracts to an single point.

Further, we find that the pseudo-divergence lines (at $\delta\mu\neq 0$)
become parallel to $\delta\mu=\pm U/2$ in the limit of
$U,\delta\mu\rightarrow\infty$ which mark the boundaries of a V-shaped region
in the phase diagram. Thus, the pseudo-divergences that stem from the loop
structure at $U>0$ remain inside this V-shaped region ($U>2|\delta\mu|$) and
the pseudo-divergences that emerge from the parabolas at $U<0$ stay outside of
it ($U<2|\delta\mu|$).

Applying the same analysis to the spin and pairing channel, one can show that
the pairing divergences are confined inside the V-shaped area while the spin
pseudo-divergences stay outside of it.

From these intuitive considerations we can schematically draw fig. 3 in the
main text.

In principle, a more rigorous treatment of the $T=0$ limit, which yield
however the same results 131313Indeed the results obtained with the simple
scaling argument of the dimensionless phase space agree well with the results
from the more rigorous treatment of the $T\rightarrow 0$ limit of the
analytical formulas, where only special care has to be taken for the
discontinuous parts of the phase space. This shows the usefulness of the
dimensionless phase space., can be performed as explained in [65]. In
particular, for the analytical expressions of the generalized susceptibility
reported in appendix D one has to take into account that the Matsubara
frequencies become continuous and can no longer be regarded as matrix indices.

Consequently for $T=0$, a sum over Matsubara frequencies becomes an integral,
and a Kronecker-Delta in Matsubara frequencies becomes a Dirac-Delta:

$\displaystyle\frac{1}{\beta}\sum_{\nu}\overset{\beta\rightarrow\infty}{\longrightarrow}\int_{-\infty}^{\infty}\frac{d\nu}{2\pi},$
(99)
$\displaystyle\beta\delta_{\nu\nu^{\prime}}\overset{\beta\rightarrow\infty}{\longrightarrow}2\pi\delta(\nu-\nu^{\prime}),$
(100) $\displaystyle\beta\delta_{\omega
0}\overset{\beta\rightarrow\infty}{\longrightarrow}2\pi\delta(\omega).$ (101)

For the remaining terms in the analytical formulas in appendix D we take the
limit $\beta\rightarrow\infty$.

##  Appendix G Additional information to vertex divergence structure

In this section, we will give some additional information about the vertex
(pseudo-)divergences phase diagram.

To get started we show the first two vertex (pseudo-)divergences line
structures in the coupled longitudinal channel for $\alpha=42^{\circ}$ in fig.
8.

Figure 8:  The phase space diagrams of the vertex (pseudo-)divergences in the
coupled longitudinal channel for the Hubbard atom are shown. Note that the
first two divergence line structures are displayed. The plotting conventions
are as in fig. 2.

Not surprisingly, in addition to the first divergence line loop, we find a
second loop at higher $U$ values for $U>0$. On the other hand, for $U<0$ we
find a second pair of parabolas. In addition, we find a second parabola for
$U>0$ at smaller $r$ values than the first parabola at $U>0$. The pseudo-