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

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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][24] beyond the standard textbook discussions, with a particular focus on their highfrequency asymptotics [25][26][27] and on their algorithmic treatments [28].Considering the specific case of the physical interpretation of the local (on-site) two-particle formalism, interesting information was obtained in Ref. [29].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 [29] unveiled intrinsic symmetry properties of the four-point * herbert.essl@tuwien.ac.at 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. [29] were limited to a very specific case: the Shiba transformation of the static (i.e., for zero frequency transfer, ω = 0) twoparticle generalized susceptibilities in the particle-hole symmetric Hubbard model, i.e. half filling (µ = 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 ω.
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 [30][31][32][33] and the physical [34][35][36][37] aspects associated to the breakdown [30][31][32] of the self-consistent perturbation theory for many-electron systems, beyond the (unrealistic) assumption of perfect particle-hole symmetry made so far in the largest part of the corresponding literature (with the notable exceptions of Refs.[33][34][35]).
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 specific 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 A. 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 [20,38].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 χ can be defined as follows [15,20]: where σ, σ ′ denote the spins, ν i are fermionic Matsubara frequencies (where the fourth frequency is set by σ annihilate (create) an electron with spin σ, T τ denotes the imaginary time ordering operator, and ⟨. . .⟩ = 1  Z Tr(e −βH . ..) the thermal expectation value.Note that for non-local χ, c ( †) i,σ acquire an additional lattice index i, and for space translation invariant systems, one of the four lattice indices can be set to zero.
Instead of using the definitions in Eqs. ( 1), (2) directly, it is often more useful to consider specific spin combinations, which correspond to physical response functions.
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): For the subsequent discussion it is convenient to view the generalized susceptibilities as (infinite dimensional) matrices in the frequencies ν, ν ′ and treat ω as an additional parameter.
To lighten the notation we are going to omit the fermionic frequencies ν and ν ′ in the following and state the bosonic frequency ω 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 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 on the symmetries of the considered model (see Appendix A for an in depth discussion).
For ω = 0, the matrices of the transversal spin channel χ Sx as well as the pairing channel χ pair can be identified as centro-Hermitian matrices [39]  in complex conjugate pairs [40].Moreover, the matrix of the coupled longitudinal channel χ L is classified as a κ−real matrix Eq. (A13), which shows the same eigenvalue properties [41].
In addition, the generalized susceptibilities for all considered channels are symmetric matrices and, hence, can be diagonalized by an orthogonal transformation [42].This, together with the centro-Hermitian respectively κ−real property, leads to the fact that eigenvectors v α corresponding to real eigenvalues λ α have a symmetric/antisymmetric real part Re v α (in ν) and an antisymmetric/symmetric imaginary part Im v α (see Eq. (A16)) [39].For the longitudinal channel eigenvectors are symmetric/antisymmetric in the charge respectively spin subspace (see Eq. (A17)).Fig. 1 shows a schematic example of a κ−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.
A concise summary of the generalized susceptibility matrix properties for ω = 0 in the different channels and under additional symmetries can be seen in Tab.I, where No Symmetry refers to time translation symmetry, spin conservation and that the Hamiltonian is a real function of c and c † .Note that the last assumption can be violated, for example, by the inclusion of an external electromagnetic potential via the Peierls substitution in lattice models and/or if spin-orbit coupling in is taken into account.All these symmetries can be derived by applying Eqs.(A6) to (A10) on the respective definition of the different channels.

B. 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. [29] and is derived by applying a partial (local) particle-hole transformation, or Shiba transformation [43], acting on the creation and annihilation operators with By considering a local single orbital Hubbard Hamiltonian (written in a symmetric form with δµ = µ − U /2) one readily sees that applying the Shiba transformation (n σ = c † σ c σ ) corresponds to interchanging the values of δµ ↔ 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 fields h.The model can be solved analytically [44]; the corresponding expressions for the one-and twoparticle 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 Hubbard model (HM) reads 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,↓ → c † i,↓ (−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 S with the components and the (local) pseudospin operator S p with the components Evidently, the Shiba transformation maps the spin S to the pseudospin S p .Since the components of S and S p commute, spin and pseudospin can be regarded as independent set of operators [38].
The pseudospin can be given the following physical interpretation: S p,x and 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 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 S if h = 0, which we will denote as SU (2) S symmetry, and with all components of S p for δµ = 0, referred to as SU (2) P symmetry.
The on-site physical susceptibilities, describing the (linear) response of the system on a local observable A when a local observable B is coupled to an external perturbation can be calculated by Fourier-transforming/analytically continuing the following expression: In Matsubara frequency space, they are related to the generalized susceptibilities via: Using Eq. ( 21) and the definitions in Eqs. ( 4) to (9) we find 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 local [46] generalized susceptibilities of the charge, longitudinal spin, transversal spin, and pairing channel (see Eqs. (4) to ( 9)) for the most general cases, i.e. arbitrary filling, external magnetic field, and arbitrary bosonic transfer frequency ω.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: where the subscript U refers to the model parameters (U, δµ, h) and the subscript -U to (−U, h, δµ), 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 ω = 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 χ L,U and χ L,-U .Thereby, we employ the following matrix Q in Matsubara space: where we divided the Matsubara frequencies into quadrants of positive + and negative − frequencies The similarity transformation Q T χ 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-skewsymmetric matrix (JAJ = −A) into two blocks along the skew diagonal.
Hence, a similarity transformation with Q separates the real χ ′ and imaginary part χ ′′ of a centro-Hermitian matrix χ CH (Jχ CH J = χ * CH ) into the symmetric and antisymmetric components respectively A detailed definition of the different blocks can be found in Appendix C.
By defining the transformations Q and T in the tensor product space of the coupled L channel (c.f.Eq. ( 10)) we can determine (see Appendix C) the transformation S, which relates the two sides of the Shiba transformation: In Appendix C we show that this relation holds for arbitrary bosonic transfer frequencies ω, when adapting the Q and T matrices by shifting the boundaries of the Matsubara frequency quadrants by (−ω/2, −ω/2) in analogy to Eq. (30).
Hence, for finite ω the susceptibility matrices can be viewed as centro-Hermitian in regard to the shifted Matsubara frequency basis (see Eq. (A12)).
From Eq. ( 33) (better visible in Eq. ( C14)) it follows that the Shiba transformation leaves the real antisymmetric part χ ′ A of each charge-spin components (χ c , χ s , χ cs , χ sc ) in χ L invariant and exchanges the real symmetric part χ ′ S between χ c ↔ χ s , and between 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: By examining the eigenvectors of the generalized susceptibility in the longitudinal channel, it follows 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. (C18), (C19)).
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 (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 neighbor[47] hopping t and (U, δµ, h) to the corresponding lattice model with the same hopping and (−U, h, δµ) [48].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) [49].
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 the breakdown of the self-consistent perturbation expansion for many-electron systems.

A. Vertex divergences and their generalization
In this section, we extend the existing studies [22,29,32,50] 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 δµ = 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 ω = 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 Γ ν,ν ′ r (ω) in a given scattering channel r (e.g., longitudinal, pairing, etc.) is formally defined [20,38,51,52] 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.
where the transversal channels has to be treated with care if H ∉ R and ω ≠ 0 (see Appendix B) Hence, the divergences of the Γ r are directly determined by the vanishing of an eigenvalue of the corresponding generalized susceptibility matrix χ ν,ν ′ , ω r in the fermionic Matsubara frequencies ν, ν ′ [53].
As mentioned in Section II A by perfect SU (2) S and SU (2) P symmetry, all local two-particle quantities for zero-transfer frequency (ω = 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 Γ νν ′ r .These sign changes, as discussed in the recent literature [29,32,[35][36][37]54], 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 [35,39,[54][55][56][57].However, if either SU (2) S (h ≠ 0) or SU (2) P (δµ ≠ 0) symmetries are broken, χ νν ′ r has no longer to be a real and symmetric matrix for all channels, but can be either a centro-Hermitian or a κ-real matrix (see Tab. I).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 [40,41].
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.We note that the occurrence of this specific behavior of the eigenvalues of the generalized charge susceptibility has been reported in the phase diagram of Ref. [33] showing the DMFT [49]/Cellular-DMFT [58,59] solution of the HM out of half-filling, as well as, in a somewhat different perspective, in Refs.[35,39,56].Indeed, these observations highlight the general relevance of such kind of evolution of the eigenvalues of the generalized susceptibility beyond the specific framework of the HA, which we are going to investigate systematically in the following.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, the concept of vertex divergences can be extended to those cases, where sign-changes of the real part of an eigenvalue pair of χ νν ′ r are not directly associated to an actual divergence of Γ νν ′ r .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 axis 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 [60,61] and/or of the perturbative expansion of the electron self-energy [10,[62][63][64] in abstract complex planes obtained, e.g. through the complexification of the electronic interaction U [10, 60, 61].

B. 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-site [65] generalized susceptibilities, whose expression are reported, e.g., Appendix D and in Ref. [66].To obtain a dimensionless representation we scale all parameters of the Hamiltonian with inverse temperature β.Formally this can be done by considering a model FIG. 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 for the two special cases of (a) no magnetic field but out of half-filling (left panel) and (b) 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 (see legend in the box below the figure): 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 ≠ 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.
Hamiltonian, which is defined through N real parameters {ε i }| N i=1 (e.g., the hopping t, the on-site interaction U , etc.), where all ε 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 ({ε i })}.As a consequence, all corresponding dimensionless quantities βE j must only be dependent on the set {βϵ 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 β 3 , the generalized susceptibility can be always expressed in terms of the parameter set {βE j } which can be fully expressed via the parameter set {βε i }.Therefore, one can rewrite the Lehmann representation as: where n, n ′ and m are the indices for ν, ν ′ and ω and α 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 {βδµ, βh, βU } 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 pseudodivergences.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 the spin channel at SU (2) S symmetry, where all three spin directions are equivalent.Eventually, for broken SU (2) S symmetry (h ≠ 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. (A15) to normalize the eigenvectors.

C. Vertex divergences of the HA: Limiting cases
We start our analysis by illustrating in Fig. 2 the results obtained for two selected 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 Fig. 2a) or the SU (2) P (δµ = 0 Fig. 2b) symmetry is preserved.Such choice allows us to showcase in the most transparent way the action of the Shiba-mapping on the vertex functions beyond particle-hole symmetry.In fact, a first glance at the geometrical shape and at the nature (encoded by the different colors) of the vertex (pseudo-)divergence lines displayed in Fig. 2 easily unveils the direct connection between the results of the repulsive model for varying δµ (upper panel of Fig. 2a) and those of the attractive model for varying h (lower panel of Fig. 2b).Consistent with the mapping relations, an analogous correspondence is also observed between our two-particle calculations for the attractive model at different δµ (lower panel of Fig. 2a) and those for the repulsive model at different h (upper panel of Fig. 2b).
Before proceeding with the detailed analysis of the geometry and the nature of vertex (pseudo-)divergence lines in Fig. 2, we recall that for the case of perfect particle-hole/SO(4) symmetry (δµ = h = 0), it has been analytically demonstrated in the literature [22,29,50] that the HA displays a discrete set of vertex divergences, marked by colored dots in Fig. 2. In particular, for U > 0, the red dots on the positive βU -axis of both panels of Fig. 2 indicate the location of the irreducible vertex divergences in the charge channel of the HA 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 for perfect particle-hole symmetry [29], one finds along the negative βU -axes of Fig. 2, exactly in mirrored positions w.r.t. the U > 0 case, alternating antisymmetric vertex divergences in the charge sector (red dots) and symmetric ones in the spin sector (bluish green dots).
We begin now our detailed discussion of the vertex divergences in the HA beyond particle-hole symmetry by examining the data shown in Fig. 2a, corresponding to the case of h = 0 and varying δµ.Here, it should be stressed that, as soon as the SU (2) P -symmetry is lifted for any δµ ≠ 0, the symmetric divergences in the charge and pairing channel, marked by the orange dots on the positive βU axes of Fig. 2, 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 δµ ≠ 0: As reported in Tab.I 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 actually diverges, without any possible occurrence of pseudodivergences and/or exceptional points (EPs).The latter are defined as the points in parameter space where two (or more) eigenvalues and eigenvectors coalesce and the matrix is no longer diagonalizable [39,42].For the same reason, the symmetric nature of its singular eigenvectors is maintained for arbitrary values of δµ.
On a more geometrical viewpoint, one can easily observe in Fig. 2a that the divergences lines of the charge sector form elliptic shaped loops in the phase space, terminating on both sides in EPs for positive and negative values of δµ.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 βU values corresponds to a hole/electron doping of about 6%, which roughly coincides [67] with the doping level at which the corresponding EPs have been found in the DMFT/Cellular DMFT calculations of the HM of Ref. [33].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 βU .In fact, consistent with Refs.[22,50,68], where the occurrence of infinitely many vertex divergences at half filling has been demonstrated, we also find infinitely many loops for increasing βU values.
Turning 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 Fig. 2a.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 βU → −∞.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 situation [69], 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 βU → −∞, at an EP, from which a pseudodivergence 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 βU , 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.Hence, one may regard the pseudo-divergences lines that cross the U = 0 axis as driven by the symmetry breaking fields (δµ and/or h) in contrast to the pseudo-divergences lines that do not cross the U = 0 axis which would then be driven by the interaction U .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 βδµ = βµ = ±(2N − 1)π (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 [70].
As we noted at the beginning of this subsection, in the case of preserved SU (2) P symmetry (i.e., δµ = 0, h ≠ 0, see Fig. 2b), the mapping relations determine a perfectly "mirrored" disposition of vertex divergences and pseudodivergences, 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 ≠ 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 Fig. 2a) are transformed in the corresponding divergences of the transversal channel in the attractive sector with SU (2) P symmetry (blue lines in Fig. 2b).Indeed, for the same reasons explained above, the generalized susceptibility of the spin transversal channels remains real and bi-symmetric for δµ = 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 scale [71], as detailed in Appendix E.Moreover, a parabola in the pairing channel (marked in a thicker yellow line in Fig. 2b) is superimposed to every second parabola of divergences in the longitudinal sector.

D. 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, it is worth to exploit our results to infer how the vertex (pseudo-)divergences evolve in the T → 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.
In particular, in the lower panels of Fig. 3 we observe a continuous distribution of vertex divergences at particlehole symmetry (δµ = 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 [22,29,50].
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 δµ = −U /2 and δµ = 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 pairing channel and pseudo-divergence in the charge channel are found simultaneously.Indeed, the latter ones stem from the "loop-like" structures in Fig. 2a, that collapse on the δµ = 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. 3 lower left).The pseudo-divergences originate from the parabola structures in Fig. 2a as they progressively move towards the δµ = 0-axis in the U < 0 sector.

FIG. 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 δµ ≠ 0; right: h ≠ 0 and δµ = 0) compared to the corresponding filling n = n↑ + n↓ (upper left) and magnetization m = n↑ − n↓ (upper right) of the system.These values can be understood by the corresponding ground-state(s) of the HA, which are explicitly depicted in the different parameter regions.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 the opposite situation of δµ = 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 δµ = −U /2 and δµ = 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 pseudodivergences in the longitudinal sector and divergences in the transversal spin channel, while pseudo-divergences in the longitudinal and pairing channel are present outside.

E. Vertex divergences of the HA: The general case
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 δµ.For the sake of compactness, we introduce the parameters α and r, as sketched in Fig. 4b, in order to facilitate the visualization of our results when the values of δµ, h are varied between the limiting cases of h = 0 (α = 0) and δµ = 0 (α = 90 ○ ) shown in Fig. 2a and Fig. 2b respectively.
Specifically, the parameter α and r are defined as the following set of cylinder coordinates: δµ = r cos α and h = r sin α (38) It is then easy to see that the sets of δµ, 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 α: whereas r remains unchanged.Further, due to Eqs. (A3) to (A5), it is enough to consider α ∈ [0 ○ , 90 ○ ], 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 studied in Section IV C.These correspond to the case of α = 0 ○ (Fig. 2a) and α = 90 ○ (Fig. 2b) and are also reproduced for reference in Fig. 4a/c respectively.Note that, for the sake of clarity, we only show here the lowest [72] loop and parabola structures and their associated pseudodivergences.
We consider first the situation where the variation of chemical potential is larger than the magnetic field, i.e. 0 ○ < α < 45 ○ , which we illustrate by showing, as an example, the case of α = 35 ○ in Fig. 4d.Interestingly, for U > 0, in addition to the loop-structures and their associated pseudo-divergences already discussed for α = 0 ○ , a parabola structure appears at large values of βU .We can also observe that the pseudo-divergence associated to this parabola at U > 0 are connected with the pseudodivergence emerging from the second parabola for U < 0.
In the reversed situation, where the magnetic field dominates over δµ (i.e. for 45 ○ < α < 90 ○ ), 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 α ′ = 90 ○ − α, with 0 ○ < α ′ < 45 ○ , but for the flipping of the U and −U sectors (see Fig. 4f).
Remarkably, for exactly α = 45 ○ , where h equals δµ, 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 as it can be seen in Fig. 4e.
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 = ν √ 1±2 cos α sin α , which is explicitly demonstrated in Eqs.(G8), (G9), while the ones associated to the loops never cross the non-interacting axis.Evidently, the condition α = 45 ○ 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.α = 35 ○ , 45 ○ , 55 ○ ) 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 particlehole 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 = ν √ (cos α) 2 −(sin α) 2 , i.e.only for 0 ○ ≤ α < 45 ○ , while those of the pairing channel cross the U = 0 axis at r = ν √ (sin α) 2 −(cos α) 2 and therefore only for 45 ○ < α ≤ 90 ○ .

F. The mapping of the eigenvectors
Crucial information [29,32,35,39,54,56] 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 particlehole symmetric results of [29], 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., α = 35 ○ for U > 0 and α = 55 ○ 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 S ) and and antisymmetric (v A ) part, while their projected weight in the charge and spin subspace (see Appendix E), 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 into 2b and 3a into 3b.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 κ-real matrix χ L (see Eq. (A17)).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 and 3b is one of the 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.Note that only one of the eigenvectors of the pseudo-divergence is shown since the other one can be reconstructed by Eq. (A17) and the projected weights of these two eigenvectors are the same.
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 S changes from charge to spin subspace and vice versa, while the anti-symmetric part v 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.

G. Finite bosonic transfer frequency
According to the state-of-the-art knowledge (cf.Ref. [32]) only irreducible vertex divergences occurring at ω = 0 can be directly linked to the breakdown of the self-consistent perturbation theory for the many-electron problem.For this reason, the illustrative study of the (pseudo-)divergences of the HA presented in this work is mostly focused on the ω = 0 case.
Notwithstanding, for the sake of completeness, it is worth to get a glimpse on some relevant modifications of the results presented so far, when considering vertex divergences structures at finite transfer frequency (ω ≠ 0).To this aim, we will focus our discussion of the finite ωcase on the (pseudo-)divergences occurring in the charge and spin sectors, when SU (2) S symmetry is preserved (i.e., δµ ≠ 0, h = 0).In this respect, our starting point will be represented by the findings of Ref. [22], where vertex divergences of the particle-hole symmetric HA (δµ = h = 0) have been shortly analyzed for specific finite frequency values.
More precisely, the analytical expressions of Ref. [22] show that, for the ph-symmetric HA, vertex divergences occur at all βU values such that (i β are the fermionic and the bosonic Matsubara frequencies, respectively.The eigenvectors associated to these divergences display an antisymmetric structure in the shifted frequency space Eq.(A12), i.e. they are proportional to δ ν,ν ′ − δ ν,−ν ′ −ω .
Since the discriminant in the above relations (i) and (ii) must be positive, anti-symmetric divergences in the spin channel can only occur for ω ≠ 0, when ν(ν + ω) < 0. At the same time, for these frequency values no divergence can occur in the charge channel.However, the fulfill- ment of ν(ν + ω) < 0 is not a sufficient condition for an anti-symmetric divergence to occur, since anti-symmetric eigenvectors require two distinct fermionic frequencies ν ≠ −ν − ω for their construction.As a result, divergences in the spin channel only appear if |m| > 1 and a new anti-symmetric divergence only appears at even m values.
Furthermore, also vertex divergences associated to symmetric eigenvectors in the shifted frequency space can be found [22], namely for all U values where is fulfilled, where the + (−) sign is for the charge (spin) channel and From Eq. ( 40) one finds [22] the occurrence of symmetric divergences in the charge channel for all values of ω if U > 0. However, we should note that the condition of Eq. ( 40) can not be fulfilled, if the corresponding eigenvector is maximal at a ν value, for which ν(ν + ω) < 0. Note that this is the same condition as for the anti-symmetric divergences.
As for the symmetric divergences in the spin channel, differently as reported in a side remark in Ref. [22], these can indeed occur at U > 0 for m ≥ 3.
In summary, in the spin channel we find ⌊m/2⌋ antisymmetric divergences, while an explicit evaluation of Eq. ( 40) also yields ⌈m/2⌉ − 1 symmetric divergences.At the same time, in the charge channel, infinitely many symmetric and anti-symmetric divergences occur, under the condition that the corresponding eigenvectors do not have a maximum at frequencies where ν(ν + ω) < 0.
The location of the first two symmetric and two antisymmetric divergences in charge and spin channel is shown in Fig. 7 for different values of ω at ph-symmetry.The location of the divergences is symmetric in ω, i.e. it is the same for ω and −ω.
For particle-hole symmetry the action of the Shiba mapping at finite ω can be evidenced by noting that (i) all antisymmetric vertex divergences occur at the same absolute U values for positive and negative U , and (ii) that f c is transformed into −f s by changing U to −U .The latter condition implies that the symmetric vertex divergences for U < 0 can be obtained by just exchanging charge and spin channel in the discussion for U > 0. Hence, as a direct consequence of the Shiba mapping at finite frequency see Eq. ( 33), the findings of Ref. [29] for the vertex divergence of the static particle-hole case remain identically applicable also at ω ≠ 0.
At the same time, a remarkable difference to the ω = 0 case is that negative real eigenvalues of the generalized charge/spin susceptibility at ph-symmetry can be found already at U = 0 (see Eq. (G1)) namely when the frequency condition ν(ν + ω) < 0 is fulfilled.However, the role played by these negative eigenvalues in triggering additional divergences at ω ≠ 0 is quite different in the charge and in the spin channel.In fact, in the former one, consistently with the discussion above, the negative eigenvalues found at U = 0 never cross zero by increasing U .In the spin channel, instead, all negative eigenvalues found at U = 0 will cross zero by increasing U , except the ones that correspond to eigenvectors which either have their maximum at the central frequency ν * = −ω/2 if m is odd or that are symmetric and have their maximum at ν = ω/2 ± π/β if m is even.Such crossings evidently trigger vertex divergences in the spin sector that were not present for ω = 0.
Eventually, merging the description of the vertex divergences at finite ω for the ph-symmetric case [22] with the knowledge of the δµ-evolution of the vertex divergences at ω = 0 (Section IV C), it becomes easy to intuitively outline their qualitative behavior as a function of a varying chemical potential.Specifically, by gradually increasing the absolute value of δµ, we find that pairs of symmetric and anti-symmetric vertex divergences get closer in parameter space, forming loop-structures in the phase diagram analogous to those shown in Fig. 2. In contrast, vertex divergences that do not have a partner at ph-symmetry, i.e. there exists no vertex divergence which has its maximum at the same frequency but has opposite symmetry, form a parabola-like structure in the phase space.Such divergences are, for example, the anti-symmetric divergences in the spin channel for m = 2, 4 at the largest βU value shown in Fig. 7.
Hence, the overall evolution of the vertex (pseudo-)divergence lines for ω ≠ 0 follows the intuitive expectation gained from the ω = 0 case.These intuitive picture is quantitatively confirmed by direct calculations (not shown).

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 [29], 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 [20,21,23,28] for different cutting-edge algorithms [22,26,28] 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 a 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 [22,50].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 [39] of the thermodynamic phase-separation instabilities emerging [35,54,56,73,74] in the proximity of a Mott-Hubbard metal-to-insulator transitions of correlated lattice systems.At the same time, as demonstrated in Ref. [33], their occurrence is not limited to purely on-site four-point correlation functions.Eventually, the EPs might play, together with the pseudo-divergences discussed in our work as well as with the pole structures [61] 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 [22,31,32,75], beyond [33] the rather specific condition of perfect particle-hole symmetry condition mostly investigated so far.
and a sign change in the magnetic field just flips all spins of the generalized susceptibility (σ → −σ, ∀σ).We find the following relations for the different channels: where M = ( 1 0 0 −1 ), and for the transversal spin and the pairing channel we find Since the transformations in Eqs.(A3) to (A5) change neither the eigenvalues nor the projected weights (see Eq. (E2)) 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 δµ > 0 and h > 0. This corresponds to the interval α ∈ [0 ○ , 90 ○ ] in the cylinder coordinates of Eq. (38).
In Eqs.(A6) to (A10) the properties of the generalized susceptibilities under the specific symmetry relations are explicitly written following Ref.[38].
Crossing symmetry and complex conjugation are fundamental "symmetries", i.e. they are always present, H ∈ R refers to the Hamiltonian being a real function of c and c † and SU (2) S /SU (2) P means symmetric with respect to spin/pseudo-spin.

Crossing Symmetry
From Eqs. (A6) to (A10) the relations written in Tab.I of the main text can be derived.
In the following, we introduce the class of centro-Hermitian matrices C which fulfill Centro-Hermitian matrices have eigenvalues that are either real or come in complex conjugate pairs [40].
By using the symmetry Eqs.(A6) to (A8) we find that all generalized susceptibilities are centro-Hermitian for ω = 0. Note that for χ ω=0 σσ ′ ,ph the centro-Hermitian property is fundamental, since only complex conjugation and crossing symmetry (which are always present) are needed, whereas for χ ω=0 σσ ′ ph/pp also H ∈ R is required (which could be violated in principle).Without this restriction χ ω=0 σσ ′ ph/pp are per-hermitian matrices [39].
Moreover, we introduce the class of κ-real matrices [41], which also have only real or complex conjugate pairs as eigenvalues.The generalized susceptibility of the coupled longitudinal channel χ L is a κ-real matrix which fulfills the relation Further, all considered channels are symmetric matrices if H ∈ R and can be diagonalized by a complex orthogonal transformation, if there is no exceptional point.The corresponding inner product of the orthogonal Euclidean quasi-norm is defined as and Eigenvectors of symmetric and centro-Hermitian matrices that are normalized with respect to Eq. (A15) have the following properties [39]: 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 ν (′) → ν (′) + ⌊m/2⌋2π/β).
For the symmetric κ-real matrix χ L we can make a similar argument as in Ref. [39] to obtain: 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, χ Sx and χ pair do not represent decoupled channels in the BSE.Even though χ Sx = χ Sy , they still couple to each other for ⟨S z ⟩ ≠ 0 [19].The same considerations apply to the real and imaginary part of the pairing field ∆, which are given by S p,x and S p,y , and couple for ⟨S p,z ⟩ ≠ 0. Therefore the S x /pairing channel generally (for arbitrary ω) 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.(B1), (B2).This is an example of how the mapping "preserves" the decoupling of the channels regarding the BSE.Instead the transverse components χ σσ ′ and χ σσ ′ ,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 χ σσ ′ directly) ) * (see Appendix A), and both the BSE of the pairing channel and the transversal spin channel decouple for ω = 0. Hence, to investigate the divergences of the two-particle irreducible vertices for ω = 0, χ Sx and χ pair can be used for the scope of this paper as it is done in Section IV.where an appropriate Q matrix has to be used depending on whether the generalized susceptibility is an even or an odd matrix.From Eq. (C14) and the fact that Q is an orthogonal transformation Eq. ( 33) in the main text is obtained.
In Eq. (C14) one can see that after the Q transformation χ L,U and χ L,-U only differ by a permutation of the different sub matrices.This reshuffling is done by the matrix T in Eq. ( 32) which needs to be adapted to Eq. (C15) for the odd-dimensional case 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 We find that (i), a symmetric/anti-symmetric block of χ L,U always maps to a symmetric/anti-symmetric block of χ L,-U and (ii), all χ ′ A blocks are invariant under the mapping.All other blocks get redistributed according to Eq. (C14), which is summarized in Eqs.(34), (35) in the main text.[76] Note that for χ L the subscript 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 and for the odd-dimensional case Eventually, we consider the action of the Shiba transformation S = Q T T Q, which is orthogonal and symmetric, on the eigenvectors and eigenvalues of χ L .
Since S is a orthogonal transformation, χ L,U and χ L,-U have the same eigenvalues.Further, if v is a eigenvector of χ L,U then Sv is an eigenvector of χ L,-U with the same eigenvalue.Transforming the eigenvector v we find for the even dimensional case and 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.
[79] 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 → 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.

FIG. 4 .
FIG. 4. (b): Sketch of the new coordinate system {U, r, α} introduced to systematically interpolate the illustration of the vertex (pseudo-)divergences of the HA for arbitrary values of h and δµ between the limiting cases shown in Fig. 2, which correspond to α = 0 ○ and 90 ○ .The corresponding (pseudo-)divergences in the coupled longitudinal sector already shown in Fig. 2 are reproduced here, for reference, in (a)/(c).(d)-(f ): 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 α planes chosen (see text).The plotting conventions are as in Fig. 2.

FIG. 5 .
FIG. 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 α-planes.The vertex (pseudo-)divergences of the coupled longitudinal channel, already shown in Fig. 4, are reproduced here in gray for reference.

FIG. 6 .
FIG. 6. Left: The phase space diagram of the vertex (pseudo-)divergences in longitudinal channel are shown for α = 35 ○ in the repulsive sector U > 0 and for α = 55 ○ 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 S ) and antisymmetric part (v A ) to make the mapping more visible.Note that (a) always corresponds to α = 35 ○ while (b) corresponds to α = 55 ○ .Right: The projected weights of the eigenvectors again split into symmetric and antisymmetric part.

FIG. 7 .
FIG. 7. Location of the first two symmetric and two antisymmetric vertex divergences in charge and spin channel at ph-symmetry for different ω = 2πm/β values.
(see Eq. (A11)), and therefore only have eigenvalues that are real or come

TABLE I .
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 † .