# Protection of Correlation-Induced Phase Instabilities by Exceptional
Susceptibilities

M. Reitner  Institute of Solid State Physics, TU Wien, 1040 Vienna, Austria
L. Crippa  Institut für Theoretische Physik und Astrophysik and Würzburg-
Dresden Cluster of Excellence ct.qmat, Universität Würzburg, 97074 Würzburg,
Germany  D. R. Fus  Institute of Solid State Physics, TU Wien, 1040 Vienna,
Austria  J. C. Budich  Institute of Theoretical Phyiscs, Technische
Universität Dresden and Würzburg-Dresden Cluster of Excellence ct.qmat, 01062
Dresden, Germany  Max Planck Institute for the Physics of Complex Systems,
Nöthnitzer Str. 38, 01187 Dresden, Germany  A. Toschi  Institute of Solid
State Physics, TU Wien, 1040 Vienna, Austria  G. Sangiovanni  Institut für
Theoretische Physik und Astrophysik and Würzburg-Dresden Cluster of Excellence
ct.qmat, Universität Würzburg, 97074 Würzburg, Germany

###### Abstract

At thermal equilibrium, we find that generalized susceptibilities encoding the
static physical response properties of Hermitian many-electron systems possess
inherent non-Hermitian (NH) matrix symmetries. This leads to the generic
occurrence of exceptional points (EPs), i.e., NH spectral degeneracies, in the
generalized susceptibilities of prototypical Fermi-Hubbard models, as a
function of a single parameter such as chemical potential. We demonstrate that
these EPs are necessary to promote correlation-induced thermodynamic
instabilities, such as phase-separation occurring in the proximity of a Mott
transition, to a topologically stable phenomenon.

Introduction. Topology has been introduced in physics to understand
robustness. A pioneering achievement of this approach is the understanding of
the quantum Hall effect [1], where the striking quantization of a transverse
conductance has been explained in terms of topological properties of Bloch
bands [2, 3, 4]. Since then, topology has conquered a wide range of physical
settings far beyond the band theory of solids [5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15]. As a novel direction within this paradigm, here we explain the
robustness of phase instabilities in Fermi-Hubbard models by revealing and
studying unique topological properties of their _generalized susceptibilities_
[16, 17]. These quantities, which encode information on the electronic
fluctuations of a many-body system, can be linked to basic observables such as
the uniform charge response (isothermal compressibility) $\chi_{\bm{q}=0}$
[18, 16].

Importantly, even for closed many-body systems described by a Hermitian
Hamiltonian, such generalized susceptibilities naturally acquire a complex
spectrum as matrices in Matsubara frequency space, and are thus subject to a
non-Hermitian (NH) topological classification approach [15, 19]. There, we
identify the emergence of inherent NH symmetries [20, 21], requiring merely
the physical assumptions of thermal equilibrium and Fermi statistics of the
constituents. Under these ubiquitous circumstances, pairs of exceptional
points (EPs), i.e., NH spectral degeneracies at which the generalized charge
susceptibility matrix $\chi^{\nu\nu^{\prime}}_{c}$ becomes non-diagonalizable
[22, 23, 24, 25, 26, 27, 28, 29], generically occur as a function of a single
tuning parameter such as chemical potential (or filling fraction). These EPs
are topologically protected by the aforementioned inherent NH symmetry and
their splitting in parameter space (see Fig. 1(b) for an illustration).

Figure 1:  Schematic illustration of the enhancement of the uniform charge
response $\chi_{\bm{q}=\bm{0}}$ (a) triggered by the eigenvalue
$\lambda_{\text{I}}$ reaching the condition $\lambda_{\text{I}}=-1/t^{2}$ (b).
The chemical potential $\mu$ at which the divergence happens is located within
a finite range, bounded by the EPs (red dots on (b)). If a divergence is
found, no small perturbation can wash it out, as the intersection between the
“rectified” $\lambda_{I}$ curve (thin gray line) and the $-1/t^{2}$ limit can
only be continuously shifted.

As a consequence, purely real eigenvalues $\lambda_{I}$ of
$\chi^{\nu\nu^{\prime}}_{c}$ occur in an extended parameter range in between
the EPs and generically trigger divergences in the uniform charge response
$\chi_{\bm{q}=0}$ [30] that are robust against small parameter changes (cf.
Fig. 1(a) for an illustration). These divergences in $\chi_{\bm{q}=0}$ signal
the propensity of the correlated system to undergo a thermodynamic phase
separation between a compressible metallic and an almost incompressible “bad
metal” phase, often occurring in the proximity of Mott metal-to-insulator
transitions [31, 32, 33, 34, 30]. This phase separation can be viewed, in many
respects, as the electronic counterpart of the liquid gas transition for water
molecules.

Non-Hermitian topology and EPs have been widely discussed in systems where the
Bloch band structure has been augmented by dissipative terms of various
physical origin, ranging from scattering rates of quasi-particles [35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45] to gain and loss in optical systems [46, 47,
48, 49, 50, 51, 52]. There, the experimental visibility of NH signatures is
oftentimes limited by the overall blurring introduced by imaginary damping
terms in the dissipative time-evolution. By contrast, since the generalized
susceptibilities at the heart of our present analysis do not relate to
effective complex energy spectra, their NH topology has a direct impact on
natural observables, independent of idealistic assumptions on temperature and
without the need for complex multi-orbital models, respectively.

Figure 2:  Panels (a) and (b), diagrammatic representation of the generalized
susceptibility, as given in Eq. 1, using
$\langle\mathcal{T}c^{\dagger}_{\nu}c^{{\phantom{\dagger}}}_{\nu^{\prime}}\rangle=\nu\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynhand \vertex(a) at (-0.5,0);
\vertex(b) at (0.5,0); \propag[fermion] (a) to (b);
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\nu^{\prime}=\delta^{\nu\nu^{\prime}}G(\mathrm{i}\nu)$
and
$\langle\mathcal{T}c^{{\phantom{\dagger}}}_{\nu}c^{\dagger}_{\nu^{\prime}}\rangle=\nu\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynhand \vertex(a) at (-0.5,
0.); \vertex(b) at (0.5,0.); \propag[antfer] (a) to (b);
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\nu^{\prime}=-\delta^{\nu\nu^{\prime}}G(\mathrm{i}\nu)$,
where $G(\mathrm{i}\nu)$ refers to the one-particle Green’s function (here, we
dropped the $\alpha_{i}$ indices for clarity). Panel (c), schematic
illustration of the centroHermitian matrix symmetry.

NH symmetries of generalized susceptibilities. The central objects of our
analysis are four-point functions describing the propagation of a particle-
hole pair (see Fig. 2(b)) in the setting of a time-independent Hamiltonian $H$
at thermal equilibrium. These are expressed as matrices of two fermionic
Matsubara frequencies $\nu$ and $\nu^{\prime}$, where
$\nu^{(\prime)}=(2n^{(\prime)}+1)\pi/\beta$, $n^{(\prime)}\in\mathbb{Z}$, and
$\beta=1/T$ is the inverse temperature. In the literature, such quantities are
referred to as _generalized_ susceptibilities, since they yield static
physical response functions when summed over both fermionic frequencies [53].
Specifically, we define them as

$\displaystyle\chi^{\nu\nu^{\prime}}_{{ph},\alpha_{1}\dots\alpha_{4}}={}$
$\displaystyle\overbrace{\big{<}\mathcal{T}c^{\dagger}_{\nu\alpha_{1}}c^{{\phantom{\dagger}}}_{\nu\alpha_{2}}c^{\dagger}_{\nu^{\prime}\alpha_{3}}c^{{\phantom{\dagger}}}_{\nu^{\prime}\alpha_{4}}\big{>}}^{G^{(2)\nu\nu^{\prime}}_{\alpha_{1}\dots\alpha_{4}}}$
(1)
$\displaystyle-\big{<}\mathcal{T}c^{\dagger}_{\nu\alpha_{1}}c^{{\phantom{\dagger}}}_{\nu\alpha_{2}}\big{>}\big{<}\mathcal{T}c^{\dagger}_{\nu^{\prime}\alpha_{3}}c^{{\phantom{\dagger}}}_{\nu^{\prime}\alpha_{4}}\big{>}$

(illustrated in Fig. 2(a)), where $\mathcal{T}$ denotes the imaginary time
ordering operator,
$\big{<}\dots\big{>}=1/Z\operatorname{Tr}(\mathrm{e}^{-\beta H}\dots)$ the
thermal expectation value,
$c^{(\dagger)}_{\nu\alpha_{i}}=\frac{1}{\sqrt{\beta}}\int^{\beta}_{0}\differential\tau\mathrm{e}^{(-)\mathrm{i}\nu\tau}\mathrm{e}^{H\tau}c^{(\dagger)}_{\alpha_{i}}\mathrm{e}^{-H\tau}$
the Fourier transform of the (creation) annihilation operators 111Here we have
shortened the notation of the Fourier transform of the time-ordered operators
to make their properties in the following derivations more transparent, in Eq.
1 the time ordering is acting first on the time arguments of the operators and
then the respective integrals are performed., and
$G^{(2)\nu\nu^{\prime}}_{\alpha_{1}\dots\alpha_{4}}$ the two-particle Green’s
function. $\alpha_{i}$ refers to, in principle, any of the degrees of freedom
of the model (momenta, spin, orbital, etc.). Some properties of
$\chi^{\nu\nu^{\prime}}_{{ph},\alpha_{1}\dots\alpha_{4}}$ have been already
analyzed in Refs. [16, 55, 53, 17, 56]. In this work, we are investigating the
topological properties of the corresponding eigenvalue spectrum.

Taking the complex conjugate of Eq. 1 and considering
$(c^{\dagger}_{\nu})^{*}=-c^{{\phantom{\dagger}}}_{-\nu}$, and
$(c^{{\phantom{\dagger}}}_{\nu})^{*}=-c^{\dagger}_{-\nu}$ _inside_ of
$\big{<}\dots\big{>}$, one obtains
$(\chi^{\nu\nu^{\prime}}_{{ph},\alpha_{1}\dots\alpha_{4}})^{*}=\chi^{-\nu^{\prime}-\nu}_{{ph},\alpha_{4}\dots\alpha_{1}}$
222A detailed derivation can be found in the supplemental material [67].. With
simple further manipulations on the indices leaving
$\big{<}\mathcal{T}\dots\big{>}$ invariant, this leads to
$(\chi^{\nu\nu^{\prime}}_{{ph},\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}})^{*}=\chi^{-\nu-\nu^{\prime}}_{{ph},\alpha_{2}\alpha_{1}\alpha_{4}\alpha_{3}}$.
The latter mathematical object has the form of a matrix with coefficients
$\chi^{\beta\beta^{\prime}}_{{ph}}$, which, crucially, satisfies the relation

$\chi^{\beta\beta^{\prime}}_{{ph}}=\sum_{\beta_{1}\beta_{2}}\Pi^{\beta\beta_{1}}(\chi^{\beta_{1}\beta_{2}}_{{ph}})^{*}\Pi^{\beta_{2}\beta^{\prime}},$
(2)

where $\Pi^{\beta\beta^{\prime}}$ is the permutation matrix
$\beta:=(\nu,\alpha_{1},\alpha_{2})\to\beta^{{}^{\prime}}:=(-\nu,\alpha_{2},\alpha_{1})$.
This property has important consequences for the eigenvalues $\lambda$ of
$\chi_{{ph}}$, which belongs to the class of $\kappa$-real matrices
$\bm{K}_{r}=\bm{\Pi}\bm{K}_{r}^{*}\bm{\Pi}$ [58], where $\bm{\Pi}$ refers to
any permutation matrix. These have been shown [58] to have a characteristic
polynomial with real coefficients and, hence, either real or complex conjugate
eigenvalues due to the fundamental theorem of algebra. A relevant subclass of
$\kappa$-real matrices are _centroHermitian_ matrices [59, 60], which are
invariant under a transformation that combines complex conjugation with
centro-symmetry, as illustrated in Fig. 2(c).

In the following, we consider the _local_ generalized charge susceptibility
$\chi^{\nu\nu^{\prime}}_{c}=\frac{1}{2}\sum_{\sigma\sigma^{\prime}}\chi^{\nu\nu^{\prime}}_{{ph},\sigma\sigma\sigma^{\prime}\sigma^{\prime}}$
of a one-orbital model that satisfies the following relation:

$(\chi^{\nu\nu^{\prime}}_{c})^{*}=\frac{1}{2}\sum_{\sigma\sigma^{\prime}}\chi^{-\nu-\nu^{\prime}}_{{ph},\sigma\sigma\sigma^{\prime}\sigma^{\prime}}=\chi^{-\nu-\nu^{\prime}}_{c}$
(3)

and is therefore a _centroHermitian_ matrix. In addition, if the Hamiltonian
possesses specific symmetries, these can impose even stricter matrix
properties. For instance, for particle-hole symmetry (PHS)
$\chi^{\nu\nu^{\prime}}_{c}$ becomes real and has only real eigenvalues [17,
56].

Minimal model for exceptional susceptibilities. To illustrate our general
findings, we consider a $2\times 2$ matrix $\chi_{ph}^{2\times 2}$ obeying the
_centroHermitian_ condition:

$\chi_{ph}^{2\times 2}=\left(\begin{aligned} a+\mathrm{i}b&&c-\mathrm{i}d\\\
c+\mathrm{i}d&&a-\mathrm{i}b\end{aligned}\right)=a\cdot\mathbb{I}+\vec{v}\cdot\vec{\sigma}$
(4)

where $a,b,c,d\in\mathbb{R}$ and
$\vec{v}=\vec{v}_{R}+i\vec{v}_{I}=(c,d,0)+i(0,0,b)$ is a complex vector. The
$a$ parameter can be safely disregarded, as it only amounts to a rigid
eigenvalue shift. EPs are globally stable for a two- or higher-dimensional
parameter space, because for the matrix to become non-diagonalizable, two
conditions ($v_{R}^{2}-v_{I}^{2}=0$, $\vec{v}_{R}\cdot\vec{v}_{I}=0$) have to
be simultaneously satisfied. It is immediate to see that the centroHermitian
property implies that the second is always fulfilled. It is then sufficient,
for the exceptional points to manifest, that $c^{2}+d^{2}-b^{2}=0$, which
implies that even in a one-dimensional space, EPs – if any are present – will
be globally robust against any perturbation representable by a matrix of the
form given in Eq. 4 [61]. Crucially, no other perturbation can arise, because
the centroHermitian condition does not originate from any further symmetry,
but it is an intrinsic consequence of the time-independence of $H$ and a
defined quantum (here: Fermi-Dirac) statistics 333See supplemental material
for a short discussion on the differences between the susceptibility matrix
properties of Fermi-Dirac and Bose-Einstein statistics [67].. The occurrence
of these stable EPs generically mark the borders of regions with complex
conjugate pairs on the one hand and real and distinct eigenvalues on the other
hand. On the contrary, if we additionally impose PHS, $\chi_{ph}^{2\times 2}$
becomes purely real and symmetric ($b,d=0$), which implies that the only
solution of the two conditions for the EPs will be for
$\vec{v}_{R}=\vec{v}_{I}=0$. This is known in literature as a diabolic point,
which is effectively concurrent with a Hermitian degeneracy [15] and, indeed,
generally requires fine-tuning.

Figure 3:  Real part (top row) and imaginary part (bottom row) of the
eigenvalues $\lambda_{i}$ of $\chi^{\nu\nu^{\prime}}_{c}$ for the atomic limit
(AL) at temperature $T=1/100$, $U=0$ (a, b) and at $U=1$ (c, d) as function of
chemical potential away from particle-hole symmetry (PHS) at $\mu=U/2$. The
ten eigenvalues which are lowest in $\operatorname{Re}\lambda_{i}$ at $U=1$
are displayed.

Analytical study of the atomic limit. As the simplest physical platform to
exemplify the spectral properties of $\chi_{c}^{\nu\nu^{\prime}}$, we now
study the exactly solvable atomic limit of the Hubbard model (AL)

$H=-\mu(n_{\uparrow}+n_{\downarrow})+Un_{\uparrow}n_{\downarrow},$ (5)

where $\mu$ is the chemical potential,
$n_{\sigma}=c^{\dagger}_{\sigma}c^{\phantom{\dagger}}_{\sigma}$ the occupation
of an electron with spin $\sigma$, and $U$ the on-site Coulomb repulsion given
in arbitrary units of energy. This model fulfills PHS if $\mu=U/2$ and is in
general $\operatorname{SU}(2)$-symmetric 444For the effect of the
$\operatorname{SU}(2)$-symmetry on the generalized susceptibilities see Refs.
[18, 16].. In the case of zero interaction $U=0$, the local generalized charge
susceptibility reads

$\chi^{\nu\nu^{\prime}}_{c}\stackrel{{\scriptstyle
U=0}}{{=}}-G(\mathrm{i}\nu)G(\mathrm{i}\nu^{\prime})\delta^{\nu\nu^{\prime}}=-\frac{\delta^{\nu\nu^{\prime}}}{(\mathrm{i}\nu+\mu)^{2}},$
(6)

where
$G(\mathrm{i}\nu)=\big{<}\mathcal{T}c^{\dagger}_{\nu\sigma}c^{{\phantom{\dagger}}}_{\nu\sigma}\big{>}$
is the one-particle Green’s function. $\chi^{\nu\nu^{\prime}}_{c}$ is
diagonal, hence the eigenvalues can be immediately read from Eq. 6. These
become doubly degenerate ($\lambda_{\nu}=\lambda_{-\nu}=1/\nu^{2}$) at PHS,
i.e. $\mu=0$, while they form complex conjugate pairs
($\lambda_{\nu}=\lambda_{-\nu}^{*}$) at finite $\mu$. In the left column of
Fig. 3 the real (a) and imaginary part (b) of $\lambda_{i}$ are shown for
different $\mu$ at finite temperature $T=1/100$.

At finite interaction $U>0$, $\chi^{\nu\nu^{\prime}}_{c}$ becomes a more
complicated expression [17, 64, 65, 66], given in the supplemental material
[67]. The crucial point is the appearance of progressively larger off-diagonal
components. The resulting eigenvalues $\lambda_{i}$ are shown in the right
column of Fig. 3. Significantly, at PHS, $\lambda_{i}$ are still purely real
but no longer degenerate. Importantly, this remains true in a finite region of
$\mu$ around $U/2$. Far away from PHS, however, the effect of the interaction
weakens, and all eigenvalues become complex conjugate pairs. To switch between
these two regimes, two eigenvalues have to coalesce: this creates a pair of
distinct EPs in $\mu$-space, which delimit and protect the real-eigenvalue
“lens”-shaped structure (Fig. 3(c)). In the 2$\times$2-picture of Eq. 4, we
can identify the interaction $U$ as responsible for the presence of the off-
diagonal finite elements $c$ and $d$ in the matrix, and the finite $\mu$ for
the diagonal element $b$, which are the two ingredients necessary to satisfy
the EP conditions. Hence, for the AL any finite value of $U$ will result in
exceptional points away from PHS and a finite-size real eigenvalue lens shape.

Figure 4:  Real part (a) and imaginary part (b) of the eigenvalues
$\lambda_{i}$ of $\chi^{\nu\nu^{\prime}}_{c}$ for the square lattice Hubbard
model (with half bandwidth $D=4t=1$) solved within DMFT as function of
chemical potential away from PHS at $\mu=U/2$. Interaction strength $U=2.4$
and temperature $T=1/53$ coincide with Ref. [30] to show the situation close
to the thermodynamic instability. Calculated data are displayed as dots, the
positive $(\mu-U/2)$-axis is mapped from the negative one, exploiting the
symmetry of the model considered. The ten eigenvalues which are lowest in
$\operatorname{Re}\lambda_{i}$ at $\mu-U/2=0$ are displayed.

Implications on correlation-induced instabilities. We now turn to a more
generic scenario, namely the single-orbital Hubbard Hamiltonian on a lattice:

$\displaystyle H=$ $\displaystyle-t\sum_{\langle
ij\rangle,\sigma}(c^{\dagger}_{i\sigma}c^{\phantom{\dagger}}_{j\sigma}+c^{\dagger}_{j\sigma}c^{\phantom{\dagger}}_{i\sigma})-\mu\sum_{i,\sigma}n_{i\sigma}$
(7) $\displaystyle+U\sum_{i}n_{i\uparrow}n_{i\downarrow}.$

with constant hopping $t$ between neighboring sites $i$ and $j$. This model is
again $\operatorname{SU}(2)$-symmetric and for $\mu=U/2$ it fulfills PHS.
Except for one or infinite spatial dimensions, the model has not been solved
analytically. In order to get a non-perturbative, albeit approximated many-
body solution, we use dynamical mean-field theory (DMFT) (which becomes exact
only in the limit of infinite dimensions) [68, 69] with a continuous-time
quantum Monte Carlo solver from w2dynamics [70]. As shown in Ref. [30], the
eigenvalues $\lambda_{i}$ and the corresponding eigenvectors $v_{i}^{\nu}$ of
the local generalized susceptibility $\chi^{\nu\nu^{\prime}}_{c}$
($\sum_{\nu^{\prime}}\chi^{\nu\nu^{\prime}}_{c}v_{i}^{\nu^{\prime}}=\lambda_{i}v_{i}^{\nu}$)
play an important role for the response functions of the whole lattice: they
lead to an enhancement and in some cases to a divergence of the _uniform_
(i.e. for zero transfer momentum $\mathbf{q}=0$) susceptibility. In
particular, for the Bethe lattice with infinite connectivity (where DMFT is
exact) the static uniform charge response, obtained by summing the generalized
susceptibility over all Matsubara frequencies
$\chi_{\mathbf{q}=0}=\sum_{\nu\nu^{\prime}}\chi^{\nu\nu^{\prime}}_{\mathbf{q}=0}$,
can be re-expressed in terms of $\lambda_{i}$ and corresponding weights
$w_{i}=(\sum_{\nu}(v_{i}^{-1})^{\nu})(\sum_{\nu^{\prime}}v_{i}^{\nu^{\prime}})$.
This leads to the following expression

$\chi_{\mathbf{q}=0}=\frac{1}{\beta}\sum_{i}\left(\frac{1}{\lambda_{i}}+t^{2}\right)^{-1}w_{i},$
(8)

which diverges – thus inducing a phase instability in the charge sector – when
one eigenvalue fulfills the condition $\lambda_{i}=-1/t^{2}$. Close to this
condition, $\lambda_{i}$ gives the dominating contribution to the charge
response and determines the stability of the physical solution [71].
Importantly, this is possible only when $\lambda_{i}$ is _real_ 555Further,
the corresponding weight must be $w_{i}\neq 0$, which is in general fulfilled
away from PHS for $\mu\neq U/2$ [30].

Although Eq. 8 is only exact in the case of the Bethe lattice, numerical
calculations have shown [30] that it holds also for a square lattice if $t$ is
replaced by a temperature- and $\mu$-dependent $t_{\mathrm{eff}}(\mu,T)$.
Here, the central role of EPs becomes apparent: their presence guarantees that
the imaginary part of $\lambda_{i}$ remains zero in the whole extended region
of the lens shape. In other words, the possibility of inducing a divergence in
$\chi_{\mathbf{q}=0}$ is not accidental and does not rely on a fine-tuning of
$U$, $T$ and $\mu$: the phase instability is in fact topologically protected.
For the square lattice, this is illustrated in Fig. 4, where we plot the real
(a) and imaginary part (b) of the eigenvalues $\lambda_{i}$ of the local
charge susceptibility $\chi^{\nu\nu^{\prime}}_{c}$ close to the critical point
of the phase separation (cf. sketch in Fig. 1). Here, the lowest eigenvalue
$\lambda_{I}$ satisfies (up to numerical accuracy) the condition
$\lambda_{I}=-1/t_{\mathrm{eff}}^{2}$ in the region of the lens shape. Hence,
the phase instability condition is also fulfilled for any further reduction of
the temperature $T$ (or for any moderate reduction of the interaction $U$). In
particular, at lower $T$, we enter a regime, where a first order phase
separation occurs. This regime is, thus, characterized by two locally stable
DMFT solutions (i.e., two coexisting values of $\lambda_{I}$), corresponding
to a less correlated metallic and a “bad metal” phase (connected by an
instable solution, where $\lambda_{I}<-1/t_{\mathrm{eff}}^{2}$ [71]). Here,
the topological robust arguments related to the condition
$\lambda_{I}=-1/t_{\mathrm{eff}}^{2}$ remain nonetheless applicable, albeit to
the two corresponding metastable solutions 666A schematic illustration of this
regime can be found in the supplemental material [67]..

Finally, let us notice that a negative eigenvalue is a necessary condition for
the instability criterion to be fulfilled [30]. Remarkably, the role of the
negative eigenvalues in the generalized charge susceptibility has been
recently related to the local moment formation [74, 75, 76, 77] and, on a more
formal level, to divergences of the irreducible vertex function and the
multivaluedness of the Luttinger-Ward functional [78, 79, 80, 81, 82, 83, 84,
85, 86, 87, 88, 89, 90, 17, 34, 91, 56, 92, 93, 30, 74, 94, 75, 95, 76, 96,
77, 97, 98]. Therefore, these negative eigenvalues can be regarded as a
feature of strong electronic correlations, which cannot be commonly described
by “perturbative” theories, e.g., the random phase approximation. However, the
considerations behind Eq. 8 are not solely restricted to negative eigenvalues.
They can be also applied to the opposite case, where a positive eigenvalue
reaching a maximum (e.g., $\lambda_{i}=1/t_{\mathrm{eff}}^{2}$) triggers a
phase instability, such as the antiferromagnetic transitions of the Hubbard
model [99]. Thus, in these strongly correlated systems, the presence of EPs is
found to generally promote phase instabilities in the ${ph}$-channel 777In
contrast, for the situation of the generalized susceptibility in the
${pp}$-channel see supplemental material [67]. to a stable phenomenon, and
thereby enables the instabilities to naturally occur for a finite range in
parameter space.

Conclusion. We have found the opening of an EP phase for the associated
eigenvalues of the static local susceptibility in the $U/\mu$ phase diagram of
models for correlated electron systems. The remarkable consequence is that the
interaction-induced charge instabilities such as the phase separation
occurring close to the Mott metal-to-insulator transition in the Hubbard model
do not need any fine-tuning but can occur in an entire finite range of
parameters. This unexpected global robustness is a consequence of the peculiar
centroHermitian form of the susceptibility matrix, which is not dictated by
some ad-hoc antiunitary symmetry but by the time-independence of $H$ and the
intrinsic nature of Fermi-Dirac statistics.

The susceptibility EPs represent a clear-cut and compelling manifestation of
non-Hermitian topology, surpassing the conventional realizations based on
spectral functions. This phenomenon is indeed ubiquitous even in the simplest
correlated fermion models and does not require any assumption on the
interaction nor any specific choice of non-Hermitian Hamiltonian terms. Our
results call for future investigations beyond the local correlation effects on
the charge sector considered here: e.g., of the spin or particle-particle
channel and including non-local correlations in the description. Further, one
could also search for higher order exceptional degeneracies in the
susceptibility spectrum and explore the respective consequences on the phase
instabilities. This may open new doors to experimentally detectable hallmarks
of non-Hermitian topology.

###### Acknowledgements.

Acknowledgments. We thank P. Chalupa-Gantner, H. Eßl, P. Oberleitner for
insightful discussions, and S. Di Cataldo, P. Kappl, P. Worm for helpful
comments. M.R. acknowledges support as a recipient of a DOC fellowship of the
Austrian Academy of Sciences and financial support from the Austrian Science
Fund (FWF), within the project I 5487. A.T. acknowledges the Austrian Science
Fund (FWF) for the project I 5868 (part of the FOR 5249 [QUAST] of the German
Science Foundation, DFG). L.C., J.C.B. and G.S. acknowledge financial support
by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy–EXC2147 “ct.qmat” (project‐id 390858490) as well
as through Project-ID 258499086 – SFB 1170 “ToCoTronics” and Project-ID
247310070 - SFB 1143. Calculations have been performed on the Vienna
Scientific Cluster (VSC).

###### Acknowledgements.

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