# On Green’s function embedding using sum-over-pole representations

Andrea Ferretti<EMAIL_ADDRESS>Centro S3, CNR–Istituto
Nanoscienze, 41125 Modena, Italy  Tommaso Chiarotti  Theory and Simulations of
Materials (THEOS) and National Centre for Computational Design and Discovery
of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, 1015
Lausanne, Switzerland  Nicola Marzari  Theory and Simulations of Materials
(THEOS) and National Centre for Computational Design and Discovery of Novel
Materials (MARVEL), École Polytechnique Fédérale de Lausanne, 1015 Lausanne,
Switzerland

###### Abstract

In Green’s function theory, the total energy of an interacting many-electron
system can be expressed in a variational form using the Klein or Luttinger-
Ward functionals. Green’s function theory also naturally addresses the case
where the interacting system is embedded into a bath. This latter can then act
as a dynamical (i.e., frequency-dependent) potential, providing a more general
framework than that of conventional static external potentials. Notably, the
Klein functional includes a term of the form
$\text{Tr}_{\omega}\text{Ln}\left\\{G_{0}^{-1}G\right\\}$, where
$\text{Tr}_{\omega}$ is the frequency integration of the trace operator. Here,
we show that using a sum-over-pole representation for the Green’s functions
and the algorithmic-inversion method one can obtain in full generality an
explicit analytical expression for
$\text{Tr}_{\omega}\text{Ln}\left\\{G_{0}^{-1}G\right\\}$. This allows one,
e.g., to derive a variational expression for the Klein functional in the
presence of an embedding bath, or to provide an explicit expression of the RPA
correlation energy in the framework of the optimized effective potential.

##  I Introduction

Electronic-structure simulations based on density-functional theory (DFT) Kohn
(1999) are today widely exploited Pribram-Jones  _et al._ (2015) in condensed-
matter physics, quantum chemistry, or materials modelling Marzari  _et al._
(2021). Even if DFT can in principle be used to access any observable of an
interacting system as a functional of the density Hohenberg and Kohn (1964);
Martin  _et al._ (2016); Marzari  _et al._ (2021), currently available
functionals and approximations are mostly limited to the ground-state total
energy (and, in turn, to its derivatives wrt external parameters) and to
observables connected to the charge density. Instead, electronic excitations
are typically addressed by extensions of the basic theory, such as time-
dependent DFT Runge and Gross (1984); Petersilka  _et al._ (1996); Ullrich
(2012) or ensemble DFT Gross  _et al._ (1988); Gould and Pittalis (2019);
Cernatic  _et al._ (2021). Notably, all these approaches are equipped with a
variational principle which allows one to determine the basic quantity of the
theory (e.g. the density in DFT or its time-dependent version in TD-DFT) for
the systems studied.

Conversely, Green’s function (GF) methods Stefanucci and van Leeuwen (2013);
Martin  _et al._ (2016) such as the GW approximation and its combination with
the Bethe-Salpeter equation (BSE) Hedin (1965); Reining (2018); Golze  _et
al._ (2019); Onida  _et al._ (2002), are commonly used to address charged and
neutral excitations. Nevertheless, the one-particle GF can also be used to
access the ground-state total energy Fetter and Walecka (1971); Martin  _et
al._ (2016) (e.g., via the Galitskii-Migdal expression). Variationality of the
total energy wrt the one-particle Green’s function can be recovered by using
the Luttinger-Ward or Klein (LWK) functionals Luttinger and Ward (1960); Klein
(1961); Baym and Kadanoff (1961); Almbladh  _et al._ (1999), which become
stationary when evaluated at the interacting Green’s function of the system.
Examples include applications to atoms and molecules Dahlen  _et al._ (2004);
Dahlen and von Barth (2004a, b); Dahlen  _et al._ (2006), to Hubbard chains
Puig von Friesen  _et al._ (2010); Di Sabatino  _et al._ (2021), or to the
homogeneous electron gas Holm and Aryasetiawan (2000); García-González and
Godby (2001); Chiarotti  _et al._ (2022). When the Klein functional is
combined with an optimized effective potential approach Casida (1995); Kümmel
and Kronik (2008) one obtains the linearized Sham-Schlüter equation Sham and
Schlüter (1983); Godby  _et al._ (1987), which can be used to derive advanced
KS-DFT functionals from diagrammatic approximations, such as the EXX+RPA
exchange-correlation functional Almbladh  _et al._ (1999); Ismail-Beigi
(2010); Ren  _et al._ (2012); Paier  _et al._ (2012); Hellgren  _et al._
(2018); Fetter and Walecka (1971); Stefanucci and van Leeuwen (2013); Martin
_et al._ (2016). Notably, the Klein functional features a term of the form
$\int\frac{d\omega}{2\pi}\text{Tr}\text{Ln}\\{G_{0}^{-1}G\\}$ (see in Sec. II
for more details), which is quite cumbersome to be evaluated numerically and
needs dedicated treatment Dahlen  _et al._ (2006); Puig von Friesen  _et al._
(2010). The LW functional displays similar issues. In passing we also note
that besides DFT-based and GF methods, other orbital-dependent or dynamical
approaches Marzari  _et al._ (2021) addressing excitations are available,
including dynamical mean field theory (DMFT) Kotliar  _et al._ (2006),
spectral potentials Gatti  _et al._ (2007); Ferretti  _et al._ (2014), or
Koopmans-compliant functionals Dabo  _et al._ (2009, 2010); Ferretti  _et al._
(2014); Nguyen  _et al._ (2018).

Importantly, dynamical potentials can be naturally employed to describe
embedding situations, where the system of interest is placed in contact with
an external bath. In these cases, for non-interacting systems, the embedded GF
can be calculated by adding an embedding self-energy Stefanucci and van
Leeuwen (2013); Martin  _et al._ (2016), which has the form of a non-local and
dynamical potential, to the pristine Hamiltonian. This approach has been
successfully exploited, e.g., in the description of semi-infinite systems
(surface Green’s function) and applied to simulations of quantum transport
through nanojunctions Meir and Wingreen (1992); Haug and Jauho (1996);
Buongiorno Nardelli (1999); Brandbyge  _et al._ (2002); Calzolari  _et al._
(2004). When particle interactions are considered, the situation becomes more
complex, but the assumption of dealing with a non-interacting bath Meir and
Wingreen (1992) allows one to treat the problem similarly to the non-
interacting case. Approaches such as DMFT Kotliar  _et al._ (2006), which is a
dynamical method targeting both total energies and spectral properties,
exploit the embedding of an interacting impurity model to describe the
electron-electron self-energy of strongly interacting systems.

In general, the use of dynamical potentials (e.g., originating from many-body
perturbation theory Stefanucci and van Leeuwen (2013); Martin  _et al._
(2016), embedding, or spectral potentials Gatti  _et al._ (2007); Ferretti
_et al._ (2014); Marzari  _et al._ (2021)) in electronic-structure methods is
a challenge by itself. Indeed, the frequency representation of propagators (or
dynamical potentials) is non trivial Chiarotti  _et al._ (2022, 2023);
Chiarotti (2023) with viable approaches ranging from discretized frequency
grids (both on the real or imaginary axis) to the use of meromorphic functions
and Padé approximants Engel  _et al._ (1991); Rojas  _et al._ (1995), or
imaginary-time treatments Rojas  _et al._ (1995). Moreover, the solution of
the resulting Dyson equation (which can be cast in the form of a non-linear
eigenvalue problem Güttel and Tisseur (2017)) adds further numerical and
conceptual complications (including multiple solutions and non-orthonormality
of the eigenvectors Golze  _et al._ (2019); Martin  _et al._ (2016); Güttel
and Tisseur (2017)). In order to address this problem, we have recently
exploited the combination of a sum-over-poles (SOP) representation for the
propagators, with the algorithmic-inversion method (AIM) Chiarotti  _et al._
(2022, 2023); Chiarotti (2023) to exactly solve the Dyson equation resulting
from dynamical potentials.

In this work, by taking advantage of the SOP-AIM approach Chiarotti  _et al._
(2022, 2023); Chiarotti (2023), we first derive an analytical expression for
terms of the form $\text{Tr}_{\omega}\left\\{G_{0}^{-1}G\right\\}$, as those
appearing in the Klein functional, that is valid in the general case of
interacting propagators. Next, we exploit this result to ($i$) recover an
exact expression Ismail-Beigi (2010) for the RPA correlation energy Almbladh
_et al._ (1999); Paier  _et al._ (2012); Hellgren  _et al._ (2018), and to
($ii$) obtain a Klein functional valid in the case of embedding where the
system of interest is coupled to a non-interacting bath.

The paper is organized as follows. In Sec. II we present the theoretical
framework used throughout the work. Next, in Sec. III we derive an analytical
expression for $\text{Tr}_{\omega}\left\\{G_{0}^{-1}G\right\\}$. Finally, in
Sec. IV we apply the newly derived result first to evaluate the RPA
correlation energy, and then to the embedding of the Klein functional.
Complementary details about Green’s function embedding and TrLn terms are
provided in Appendix A and Appendix B, respectively.

##  II Theoretical framework

In this Section we present the theoretical framework underpinning the use of
Green’s function methods to describe an interacting system in the presence of
a non-interacting bath; additional details are provided in Appendix A. We
consider a closed quantum system $C$ that is partitioned into two subsystems,
$S$ and $B$, such that, in terms of degrees of freedom, one has $C=S\cup B$.
Particle interactions are present but limited to subsystem $S$ only, leaving
subsystem $B$ as a non-interacting bath. All single particle operators,
including Hamiltonians, self-energies, and Green’s function, become 2$\times$2
block matrices, indexed according to the $S$ and $B$ subsystems. As detailed
in Fig. 1, $h_{0}$ represents the non-interacting Hamiltonian of the two
systems without coupling, while $H_{0}$ is the non-interacting Hamiltonian of
$C$ when the coupling $V$ is included. Eventually, self-energy terms
accounting for the particle-particle interaction are included. As discussed in
App. A, since interactions are only present within $S$, one can show that the
corresponding self-energy is limited to the same subsystem. Moreover, since
$h_{0B}$ is non-interacting, without loss of generality we may take it
diagonal on the chosen basis, such that
$h_{0B}=\text{diag}(\Omega_{1},\dots,\Omega_{n},\dots)$.

$\begin{array}[]{ccc}h_{0},g_{0}&H_{0},G_{0}&H,G\\\\[7.0pt]
\left[\begin{array}[]{cc}h_{0S}&0\\\
0&h_{0B}\end{array}\right]&\left[\begin{array}[]{cc}h_{0S}&V\\\
V^{\dagger}&h_{0B}\end{array}\right]&\left[\begin{array}[]{cc}h_{0S}+\Sigma&V\\\
V^{\dagger}&h_{0B}\end{array}\right]\end{array}$

Figure 1: Upper panel: Partitioning of the closed system $C$ into the subparts
$S$ (interacting, as indicated by the wiggly line) and $B$ (non-interacting).
The Hamiltonian and self-energy blocks and the coupling $V$ of the two
subsystems are also indicated. Bottom panel: Sketch view of the three
different Hamiltonians and Green’s functions involved in the discussion of
embedding. Left: $S$ and $B$ are non-interacting and uncoupled; Central: $S$
and $B$ are non-interacting but coupled; Right: $S$ is interacting and coupled
to the non-interacting $B$.

Within the above definitions, and following Fig. 1, one can define the Green’s
functions for the whole system $C$, at different levels of description (non-
interacting and uncoupled, non-interacting and coupled, interacting in $S$ and
coupled), according to:

$\displaystyle g_{0}(\omega)$ $\displaystyle=$ $\displaystyle\left[\omega
I-\text{diag}(h_{0S},h_{0B})\right]^{-1}=\left[\omega I-h_{0}\right]^{-1}$
$\displaystyle G_{0}(\omega)$ $\displaystyle=$ $\displaystyle\left[\omega
I-H_{0}\right]^{-1}$ $\displaystyle G(\omega)$ $\displaystyle=$
$\displaystyle\left[\omega I-H_{0}-\Sigma(\omega)\right]^{-1}$ (1)

(time-ordered offsets from the real axis are left implicit). We note that when
$G$ is the physical GF, then $\Sigma=\Sigma_{\text{Hxc}}$ is the interaction
self-energy (accounting for Hartree, exchange, and correlation terms).
Nevertheless, in the following we will also consider cases where $G$ is a
trial GF, as discussed, e.g., in Sec. III.2. In these cases,
$\Sigma=\widetilde{\Sigma}$ just collects a set of degrees of freedom useful
to represent $G$ via

$G=G_{0}+G_{0}\widetilde{\Sigma}G.$ (2)

Within this construction, the self-energy will also be constrained to have
non-zero matrix elements only within subsystem $S$, which can be seen as a
domain definition for the set of trial $G$’s.

By focusing on the subsystem $S$ and making reference to the theory of Green’s
function embedding, the $S$ blocks of the above GFs are obtained as:

$\displaystyle g_{0S}(\omega)$ $\displaystyle=$ $\displaystyle\left[\omega
I_{S}-h_{0S}\right]^{-1},$ $\displaystyle G_{0S}(\omega)$ $\displaystyle=$
$\displaystyle\left[\omega I_{S}-h_{0S}-\Delta v_{S}(\omega)\right]^{-1},$
$\displaystyle G_{S}(\omega)$ $\displaystyle=$ $\displaystyle\left[\omega
I_{S}-h_{0S}-\Delta v_{S}(\omega)-\Sigma(\omega)\right]^{-1},$ (3)

where $\Delta v_{S}$ is an embedding self-energy due to the bath $B$ Haug and
Jauho (1996); Stefanucci and van Leeuwen (2013); Martin  _et al._ (2016);
Buongiorno Nardelli (1999); Calzolari  _et al._ (2004):

$\Delta
v_{S}(\omega)=Vg_{0B}(\omega)V^{\dagger}=\sum_{n}\frac{R_{n}}{\omega-\Omega_{n}\pm
i0^{+}},$ (4)

which acts as a correction to the external potential of $S$.

The total energy of the closed system $C$ can be obtained variationally, e.g.,
via the Klein functional Klein (1961); Almbladh  _et al._ (1999), reading

$\displaystyle E^{K}[G]$ $\displaystyle=$
$\displaystyle\text{Tr}_{\omega}\text{Ln}\left\\{G_{0}^{-1}G\right\\}+\text{Tr}_{\omega}H_{0}G_{0}$
$\displaystyle+$
$\displaystyle\text{Tr}_{\omega}\left[I-G_{0}^{-1}G\right]+\Phi_{\text{Hxc}}[G],$

where $\Phi_{\text{Hxc}}[G]$ is a functional Luttinger and Ward (1960); Klein
(1961); Baym and Kadanoff (1961) to be approximated that is related to the
interaction self-energy as

$\frac{\delta\Phi_{\text{Hxc}}[G]}{\delta G}=\frac{1}{2\pi
i}\Sigma_{\text{Hxc}}[G].$ (6)

With the above definitions, one can show Stefanucci and van Leeuwen (2013);
Martin  _et al._ (2016) that the gradient of the Klein functional is zero for
the GF $G$ that satisfies the self-consistent Dyson equation

$G=G_{0}+G_{0}\Sigma_{\text{Hxc}}[G]G.$ (7)

###  II.1 Sum-over-poles and algorithmic inversion

In order to make progress in the numerical exploitation of the above described
techniques, in the following we make use of the concept of sum-over-poles
(SOP) Chiarotti  _et al._ (2022, 2023); Engel  _et al._ (1991); Puig von
Friesen  _et al._ (2010); Di Sabatino  _et al._ (2021) to represent
propagators, combined with that of the algorithmic-inversion method (AIM) to
solve Dyson-like equations. In practice, this amounts to writing propagators
and self-energies using discrete poles and residues (meromorphic
representation Puig von Friesen  _et al._ (2010)) as

$\displaystyle G_{0}(\omega)$ $\displaystyle=$
$\displaystyle\sum_{n}\frac{A^{0}_{n}}{\omega-\epsilon^{0}_{n}\pm i0^{+}},$
(8) $\displaystyle G(\omega)$ $\displaystyle=$
$\displaystyle\sum_{s}\frac{A_{s}}{\omega-\epsilon_{s}\pm i0^{+}},$ (9)
$\displaystyle\Sigma(\omega)$ $\displaystyle=$
$\displaystyle\Sigma_{0}+\sum_{n}\frac{\Gamma_{n}}{\omega-\omega_{n}\pm
i0^{+}},$ (10)

which could be seen also as discrete Lehmann representations Engel  _et al._
(1991). Recently, SOPs have also been used to represent the screened Coulomb
interaction in the context of GW leading to the multi-pole approximation (MPA)
Leon  _et al._ (2021, 2023). For simplicity, in this work we assume all
residues and poles to be Hermitian and real, respectively. In the above
expressions, $G_{0}$ is a non-interacting Green’s function (GF) obtained from
the single-particle Hamiltonian $h_{0}$,

$\displaystyle
h_{0}|\phi^{0}_{n}\rangle=\epsilon^{0}_{n}|\phi^{0}_{n}\rangle,\qquad
A^{0}_{n}=|\phi^{0}_{n}\rangle\langle\phi^{0}_{n}|,$ (11)

while $G$ is an interacting or embedded GF, obtained from $G_{0}$ by a Dyson
equation involving $\Sigma$, i.e. $G=[\omega I-h_{0}-\Sigma(\omega)]^{-1}$.

Having assumed discrete and real poles poles for $\Sigma$ and $G_{0}$ (and
Hermitian residues) implies Chiarotti  _et al._ (2023); Güttel and Tisseur
(2017) that also $G$ has real discrete poles and that the residues can be
written as

$\displaystyle\big{[}h_{0}+\Sigma(\epsilon_{s})\big{]}|f_{s}\rangle=\epsilon_{s}|f_{s}\rangle,\qquad
A_{s}=|f_{s}\rangle\langle f_{s}|,$ (12)

where the normalization of $|f_{s}\rangle$ is defined according to

$\displaystyle\langle f_{s}|f_{s}\rangle$ $\displaystyle=$ $\displaystyle
Z_{s}=1+\langle f_{s}|\dot{\Sigma}(\epsilon_{s})|f_{s}\rangle\leq 1,$ (13)
$\displaystyle\sum_{s}|f_{s}\rangle\langle f_{s}|$ $\displaystyle=$
$\displaystyle I,$ (14)

i.e., the $|f_{s}\rangle$ are complete though not linearly independent nor
orthonormalized (see also Ref. Chiarotti (2023)), where we have used
$\dot{\Sigma}(\omega)=\partial\Sigma(\omega)/\partial\omega$. In writing the
expressions above the Dyson equation has been mapped to a non-linear
eigenvalue problem involving rational functions Chiarotti  _et al._ (2023);
Güttel and Tisseur (2017). Moreover, noting that the residues of $G$ in Eq.
(9) are positive semi-definite (PSD) by construction, the residues
$\Gamma_{n}$ of $\Sigma$ are also forced to be PSD Hermitian operators. In
fact, given

$\displaystyle A(\omega)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\left[G(\omega)-G^{\dagger}(\omega)\right]\text{sign}(\mu-\omega),$
$\displaystyle\Gamma(\omega)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\left[\Sigma(\omega)-\Sigma^{\dagger}(\omega)\right]\text{sign}(\mu-\omega),$
(15) $\displaystyle A(\omega)$ $\displaystyle=$ $\displaystyle
G(\omega)\Gamma(\omega)G^{\dagger}(\omega),$ (16)

(the last identity coming from the Dyson equation), the positive semi-
definiteness of $A$ is equivalent Stefanucci and van Leeuwen (2013);
Stefanucci  _et al._ (2014) (i.e., if and only if) to that of $\Gamma$.

Next, given $G_{0}$ and $\Sigma$ represented as SOPs, it is possible to
explicitly evaluate the coefficients of the GF $G$ solving the related Dyson
equation. This approach, termed algorithmic-inversion method (AIM) Chiarotti
_et al._ (2023), maps the non-linear eigenvalue problem of the Dyson equation
into a linear eigen-problem in a larger space. Algebraically, this can be seen
as the consequence of identifying the interaction self-energy as an embedding
self-energy [see Eqs. (29-37)], and then solving the Hamiltonian problem in
the larger subspace; details are provided in Ref. Chiarotti  _et al._ (2023).
We also note that similar techniques have been used in the context of
dynamical mean-field theory Savrasov  _et al._ (2006); Budich  _et al._
(2012); Wang  _et al._ (2012), lattice Hamiltonians Puig von Friesen  _et al._
(2010), and, more recently, within the GW and Bethe-Salpeter equation
formalism Bintrim and Berkelbach (2021, 2022).

##  III Analytical evaluation of TrLn terms

As a technical prerequisite for this work, and as a relevant result in itself,
in this Section we focus on integrals of the form:

$\displaystyle\Delta E_{K}$ $\displaystyle=$
$\displaystyle\text{Tr}_{\omega}\text{Ln}\left\\{G_{0}^{-1}G\right\\}$ (17)
$\displaystyle=$ $\displaystyle\int\frac{d\omega}{2\pi i}\,e^{i\omega
0^{+}}\,\text{Tr}\text{Ln}\left\\{G_{0}^{-1}(\omega)G(\omega)\right\\}.$

By representing the Green’s functions $G_{0}$ and $G$ in the above equation as
SOPs according to Eqs. (8-9), one can derive a general analytical expression
for $\Delta E_{K}$ of Eq. (17), as shown below.

In order to do this, we will make use of some common operator and matrix
identities, that we report below for completeness. For instance, we will use
the following identity:

$\text{Tr}\text{Ln}(A)=\text{Ln}\det(A).$ (18)

Bearing Eq. (18) in mind, the following relations also hold:

$\displaystyle\det(AB)$ $\displaystyle=$ $\displaystyle\det(A)\det(B),$ (19)
$\displaystyle\text{Tr}\text{Ln}(AB)$ $\displaystyle=$
$\displaystyle\text{Tr}\text{Ln}(A)+\text{Tr}\text{Ln}(B).$ (20)

Moreover, given an operator $A$ represented in the form

$A=\left[\begin{array}[]{cc}S&V_{1}\\\ V_{2}^{\dagger}&B\end{array}\right],$
(21)

its determinant can be expressed according to Brualdi and Schneider (1983):

$\text{det}(A)=\text{det}(B)\cdot\text{det}(S-V_{1}B^{-1}V_{2}^{\dagger}),$
(22)

which is a result reminiscent of techniques used in GF embedding, presented in
Sec. II.

###  III.1 Special case: non-interacting $G$

As a first step, we consider the case of both $G_{0}$ and $G$ in Eq. (17)
being non-interacting GFs corresponding to mean-field Hamiltonians $h_{0}$ and
$h_{1}$, defined as:

$h_{i}=\sum_{m}\,|\phi^{i}_{m}\rangle\epsilon^{i}_{m}\langle\phi^{i}_{m}|.$
(23)

This means that both $G_{0}$ and $G$ are diagonal on single-particle
orthonormal basis sets ($|\phi^{0}_{m}\rangle$ and $|\phi^{1}_{m}\rangle$),
that can be used to evaluate the traces. Importantly, we assume that the
number of occupied electrons is the same for $G_{0}$ and $G$. By considering
Eq. (20) and taking $A=G_{0}^{-1}$ and $B=G$, one can write the $\Delta E_{K}$
integral as

$\displaystyle\Delta E_{K}$ $\displaystyle=$
$\displaystyle\int\frac{d\omega}{2\pi i}e^{i\omega
0^{+}}\left[-\text{Tr}\text{Ln}G_{0}+\text{Tr}\text{Ln}G\right],$ (24)
$\displaystyle=$ $\displaystyle\int\frac{d\omega}{2\pi i}e^{i\omega
0^{+}}\Big{[}\text{Ln}\frac{\Pi_{m}^{\text{all}}(\omega-\epsilon^{0}_{m}\pm
i0^{+})}{\Pi_{m}^{\text{all}}(\omega-\epsilon^{1}_{m}\pm i0^{+})}\Big{]}.$
(25)

The label “all” in the product means that both occupied and empty poles are
considered. In order to evaluate the integral using residues, the contour
needs to be closed in the upper half plane, the enclosed poles corresponding
to occupied states of both $G_{0}$ and $G$. Since the number of occupied poles
of both systems is the same, the integral $\Delta E_{K}$ can be re-written as

$\displaystyle\Delta E_{K}$ $\displaystyle=$
$\displaystyle\sum_{m}^{\text{occ}}\,\oint_{\Gamma_{m}}\frac{dz}{2\pi
i}\text{Ln}\,\frac{z-\epsilon^{0}_{m}-i0^{+}}{z-\epsilon^{1}_{m}-i0^{+}},$
(26)

with an example of a $\Gamma_{m}$ contour represented in Fig. 3 of App. B.1.
The analytical expression for contour integrals as those appearing in Eq. (26)
is provided in Eq. (73) of App. B.1. Taking advantage of that expression, we
recover the well-known result Stefanucci and van Leeuwen (2013); Martin  _et
al._ (2016); Dahlen and von Barth (2004b); Ismail-Beigi (2010):

$\Delta
E_{K}=\sum_{m}^{\text{occ}}\,\left[n^{1}_{m}\epsilon^{1}_{m}-n_{m}^{0}\epsilon^{0}_{m}\right],$
(27)

where we have made the eigenvalue multiplicities $n_{m}^{i}$ explicit and
limited the sum to distinct multiplets.

###  III.2 General case: interacting $G$

Next, in this Section we consider the case of Eq. (17) with a fully
interacting $G$. Without loss of generality, we can define a self-energy
connecting $G$ and $G_{0}$ by a Dyson equation, by writing:

$\Sigma(\omega)=G_{0}^{-1}-G^{-1}.$ (28)

It is important to note that such self-energy is not necessarily physical
(i.e. it may not originate from perturbation theory or from a functional
formulation), but rather an auxiliary mathematical object. Since $G_{0},G$ and
$\Sigma$ are connected by a Dyson equation, and having assumed discrete poles
for both $G_{0}$ and $G$ (which then result meromorphic functions of the
frequency), also $\Sigma$ has discrete poles. We are therefore in the
condition to use the SOP representations given in Eqs. (8-10). In what follows
we assume to represent single-particle operators on a truncated basis set,
thereby mapping them to finite dimension matrices.

As discussed in Sec. II.1, the residues $\Gamma_{n}$ of $\Sigma$ are semi-
positive definite (stemming from the SPD of the spectral function of $G$) and,
following Refs. Chiarotti  _et al._ (2022, 2023), one can introduce $V_{n}$
such that

$\Gamma_{n}=V^{\dagger}_{n}V_{n}.$ (29)

In doing so, $V_{n}$ can be taken, e.g., to be the square root of $\Gamma_{n}$
or to be a lower-rank rectangular matrix (when represented on a basis) if
$\Gamma_{n}$ is low-rank. By doing this, $G$ can be seen as the GF of an
embedded system (index 0, below), coupled to an external bath. Indeed, by
defining the inverse resolvent ($\omega I-\mathcal{H}$) of the whole auxiliary
system as

$\displaystyle\omega I-\mathcal{H}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c c c c c}\omega
I-h_{0}&V_{1}&V_{2}&\dots&\\\ V_{1}^{\dagger}&(\omega-\omega_{1})I&&&\\\
V_{2}^{\dagger}&&(\omega-\omega_{2})I&&\\\ \vdots&&&\ddots&\\\
\end{array}\right],$ (34) $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c c}S&V\\\
V^{\dagger}&B\end{array}\right],$ (37)

one can immediately verify that the self-energy in Eq. (10) is the embedding
self-energy for the zeroth-block subsystem $S$ (in the following, calligraphic
operators such as $\mathcal{H}$ refer to the enlarged auxiliary space). This
construction is the same used in the framework of the algorithmic inversion
method Chiarotti  _et al._ (2022, 2023), used to solve Dyson equations
involving propagators represented as SOP and presented in Sec. II.1.

We can now apply the identity in Eq. (22) to the matrix in Eq. (37),
obtaining:

$\displaystyle\text{det}(\omega I-\mathcal{H})$ $\displaystyle=$
$\displaystyle\text{det}(B)\times\text{det}\left(S-VB^{-1}V^{\dagger}\right)$
$\displaystyle=$
$\displaystyle\prod_{n}(\omega-\omega_{n})^{r_{n}}\times\text{det}(\omega
I-h_{0}-\Sigma),$

where $r_{n}$ is the rank of the $\Gamma_{n}$ matrix. The above equation can
be recast in the following form:

$\displaystyle\text{det}\ G(\omega)$ $\displaystyle=$
$\displaystyle\prod_{n}(\omega-\omega_{n})^{r_{n}}\times\text{det}(\omega
I-\mathcal{H})^{-1},$ (39) $\displaystyle=$
$\displaystyle\frac{\prod^{\text{all}}_{n}(\omega-\omega_{n})^{r_{n}}}{\prod^{\text{all}}_{s}(\omega-\epsilon_{s})^{n_{s}}},$
(40)

where we have exploited the fact that the poles of $G$ are also eigenvalues of
$\mathcal{H}$ for the whole system, and made the multiplicities $n_{s}$
explicit.

Combining Eq. (24) with the identity connecting TrLn to Ln det, Eq. (18), we
obtain:

$\displaystyle\Delta E_{K}$ $\displaystyle=$
$\displaystyle\int\frac{d\omega}{2\pi i}e^{i\omega 0^{+}}\left[\text{Ln}\det
G-\text{Ln}\det G_{0}\right],$ $\displaystyle=$
$\displaystyle\int\frac{d\omega}{2\pi i}e^{i\omega
0^{+}}\text{Ln}\left[\frac{\prod^{\text{all}}_{n}(\omega-\omega_{n})^{r_{n}}\prod^{\text{all}}_{m}(\omega-\epsilon^{0}_{m})^{n_{m}^{0}}}{\prod^{\text{all}}_{s}(\omega-\epsilon_{s})^{n_{s}}}\right].$
$\displaystyle=$ $\displaystyle\int\frac{d\omega}{2\pi i}\,e^{i\omega
0^{+}}\,\text{Tr}\text{Ln}\left\\{\mathcal{G}_{0}^{-1}(\omega)\mathcal{G}(\omega)\right\\}.$
(42)

In the last equation, $\mathcal{G}$ and $\mathcal{G}_{0}$ are the GFs of the
auxiliary system obtained with and without including the coupling matrices $V$
in $\mathcal{H}$, respectively. A counting of the degrees of freedom shows
that the cardinality of $\\{\epsilon_{s}\\}$ is equal to that of
$\\{\epsilon^{0}_{m}\\}\cup\\{\omega_{n}\\}$, as also shown by the embedding
construction in Eq. (37). Nevertheless, only occupied poles (i.e. poles above
the real axis) count in the integral.

If the number of such poles in the numerator and in the denominator is the
same, by exploiting Eq. (26) we obtain the final result:

$\Delta
E_{K}=\sum_{s}^{\text{occ}}n_{s}\epsilon_{s}-\left[\sum^{\text{occ}}_{m}n_{m}^{0}\epsilon_{m}^{0}+\sum^{\text{occ}}_{n}r_{n}\omega_{n}\right].$
(43)

This expression is the first key result of the present work. The condition of
having the same number of occupied states in the numerator and denominator in
the second line of Eq. (III.2) is equivalent to having the same number of
occupied states before and after the switch-on of the coupling matrix elements
$V$. This condition, therefore, encodes charge conservation within the closed
system $C=S\cup B$. In App. B.3 we also provide a generalization of Eq. (43)
where both propagators in the TrLn term are interacting (or embedded).

At this point it is worth discussing alternative approaches existing in the
literature aimed at evaluating terms of the form
$\text{Tr}_{\omega}\text{Ln}\left\\{G_{0}^{-1}G_{1}\right\\}$. For instance,
in a series of papers, Dahlen and co-workers Dahlen  _et al._ (2006); Dahlen
and von Barth (2004a, b) first re-write the TrLn term of the Luttinger-Ward
functional by factorizing the static part of the self-energy $\Sigma_{x}$, and
then recasting Dahlen  _et al._ (2006) the integral for numerical integration
over the imaginary axis. Along the same lines, in App. B.2 we provide a scheme
for numerical integration of the TrLn terms that we have used in the present
work to numerically validate analytical expressions such as Eq. (43). In Ref.
Puig von Friesen  _et al._ (2010), Friesen and co-workers (which also adopt a
meromorphic, i.e. SOP in our language, representation for the propagators)
first handle the $\Sigma_{x}$ term as in Refs. Dahlen  _et al._ (2006); Dahlen
and von Barth (2004a, b) and then numerically evaluate the residual
contribution to the integral using a coupling-constant integration. In Ref.
Ismail-Beigi (2010), Ismail-Beigi discusses the RPA correlation energy in the
context of Green’s function theory, and, exploiting algebraic techniques
similar to those employed in this work, provides an analytical expression
involving the poles of the independent-particle and RPA response functions. We
discuss the RPA correlation energy in Sec. IV.1 where we re-derive Ismail-
Beigi’s expression by means of the present formalism. Additionally,
Aryasetiawan et al. Aryasetiawan  _et al._ (2002) write the RPA correlation
energy in a form similar to that of Ref. Dahlen  _et al._ (2006) and App. B.2
for numerical evaluation along the imaginary axis.

##  IV Applications

Having derived an analytical expression for the TrLn terms defined by Eq.
(17), in this Section we present two applications. First we focus on the
calculation of the RPA correlation energy, providing a re-derivation of a
result already known in the literature Ismail-Beigi (2010), and then apply the
formalism to analyze and partition the Klein functional in the presence of
embedding.

###  IV.1 RPA correlation energy and plasmons

Figure 2:  RPA exchange and correlation energy represented by means of Feynman
diagrams.

In the context of Green’s function methods, the RPA correlation energy is
written as Almbladh  _et al._ (1999); Ismail-Beigi (2010); Ren  _et al._
(2012); Paier  _et al._ (2012); Hellgren  _et al._ (2018); Fetter and Walecka
(1971); Stefanucci and van Leeuwen (2013); Martin  _et al._ (2016):

$\displaystyle P(\mathbf{x}_{1},\mathbf{x}_{2},\omega)$ $\displaystyle=$
$\displaystyle\int\frac{d\omega^{\prime}}{2\pi
i}\,G(\mathbf{x}_{1},\mathbf{x}_{2},\omega+\omega^{\prime})G(\mathbf{x}_{2},\mathbf{x}_{1},\omega^{\prime})$
$\displaystyle\Phi^{\text{RPA}}[P]$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\text{Tr}_{\omega}\Big{\\{}\sum_{n=2}^{\infty}\frac{1}{n}\left[vP(\omega)\right]^{n}\Big{\\}}$
$\displaystyle=$
$\displaystyle+\frac{1}{2}\text{Tr}_{\omega}\text{Ln}\big{\\{}I-vP(\omega)\big{\\}}$
$\displaystyle+\frac{1}{2}\text{Tr}_{\omega}\left\\{vP\right\\},$
$\displaystyle=$
$\displaystyle\Delta\Phi^{\text{RPA}}_{1}+\Delta\Phi^{\text{RPA}}_{2},$

where the irreducible polarizability $P$ is either evaluated using the Kohn-
Sham Green’s function $G_{s}$ in the optimized-effective-potential (OEP)
method Casida (1995), or by an interacting Green’s function (e.g. at the level
of self-consistent GW) when making stationary the Klein or Luttinger-Ward
functionals Klein (1961); Luttinger and Ward (1960); Baym and Kadanoff (1961);
Stefanucci and van Leeuwen (2013); Martin  _et al._ (2016). By considering the
Dyson equation

$\chi(\omega)=P(\omega)+P(\omega)v\chi(\omega),$ (46)

connecting the irreducible and reducible polarizabilities ($P$ and $\chi$,
respectively), one obtains

$I-vP=\epsilon=\chi^{-1}P,$ (47)

which can be used in the first term $\Delta\Phi^{\text{RPA}}_{1}$of Eq.
(IV.1), leading to:

$\Phi^{\text{RPA}}[P]=-\frac{1}{2}\text{Tr}_{\omega}\text{Ln}\big{\\{}\chi
P^{-1}\big{\\}}+\frac{1}{2}\text{Tr}_{\omega}\left\\{vP\right\\}.$ (48)

By considering the $\chi$ and $P$ as two interacting single particle
propagators, we can apply Eqs. (82-83) with $\Sigma_{21}=v$ in view of Eq.
(46). This means that the poles of the two self-energies need to cancel out
identically and therefore do not contribute to the evaluation of the Tr Ln
term. In turn, we obtain:

$\displaystyle\Delta\Phi^{\text{RPA}}_{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\sum_{p}^{\text{occ}}\left[n_{p}\Omega_{p}-n_{p}^{0}\Omega_{p}^{0}\right],$
(49) $\displaystyle=$
$\displaystyle\phantom{+}\frac{1}{2}\sum_{p}^{\Omega_{p}>0}\left[n_{p}\Omega_{p}-n_{p}^{0}\Omega_{p}^{0}\right],$

where $\Omega_{p}$ and $\Omega^{0}_{p}$ are the poles of $\chi$ and $P$
respectively, and we have considered that each time-ordered polarizability has
poles at $\pm|\Omega^{(0)}_{p}|$, the negative ones being those above the real
axis and contributing to the integral. Degeneracies of the poles ($n_{p}$ and
$n_{p}^{0}$), have been marked explicitly.

We now turn to the evaluation of the second term,
$\Delta\Phi^{\text{RPA}}_{2}$ in Eq. (IV.1). The irreducible polarizability
$P$ can be represented as a sum-over-poles according to:

$P(\omega)=\sum_{p}^{\Omega_{p}>0}\left[\frac{|t_{p}\rangle\langle
t_{p}|}{\omega-\Omega^{0}_{p}+i0^{+}}-\frac{|t_{p}\rangle\langle
t_{p}|}{\omega+\Omega^{0}_{p}-i0^{+}}\right],$ (50)

where
$\langle\mathbf{x}|t\rangle=\phi_{c}(\mathbf{x})\phi_{v}^{*}(\mathbf{x})$,
$(c,v)$ referring to conduction and valence single particle orbitals,
respectively. With the above definitions, one obtains:

$\displaystyle\Delta\Phi^{\text{RPA}}_{2}=-\frac{1}{2}\sum_{p}^{\Omega_{p}>0}\langle
t_{p}|v|t_{p}\rangle,$ (51)

which completes the evaluation of the RPA correlation energy, consistently
with existing literature. In particular, we have recovered Eq. (23) of Ref.
Ismail-Beigi (2010).

###  IV.2 Embedding of the Klein functional

The main goal of the present Section is to study the Klein functional in the
presence of an embedding scheme as the one described in Sec. II and App. A, in
order to derive, as demonstrated below, a variational partition of the total
energy. In order to do so we begin by partitioning each term appearing in the
Klein functional given by Eq. (II). Notably, the functional depends on a trial
Green’s function $G$ that, according to Eq. (2), we represent by means of a
self-energy $\widetilde{\Sigma}$ constrained to be localized on the subsystem
$S$. As discussed in Sec. II, this represents a definition for the domain of
the trial GF $G$.

For what concern $\Phi_{\text{Hxc}}$, the partition is already in place since
the particle-particle interaction is only present in $S$. Therefore one has

$\Phi_{\text{Hxc}}[G]=\Phi_{\text{Hxc}}[G_{S}].$ (52)

This can be understood, e.g., diagrammatically, since the bare interaction
lines only connect points in the $S$ subsystem, making each vertex located in
$S$. This is further discussed in App. A. Next we consider the
$\text{Tr}_{\omega}H_{0}G_{0}$ term, which is the non-interacting energy of
the closed $C=S\cup B$ system, and can be partitioned as

$\displaystyle\text{Tr}_{\omega}\left\\{H_{0}G_{0}\right\\}$ $\displaystyle=$
$\displaystyle\text{Tr}^{S}_{\omega}\left\\{(h_{0S}+\Delta
v_{S})G_{0S}\right\\}$ (53) $\displaystyle+$
$\displaystyle\text{Tr}^{B}_{\omega}\left\\{(h_{0B}+\Delta
v_{B})G_{0B}\right\\}$ $\displaystyle=$
$\displaystyle\sum_{s}^{\text{occ}}\epsilon_{s}^{0},$ (54)

where $\epsilon_{s}^{0}$ are the eigenvalues of the non-interacting problem
for $C$, $H_{0}|\phi_{s}\rangle=\epsilon_{s}^{0}|\phi_{s}\rangle$.

Coming to the next term, the following chain of identities also holds

$\displaystyle\text{Tr}_{\omega}\left\\{I-G_{0}^{-1}G\right\\}$
$\displaystyle=$
$\displaystyle-\text{Tr}_{\omega}\left\\{\widetilde{\Sigma}G\right\\}$ (55)
$\displaystyle=$
$\displaystyle-\text{Tr}^{S}_{\omega}\left\\{\widetilde{\Sigma}_{S}G_{S}\right\\},$
$\displaystyle=$
$\displaystyle\phantom{+}\text{Tr}^{s}_{\omega}\left\\{I_{S}-G_{0S}^{-1}G_{S}\right\\},$

where we have represented the trial $G$ according to Eq. (2), and limiting
$\widetilde{\Sigma}$ to have non-zero matrix elements only in $S$ and to have
a regular propagator-like analytical structure featuring time-ordering and
simple (first order) poles. Indeed, the last step is valid because of the
following equation:

$G_{S}=G_{0S}+G_{0S}\widetilde{\Sigma}_{S}G_{S}.$ (56)

The last and most interesting term in Eq. (II) is
$\text{Tr}_{\omega}\text{Ln}G_{0}^{-1}G$, which can be evaluated using Eq.
(43):

$\displaystyle\text{Tr}_{\omega}\text{Ln}\left\\{G_{0}^{-1}G\right\\}$
$\displaystyle=$
$\displaystyle\sum^{\text{occ}}_{s}\epsilon_{s}-\sum_{n}^{\text{occ}}\epsilon^{0}_{n}-\sum_{n}^{\text{occ}}\text{poles}(\widetilde{\Sigma})$
(57) $\displaystyle=$
$\displaystyle\text{Tr}^{S}_{\omega}\text{Ln}\left\\{G_{0S}^{-1}G_{S}\right\\},$

where we have used the fact that
$\sum_{s}\epsilon_{s}=\sum\text{poles}(G_{S})$,
$\sum_{n}\epsilon^{0}_{n}=\sum\text{poles}(G_{0S})$. Using the notation
introduced in Eqs. (3-4) where $\Omega_{n}$ are the poles of the embedding
self-energy, one can show that the term
$\sum_{n}\Omega_{n}=\sum\text{poles}(\Delta v_{S})$ does not explicitly appear
because the embedding self-energy is used in the evaluation of both the
$G_{0S}$ and $G_{S}$ Green’s functions. Multiplicities have been kept implicit
in the sums over eigenvalues.

Alternatively, the same result can be obtained directly from the use of Eq.
(18) and the identity concerning the determinant of block matrices, Eq. (22).
In particular, from

$G^{-1}(\omega)=\left[\begin{array}[]{cc}\omega
I_{S}-h_{0S}-\widetilde{\Sigma}&-V\\\ -V^{\dagger}&\omega
I_{B}-h_{0B}\end{array}\right]$ (58)

one gets

$\displaystyle\text{det}G^{-1}$ $\displaystyle=$
$\displaystyle\text{det}g^{-1}_{0B}\times\text{det}G_{S}^{-1},$ (59)
$\displaystyle\text{det}G^{-1}_{0}$ $\displaystyle=$
$\displaystyle\text{det}g^{-1}_{0B}\times\text{det}G_{0S}^{-1},$ (60)

which gives

$\displaystyle\text{Tr}_{\omega}\left\\{G_{0}^{-1}G\right\\}$ $\displaystyle=$
$\displaystyle-\text{Ln det}g^{-1}_{0B}-\text{Ln det}G^{-1}_{S}$
$\displaystyle+\text{Ln det}g^{-1}_{0B}+\text{Ln det}G^{-1}_{0S}$
$\displaystyle=$ $\displaystyle\phantom{+}\text{Ln det}G^{-1}_{0S}G_{S},$

the last line being equivalent to the result to be proven.

We are now in the position to put all terms together to obtain:

$\displaystyle E_{K}[G]$ $\displaystyle=$
$\displaystyle\text{Tr}^{S}_{\omega}\text{Ln}\left\\{G_{S}G^{-1}_{0S}\right\\}+\sum_{s}^{\text{occ}}\epsilon_{s}^{0}$
$\displaystyle+$
$\displaystyle\text{Tr}^{S}_{\omega}\left\\{I_{S}-G_{0S}^{-1}G_{S}\right\\}+\Phi_{\text{Hxc}}[G_{S}].$

Next, the first term on the rhs can be further rewritten using:

$\displaystyle\text{Tr}^{S}_{\omega}\text{Ln}\left\\{G_{S}G^{-1}_{0S}\right\\}$
$\displaystyle=$
$\displaystyle\text{Tr}^{S}_{\omega}\text{Ln}\left\\{G_{S}g_{0S}^{-1}\,g_{0S}G^{-1}_{0S}\right\\}$
(63) $\displaystyle=$
$\displaystyle\text{Tr}^{S}_{\omega}\text{Ln}\left\\{G_{S}g^{-1}_{0S}\right\\}$
$\displaystyle-$
$\displaystyle\text{Tr}^{S}_{\omega}\text{Ln}\left\\{G_{0S}g^{-1}_{0S}\right\\},$
$\displaystyle\text{Tr}^{S}_{\omega}\text{Ln}\left\\{G_{0S}g^{-1}_{0S}\right\\}$
$\displaystyle=$
$\displaystyle\sum_{s}^{\text{occ}}\epsilon_{s}^{0}-\sum_{s}^{\text{occ}}\bar{\epsilon}_{s}^{0}-\sum_{n}^{\text{occ}}\Omega_{n}$
$\displaystyle=$
$\displaystyle\sum_{s}^{\text{occ}}\epsilon_{s}^{0}-\text{Tr}^{S}_{\omega}\left\\{h_{0S}g_{0S}\right\\}$
$\displaystyle-$
$\displaystyle\text{Tr}^{B}_{\omega}\left\\{h_{0B}g_{0B}\right\\},$ (65)

where the eigenvalues $\bar{\epsilon}_{s}^{0}$ refer to susbsystem $S$ in the
absence of coupling to $B$.

Eventually, this leads to the final result for the partitioning of the Klein
energy functional:

$\displaystyle E_{K}[G]$ $\displaystyle=$ $\displaystyle
E^{S}_{K}[G_{S}]+\text{Tr}^{B}_{\omega}\left\\{h_{0B}g_{0B}\right\\},$ (66)
$\displaystyle E^{S}_{K}[G_{S}]$ $\displaystyle=$
$\displaystyle\text{Tr}^{S}_{\omega}\text{Ln}\left\\{G_{S}g^{-1}_{0S}\right\\}+\text{Tr}^{S}_{\omega}\left\\{h_{0S}g_{0S}\right\\}$
(67) $\displaystyle+$
$\displaystyle\text{Tr}^{S}_{\omega}\left\\{I_{S}-g_{0S}^{-1}G_{S}\right\\}+\text{Tr}^{S}_{\omega}\left\\{\Delta
v_{S}G_{S}\right\\}$ $\displaystyle+$ $\displaystyle\Phi_{\text{Hxc}}[G_{S}].$

This is the second key result of the present paper, implying that
$E^{S}_{K}[G_{S}]$ is stationary for the $G_{S}$ that solve the embedding
Dyson equation, namely:

$2\pi i\frac{\delta E^{S}_{K}[G_{S}]}{\delta
G_{S}}=G_{S}^{-1}-g_{0S}^{-1}+\Delta v_{S}+\Sigma_{\text{Hxc}}[G_{S}]=0,$ (68)

showing that the partition of the Klein energy is exact and also variational
for what concern subsystem $S$.

Interestingly, we note that an equation formally equivalent to Eq. (67) has
been used by Savrasov and Kotliar in Refs. Savrasov and Kotliar (2004);
Kotliar  _et al._ (2006) to express the grand-potential of a quantum system in
the presence of an external local and dynamical potential coupled to the local
Green’s function. In the present context, that term is played by $\Delta
v_{S}$, here originating from an embedding procedure. Interestingly, the
embedding construction allows us to further inspect the physical nature of the
energy terms in Eqs. (66-67). In particular, the complement energy
$\text{Tr}^{B}_{\omega}\left\\{h_{0B}g_{0B}\right\\}$ (i.e. the energy that
needs to be summed to $E^{S}_{K}[G_{S}]$ to give the total energy of the
closed system $C$, $E_{K}[G]$) is that of the non-interacting and uncoupled
bath. This means that all effects of the coupling $V$ need to be absorbed in
$E^{S}_{K}[G_{S}]$ to allow for variationality. This is at variance with other
possible partitions of the $C$ total energy (such as, e.g., those suggested by
the Galitskii-Migdal expression).

##  V Conclusions

In this work, and within the framework of Green’s function methods, we address
the use of the Klein functional when embedding an interacting system $S$ into
a non-interacting bath $B$. Exploiting a sum-over-pole (SOP) representation
for the propagators, and taking advantage of the algorithmic-inversion method
(AIM) introduced to solve Dyson-like equations involving SOP propagators
Chiarotti  _et al._ (2022, 2023), we have first derived an exact analytical
expression to evaluate terms of the form
$\text{Tr}_{\omega}\text{Ln}\left\\{G_{0}^{-1}G\right\\}$. Notably, such terms
appear in the Klein and Luttinger-Ward functionals Klein (1961); Luttinger and
Ward (1960); Baym and Kadanoff (1961); Stefanucci and van Leeuwen (2013);
Martin  _et al._ (2016) as well as in other common maby-body terms such as the
RPA correlation energy Almbladh  _et al._ (1999); Ismail-Beigi (2010); Ren
_et al._ (2012); Paier  _et al._ (2012); Hellgren  _et al._ (2018); Fetter and
Walecka (1971); Stefanucci and van Leeuwen (2013); Martin  _et al._ (2016). In
this respect, the analytical expression obtained represents the first key
result of the paper.

Next, we have used the above analytical result to partition the Klein
functional of an embedded system intro two contributions, one associated to
the subsystem $S$ and one to the non-interacting bath $B$. Importantly, the
energy associated to $S$ is also variational as a functional of the $S$
Green’s function $G_{S}$, with the functional gradient becoming zero for the
physical embedded $G_{s}$. This is the second main result of the work. Last,
we have also exploited the analytical result for the TrLn terms to recover an
exact analytical expression for the RPA correlation energy known in the
literature Ismail-Beigi (2010).

##  VI Acknowledgments

We thank Prof. Marco Gibertini and Prof. Lucia Reining for useful discussions
on the subject. We also thank Matteo Quinzi for reading the manuscript and for
providing further numerical validation for some of the analytical results
presented.

##  Appendix A Green’s function embedding and perturbation theory

In this Appendix we discuss the building of many-body perturbation theory
(MBPT), to include particle interaction effects in the Green’s function in the
presence of embedding. We consider the case of fermions at $T=0$, for
simplicity. As mentioned in Sec. II and sketched in Fig. 1, we consider a
closed quantum system $C$ partitioned into two sub-units, $C=S\cup B$,
interacting via a coupling potential $V$, with particle interactions confined
to the $S$ region, with $B$ being a non-interacting bath. The particle-
particle interaction can be written in the usual form of a two-body potential:

$\displaystyle V_{ee}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d\mathbf{x}d\mathbf{x}^{\prime}\,\hat{\psi}^{\dagger}(\mathbf{x})\hat{\psi}^{\dagger}(\mathbf{x}^{\prime})\,v_{\text{int}}(\mathbf{x},\mathbf{x}^{\prime})\,\hat{\psi}(\mathbf{x}^{\prime})\hat{\psi}(\mathbf{x}),$
(69) $\displaystyle v_{\text{int}}(\mathbf{x},\mathbf{x}^{\prime})\neq
0\qquad\text{for}\quad\mathbf{x},\mathbf{x}^{\prime}\in S,$

where the constraint on $v_{\text{int}}(\mathbf{x},\mathbf{x}^{\prime})$
expresses the fact that the interaction is present only in the $S$ region.

Within the above definitions, the perturbation expansion for the Green’s
function of the closed system $C$ leads to Fetter and Walecka (1971);
Stefanucci and van Leeuwen (2013); Martin  _et al._ (2016):

$iG(\mathbf{x},t;\mathbf{x}^{\prime},t^{\prime})=\sum_{n=0}^{\infty}\,\frac{(-i)^{n}}{n!}\int_{-\infty}^{+\infty}dt_{1}\dots
dt_{n}\\\
\times\frac{\langle\Phi_{0}|\mathcal{T}\Big{[}\hat{V}_{ee}(t_{1})\dots\hat{V}_{ee}(t_{n})\,\hat{\psi}(\mathbf{x},t)\hat{\psi}^{\dagger}(\mathbf{x}^{\prime},t^{\prime})\Big{]}|\Phi_{0}\rangle}{\langle\Phi_{0}|\hat{S}|\Phi_{0}\rangle},$
(70)
${\langle\Phi_{0}|\hat{S}|\Phi_{0}\rangle}=\sum_{n=0}^{\infty}\,\frac{(-i)^{n}}{n!}\int_{-\infty}^{+\infty}dt_{1}\dots
dt_{n}\\\
\times\langle\Phi_{0}|\mathcal{T}\Big{[}\hat{V}_{ee}(t_{1})\dots\hat{V}_{ee}(t_{n})\,\Big{]}|\Phi_{0}\rangle.$
(71)

First we focus on $G^{S}$, i.e., on the case when
$\mathbf{x},\mathbf{x}^{\prime}$ are located in $S$. Since $\hat{V}_{ee}$ only
contains field operators related to subspace $S$, all self-energy diagrams
resulting from Eq. (70) have only vertexes within the subsystem $S$.
Similarly, if we consider $G$ in the general case (end points either in $B$ or
$S$), $B$ points will be present only in disconnected diagrams (to be dropped)
or in the external ends of the connected diagrams, which do not show in the
proper self-energy. Therefore, the interaction self-energy is zero for matrix
elements out of the $S$ block, as shown in Fig. 1.

So far, perturbation theory in terms of the bare Green’s function $G_{0}$ has
been addressed, with $\Sigma^{S}[G_{0}]=\Sigma^{S}[G_{0}^{S}]$. Nevertheless,
one can perform the usual steps Fetter and Walecka (1971); Stefanucci and van
Leeuwen (2013); Martin  _et al._ (2016) in passing from bare diagrams
involving $G_{0}$ to skeleton diagrams involving $G$, leading to:

$\Sigma^{S}[G]=\Sigma^{S}[G^{S}],$ (72)

where we can substitute $G^{S}$ to $G$ because of the localization of the bare
interaction, Eq. (69). A similar reasoning can be applied to the $\Phi$
functional to obtain $\Phi_{\text{Hxc}}[G]=\Phi_{\text{Hxc}}[G^{S}]$. In
summary, within the non-interacting bath condition, the interaction self-
energy $\Sigma^{S}$ has a perturbation expansion structurally identical to the
one usually developed for closed systems Fetter and Walecka (1971); Stefanucci
and van Leeuwen (2013); Martin  _et al._ (2016), and does not make any
reference to the $B$ unit, i.e. all diagrams develop within $S$, as if $S$
were disconnected from $B$. Of course, $G^{S}$ is then calculated in the
presence of the bath, e.g. via embedding self-energies, which in turn make the
effect of the interaction spread all over the system. Notably, the Anderson
impurity model Anderson (1961); Kotliar  _et al._ (2006); Martin  _et al._
(2016) can be seen as a special case of the above setting. Indeed, the exact
electron-electron self-energy of the model is localized on the impurity
Anderson (1961) ($S$ in our notation), and can be computed, e.g., using bare
perturbation theory Yosida and Yamada (1970); Yamada (1975); Horvatić  _et
al._ (1987); Kotliar  _et al._ (2006) involving $G^{S}_{0}$.

As a relevant point for the present discussion, the use of the skeleton
perturbation theory and the Luttinger-Ward functional has been recently
questioned Kozik  _et al._ (2015); Eder (2014); Stan  _et al._ (2015); Rossi
and Werner (2015); Schäfer  _et al._ (2016); Tarantino  _et al._ (2017),
leading to a discussion about the domain of the trial $G$ and the rise of
multiple solutions of the non-linear Dyson equation involving $\Sigma[G]$ (see
e.g. Ref. [Tarantino  _et al._ , 2017] for additional details). For the sake
of the present work, we assume to be in the situation where perturbation
theory does not pose convergence problems and one is able to discriminate
between physical from unphysical solutions when needed.

##  Appendix B Complements on TrLn terms

###  B.1 Notable integrals

Figure 3:  Illustration of the contour used in Eq. (73) and its decomposition
in simple paths, $\Gamma_{1}-\Gamma_{4}$.

In this Section we provide a detailed derivation of the expression

$I=\oint_{\Gamma}\frac{dz}{2\pi i}\,\text{Ln}\,\frac{z-a}{z-b}=b-a,$ (73)

where both $a,b$ are assumed to be real numbers. Making reference to Fig. 3,
the contour integral can be split into four contributions, labelled
$\Gamma_{1}-\Gamma_{4}$, such that $I=I_{1}+I_{2}+I_{3}+I_{4}$, with
$I_{i}=\int_{\Gamma_{i}}[...]$.

Let us first consider $I_{1}$, where we assume that $\Gamma_{1}$ corresponds
to the pole in $a$. Using the parametrization $z=Re^{i\theta}$ one has:

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\int_{\Gamma_{1}}\frac{dz}{2\pi i}\text{Ln}\frac{z-a}{z-b},$
(74) $\displaystyle=$
$\displaystyle-R\int_{0}^{2\pi}\frac{d\theta}{2\pi}e^{i\theta}\text{Ln}\left[1+\frac{a-b}{Re^{i\theta}}\right],$

which goes to zero in the limit $R\to 0$, e.g. in view of $R\text{ln}(1/R)\to
0$. A similar argument holds for $I_{3}$, so that we have $I_{1,3}\to 0$ when
$R\to 0$. Coming to remaining paths, we have

$\displaystyle I_{2+4}=\frac{1}{2\pi
i}\left[-\int_{a+R}^{b-R}\\!\\!\\!dz^{+}+\int_{a+R}^{b-R}\\!\\!\\!dz^{-}\right]\,\text{Ln}\frac{z-a}{z-b},$
(75)

where $dz^{+}$ and $dz^{-}$ refer to the upper ($\Gamma_{4}$) and lower
($\Gamma_{2}$) branch, respectively. The real part of the logarithm function
does not contribute (the two branches cancel out), while the imaginary part
does. Indeed, choosing the branch cut of the complex Log going from 0 to
$+\infty$, one obtains:

$I_{2+4}=\frac{1}{2\pi}(\pi-0+2\pi-\pi)(b-a)=b-a,$ (76)

which completes the derivation of Eq. (73).

###  B.2 Computational evaluation of TrLn terms

In order to develop a form of Eq. (17) suitable for numerical evaluation, that
we have used e.g. to compare with the analytical results of this work, we
follow some of the ideas from the App. B of Ref. [Dahlen  _et al._ , 2006]. We
start by re-writing Eq. (III.2) by rotating the integration over the imaginary
axis:

$\displaystyle\Delta E_{K}$ $\displaystyle=$
$\displaystyle\int_{+i\infty}^{-i\infty}\frac{dz}{2\pi
i}\text{Ln}\left[\frac{\det G(z)}{\det G_{0}(z)}\right],$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}\frac{dx}{2\pi}\text{Ln}\left[\frac{\det
G(ix)}{\det G_{0}(ix)}\right],$ $\displaystyle=$
$\displaystyle\int_{0}^{+\infty}\frac{dx}{2\pi}\bigg{[}\text{Ln}\det
G(ix)+\text{Ln}\,{\det}^{*}G(ix)$ $\displaystyle\quad\quad\quad\ \
-\text{Ln}\,{\det}G_{0}(ix)-\text{Ln}\,{\det}^{*}G_{0}(ix)\bigg{]},$
$\displaystyle=$
$\displaystyle\int_{0}^{+\infty}\frac{dx}{2\pi}\bigg{[}\,\text{ln}{\big{|}\det
G(ix)\big{|}^{2}}-\text{ln}{\big{|}{\det}G_{0}(ix)\big{|}^{2}}\bigg{]}.$

In deriving these equations we have made use of the relations
$G(-ix)=G(ix)^{\dagger}$ and $\det M^{\dagger}=(\det M)^{*}$. The last
expression is suited for numerical evaluation, that we performed using a
tangent grid on the imaginary axis.

###  B.3 TrLn term with two interacting Green’s functions

As anticipated in Sec. III.2, Eq. (43) can be further generalized to the case
of TrLn computed for two interacting GFs, $G_{1}$ and $G_{2}$. As a first step
we make reference to an arbitrary non-interacting $G_{0}$ by exploit the
identity in Eq. (20),

$\displaystyle\text{Tr}_{\omega}\text{Ln}\left\\{G_{1}^{-1}G_{2}\right\\}$
$\displaystyle=$
$\displaystyle\text{Tr}_{\omega}\text{Ln}\left\\{G_{0}^{-1}G_{2}\right\\}$
(79) $\displaystyle-$
$\displaystyle\text{Tr}_{\omega}\text{Ln}\left\\{G_{0}^{-1}G_{1}\right\\}.$

Next we can connect $G_{1,2}$ to $G_{0}$ via Dyson-like equations, by writing:

$\displaystyle G_{1}$ $\displaystyle=$ $\displaystyle
G_{0}+G_{0}(\Sigma_{1}-v_{0})G_{1}$ (80) $\displaystyle G_{2}$
$\displaystyle=$ $\displaystyle G_{0}+G_{0}(\Sigma_{2}-v_{0})G_{2}$ (81)

where $\Sigma_{i}$ are suitable self-energy operators. Upon defining
$\Sigma_{21}=\Sigma_{2}-\Sigma_{1}$, the above equations give:

$G_{2}=G_{1}+G_{1}\Sigma_{21}G_{2}.$ (82)

We can now evaluate Eq. (79) by means of Eq. (43), obtaining:

$\displaystyle\Delta E_{K}$ $\displaystyle=$
$\displaystyle\left[\sum_{s}^{\text{occ}}n_{s}^{(2)}\epsilon_{s}^{(2)}-\sum^{\text{occ}}\text{poles}(\Sigma_{2})\right]$
(83) $\displaystyle-$
$\displaystyle\left[\sum_{s}^{\text{occ}}n_{s}^{(1)}\epsilon_{s}^{(1)}-\sum^{\text{occ}}\text{poles}(\Sigma_{1})\right].$

## References

  * Kohn (1999) W. Kohn, Rev. Mod. Phys. 71, 1253 (1999).
  * Pribram-Jones  _et al._ (2015) A. Pribram-Jones, D. A. Gross, and K. Burke, Annu. Rev. Phys. Chem. 66, 283–304 (2015).
  * Marzari  _et al._ (2021) N. Marzari, A. Ferretti, and C. Wolverton, Nat. Mater. 20, 736–749 (2021).
  * Hohenberg and Kohn (1964) P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
  * Martin  _et al._ (2016) R. M. Martin, L. Reining, and D. Ceperley,  _Interacting Electrons Theory and Computational Approaches_ (Cambridge University Press, 2016).
  * Runge and Gross (1984) E. Runge and E. K. U. Gross, Phys. Rev. Lett. 52, 997 (1984).
  * Petersilka  _et al._ (1996) M. Petersilka, U. J. Gossmann, and E. K. U. Gross, Phys. Rev. Lett. 76, 1212 (1996).
  * Ullrich (2012) C. A. Ullrich,  _Time-Dependent Density-Functional Theory: Concepts and Applications_ (Oxford University Press, 2012).
  * Gross  _et al._ (1988) E. K. U. Gross, L. N. Oliveira, and W. Kohn, Phys. Rev. A 37, 2809 (1988).
  * Gould and Pittalis (2019) T. Gould and S. Pittalis, Phys. Rev. Lett. 123, 016401 (2019).
  * Cernatic  _et al._ (2021) F. Cernatic, B. Senjean, V. Robert, and E. Fromager, Top. Curr. Chem. (Z) 380, 4 (2021).
  * Stefanucci and van Leeuwen (2013) G. Stefanucci and R. van Leeuwen,  _Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction_ (Cambridge University Press, 2013).
  * Hedin (1965) L. Hedin, Phys. Rev. 139, A796 (1965).
  * Reining (2018) L. Reining, WIREs Computational Molecular Science 8, e1344 (2018).
  * Golze  _et al._ (2019) D. Golze, M. Dvorak, and P. Rinke, Frontiers in Chemistry 7, 377 (2019).
  * Onida  _et al._ (2002) G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (2002).
  * Fetter and Walecka (1971) A. L. Fetter and J. D. Walecka,  _Quantum theory of many-particle systems_ (McGraw-Hill, New York, 1971).
  * Luttinger and Ward (1960) J. M. Luttinger and J. C. Ward, Phys. Rev. 118, 1417 (1960).
  * Klein (1961) A. Klein, Phys. Rev. 121, 950 (1961).
  * Baym and Kadanoff (1961) G. Baym and L. P. Kadanoff, Phys. Rev. 124, 287 (1961).
  * Almbladh  _et al._ (1999) C.-O. Almbladh, U. von Barth, and R. van Leeuwen, Intl. J. Mod. Phys. B 13, 535 (1999).
  * Dahlen  _et al._ (2004) N. E. Dahlen, R. van Leeuwen, and U. von Barth, Int. J. Quantum Chem. 101, 512–519 (2004).
  * Dahlen and von Barth (2004a) N. E. Dahlen and U. von Barth, J. Chem. Phys. 120, 6826–6831 (2004a).
  * Dahlen and von Barth (2004b) N. E. Dahlen and U. von Barth, Phys. Rev. B 69, 195102 (2004b).
  * Dahlen  _et al._ (2006) N. E. Dahlen, R. van Leeuwen, and U. von Barth, Phys. Rev. A 73, 012511 (2006).
  * Puig von Friesen  _et al._ (2010) M. Puig von Friesen, C. Verdozzi, and C.-O. Almbladh, Phys. Rev. B 82, 155108 (2010).
  * Di Sabatino  _et al._ (2021) S. Di Sabatino, P.-F. Loos, and P. Romaniello, Front. Chem. 9, 751054 (2021).
  * Holm and Aryasetiawan (2000) B. Holm and F. Aryasetiawan, Phys. Rev. B 62, 4858-4865 (2000).
  * García-González and Godby (2001) P. García-González and R. W. Godby, Phys. Rev. B 63, 075112 (2001).
  * Chiarotti  _et al._ (2022) T. Chiarotti, N. Marzari, and A. Ferretti, Phys. Rev. Res. 4, 013242 (2022).
  * Casida (1995) M. E. Casida, Phys. Rev. A 51, 2005-2013 (1995).
  * Kümmel and Kronik (2008) S. Kümmel and L. Kronik, Rev. Mod. Phys. 80, 3 (2008).
  * Sham and Schlüter (1983) L. J. Sham and M. Schlüter, Phys. Rev. Lett. 51, 1888 (1983).
  * Godby  _et al._ (1987) R. W. Godby, M. Schlüter, and L. J. Sham, Phys. Rev. B 36, 6497 (1987).
  * Ismail-Beigi (2010) S. Ismail-Beigi, Phys. Rev. B 81, 195126 (2010).
  * Ren  _et al._ (2012) X. Ren, P. Rinke, C. Joas, and M. Scheffler, J. Mater. Sci. 47, 7447–7471 (2012).
  * Paier  _et al._ (2012) J. Paier, X. Ren, P. Rinke, G. E. Scuseria, A. Grüneis, G. Kresse, and M. Scheffler, New J. Phys. 14, 043002 (2012).
  * Hellgren  _et al._ (2018) M. Hellgren, N. Colonna, and S. de Gironcoli, Phys. Rev. B 98, 045117 (2018).
  * Kotliar  _et al._ (2006) G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78, 865-951 (2006).
  * Gatti  _et al._ (2007) M. Gatti, F. Bruneval, V. Olevano, and L. Reining, Phys. Rev. Lett. 99, 266402 (2007).
  * Ferretti  _et al._ (2014) A. Ferretti, I. Dabo, M. Cococcioni, and N. Marzari, Phys. Rev. B 89, 195134 (2014).
  * Dabo  _et al._ (2009) I. Dabo, M. Cococcioni, and N. Marzari, arXiv (2009), arXiv:0901.2637v1 [cond-mat.mtrl-sci] .
  * Dabo  _et al._ (2010) I. Dabo, A. Ferretti, N. Poilvert, Y. Li, N. Marzari, and M. Cococcioni, Phys. Rev. B 82, 115121 (2010).
  * Nguyen  _et al._ (2018) N. L. Nguyen, N. Colonna, A. Ferretti, and N. Marzari, Phys. Rev. X 8, 021051 (2018).
  * Meir and Wingreen (1992) Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512-2515 (1992).
  * Haug and Jauho (1996) H. Haug and A.-P. Jauho,  _Transport and Optics of Semiconductors_ (Springer, Berlin, 1996).
  * Buongiorno Nardelli (1999) M. Buongiorno Nardelli, Phys. Rev. B 60, 7828-7833 (1999).
  * Brandbyge  _et al._ (2002) M. Brandbyge, J.-L. Mozos, P. Ordejón, J. Taylor, and K. Stokbro, Phys. Rev. B 65, 165401 (2002).
  * Calzolari  _et al._ (2004) A. Calzolari, N. Marzari, I. Souza, and M. Buongiorno Nardelli, Phys. Rev. B 69, 035108 (2004).
  * Chiarotti  _et al._ (2023) T. Chiarotti, A. Ferretti, and N. Marzari, arXiv (2023), arXiv:2302.12193v1 [cond-mat.mtrl-sci] .
  * Chiarotti (2023) T. Chiarotti,  _Spectral and thermodynamic properties of interacting electrons with dynamical functionals_ , Ph.D. thesis, EDMX Doctoral Program, École polytechnique fédérale de Lausanne (EPFL), Switzerland (2023).
  * Engel  _et al._ (1991) G. E. Engel, B. Farid, C. M. M. Nex, and N. H. March, Phys. Rev. B 44, 13356-13373 (1991).
  * Rojas  _et al._ (1995) H. N. Rojas, R. W. Godby, and R. J. Needs, Phys. Rev. Lett. 74, 1827-1830 (1995).
  * Güttel and Tisseur (2017) S. Güttel and F. Tisseur, Acta Numerica 26, 1–94 (2017).
  * Leon  _et al._ (2021) D. A. Leon, C. Cardoso, T. Chiarotti, D. Varsano, E. Molinari, and A. Ferretti, Phys. Rev. B 104, 115157 (2021).
  * Leon  _et al._ (2023) D. A. Leon, A. Ferretti, D. Varsano, E. Molinari, and C. Cardoso, Phys. Rev. B 107, 155130 (2023).
  * Stefanucci  _et al._ (2014) G. Stefanucci, Y. Pavlyukh, A.-M. Uimonen, and R. van Leeuwen, Phys. Rev. B 90, 115134 (2014).
  * Savrasov  _et al._ (2006) S. Y. Savrasov, K. Haule, and G. Kotliar, Phys. Rev. Lett. 96, 036404 (2006).
  * Budich  _et al._ (2012) J. C. Budich, R. Thomale, G. Li, M. Laubach, and S.-C. Zhang, Phys. Rev. B 86, 201407(R) (2012).
  * Wang  _et al._ (2012) L. Wang, H. Jiang, X. Dai, and X. C. Xie, Phys. Rev. B 85, 235135 (2012).
  * Bintrim and Berkelbach (2021) S. J. Bintrim and T. C. Berkelbach, J. Chem. Phys. 154, 041101 (2021).
  * Bintrim and Berkelbach (2022) S. J. Bintrim and T. C. Berkelbach, J. Chem. Phys. 156, 044114 (2022).
  * Brualdi and Schneider (1983) R. A. Brualdi and H. Schneider, Linear Algebra and its Applications 52–53, 769 (1983).
  * Aryasetiawan  _et al._ (2002) F. Aryasetiawan, T. Miyake, and K. Terakura, Phys. Rev. Lett. 88, 166401 (2002).
  * Savrasov and Kotliar (2004) S. Y. Savrasov and G. Kotliar, Phys. Rev. B 69, 245101 (2004).
  * Anderson (1961) P. W. Anderson, Phys. Rev. 124, 41–53 (1961).
  * Yosida and Yamada (1970) K. Yosida and K. Yamada, Progress of Theoretical Physics Supplement 46, 244–255 (1970).
  * Yamada (1975) K. Yamada, Progress of Theoretical Physics 53, 970–986 (1975).
  * Horvatić  _et al._ (1987) B. Horvatić, D. Sokcević, and V. Zlatić, Phys. Rev. B 36, 675-683 (1987).
  * Kozik  _et al._ (2015) E. Kozik, M. Ferrero, and A. Georges, Phys. Rev. Lett. 114, 156402 (2015).
  * Eder (2014) R. Eder, arXiv (2014), arXiv:1407.6599v1 [cond-mat.str-el] .
  * Stan  _et al._ (2015) A. Stan, P. Romaniello, S. Rigamonti, L. Reining, and J. A. Berger, New J. Phys. 17, 093045 (2015).
  * Rossi and Werner (2015) R. Rossi and F. Werner, J. Phys. A: Math. Theor. 48, 485202 (2015).
  * Schäfer  _et al._ (2016) T. Schäfer, S. Ciuchi, M. Wallerberger, P. Thunström, O. Gunnarsson, G. Sangiovanni, G. Rohringer, and A. Toschi, Phys. Rev. B 94, 235108 (2016).
  * Tarantino  _et al._ (2017) W. Tarantino, P. Romaniello, J. A. Berger, and L. Reining, Phys. Rev. B 96, 045124 (2017).


Since the action of parabolic Cayley transform on cycles does not preserve
zero-radius cycles one shall better use infinitesimal-radius cycles from
Section 6.1 instead. First of all images of infinitesimal cycles under
parabolic Cayley transform are infinitesimal cycles again [47, § 3.6.4],
secondly Lemma 6.5.2 provides a useful expression of concurrence with
infinitesimal cycle focus through f-orthogonality. Although f-orthogonality is
not preserved by the Cayley transform (p)(p) for generic cycles it did for the
infinitesimal ones, see [47, § 3.6.4]:

######  Lemma 8.9.

An infinitesimal cycle ${C^{a}_{\breve{\sigma}}}$ (6.1) is f-orthogonal $($in
the sense of Lemma 6.5.2) to a cycle ${\tilde{C}^{a}_{\breve{\sigma}}}$ if and
only if the Cayley transform (p)(p) of ${C^{a}_{\breve{\sigma}}}$ is
f-orthogonal to the Cayley transform of ${\tilde{C}^{a}_{\breve{\sigma}}}$.

We main observation of this paper is that the potential of the Erlangen
programme is still far from exhausting even for two-dimensional geometry.

### Acknowledgements

This paper has some overlaps with the paper [52] written in collaboration with
D. Biswas. However the present paper essentially revises many concepts (e.g.
lengths, orthogonality, the parabolic Cayley transform) introduced in [52],
thus it was important to make it an independent reading to avoid confusion
with some earlier (and naïve!) guesses made in [52].

The author is grateful to Professors S. Plaksa, S. Blyumin and N. Gromov for
useful discussions and comments. Drs. I.R. Porteous, D.L. Selinger and J.
Selig carefully read the previous paper [52] and made numerous comments and
remarks helping to improve this paper as well. I am also grateful to D. Biswas
for many comments on this paper.

The extensive graphics in this paper were produced with the help of the GiNaC
[5, 45] computer algebra system. Since this tool is of separate interest we
explain its usage by examples from this article in the separate paper [47].
The noweb [65] wrapper for C++ source code is included in the arXiv.org files
of the papers [47].

## References

  * [1]
  * [2] Arveson W., An invitation to $C^{*}$-algebras, Graduate Texts in Mathematics, Vol. 39, Springer-Verlag, New York – Heidelberg, 1976.
  * [3] Baird P., Wood J.C., Harmonic morphisms from Minkowski space and hyperbolic numbers, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 52(100) (2009), 195–209.
  * [4] Barker W., Howe R., Continuous symmetry. From Euclid to Klein, American Mathematical Society, Providence, RI, 2007.
  * [5] Bauer C., Frink A., Kreckel R., Vollinga J, GiNaC is Not a CAS, http://www.ginac.de/.
  * [6] Beardon A.F., The geometry of discrete groups, Graduate Texts in Mathematics, Vol. 91, Springer-Verlag, New York, 1995.
  * [7] Beardon A.F., Algebra and geometry, Cambridge University Press, Cambridge, 2005\.
  * [8] Bekkara E., Frances C., Zeghib A., On lightlike geometry: isometric actions, and rigidity aspects, C. R. Math. Acad. Sci. Paris 343 (2006), 317–321.
  * [9] Benz W., Classical geometries in modern contexts. Geometry of real inner product spaces, 2nd ed., Birkhäuser Verlag, Basel, 2007.
  * [10] Berger M., Geometry. II, Springer-Verlag, Berlin, 1987.
  * [11] Boccaletti D., Catoni F., Cannata R., Catoni V., Nichelatti E., Zampetti P., The mathematics of Minkowski space-time and an introduction to commutative hypercomplex numbers, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2008.
  * [12] Catoni F., Cannata R., Nichelatti E., The parabolic analytic functions and the derivative of real functions, Adv. Appl. Clifford Algebr. 14 (2004), 185–190.
  * [13] Catoni F., Cannata R., Catoni V., Zampetti P., $N$-dimensional geometries generated by hypercomplex numbers, Adv. Appl. Clifford Algebr. 15 (2005), 1–25.
  * [14] Cerejeiras P., Kähler U., Sommen F., Parabolic Dirac operators and the Navier–Stokes equations over time-varying domains, Math. Methods Appl. Sci. 28 (2005), 1715–1724.
  * [15] Chern S.-S., Finsler geometry is just Riemannian geometry without the quadratic restriction, Notices Amer. Math. Soc. 43 (1996), 959–963.
  * [16] Cnops J., Hurwitz pairs and applications of Möbius transformations, Habilitation Dissertation, University of Gent, 1994.
  * [17] Cnops J., An introduction to Dirac operators on manifolds, Progress in Mathematical Physics, Vol. 24, Birkhäuser Boston, Inc., Boston, MA, 2002.
  * [18] Coxeter H.S.M., Greitzer S.L., Geometry revisited, Random House, New York, 1967.
  * [19] Davis M., Applied nonstandard analysis, John Wiley & Sons, New York, 1977\.
  * [20] Eelbode D., Sommen F., Taylor series on the hyperbolic unit ball, Bull. Belg. Math. Soc. Simon Stevin 11 (2004), 719–737.
  * [21] Eelbode D., Sommen F., The fundamental solution of the hyperbolic Dirac operator on $\mathbb{R}^{1,m}$: a new approach, Bull. Belg. Math. Soc. Simon Stevin 12 (2005), 23–37.
  * [22] Fjelstad P., Gal S.G., Two-dimensional geometries, topologies, trigonometries and physics generated by complex-type numbers, Adv. Appl. Clifford Algebr. 11 (2001), 81–107.
  * [23] Garas’ko G.I., Elements of Finsler geometry for physicists, TETRU, Moscow, 2009 (in Russian), available at http://hypercomplex.xpsweb.com/articles/487/ru/pdf/00-gbook.pdf.
  * [24] Gromov N.A., Contractions and analytic extensions of classical groups. Unified approach, Akad. Nauk SSSR Ural. Otdel. Komi Nauchn. Tsentr, Syktyvkar, 1990 (in Russian).
  * [25] Gromov N.A., Kuratov V.V., Noncommutative space-time models, Czechoslovak J. Phys. 55 (2005), 1421–1426, hep-th/0507009.
  * [26] Herranz F.J., Santander M., Conformal compactification of spacetimes, J. Phys. A: Math. Gen. 35 (2002), 6619–6629, math-ph/0110019.
  * [27] Herranz F.J., Santander M., Conformal symmetries of spacetimes, J. Phys. A: Math. Gen. 35 (2002), 6601–6618, math-ph/0110019.
  * [28] Howe R., Tan E.-C., Non-abelian harmonic analysis. Applications of ${SL(2,\mathbb{R}{})}$, Springer-Verlag, New York, 1992.
  * [29] Khrennikov A.Yu., Hyperbolic quantum mechanics, Dokl. Akad. Nauk 402 (2005), 170–172 (in Russian).
  * [30] Khrennikov A., Segre G., Hyperbolic quantization, in Quantum Probability and Infinite Dimensional Analysis, QP–PQ: Quantum Probab. White Noise Anal., Vol. 20, World Sci. Publ., Hackensack, NJ, 2007, 282–287.
  * [31] Kirillov A.A., Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften, Vol. 220, Springer-Verlag, Berlin – New York, 1976.
  * [32] Kirillov A.A., A tale of two fractals, see http://www.math.upenn.edu/~kirillov/MATH480-F07/tf.pdf.
  * [33] Kisil A.V., Isometric action of ${\rm SL}_{2}(\mathbb{R})$ on homogeneous spaces, Adv. Appl. Clifford Algebr. 20 (2010), 299–312, arXiv:0810.0368.
  * [34] Kisil V.V., Construction of integral representations for spaces of analytic functions, Dokl. Akad. Nauk 350 (1996), 446–448 (Russian).
  * [35] Kisil V.V., Möbius transformations and monogenic functional calculus, Electron. Res. Announc. Amer. Math. Soc. 2 (1996), 26–33.
  * [36] Kisil V.V., Towards to analysis in ${\mathbb{R}}^{pq}$, in Proceedings of Symposium Analytical and Numerical Methods in Quaternionic and Clifford Analysis (Seiffen, Germany, 1996), Editors W. Sprößig and K. Gürlebeck, TU Bergakademie Freiberg, Freiberg, 1996, 95–100.
  * [37] Kisil V.V., How many essentially different function theories exist?, in Clifford Algebras and Their Application in mathematical physics (Aachen, 1996), Fund. Theories Phys., Vol. 94, Kluwer Acad. Publ., Dordrecht, 1998, 175–184.
  * [38] Kisil V.V., Analysis in $\mathbf{R}^{1,1}$ or the principal function theory, Complex Variables Theory Appl. 40 (1999), 93–118, funct-an/9712003.
  * [39] Kisil V.V., Relative convolutions. I. Properties and applications, Adv. Math. 147 (1999), 35–73, funct-an/9410001.
  * [40] Kisil V.V., Two approaches to non-commutative geometry, in Complex Methods for Partial Differential Equations (Ankara, 1998), Int. Soc. Anal. Appl. Comput., Vol. 6, Kluwer Acad. Publ., Dordrecht, 1999, 215–244, funct-an/9703001.
  * [41] Kisil V.V., Wavelets in Banach spaces, Acta Appl. Math. 59 (1999), 79–109, math.FA/9807141.
  * [42] Kisil V.V., Spaces of analytical functions and wavelets – Lecture notes, 2000–2002, math.CV/0204018.
  * [43] Kisil V.V., Meeting Descartes and Klein somewhere in a noncommutative space, in Highlights of Mathematical Physics (London, 2000), Amer. Math. Soc., Providence, RI, 2002, 165–189, math-ph/0112059.
  * [44] Kisil V.V., Spectrum as the support of functional calculus, in Functional Analysis and Its Applications, North-Holland Math. Stud., Vol. 197, Elsevier, Amsterdam, 2004, 133–141, math.FA/0208249.
  * [45] Kisil V.V., An example of Clifford algebras calculations with GiNaC, Adv. Appl. Clifford Algebr. 15 (2005) 239–269, cs.MS/0410044.
  * [46] Kisil V.V., Starting with the group ${\rm SL}_{2}({\bf R})$, Notices Amer. Math. Soc. 54 (2007), 1458–1465, math.GM/0607387.
  * [47] Kisil V.V., Fillmore–Springer–Cnops construction implemented in GiNaC, Adv. Appl. Clifford Algebr. 17 (2007), 59–70, cs.MS/0512073.
  * [48] Kisil V.V., Erlangen program at large-2 1/2: induced representations and hypercomplex numbers, arXiv:0909.4464.
  * [49] Kisil V.V., Erlangen program at large-2: inventing a wheel. The parabolic one, Proc. Inst. Math. of the NAS of Ukraine, to appear, arXiv:0707.4024.
  * [50] Kisil V.V., Erlangen programme at large 3.1: hypercomplex representations of the Heisenberg group and mechanics, arXiv:1005.5057.
  * [51] Kisil V.V., Erlangen program at large: outline, arXiv:1006.2115.
  * [52] Kisil V.V., Biswas D., Elliptic, parabolic and hyperbolic analytic function theory-0: geometry of domains, in Complex Analysis and Free Boundary Flows, Proc. Inst. Math. of the NAS of Ukraine 1 (2004), no. 3, 100–118, math.CV/0410399.
  * [53] Konovenko N., Projective structures and algebras of their differential invariants, Acta Appl. Math. 109 (2010), 87–99.
  * [54] Konovenko N.G., Lychagin V.V., Differential invariants of nonstandard projective structures, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki (2008), no. 11, 10–13 (in Russian).
  * [55] Kurucz A., Wolter F., Zakharyaschev M., Modal logics for metric spaces: open problems, in We Will Show Them: Essays in Honour of Dov Gabbay, Editors S. Artemov, H. Barringer, A.S. d’Avila Garcez, L.C. Lamb and J. Woods, College Publications, 2005, Vol. 2, 193–208.
  * [56] Lang S., ${\rm SL}_{2}({\bf R})$, Graduate Texts in Mathematics, Vol. 105, Springer-Verlag, New York, 1985\.
  * [57] Lavrent’ev M.A., Shabat B.V., Problems of hydrodynamics and their mathematical models, 2nd ed., Nauka, Moscow, 1977 (in Russian).
  * [58] McRae A.S., Clifford algebras and possible kinematics, SIGMA (2007), 079, 29 pages, arXiv:0707.2869.
  * [59] Mirman R., Quantum field theory, conformal group theory, conformal field theory, in Mathematical and Conceptual Foundations, Physical and Geometrical Applications, Nova Science Publishers, Inc., Huntington, NY, 2001.
  * [60] Motter A.E., Rosa M.A.F., Hyperbolic calculus, Adv. Appl. Clifford Algebr. 8 (1998), 109–128.
  * [61] Olver P.J., Classical invariant theory, London Mathematical Society Student Texts, Vol. 44, Cambridge University Press, Cambridge, 1999\.
  * [62] Pavlov D.G., Symmetries and geometric invariants, Hypercomplex Numbers Geom. Phys. 3 (2006), no. 2, 21–32 (in Russian).
  * [63] Pimenov R.I., Unified axiomatics of spaces with maximal movement group, Litovsk. Mat. Sb. 5 (1965), 457–486 (in Russian).
  * [64] Porteous I.R., Clifford algebras and the classical groups, Cambridge Studies in Advanced Mathematics, Vol. 50, Cambridge University Press, Cambridge, 1995\.
  * [65] Ramsey N., Noweb – a simple, extensible tool for literate programming, available at http://www.eecs.harvard.edu/~nr/noweb/.
  * [66] Rozenfel’d B.A., Zamakhovskiĭ M.P., Geometry of Lie groups. Symmetric, parabolic, and periodic spaces, Moskovskiĭ Tsentr Nepreryvnogo Matematicheskogo Obrazovaniya, Moscow, 2003 (in Russian).
  * [67] Segal I.E., Mathematical cosmology and extragalactic astronomy, Pure and Applied Mathematics, Vol. 68, Academic Press, New York, 1976.
  * [68] Sharpe R.W., Differential geometry. Cartan’s generalization of Klein’s Erlangen program, Graduate Texts in Mathematics, Vol. 166, Springer-Verlag, New York, 1997\.
  * [69] Taylor M.E., Noncommutative harmonic analysis, Mathematical Surveys and Monographs, Vol. 22, American Mathematical Society, Providence, RI, 1986.
  * [70] Uspenskiĭ V.A., What is non-standard analysis?, Nauka, Moscow, 1987 (in Russian).
  * [71] Vignaux J.C., Durañona y Vedia A., Sobre la teoría de las funciones de una variable compleja hiperbólica, Univ. nac. La Plata. Publ. Fac. Ci. fis. mat. 104 (1935), 139–183 (in Spanish).
  * [72] Wildberger N.J., Divine proportions. Rational trigonometry to universal geometry, Wild Egg, Kingsford, 2005.
  * [73] Yaglom I.M., A simple non-Euclidean geometry and its physical basis, Springer-Verlag, New York, 1979.
  * [74] Zejliger D.N., Complex lined geometry. Surfaces and congruency, GTTI, Leningrad, 1934 (in Russian).