Many-body perturbation theory for the superconducting quantum dot: Fundamental role of the magnetic field

Abstract

We develop the general many-body perturbation theory for a superconducting quantum dot represented by a single-impurity Anderson model attached to superconducting leads. We build our approach on a thermodynamically consistent mean-field approximation with a two-particle self-consistency of the parquet type. The two-particle self-consistency leading to a screening of the bare interaction proves substantial for suppressing the spurious transitions of the Hartree-Fock solution. We demonstrate that the magnetic field plays a fundamental role in the extension of the perturbation theory beyond the weakly correlated $0$-phase. It controls the critical behavior of the $0-\pi$ quantum transition, lifts the degeneracy in the $\pi$-phase, where the limits to zero temperature and zero magnetic field do not commute. The response to the magnetic field is quite different in $0$- and $\pi$-phases. While the magnetic susceptibility vanishes in the $0$-phase it becomes of the Curie type and diverges in the $\pi$-phase at zero temperature.

Full text

Many-body perturbation theory for the superconducting quantum dot: Fundamental
role of the magnetic field
V´aclav Janiˇs∗and Jiawei Yan
Institute of Physics, Academy of Sciences of the Czech Republic,
Na Slovance 2, CZ-18221 Praha 8, Czech Republic
(Dated: February 26, 2021)
We develop the general many-body perturbation theory for a superconducting quantum dot rep-
resented by a single-impurity Anderson model attached to superconducting leads. We build our
approach on a thermodynamically consistent mean-field approximation with a two-particle self-
consistency of the parquet type. The two-particle self-consistency leading to a screening of the bare
interaction proves substantial for suppressing the spurious transitions of the Hartree-Fock solution.
We demonstrate that the magnetic field plays the fundamental role in the extension of the pertur-
bation theory beyond the weakly correlated 0-phase. It controls the critical behavior of the 0 −π
quantum transition, lifts the degeneracy in the π-phase, where the limits to zero temperature and
zero magnetic field do not commute. The response to the magnetic field is quite different in 0- and
π-phases. While the magnetic susceptibility vanishes in the 0-phase it becomes of the Curie type
and diverges in the π-phase at zero temperature.
PACS numbers: 72.10.Fk73.63.Rt,74.40.Kb,74.81.-g
I. INTRODUCTION
Nanostructures with well separated localized energy
levels are objects that can be isolated in regions of a few
nanometers or microns. They can be experimentally re-
alized by magnetic impurities on metallic surfaces [1–6],
semiconducting quantum dots [7] nanowires [8, 9], car-
bon nanotubes [10–28] or single C60 molecules [29]. They
are ideal systems for studying elementary quantum me-
chanical phenomena according to the substrates on which
they are grown or in which they are embedded due to
a detailed control of the relevant microscopic parame-
ters. When the impurity atoms with unpaired correlated
electrons are placed in metals one observes the Kondo
effect [4–6]. The correlated quantum nanostructres at-
tached to superconductors represent tunable microscopic
Josephson junctions [10, 12, 30]. The simultaneous pres-
ence of strong electron correlations on semiconducting
impurities and proximity of superconductors allow us to
observe and analyze the interplay between the Kondo ef-
fect and the formation of the Cooper pairs carrying the
Josephson current through the semiconducting nanode-
vices [8, 15–17, 21, 31–45].
Strong Coulomb repulsion on quantum dots attached
to superconducting leads may cause a local quantum crit-
ical point at which the lowest many-body eigenstates of
the system cross and a spin-singlet ground state with
the positive supercurrent (0-phase) goes over to a spin-
doublet state with a small negative supercurrent (π-
phase) [8, 15, 16, 25, 31–33, 37, 38, 41, 44, 46–49]. This
transition is associated with crossing of the Andreev
bound states (ABS) at the Fermi energy as has also been
observed experimentally [9, 23, 26].
∗janis@fzu.cz
A number of theoretical techniques have been used
to address the 0 −πtransition and related properties
of superconducting quantum dots. A very good quan-
titative agreement with the experiments [26, 28, 44]
can be obtained in a wide range of parameters using
heavy numerics such as numerical renormalization group
(NRG) [26, 38, 39, 45, 46, 50–53] and quantum Monte
Carlo (QMC) [28, 37, 44, 54, 55]. However, both NRG
and QMC demand extensive time and computational re-
sources and they do not disclose the microscopic origin of
this quantum critical behavior. They are also unable to
distinguish the physically different properties of the in-
gap states in the 0- and π-phases. Alternatively, analytic
approaches have been used mostly based on perturba-
tion expansions, either in the strength of the Coulomb
repulsion [56–61] or around the atomic limit [62–65].
The perturbation expansion in a small parameter can-
not describe any collective behavior. A self-consistent
summation of infinite series must be included to inter-
polate between weak and strong couplings needed to
describe the 0 −πtransition. Summations via self-
consistences are used both in the expansion in the
Coulomb repulsion and around the atomic limit [66]. Per-
turbation expansions around the atomic limit miss the
strong-coupling Kondo effect for narrow superconducting
gaps. The expansion in the Coulomb repulsion is well de-
fined only at zero temperature and in the weakly-coupled
spin-symmetric state of the 0-phase. The standard way
to include critical behavior and phase transitions is to use
a mean-field approximation with spin-polarized states as
a starting point for the perturbation expansion [57]. Al-
though the mean-field, Hartree-Fock approximation may
give reasonably good quantitative predictions for weak
and moderate coupling [67] it is conceptually unaccept-
able, since the real 0 −πtransition is a consequence of a
spurious critical transition to the magnetic state [66].
There is a way to improve upon the improper start of
arXiv:2102.13035v1 [cond-mat.str-el] 25 Feb 2021

2
the perturbation expansion in the interaction strength.
One has to replace the weak-coupling mean-field approx-
imation with an advanced one that is able to interpo-
late consistently between the weak and strong couplings.
It must be a self-consistent theory suppressing the spu-
rious transition to the magnetic state and reproducing
the Kondo strong-coupling regime in the impurity mod-
els. One of the authors proposed a mean-field theory
with a two-particle self-consistency that is free of any un-
physical behavior and qualitatively correctly reproduces
the Kondo limit of the single-impurity Anderson model
(SIAM) [68–72]. This mean-field approximation is a ther-
modynamically consistent extension of the weak-coupling
theory to the whole range of the input parameters.
It is the aim of this paper to apply the mean-field
approximation from references [68–72] on the Anderson
impurity attached to superconducting leads. The super-
conducting leads induce a gap on the impurity with no
states at the Fermi energy. Instead, discrete in-gap states
emerge the position of which depends on the interaction
strength and the phase difference between the attached
superconducting electrodes. The theory developed for
the SIAM with non-zero density of states at the Fermi
energy will be appropriately modified to offer a reliable
description of the models with a gap. The extension of
the mean-field approach to the singlet phase of the su-
perconducting quantum dot seems straightforward, since
it is the ground state in weak coupling. An extension to
the doublet phase with a degenerate ground state and no
weak-coupling regime appears to be more elaborate.
The many-body perturbation theory for low-energy
excitations can be used only with a unique, non-
degenerate many-body ground state. It means that a de-
generacy of the ground state in the doublet phase must be
lifted before we can apply the many-body Green function
technique and the diagrammatic expansion. The doublet
ground state is degenerate with respect to the spin reflec-
tion. We must then use a small magnetic field on the im-
purity to lift the degeneracy. We hence need to formulate
the mean-field approximation for the dot in an external
magnetic field. The properties of the superconducting
quantum dot with a Zeeman field were recently studied
experimentally [73–76] and also theoretically [65, 77–83].
The role of the Zeeman field in the perturbation theory
of the superconducting quantum dot is crucial. It al-
lows us to circumvent the quantum critical point of the
0−πtransition and to extend the many-body approach
from weak to strong coupling regimes at zero tempera-
ture. The limits to zero filed and zero temperature do
not commute in the π-phase. Moreover, the response to
the magnetic field is crucial for distinguishing between
the 0-phase with bound singlet Cooper pairs and the π-
phase with in-gap fermionic excitations carrying a local
magnetic moment. This will be demonstrated on the be-
havior of the magnetic susceptibility. This feature has
not yet been disclosed because a full consistent many-
body theory of the superconducting quantum dot with
the Zeeman field and at arbitrary temperature is still
missing.
The layout of the paper is the following. We intro-
duce the model and the Nambu formalism of the corre-
lated impurity attached to superconducting leads in Sec.
II. We introduce the basic ingredients of standard many-
body perturbation theory in Sec. III. The core of our
thermodynamically consistent mean-field approximation
with a two-particle self-consistency is presented in Sec.
IV. We apply our mean-field approximation to study the
behavior of the in-gap states and the 0 −πtransition in
Sec. V. Explicit calculations are performed in the asymp-
totic atomic limit of the infinite superconducting gap in
Sec. VI. Numerical results are presented in Sec. VII and
Sec. VIII brings concluding remarks. Less important and
elucidating technical details are presented in Appendices
A-C.
II. MODEL HAMILTONIAN AND ANDREEV
BOUND STATES
Standardly, a single impurity is used to simulate the
nanowire with separated energy levels connecting super-
conducting leads in the experimental setup. The Hamil-
tonian of the system consisting of a single impurity at-
tached to BCS superconducting leads is
H=Hdot +X
s=R,L
(Hs
lead +Hs
c),(1)
where the impurity Hamiltonian is a single-level atom
with the level energy ±for single electron (hole) and
Coulomb repulsion Uin the Zeeman magnetic field h
Hdot =X
σ=±1
(−σh)d†
σdσ+Ud†
↑d↑d†
↓d↓,(2)
where σ=±1 corresponds to spin up/down (paral-
lel/antiparallel to the applied magnetic field).
The Hamiltonian of the leads is
Hs
lead =X
kσ
(k)c†
skσcskσ
−∆sX
k
(eiΦsc†
sk↑c†
s−k↓+ H.c.) (3)
with s=L, R denoting left, right lead. Finally, the hy-
bridization term for the contacts reads
Hs
c=−tsX
kσ
(c†
skσdσ+ H.c.) .(4)
We use the Nambu spinor formalism to describe
the Cooper pairs and the anomalous functions related
with the superconducting order parameters and break-
ing charge conservation. The Nambu spinors in the su-
perconducting leads are
bϕskσ= cskσ
c†
s¯
k¯σ!,bϕ†
skσ=c†
skσcs¯
k¯σ,(5)

3
where we introduced ¯
k=−kand ¯σ=−σ.
Due to the hybridization the Cooper pairs can pene-
trate onto the impurity giving rise to anomalous impurity
Green functions. Hence, we introduce the Nambu spinors
also for the impurity (local) operators
b
φσ=dσ
d†
¯σ,b
φ†
σ=d†
σd¯σ.(6)
The individual degrees of freedom of the leads are
unimportant for the impurity quantities and we integrate
them out leaving only the impurity variables dynamical.
The fundamental function after projecting the lead de-
grees of freedom is the one-electron impurity Green func-
tion measuring (imaginary) time fluctuations that in the
Nambu formalism is a 2 ×2 matrix
b
Gσ(τ−τ0)
=− hTdσ(τ)d†
σ(τ0)i,hT[dσ(τ)d¯σ(τ0)]i
hThd†
¯σ(τ)d†
σ(τ0)ii,hThd†
¯σ(τ)d¯σ(τ0)ii!
=Gσ(τ−τ0),Gσ(τ−τ0)
¯
Gσ(τ−τ0),¯
Gσ(τ−τ0)(7)
correlating appearance of electrons and holes with spe-
cific spin on the impurity. We introduced normal parti-
cle and hole Gσ,¯
Gσpropagators that conserve spin and
anomalous Gσand ¯
GσGreen functions that create and
annihilate singlet Copper pairs in the spin-polarized so-
lution.
The electron and hole functions are connected by
symmetry relations ¯
Gσ(τ) = −G¯σ(−τ) = −G¯σ(τ)∗,
¯
Gσ(τ) = G¯σ(−τ)∗, and Gσ(τ) = −G¯σ(−τ).
The problem can be exactly solved for an impurity
without the onsite interaction, U= 0. In this case the in-
verse unperturbed propagator for the spin-polarized sit-
uation can be represented in the Nambu formalism as a
matrix. We use identical left and right hybridizations to
superconductors, tL=tR=twithout loss of general-
ity. The asymmetric situation can be transformed to a
symmetric one [84]. Due to energy conservation it is con-
venient to use Fourier transform from (imaginary) time
to frequency (energy) where the Green function can an-
alytically be continued to complex values. The matrix of
the inverse Green function for a complex energy zreads
b
G−1
σ(z) = z[1 + s(z)] + σh − , ∆ cos(Φ/2)s(z)
∆ cos(Φ/2)s(z), z[1 + σ(z)] + σh +,
(8)
where
s(z) = iΓ0
ζsgn(=z).(9)
is the “hybridization self-energy” σ(z), that is, a dynam-
ical renormalization of the impurity energy level due to
the hybridization to the superconducting leads. We ap-
proximated the Green function in the leads by its value
at the Fermi energy and denote Γ0= 2πt2ρ0being the ef-
fective hybridization strength. We denoted Φ = ΦL−ΦR
the difference between the phases of the attached super-
conducting leads and ρ0the density of states of the lead
electrons at the Fermi energy. To represent explicitly
the hybridization self-energy we introduced a new com-
plex number ζ=ξ+iη derived from the complex energy
z=x+iy by a quadratic equation ζ2=z2−∆2. Thereby
the following convention for the complex square root has
been used
ξη =xy, sgn(ξ) = sgn(x),sgn(η) = sign(y).(10)
The renormalized energy ζalong the real axis z=x±
i0 is real outside the energy gap (−∆,∆) and imaginary
within it
ζ= sgn(x)px2−∆2for |x|>∆,
ζ=±ip∆2−x2for |x|<∆.
(11)
Accordingly the hybridization self-energy is purely imag-
inary outside the gap and real within it
s(x±i0) = ±iΓ0sgn(x)
√x2−∆2for |x|>∆,
s(x±i0) = Γ0
√∆2−x2for |x|<∆.
(12)
With the above definitions the unperturbed (U= 0)
impurity Green function is
b
G(0)
σ(z) = 1
Dσ(z)
×z[1 + s(z)] + σh + , −cΦ∆s(z)
−cΦ∆s(z), z[1 + s(z)] + σh −.(13)
where we denoted cΦ= cos(Φ/2) and introduced
Dσ(z)=[z(1 + s(z)) + σh]2−2−c2
Φ∆2s(z)2
the determinant of the matrix of the inverse unperturbed
impurity Green function. It is decisive for the determi-
nation of the gap states. This determinant is real within
the gap and can goes through zero determining the gap
states that are simultaneously the Andreev states. They
are four of them ±ωσin the external magnetic field. We
denote the two independent
ωσ(1 + sσ) = −σh ±q2+c2
Φ∆2s2
σ.(14)
where we used Eq. (13) and denoted sσ=s(ωσ).
III. PERTURBATION EXPANSION:
DIAGRAMMATIC REPRESENTATION
The best way to represent the many-body perturba-
tion expansion is to use a graphical, diagrammatic rep-
resentation that can be introduced also in the Nambu

4
formalism. We start with the diagrammatic represen-
tation of the Nambu spinor of the impurity propagator
to which we assign solid lines decorated with arrows as
follows
Gσ(iωn),Gσ(iωn)
¯
Gσ(iωn),¯
Gσ(iωn)
=Gσ(iωn),Gσ(iωn)
G∗
¯σ(−iωn),−G¯σ(−iωn)
=







(15)
We used the symmetry relations of the unperturbed
Green functions that remain generally valid and read in
(complex) energy representation
¯
Gσ(iωn) = −G¯σ(−iωn) = −G∗
¯σ(iωn),(16a)
¯
Gσ(iωn) = G∗
¯σ(−iωn) = −G∗
¯σ(iωn).(16b)
We keep the time (charge) propagation (from left to
right) in the diagrammatic representation and attach the
spin up/down to the upper/lower line. Anomalous prop-
agators do not conserve charge by annihilating two elec-
trons with opposite spins (arrows against each other)
or create a Cooper pair (arrows from each other). We
can construct standard Feynman many-body diagrams
for processes induced by the Coulomb interaction of the
electrons on the impurity between two superconducting
leads. The Coulomb interaction will be represented via a
wavy line. Since the interaction is static, the interaction
wavy line is always vertical. Before we start to analyze
the diagrammatic contributions from the perturbation
expansion we resume basic exact relations.
The impact of the Coulomb repulsion on the one-
electron Green function is included in a matrix self-
energy ˆ
Σ(z) so that the full inverse propagator in the
spin-polarized situation reads b
G−1(iωn) = b
G−1
0(iωn)−
b
Σ(iωn). Its explicit component representation is
b
Gσ(iωn)
=1
Dσ(iωn)−X¯σ(−iωn),−cΦY(iωn)
−cΦY∗(−iωn), Xσ(iωn)(17)
with
Xσ(iωn) = iωn[1 + s(iωn)] + σ[h−∆Σ(iωn)]
−−Σ(iωn),(18a)
Y(iωn) = s(iωn)∆ − S(iωn).(18b)
We denoted Σ(iωn), ∆Σ(iωn) the even and odd parts
of the normal self-energy with respect to the magnetic
field and S(iωn) the anomalous superconducting part
of the interaction-induced self-energy. The even, spin-
symmetric self energy, Σ(iωn) and the anomalous one,
S(iωn), will be determined form the dynamical spin-
symmetric Schwinger-Dyson equation. The odd self-
energy ∆Σ(iωn) generalizes the classical order parame-
ters and will be related with the two-particle irreducible
vertex via a linearized Ward identity [72].
The spin-dependent determinant of the inverse of the
matrix propagator in this notation is
Dσ(iωn)
=−Xσ(iωn)X¯σ(−iωn)−c2
ΦY(iωn)Y∗(−iωn),(19)
with the electron-hole symmetry Dσ(iωn) = D−σ(−iωn).
The normal spin-dependent impurity propagators are
Gσ(iωn) = −X¯σ(−iωn)
Dσ(iωn),(20a)
¯
Gσ(iωn) = Xσ(iωn)
Dσ(iωn),(20b)
while the anomalous propagators are
Gσ(iωn) = −cΦ
Y(iωn)
Dσ(iωn),(21a)
¯
Gσ(iωn) = −cΦ
Y∗(−iωn)
Dσ(iωn).(21b)
The existence and positions of the Andreev states are
again determined from zeros of determinant Dσ(iωn).
They depend on the behavior of the normal and anoma-
lous self-energies for which we introduce a diagrammatic
expansion. We first formulate the perturbation expan-
sion in the thermodynamic language using the Matsub-
ara representation. Only after having constructed con-
tributions to the perturbation expansion and within the
selected approximations we perform analytic continua-
tion to real frequencies so that to control the behavior of
the Andreev bound states (ABS).
IV. PERTURBATION EXPANSION: REDUCED
PARQUET EQUATIONS
The basic element of the many-body perturbation ex-
pansion is the one-particle propagator. Knowing it we
determine all the physical quantities. The Dyson equa-
tion introduces the self-energy containing the whole im-
pact of the particle interactions on the one-particle prop-
agator. That is why most of the theoretical approaches
focus on the self-energy. It is, however, not the best way
to control the critical and crossover behavior from weak
to strong coupling regimes. Although it is more elaborate
and complex in its analytic structure, perturbation the-
ory applied directly to two-particle functions has gained
on popularity in recent years. The idea to extend the per-
turbation theory and its renormalizations to two-particle

5
functions is rather old [85, 86]. Presently, this general ap-
proach is used within the so-called parquet equations that
add a two-particle self-consistency [87, 88]. Generally,
the full unrestricted approximations at the two-particle
level can be solved only numerically and in the Matsu-
ubara formalism at non-zero temperatures. One has to
resort to simplifications if the critical behavior should be
controlled analytically. We developed the so-called re-
duced parquet equations to reach this objective [68–72].
The fundamental idea of this two-particle approach is to
treat approximate two-particle vertex functions and one-
particle self-energy separately and match them at the end
so that to keep the theory thermodynamically consistent
and conserving.
A. Two-particle vertex: Effective interaction
The fundamental element in the two-particle pertur-
bation theory is the two-particle vertex Γ. It has gen-
erally three dynamical variables, two fermionic iωn, iωn0,
one bosonic iνm, and two spin indices σ, σ0. An irre-
ducible vertex Λ plays the role of the two-particle self-
energy. The two-particle irreducibility is not uniquely
defined and hence there is not a unique way to select
the irreducible vertex [87]. The most important one is,
however, that from the two-particle scattering channel
leading to a singularity and a critical behavior in inter-
mediate coupling. It is the spin-singlet electron-hole scat-
tering channel. The full vertex then can be decomposed
into its irreducible Λ and reducible, K, parts
Γ↑↓(iωn, iωn0;iνm)=Λ↑↓ (iωn, iωn0;iνm)
+K↑↓(iωn, iωn0;iνm),(22)
where ωnand ωn0are energies of the incoming and outgo-
ing electron, respectively, and νmis the energy difference
between the electron and the hole that is conserved in
the multiple singlet electron-hole scatterings.
Generally, the reducible vertex in one scattering chan-
nel becomes irreducible in the other scattering chan-
nels. The parquet equations self-consistently intertwine
them to determine both irreducible and reducible parts
of the full vertex. Our approximation resorts to a two-
channel version of the parquet equations with only sin-
glet electron-hole and electron-electron multiple scatter-
ings. The sum of the series of the repeated scatterings
of particle pairs are mathematically represented by the
Bethe-Salpeter equations. The Bethe-Salpeter equation
in the electron-hole channel determines the reducible ver-
tex Kas a functional of the irreducible one Λ and is dia-
grammatically represented in Fig. 1. The irreducible ver-
tex in our approximation is determined from a reduced
Bethe-Salpeter equation that is diagrammatically repre-
sented in Fig. 2. The reduction of the full Bethe-Salpeter
equation in the electron-electron channel consists in sup-
pressing convolution of two diverging reducible vertices
KGGKso that not to destroy the possible quantum crit-
icality in the strong-coupling regime of the full vertex
Γ. The suppressed term is expected to be compensated
by higher-order terms not included in the two-channel
approximation [72].
The mean-field approximation enters these reduced
parquet equations by replacing the irreducible vertex by
a frequency and spin-independent constant Λ that then
plays a role of an effective interaction. The reducible
vertex determined by the equation of Fig 1 is
Kσ(iνm) = −Λ2φσ(iνm)
1+Λφσ(iνm),(23)
where the fermionic frequencies are ωn= (2n+1)πT and
ωn0= (2n0+ 1)πT , and the bosonic is νm= 2mπ T . We
denoted the full electron-hole bubble
φσ(iνm) = 1
βX
ωn
[G¯σ(iωn+iνm)Gσ(iωn)
+G¯σ(iωn+iνm)Gσ(iωn)] .(24)
The reduced parquet equations are justified in the
critical region of the magnetic transition, that is, in the
spin symmetric case where G↑=G↓. The mean-field
approximation must be, however, defined in the whole
representation space, including the spin-polarized state.
Since we introduced only a spin-independent renormal-
ization of the bare interaction strength, we replace the
spin-dependent bubble with its symmetric form, that is,
φσ(iνm)→φ(iνm)=(φ↑(iνm)+φ↓(iνm))/2 to determine
the effective interaction Λ. Inserting this function into
the reduced Bethe-Salpeter equation from Fig 2 leads to
"1 + 1
βX
νmK(−iνm)G↑(iωn+m)
×G↓(iωn0−m)Λ = U , (25)
which cannot, however, be satisfied for all fermionic fre-
quencies. An approximate treatment of this equation is
necessary to close the mean-field scheme.
The dominant contribution in metallic systems to ver-
tex Λ comes from the lowest Matsubara frequencies close
to the Fermi energy, that is |n|≈|n0| ≈ 0. We can
then take the lowest values near the Fermi energy at
low-temperatures as we did in the SIAM [70–72]. The
Fermi energy of the superconducting quantum dot lies in
the gap and there is no contribution from small fermionic
frequencies to screening of the interaction. The fluctu-
ations in the fermionic Matsubara frequencies may shift
the value of the critical interaction but do not affect the
universal critical behavior. We can use averaging over the
fermionic Matsubara frequencies to obtain a mean-field
(static) renormalization of the bare interaction strength
at any temperature within the same universality class
[68]. The averaging is not uniquely defined and the opti-
mal one, producing the most accurate result, depends on
the studied problem. We found that the most suitable

6
K=−Λ



+







Λ+K




FIG. 1. Diagrammatic representation of the Bethe-Salpeter equation for the reducible vertex in the electron-hole channel. The
electron-hole propagator contains simultaneous normal and anomalous propagators. The lines from the central part in the
brackets are attached to the left and right vertices (the three parts separated by brackets are mathematically multiplied) to
form two connected diagrams.
Λ=−KΛ
FIG. 2. The reduced Bethe-Salpeter equation as explained
in the text for the irreducible vertex from the electron-hole
scattering channel. The electron-electron propagator does not
contain an anomalous part due to conservation laws.
averaging scheme here is to multiply Eq. (25) by a prod-
uct G↑(−iωn0) exp(−iωn00+)G↓(−iωn) exp(−iωn0+) and
sum over the fermionic frequencies. The resulting equa-
tion for the effective interaction Λ then is
Λ = Un↑n↓
n↑n↓+ Λ2X(26)
where nσis the density electrons with spin σand
X=−1
βX
νm
ψ(iνm)ψ(−iνm)φ(−iνm)
1+Λφ(−iνm).(27)
We introduced the electron-electron bubble
ψ(iνm) = 1
βX
ωn
G↓(iωm+n)G↑(−iωn)
=1
βX
ωn
G↑(iωm+n)G↓(−iωn),(28)
which is spin independent.
Equation (26) determines the effective interaction for
the known densities nσand the screening integral X. The
explicit solution for Λ can be obtained by a substitution
with an auxiliary variable w
Λ = w−n2−m2
12wX,(29a)
where we used nσ= (n+σm)/2 with nand mbeing the
total charge and spin density, respectively. The cube of
the new variable w3satisfies a quadratic equation with a
single positive root
w3=Un2−m2
8X"1 + r1 + 1
27
n2−m2
U2X#.(29b)
The known value of wdetermined the effective interac-
tion Λ from Eq. (29a). The consistency condition for
positivity of the effective interaction is
12w2X ≥ n2−m2.(30)
Equation (29) does not, however, determine the effec-
tive interaction explicitly since integral Xdepends on
the solution. The final solution can be reached only via
iterations.
B. Thermodynamic propagators: Static
self-energies
One cannot close the equation for the two-particle ver-
tex Λ without connecting it with the one-particle den-
sities. It means that we must determine how the self-
energy in the one-particle propagators determining the
charge and spin densities is related with the two-particle
vertex from Eq. (26). We introduce two self-energies ac-
cording to their symmetry with respect to the spin reflec-
tion to keep the theory conserving and thermodynami-
cally consistent. We split the self-energy to two. One
with odd and the other with even symmetry with re-
spect to the symmetry-breaking field of the critical point
of the two-particle vertex. They will be related to the
two-particle vertex differently in approximate schemes.
The odd self-energy stands for the order parameter
emerging below the critical point with a diverging vertex.
The system with the repulsive interaction is driven in in-
termediate coupling towards a magnetic order. The odd
self-energy must then enter the Ward identity in order to
keep thermodynamic consistency between the criticality
in the two-particle vertex and the order parameter in the
symmetry-broken phase. We argued in previous publica-
tions [70–72] that it is sufficient to obey the Ward identity
only in the leading linear order in the symmetry-breaking
(magnetic) field to describe the critical behavior qualita-
tively correctly. The odd self-energy determined from the
static irreducible vertex Λ satisfying the linearized Ward
identity is
∆Σ = −Λ
2m . (31)
There is no critical behavior in the charge sector and
the even self-energy does not affect the critical behavior

7
near the transition to the magnetic state. It need not
be related to the two-particle irreducible vertex via the
Ward identity. It is responsible for the charge dynamics
and should obey the Schwinger-Dyson equation of mo-
tion. Its mean-field (static) version is just the Hartree-
Fock spin symmetric approximation. We then have
Σ0(ω) = U
2n . (32a)
Analogously the anomalous self energy, that has no odd
part, is
S0(ω) = Uν , (32b)
is proportional to the density of the Cooper pairs on the
impurity.
The components determining the one-electron Green
function of the superconducting quantum dot in the
mean-field approximation are
Xσ(ω) = ω[1 + s(ω)]
+σh+Λ
2m−+U
2n,(33a)
Y(ω)=∆s(ω)−Uν . (33b)
The charge and spin densities are
n=1
βX
ωn
eiωn0+[G↑(iωn) + G↓(iωn)] ,(34a)
m=1
βX
ωn
eiωn0+[G↑(iωn)−G↓(iωn)] ,(34b)
and the density of the Cooper pairs on the dot is
νcΦ=1
2βX
ωn
eiωn0+[G↑(iωn) + G↓(iωn)] .(34c)
The equations for the effective interaction Λ, the den-
sity of Cooper pairs ν, the charge density n, and mag-
netization mclose our mean-field approximation. It is
free of the unphysical and spurious finite-temperature
transition to the magnetic state due to the two-particle
self-consistency. It qualitatively correctly describes the
behavior of the quantum dot in weak as well as in strong
coupling, including the Kondo regime for the dot at-
tached to metallic leads. It can be applied at all tem-
peratures and also in an arbitrary magnetic field. This
mean-field approximation serves as the proper starting
point for the perturbation expansion to include dynami-
cal corrections. The mean-field one-particle Green func-
tions replace the bare propagators in the perturbation
expansion around the mean-field solution. We call them
thermodynamic propagators.
C. Spectral representation
The whole mean-field approximation can be fully
solved in the Matsubara formalism. What cannot be de-
termined from the Matsubara frequencies are the spectral
properties of the one and two-particle Green functions.
To determine also the spectral properties one has to per-
form analytic continuation to the real frequencies. One
needs to rewrite the sum over Matsubara frequencies to
integrals with Fermi and Bose distribution functions.
The one-electron Green functions have a gap around
the Fermi energy. Since the hybridization self-energy s(z)
has a square-root singularity at the gap/band edges, the
gap is fixed in the one-electron Green function and does
not depend on the interaction strength. The poles and
the band edges of the higher-order Green functions do,
however, depend on the interaction strength. We hence
must be careful when treating the two-particle functions
in the spectral representation.
The sum over the fermionic Matsubara frequencies
for the one-particle function can then be rewritten in the
spectral representation
1
βX
n
F(iωn)eiωn0+→X
i
f(ωi) Res[F, ωi]
−"Z−∆
−∞
+Z∞
∆#dω
πf(ω)=F(ω+i0) .(35)
Functions with bosonic symmetry have no gap in their
spectra at non-zero temperatures with discrete Matsub-
ara frequencies.
The Andreev bound states are determined from zeros
of the denominator Dσfrom Eq. (19). The frequencies
of the poles of the one-electron Green function in the
mean-field approximation are
ωσ(1 + sσ) = −σh+Λ
2m
+s+U
2n2
+c2
Φ(sσ∆−Uν)2.(36)
The other two frequencies of the gap states are symmet-
rically situated on the other side of the Fermi energy.
The spectral representation for the densities are
n=ng+nb=X
α,σ
f(ασωσ) Res[Gσ, ασωσ]
−X
σ"Z−∆
−∞
+Z∞
∆#dω
πf(ω)=Gσ(ω+),(37a)
m=mg+mb=X
α,σ
σf (ασωσ) Res[Gσ, ασωσ]
−X
σ
σ"Z−∆
−∞
+Z∞
∆#dω
πf(ω)=Gσ(ω+),(37b)
cΦν=cΦ(νg+νb) = 1
2X
α,σ
f(ασωσ) Res[Gσ, ασωσ]
−1
2X
σ"Z−∆
−∞
+Z∞
∆#dω
πf(ω)=Gσ(ω+).(37c)

8
We abbreviated the notation of the frequency with an
infinitesimal imaginary part ω+i0+=ω+. We split the
contributions to the densities to those from the in-gap
states, subscript gand from the band states, subscript b.
Notice that the density of the Cooper pairs from the gap
and band states is now spin dependent.
The residues of the one-electron Green function are
Res [Gσ, σσ0ωσ0] = 1
Kσ0Xσ0+σσ0+U
2n,(38a)
Res [Gσ, σσ0ωσ0] = −σ0cΦ
Kσ0
[sσ0∆−Uν],(38b)
with
Xσ=s+U
2n2
+c2
Φ(sσ∆−Uν)2,(39a)
Kσ= 2Xσ1 + ∆2sσ
∆2−ω2
σ
−2c2
Φ(sσ∆−Uν)ωσsσ∆
∆2−ω2
σ
.(39b)
The analytic representation of the two-particle Green
and vertex functions is more complex. The integrand of
the screening integral has no gap at non-zero tempera-
tures with a simple analytic representation of the sum
over bosonic Matsubara frequencies
X=−PZ∞
−∞
dx
πb(x)=ψ(x+)ψ(−x+)φ(−x+)
1+Λφ(−x+).(40)
The explicit analytic representations separating the gap
and band contributions of the electron-hole φσ(ω+) and
electron-electron ψ(ω+) are presented in Appendices A
and B.
D. Full Green function and the spectral self-energy
The spectral representation is necessary not only to
determine the positions of the in-gap states. It is gen-
erally needed to disclose the whole spectral structure of
the interacting system when we go beyond the mean-field
approximation in the perturbation expansion. The first
step beyond the static theory are dynamical corrections
to the static self-energy. The even self-energy is deter-
mined from the dynamical Schwinger-Dyson equation of
motion. Its form with the static irreducible vertex Λ is
for the normal part
ΣSp (ω+) = −UZ∞
−∞
dx
πf(x)=¯
GSp (x+)
1+Λφ(x−ω+)
−b(x)¯
GSp (ω++x)=1
1+Λφ(x+) (41a)
and analogously for the anomalous self-energy
cΦSSp (ω+) = −UZ∞
−∞
dx
πf(x)=¯
GSp (x+)
1+Λφ(x−ω+)
−b(x)¯
GSp (ω++x)=1
1+Λφ(x+).(41b)
The integrand in the Schwinger-Dyson equation con-
tains two parts, the two-particle and one-particle ones.
The former part, consisting of the electron-hole bub-
ble φ(ω+) and vertex Λ, controls the thermodynamic
response and the critical behavior. It hence must be
the same as used to determine the two-particle irre-
ducible vertex Λ and the odd self-energy ∆Σ. The
one-particle propagators GSp (ω+) and GSp (ω+) in the
Schwinger-Dyson equation carry information about the
spectral properties. Its odd self-energy ∆Σ must be
identical with that used to determine the two-particle
vertex. Its noncritical even self-energy ΣSp(ω+) can
be selected self-consistently containing the spectral self-
energy, a solution of the Schwinger-Dyson equation.
Since the Schwinger-Dyson equation determines only the
spin-symmetric self-energy we used the spin-averaged
propagators ¯
GSp (x+) = GSp
↑(x+) + GSp
↓(x+)/2 and
¯
GSp (x+) = GSp
↑(x+) + GSp
↓(x+)/2.
The one-particle propagators GSp
σand GSp
σused in the
Schwinger-Dyson equation then are
GSp
σ(ω+) = ω+σ(h−∆Σ) + + ΣSp(−ω+)
DSp
σ(ω+),(42a)
GSp
σ(ω+) = −cΦ
s(ω+)∆ − SS p (ω+)
DSp
σ(ω+)(42b)
with the denominator
DSp
σ(ω+) = ω++σ(h−∆Σ) −−ΣSp (ω+)
×ω++σ(h−∆Σ) + + ΣSp (−ω+)
−c2
Φs(ω+)∆ − SS p (ω+)2,(43)
where ∆Σ = −ΛmT/2 and mTis the magnetization cal-
culated with the thermodynamic propagator determining
the effective interaction Λ.
The normal dynamical self-energy ΣSp (ω+) and the
anomalous one SSp (ω+) from the Schwinger-Dyson equa-
tion (41) and the odd one ∆Σ from the Ward iden-
tity, Eq. (31) are the physical self-energies. It means
that a mean-field approximation at the two-particle
level generates nontrivial dynamical contributions to the
one-particle self-energy in an analogous manner as the
random-phase approximation generates a dynamical self-
energy for the Hartree-Fock mean-field thermodynam-
ics. We will analyze the dynamical corrections from the
Schwinger-Dyson equation in a separate paper.

9
V. GAP STATES AND 0−πTRANSITION
The spectral representation is needed for the deter-
mination of the positions of the in-gap states and finding
the point of their crossing signaling the 0−πtransition at
zero temperature. We need to keep the applied magnetic
field positive in order to be able to continue the solution
from the weak-coupling 0-phase to the strong-coupling π-
phase. We resort to the static mean-field approximation
to determine the 0 −πtransition.
We split the contributions from the band and gap
states and introduce the following abbreviations
U=+U
2n , (44a)
Γσ=sσ∆−Uν . (44b)
The one-electron parameters are
n−nb=ng=1
K↑K↓X
σ
Kσ[X¯σ−U∆f¯σ],(45)
m−mb=mg=1
K↑K↓X
σ
Kσ[X¯σ∆f¯σ−U],(46)
ν−νb=νg=1
2K↑K↓X
σ
KσΓ¯σ∆f¯σ,(47)
where the subscripts b, g refer to the band and gap con-
tributions, respectively. We denoted ∆fσ=f(−ωσ)−
f(ωσ). We recall that the poles of the mean-field prop-
agators Gσ(ω) and Gσ(ω) are ωσand −ω¯σ. The equa-
tions for the in-gap-state frequencies are determined in
Eq. (36).
The 0 −πtransition in the external magnetic field in
a spin-polarized state happens at ω↑= 0, that is
h+Λ
2m=s+U
2n2
+c2
Φ(sσ∆−Uν)2.(48)
This equation tells us that the effective interaction Λ af-
fects the transition only in the spin-polarized solution
with h > 0. The transition in our mean-field approxima-
tion with no spectral self-energy at h= 0 coincides with
the Hartree-Fock result. It is, however, important to re-
alize that unlike the Hartree-Fock solution the mean-field
approximation with an effective interaction Λ is free of
the spurious transition to the magnetic state at non-zero
temperatures and is thermodynamically consistent in the
whole range of the input parameters. Notice, however,
that the effective interaction does affect the position of
the 0 −πtransition in the spin-symmetric state if we
employ the spectral self-energy from Eq. (41).
A. Spin-symmetric state
We first approach the 0 −πtransition from the weak-
coupling regime in the spin-symmetric state. We then
have X0=q2
U+c2
Φ(s0∆−Uν)2with s0= Γ/(∆2−ω2
0)
and ω0(1 + s0) = X0. Further on,
K0= 2κ0X0= 2 1 + s0∆
(1 + s0) (∆2−ω2
0)
×∆ + c2
ΦUν +s0∆1−c2
ΦX0.(49)
The equation for the positive frequency of the gap state
is
(1 + s0)κ0ω0+U
2tanh βω0
22
=κ0b+U
22
+c2
Φκ2
0Γ2
0b,(50)
where we used an identity ∆f(ω) = tanh (βω/2). We
denoted b=+Unb/2 and Γ0b=s(ω0)∆ −U νb. There
is aways a solution for ω0>0 for arbitrary Uat non-
zero temperature. There is hence no crossing of the in-
gap states at non-zero temperature in the spin-symmetric
state as already observed in Ref. [59].
The in-gap-state frequency reaches the Fermi energy,
that is ω0= 0, at the 0 −πtransition only at zero tem-
perature. The spin-symmetric state can reach the critical
interaction strength Ucof the 0 −πtransition only for
βω0=∞defining a quantum critical point. The equa-
tion for the critical interaction reads
U2
c
4=1 + Γ
∆b+Uc
22
+c2
Φ1 + Γ
∆2
Γ2
0b.(51)
The equilibrium spin-symmetric solution must be sta-
ble with respect to the perturbations caused by a small
magnetic field. Its local stability is determined from the
static magnetic susceptibility. It is critically dependent
on the effective interaction of the mean-field approxima-
tion. The mean-field static susceptibility has the Stoner
form
χ=−2φ(0)
1+Λφ(0) .(52)
The denominator on the right-hand side of Eq. (52)
is positive at any temperature and non-diverging at
non-zero temperatures due to the appropriately chosen
screening of the interaction strength in the self-consistent
equation (26). The susceptibility can diverge only at zero
temperature in the π-phase as we demonstrate later.
B. Magnetic state
We introduce an effective magnetic field containing
the entire effect of the applied magnetic field in the mean-
field approximation to simplify the notation
hΛ=h1 + Λm
2h.(53)

10
The crossing of the in-gap states takes place when
ω↑= 0 for which ω↓(1 + s↓)=2hΛ. Consequently, K↑=
4hΛ(1 + Γ/∆), X2
↑=2
U+c2
Φ(Γ −Uν)2and
X2
↓=2
U+c2
Φ
Γ∆
q∆2−ω2
↓−Uν

2
,(54a)
K↓= 4hΛ



1 + Γ∆
∆2−ω2
↓3/2
×
∆−c2
Φ
Γ∆
q∆2−ω2
↓−Uν



(54b)
We further have ∆f↑= 0 at the crossing point at
non-zero temperatures and hence
U=U[K↓∆ + 2X↓(Γ + ∆)] + 4X↓(Γ + ∆)b
(Γ + ∆) (2K↓+U∆f↓),(55a)
Γ−Uν =2K↓Γb+U∆f↓(Γ −s↓∆)
2K↓+U∆f↓
.(55b)
The equation for frequency ω↓at the crossing is
∆22K↓+Utanh βω↓
22
=UK↓+ 4X↓1 + Γ
∆b+U
22
+c2
Φ2K↓Γb+Utanh βω↓
2(Γ −s↓∆)2
.(56)
The crossing leads to the 0-πtransition only at zero tem-
perature and zero magnetic field and it is a quantum crit-
ical point with a diverging magnetic susceptibility when
approached from the spin-symmetric state.
The solution of Eq. (56) shows a universal behavior
for non-zero magnetic field. We can divide all energy
variables T, U, Λ,∆, , K, X, Γ, by 2h > 0 to turn them
dimensionless. The dimensionless solution for ¯ω↓= 1 +
¯
Λmwill then become universal, independent of the actual
value of the applied magnetic field.
VI. ASYMPTOTIC ATOMIC LIMIT
The important test of reliability of the approxima-
tions is a comparison with the existing exact solu-
tions in specific limiting situations. The present two-
particle approximation with the effective interaction Λ
from Eq. (26) and the spectral self-energy from Eq. (41)
was shown to reproduce qualitatively correctly the Kondo
regime of the exact solution of the SIAM for ∆ = 0.
The opposite asymptotic atomic limit ∆ → ∞ can also
be exactly solved [33, 49, 50, 57, 66]. Here we compare
the predictions of the presented mean-field approxima-
tion with the exact results of the atomic limit with no
hybridization to the band electrons. The exact results of
the atomic limit are summarized in Appendix C.
The normal and anomalous Green functions in the
atomic limit are
Gσ(ω) = 1
2X0X0+U
ω−ωσ
+X0−U
ω+ω¯σ(57a)
Gσ(ω) = −cΦΓ
2X01
ω−ωσ−1
ω+ω¯σ.(57b)
where ¯σ=−σand
ωσ= ¯σhΛ+X0,(58a)
X0=q2
U+c2
Φ(Γ −Uν)2.(58b)
The explicit value of the electron-hole bubble is
φ(ω+) = φ↑↓∆ω
ω2
+−∆ω2,(59)
where we denoted φ↑↓ = (∆f↓−∆f↑)/2 = −φ(0)∆ω=m
and ∆ω=ω↓−ω↑. Further on,
φ(ω+)
1+Λφ(ω+)=φ↑↓ ∆ω
ω2
+−ω2
φ
,(60)
where the poles ±ωφof this function are
ω2
φ= ∆ω[∆ω−Λφ↑↓]=∆ω2[1 + Λφ(0)] .(61)
The particle-particle bubble in the atomic limit is
ψ(ω)
=(∆f↑+ ∆f↓)
8X2
0"(X0−U)2
ω+ 2X0−(X0+U)2
ω−2X0#.(62)
We further use the following identities
1
1+Λφ(0) = 1 + Λm
2h,(63a)
∆ω= 2h1 + Λm
2h(63b)
ωφ= 2hr1 + Λm
2h(63c)
to represent the screening integral

11
X=(∆f↓+ ∆f↑)2m
64X4
04X2
0−ω2
φr1 + Λm
2h((X0+U)4+ (X0−U)4
4X02X0coth βωφ
2−ωφcoth(βX0)
+X2
0−2
U2
4X2
0−ω2
φ
4X2
0+ω2
φcoth βωφ
2−4X0ωφcoth(βX0)−
βωφ4X2
0−ω2
φ
2 sinh2(βX0)


.(64)
We used ∆b(ω) = b(ω)−b(−ω) = coth(βω/2). The screening integral at half filling, U= 0, reduces to
X=(∆f↓+ ∆f↑)2m
256X3
0r1 + Λm
2h"2X0coth βhr1 + Λm
2h!−2hr1 + Λm
2h
cosh (βX0) + βX0
2 sinh2(βX0)#(65)
and in the spin-symmetric state, ω↑%ω↓=X0to
X=T χ
128X6
0
tanh2βX0
2h(X0+U)4
+ (X0−U)4+ 2 X2
0−2
U2i,(66)
where
χ= lim
h→0
m
h=2f(X0)(1 −f(X0))
T−2Λf(X0)(1 −f(X0)) (67)
is the magnetic susceptibility.
The spin symmetric solution is locally stable at all
non-zero temperatures. If we introduce a generalized
Kondo scale a= 1 + Λφ(0) and assume that the solution
approaches the critical point a&0 then the asymptotic
critical solution is
Λ = T
2f0(1 −f0),(68a)
a=T3C
128UX2
0n2f2
0(1 −f0)2(68b)
and
C=1
X4
0
tanh2βX0
2h(X0+U)4
+ (X0−U)4+ 2 X2
0−2
U2i.(68c)
The magnetic susceptibility can diverge only at zero
temperature. The critical region can, however, be
reached only if the product f0(1 −f0)≥T/U with the
decreasing temperature so that U≥Λ>0.
The spin-symmetric solution is identical with the
Hartree-Fock one. It becomes exact at zero temperature.
The boundary for the 0-phase at zero temperature form
Eq. (51) is
U2
c
4=+U
22
+c2
ΦΓ2,(69)
which is the exact result for the 0 −πtransition in the
atomic limit [49, 57].
Resolving the particle and Cooper-pair densities in
the spin-polarized state we obtain
n= 2 2X0−(∆f↓+ ∆f↑)
4X0+U(∆f↓+ ∆f↑),(70a)
ν=Γ (∆f↓+ ∆f↑)
4X0+U(∆f↓+ ∆f↑).(70b)
U=4X0δ
4X0+U(∆f↑+ ∆f↓)
=δ1−U
2Uc
(∆f↑+ ∆f↓),(71a)
Γ−Uν =4X0Γ
4X0+U(∆f↑+ ∆f↓),(71b)
where we denoted δ=+U/2 and Uc= 2pδ2+c2
ΦΓ2.
The equation for X0needed to obtain nand νis
2X0+U
2(∆f↑+ ∆f↓)2
= 4 δ2+c2
ΦΓ2=U2
c.(72)
Since X0≥0 then
2X0=Uc−U
2(∆f↑+ ∆f↓)≥0.(73)
Using this solution we obtain
n= 1 −δ
Uc
(∆f↑+ ∆f↓),(74a)
ν=Γ (∆f↑+ ∆f↓)
2Uc
,(74b)
and
m=1
2(∆f↓−∆f↑).(75)
Since ω↑=−(h+Λm/2)+X0and ω↓= (h+ Λm/2) +
X0, we have always three independent variables to de-
termine self-consistently, Λ, X0, and m. The three cou-
pled equations determining these variables are Eq. (29),

12
Eq. (73), and Eq. (75). The charge density and the
density of the Cooper pairs are then calculated from
Eqs. (74).
The equation determining the crossing of the in-gap
states in the applied magnetic field at non-zero temper-
ature is ω↑= 0 and ∆f↑= 0, hence
Uc−U
2tanh βω↓
2= 2h1 + Λm
2h=ω↓.(76)
There is always a crossing of the in-gap states for arbi-
trary interaction Uat non-zero temperature at an appro-
priate magnetic field.
The zero-temperature solution behaves differently.
An infinitesimally small magnetic perturbation generates
a fully polarized magnetic state in the π-phase, U > Uc.
The π-phase is bounded by ω↑<0, where m= 1 from
Eq. (75), n= 1, and ν= 0 from Eqs. (74) for β=∞.
The effective interaction Λ = Uat zero temperature and
the screening integral from Eq. (26) is proportional to
n↑n↓= (n−m)2/4 in the π-phase. It then means that
ω↓=Uc+Uwhile ω↑=Uc−Uwith h&0. The
Hartree-Fock solution at zero temperature is exact also
in the π-phase at zero temperature.
There is a fundamental difference between the 0-phase
and the π-phase and we can distinguish the two phases
by the low-temperature asymptotics of the magnetic sus-
ceptibility. We have 2X0=Uc−Uin the leading
low-temperature asymptotics in the 0-phase, that is, for
U < Uc. Consequently, the magnetic susceptibility from
Eq. (67) at low temperatures is
χ.
=8
Te−β(Uc−U),(77)
which reflects the Meissner effect due to the presence of
the singlet Bound states (ABS).
The situation in the π-phase, U > Uc, is quite differ-
ent. Equation (73) leads at low temperatures to
βX0= arctan Uc
U1−4T U
U2−U2
c.(78)
The magnetic susceptibility is
χ=U2−U2
c
2U2
1
a(79)
with the effective interaction and the Kondo scale from
Eqs. (68) are
Λ = 2UT
U2−U2
c
,(80)
a=UU 2
c
2 (U2−U2
c)2arctan−2Uc
UT . (81)
The magnetic susceptibility follows the Curie law due
to the presence of the local magnetic moment of the
fermionic excitations on the dot. The non-universal Curie
constant
C=U2−U2
c3
2U3U2
c
arctan2Uc
U(82)
is in the static mean-field approximation overestimated
and grows with the increasing interaction strength. It
lies above the exact value and will be corrected by the
dynamical spectral self-energy.
The observed behavior of the in the strong-coupling
limit for U > Ucdiscloses another feature of the solu-
tion in the external magnetic filed. We proved that the
limit to zero magnetic field and to zero temperature do
not commute. If we keep the magnetic field non-zero
and limit the temperature to zero the solution behaves
analytically and continuously reaches the fully saturated
state at zero temperature. On the other hand, if we keep
the temperature non-zero and switch off the magnetic
field we stay in the spin-symmetric states down to zero
temperature where the magnetic susceptibility diverges
and an infinitesimally small magnetic field lifts the de-
generacy to a magnetically saturated state.
VII. NUMERICAL RESULTS
We apply the mean-field approximation in the atomic
limit to show its similarities and stress the substantial
differences to the Hartree-Fock solution and to test the
reliability of our mean-field approximation in different
regimes. The major asset of the mean-field approxima-
tions is that they can be used in the whole range of the
model parameters. They are qualitatively reliable if they
do not lead to an unphysical and spurious behavior. We
know that the reduced parquet equations with a two-
particle self-consistency reproduce qualitatively correctly
the limit of the zero gap. It is instructive to apply it
in the opposite limit of the infinite superconducting gap
which is the least fitting situation for the application of
the many-body construction. Many of the qualitative
features of the solution in the atomic limit are generic
and mimic the behavior of the finite-gap model except
for the Kondo limit of the vanishing gap.
A. Spin-symmetric solution
Our static mean-field solution in the spin-symmetric
state, that is in the absence of the magnetic field, coin-
cides with the Hartree-Fock approximation. This may
seem a limiting factor, but it holds only at the one-
particle level when the dynamical corrections in the spec-
tral self-energy from the Schwinger-Dyson equation are
neglected. The first, and the most important difference
between our mean-field theory and the Hartree-Fock ap-
proximation in the spin-symmetric state is the stability
with respect to small fluctuations of the magnetic field,
which is reflected in the magnetic susceptibility.

13
012345
T
/
0.0
0.2
0.4
0.6
0.8
1.0
U
=Exact
HF
RPE
012345
T
/
100
102
104
U
= 4 Exact
HF
RPE
FIG. 3. Magnetic susceptibility as a function of tempera-
ture in the spin-symmetric state at half-filling in the 0- phase
(U= Γ) and the π-phase (U= 4Γ) for the phase differ-
ence Φ = 0. Hartree-Fock (HF), reduced parquet equations
(RPE) and exact (EXACT) solutions are compared. The un-
physical instability with the diverging susceptibility makes the
Hartree-Fock mean-field solution in strong coupling unreliable
at low temperature@‘s.
012345
T
/
0.95
0.96
0.97
0.98
0.99
1.00
U
=
RPE
HF
012345
T
/
0
1
2
3
4
U
= 4
RPE
HF
FIG. 4. Different temperature behavior of the vertex renor-
malization in the spin-symmetric state at half filling in the
0-phase and the π-phase for the phase difference Φ = 0.
We plotted the magnetic susceptibility of the spin-
symmetric state at half filling as a function of tempera-
ture in Fig. 3. We compared the two mean-field approx-
imations, our, based on the reduced parquet equations
(RPE), and the Hartree-Fock one (HF), with the exact
solution in the atomic limit. There is no big difference in
the 0-phase where all solutions asymptotically approach
zero at zero temperature. Quite a different behavior is,
however, observed in the π-phase. Both the exact and
RPE solutions lead to a divergent susceptibility at zero
temperature, while the HF solution predicts an unphys-
ical critical point with diverging susceptibility at a tem-
perature of order of the hybridization strength Γ that is
taken as the energy unit. The magnetic susceptibility is
a physical, measurable quantity being able to distinguish
the character of the in-gap states. The in-gap states in
the 0-phase are bound pairs, ABS singlets being insensi-
tive to small magnetic perturbations. The in-gap states
in the π-phase carry a local magnetic moment and react
strongly on magnetic perturbations. The whole π-phase
displays a Curie susceptibility diverging at zero temper-
ature.
The reason why the RPE suppress the HF instabil-
ity is the two-particle self-consistency renormalizing the
bare interaction strength Uto a screened effective one Λ.
012345
T
/
0.1
0.2
0.3
0.4
0.5
U
=Exact
HF & RPE
012345
T
/
0.00
0.05
0.10
0.15
0.20
0.25
U
= 4 Exact
HF & RPE
FIG. 5. Density of the Cooper pairs νin the spin-symmetric
state at half filling as a function of temperature in the 0-phase
and the π-phase for the phase difference Φ = 0.
012345
T
/
0.0
0.2
0.4
0.6
0.8
1.0
0
U
=
U
= 4
U
= 8
FIG. 6. Temperature dependence of the positive energy of
the in-gap state in the spin-symmetric solution for weak and
strong interactions for the phase difference Φ = 0.
We plotted its temperature dependence at half filling in
Fig. 4. The renormalization gets stronger with the de-
creasing temperature but starts abating in the zero phase
and dies out at zero temperature. The effective interac-
tion approaches zero, maximizing the renormalization of
the interaction, in the π-phase consistent the divergence
of the magnetic susceptibility of the exact solution.
The thermodynamic mean-field solution with a static
self-energy produces good results for quantities with odd
symmetry and sensitive to the symmetry-breaking field.
It is quantitatively less accurate in determining the spin-
symmetric one-particle quantities with even symmetry
with respect to spin flips. We plotted the temperature
dependence of the Cooper-pair density νat half filling in
the RPE/HF approximation together with the exact re-
sult in Fig. 5. We can see how the static spin-symmetric
value deviates from the exact value at low temperatures
of the π-phase. Unlike the HF mean-field, the RPE offer
a direct improvement by including the dynamical cor-
rections from the Schwinger-Dyson equation (41). It
uses the renormalized interaction and this interaction
is strongly renormalized at low temperatures of the π-
phase. The Cooper-pair density will then better follow
the dependence of the effective interaction as observed
in the 0-phase in Fig. 4. We discuss the impact of the
dynamical corrections to the static self-energy in detail
elsewhere.

14
0.0 0.2 0.4 0.6 0.8 1.0
1.00
0.75
0.50
0.25
0.00
0.25
0.50
0.75
1.00
T
= 0
+
+
0.0 0.2 0.4 0.6 0.8 1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
T
= 0.05
+
+
0.0 0.2 0.4 0.6 0.8 1.0
0.4
0.2
0.0
0.2
0.4
T
= 0.1
+
+
0.0 0.2 0.4 0.6 0.8 1.0
0.75
0.50
0.25
0.00
0.25
0.50
0.75
T
=
+
+
FIG. 7. In-gap-state energies as a function of the phase differ-
ence Φ between the superconducting leads in a weak magnetic
field h= 0.01Γ for different temperatures at half filling and
U= Γ. The critical angle of the crossing increases with tem-
perature.
The spin-symmetric solution of the many-body
Green-function approach cannot be extended to the π-
phase at zero temperature since one has to cross the
quantum critical point. One can nevertheless circumvent
the critical point in that one extends the spin-symmetric
solution to non-zero temperatures. There is no criti-
cal point at non-zero temperature and the solution can
be extended continuously from weak to strong coupling
[59]. No crossing happens and the energy of the in-gap
state remains positive and approaches the quantum criti-
cal point at zero temperature, as demonstrated in Fig. 6.
The spin-symmetric solution in the π-phase becomes un-
stable there and decays into the degenerate spin doublet
with a saturated local magnetic moment.
B. Zeeman field
The magnetic field acting on the spin of the electrons
(Zeeman field) plays an essential role in the application
of the many-body Green functions in the superconduct-
ing quantum dot. It is needed to approach the zero-
temperature solution in the π-phase and to see the cross-
ing of the in-gap states at non-zero temperatures. The
doublet ground state, π-phase, is degenerate and the Zee-
man field is the means to lift the degeneracy. Here we
analyze the properties of the low-temperature solution
with an applied magnetic field.
We plotted the dependence of the in-gap-state ener-
gies on the phase difference between the attached super-
conducting leads in Fig. 7 and on the interaction strength
in Fig. 8 for a very small magnetic field h= 0.01Γ at dif-
ferent temperatures, The value of the magnetic field at
the crossing increases with temperature. The curves of
the in-gap-state energies are continuous due the presence
of the symmetry-breaking magnetic field. We also plot-
0 2 4 6 8 10 12
6
4
2
0
2
4
6
T
= 0
+
+
0 2 4 6 8 10 12
6
4
2
0
2
4
6
T
= 0.1
+
+
0 2 4 6 8 10 12
U
/
6
4
2
0
2
4
6
T
= 0.5
+
+
0 2 4 6 8 10 12
U
/
1.0
0.5
0.0
0.5
1.0
T
=
+
+
FIG. 8. In-gap-state energies as a function of the Coulomb
repulsion Uin a weak magnetic field h= 0.01Γ for different
temperatures at half filling and phase difference Φ = 0. The
critical angle of the crossing increases with temperature.
15 10 5 0 5 10 15
( +
U
/2)/
10
5
0
5
10
+
+
FIG. 9. In-gap-state energies as a function of the impurity
energy level for a low temperature, T= 0.05Γ, Zeeman
field h= 0.2Γ, interaction U= 8Γ, and the phase difference
Φ = π/2 .
ted the dependence of the in-gap-state energies on the
impurity energy level +U/2 for strong coupling, U= 8Γ
and phase difference Φ = π/2. We used a small magnetic
field h= 0.2Γ to demonstrate the expected behavior in
the π-phase, cf. Fig. 9. Note that the RPE solution re-
produces the exact positions of the in-gap states in the
limit T→0 followed by the limit h→0.
The low-temperature asymptotics of magnetization m
shows just the opposite dependence on temperature than
the Cooper-pair density νin the weak Zeeman field (h=
0.1Γ) in both the 0-phase and the π-phase, see Fig. 10.
The magnetization vanishes and νsaturates in the 0-
phase in both mean-field approximations as well as in the
exact one. In the π-phase the temperature asymptotics
is inverted in both quantities in all solutions. The HF
mean-field better simulates the exact dependence of the
Cooper-pair density νwhile the RPE mean-field then
better fits the exact magnetization curve.
The renormalized interaction Λ in the Zeeman field

15
012345
0.00
0.02
0.04
0.06
0.08
0.10
m
U
=Exact
HF
RPE
012345
0.0
0.2
0.4
0.6
0.8
1.0
U
= 4 Exact
HF
RPE
012345
T
/
0.1
0.2
0.3
0.4
0.5
U
=Exact
HF
RPE
012345
T
/
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
U
= 4
Exact
HF
RPE
FIG. 10. Temperature dependence of magnetization mand
the Cooper-pair density νat half filling, h= 0.1Γ, and Φ = 0
in the 0-phase (U= Γ) and the π-phase (U= 4Γ) for the
mean-field solutions RPE and HF compared with the exact
behavior. We see that the exact behavior of the magnetiza-
tion, with odd symmetry with respect to spin flip, is better
reproduced by the RPE while the Cooper-pair density, with
even symmetry, is better reproduced by the HF approxima-
tion.
012345
T
/
0.95
0.96
0.97
0.98
0.99
1.00
U
=
RPE
HF
012345
T
/
2.5
3.0
3.5
4.0
U
= 4
RPE
HF
FIG. 11. The temperature dependence of the renormalized in-
teraction strength Λ at half filling in a Zeeman field h= 0.1Γ,
phase difference Φ = 0 in weak and strong couplings. The ver-
tex Λ shows the same rescaled dependence demonstrating that
there is no difference between weak and strong interaction in
the presence of the magnetic field.
has a different low-temperature asymptotics in the π-
phase than in the spin-symmetric case, see Fig. 11. Once
the magnetic field is kept non-zero down to zero temper-
ature the effective interaction approaches the bare value
and the exact solution is reproduced.
Although the HF solution quantitatively better ap-
proximates the exact behavior of the thermodynamic
quantities with even symmetry with respect to spin flips,
it fails to reproduce the exact limit of the vanishing Zee-
man field since it predicts a non-zero magnetization at
zero field below its critical transition to the magnetic
state as documented in Fig. 12. It is our mean-field that
qualitatively correctly reproduces all the limits of both
one and two-particle thermodynamic quantities.
0.0 0.2 0.4 0.6 0.8 1.0
h
/
0.0
0.2
0.4
0.6
m
U
=
Exact
HF
RPE
0.0 0.2 0.4 0.6 0.8 1.0
h
/
0.0
0.2
0.4
0.6
0.8
1.0
U
= 4
Exact
HF
RPE
FIG. 12. Magnetic-field dependence of magnetization min
weak (U= Γ) and strong (U= 4Γ) couplings for Φ = 0 and
T= 0.5Γ and half filling. The strong-coupling HF solution is
below its magnetic critical point is completely off in the limit
to zero magnetic field unlike the RPE solution.
VIII. CONCLUSIONS
The quantum dot attached to superconducting leads
poses a challenging problem to the perturbation theory
with many-body Green functions. First, electron corre-
lations on the dot lead to a line of first-order transitions
from the spin-singlet to the spin-doublet state that ends
up at a quantum critical point at zero temperature where
the in-gap states reach the Fermi energy. Moreover, the
doublet state is degenerate and it cannot be continuously
approached from the weak-coupling spin-singlet state.
The basic assumptions of the applicability of the many-
body perturbation theory is a non-degenerate ground
state and the existence of an analyticity region from the
weak-coupling limit within which it can be applied. It
cannot cope with two independent many-body equilib-
rium states with a first order transition between them. It
may lead to a new equilibrium state, phase, only if there
is a divergence accompanied by a continuous symmetry
breaking. It means that the many-body perturbation ex-
pansion cannot reliably be applied to the superconduct-
ing quantum dot at low temperatures around the quan-
tum critical point unless a self-consistency is introduced.
A self-consistency must be introduced in the many-body
perturbation expansion in order to deal with the quan-
tum critical behavior. A static mean-field approximation
is the simplest way to achieve this goal.
It has been known for long that the weak-coupling
Hartree-Fock self-consistency is not appropriate to deal
with the quantum critical point of the superconducting
quantum dot since it fails at non-zero temperatures where
it leads to a spurious transition to a magnetically ordered
state without the external magnetic field. We added a
two-particle self-consistency to the HF solution in that
we replaced the bare interaction with a renormalized,
screened one. We thereby suppressed the HF spurious
transition to the magnetic state and produced a fully
thermodynamically consistent mean-field approximation
applicable in the whole range of the input parameters.
We demonstrated that it is able to deal qualitatively cor-
rectly with the quantum critical behavior of the 0 −π

16
transition in the superconducting dot as well as with the
Kondo limit of the dot attached to metallic leads.
The most important finding of our mean-field theory
is the manifestation of the fundamental role of the Zee-
man field in the analytic description of the 0 −πtransi-
tion and in distinguishing the spin-singlet from the spin-
doublet. The magnetic field not only lifts the degener-
acy of the π-phase it allows us to determine the different
character of the in-gap states in the two phases. The
0−πtransition is signaled by a crossing of the energies
of the in-gap states. The in-gap states in the spin-singlet
phase are the genuine Andreev bound states of two elec-
trons with opposite spins that are insensitive to small
magnetic perturbations. The low-lying excitations in the
spin-doublet phase are fermions, carry a local magnetic
moment, and are sensitive to the magnetic filed. The
magnetic susceptibility vanishes in the 0-phase and di-
verges in the π-phase at zero temperature. The equi-
librium state at non-zero temperatures turns magnetic
only when the Zeeman field is applied. Consequently, the
weak-coupling spin-symmetric solution can continuously
be extended to strong coupling at non-zero temperatures
without crossing any critical point. The limit to zero field
leads to a magnetic state only at zero temperature above
the critical interaction strength of the 0 −πtransition.
The limits to zero magnetic field and to zero temperature
do not commute and the results depend on their order
in which they are performed. The Zeeman field plays
the role of a symmetry-breaking field we know from con-
tinuous phase transitions and the π-phase beyond the
quantum critical point mimics the ordered phase in the
lattice models. It means that the 0−πtransition happens
only at zero temperature and zero magnetic field and it
is a true local quantum phase transition. The crossing
of the in-gap states in the magnetic field or at non-zero
temperatures is noncritical with no phase transition.
The mean-field theory presented in this paper is the
first fully consistent analytic approximation that can de-
scribe not only the critical behavior of the 0−πtransition
but it can qualitatively correctly and reliably reproduce
the behavior of the quantum dot attached to both super-
conducting and normal leads in the whole range of the
model parameters. It was derived within the perturba-
tion expansion for two-particle vertices to control their
critical behavior. It offers a starting point for adding dy-
namical corrections in a systematic way. The first step,
without changing the static irreducible vertex, is to use
the Schwinger-Dyson equation to determine the spectral
properties of the model. This opens a new way to in-
clude dynamical fluctuation in the thermodynamic and
spectral properties with the controlled renormalizations
of the one- and two-particle functions.
ACKNOWLEDGMENT
Research on this problem was supported in part by
Grants 19-13525S of the Czech Science Foundation. VJ
thanks the INTER-COST LTC19045 Program of the
Czech Ministry of Education, Youth and Sports for finan-
cial support. We thank Tom´aˇs Novotn´y for illuminating
discussions.
Appendix A: Spectral representation - Electron-hole
bubble
The thermodynamic quantities including the effec-
tive interaction can be calculated entirely in the Mat-
subara formalism without the necessity to continue ana-
lytically to real frequencies. If we want, however, to use
the Schwinger-Dyson equation and determine the spec-
tral properties we need a spectral representation of the
two-particle bubbles, at least the electron-hole one.
We decompose the imaginary part of the electron-hole
bubble φ(ω+) to a sum of three contributions =φ(ω+) =
=φbb(ω+) + =φbg (ω+) + =φgg(ω+), according to whether
the arguments of the Green functions of the integrand
lie both within the band, one within the band and one
within the gap, and both within the gap, respectively.
We have for ω > 0
=φbb(ω+) = −X
σ"Z−∆−ω
−∞
+Z−∆
min(∆−ω,−∆)
+Z∞
∆#dx
2π[f(x)−f(x+ω)] [=Gσ(x++ω)=G−σ(x+)
+=Gσ(x++ω)=G−σ(x+)] ,(A1a)
=φbg(ω+) = −X
σ"Zmin(∆−ω,−∆)
−∆−ω
+Z∆
max(∆−ω,−∆)#dx
2π[f(x)−f(x+ω)] [=Gσ(x++ω)=G−σ(x+)
+=Gσ(x++ω)=G−σ(x+)] ,(A1b)
=φgg (ω+) = −X
σZmax(∆−ω,−∆)
−∆
dx
2π[f(x)−f(x+ω)] [=Gσ(x++ω)=G−σ(x+) + =Gσ(x++ω)=G−σ(x+)] ,(A1c)

17
and for ω < 0
=φbb(ω+) = −X
σ"Z−∆
−∞
+Zmax(∆,−∆−ω)
∆
+Z∞
∆−ω#dx
2π[f(x)−f(x+ω)] [=Gσ(x++ω)=G−σ(x+)
+=Gσ(x++ω)=G−σ(x+)] ,(A2a)
=φbg(ω+) = −X
σ"Zmin(∆,−∆−ω)
−∆
+Z∆−ω
max(∆,−∆−ω)#dx
2π[f(x)−f(x+ω)] [=Gσ(x++ω)=G−σ(x+)
+=Gσ(x++ω)=G−σ(x+)] ,(A2b)
=φgg (ω+) = −X
σZ∆
min(∆,−∆−ω)
dx
2π[f(x)−f(x+ω)] [=Gσ(x++ω)=G−σ(x+) + =Gσ(x++ω)=G−σ(x+)] .(A2c)
The subscript at ω+=ω+i0+denotes the way the real
axis is reached from the complex plane.
The real part of the bubble is then determined from
the Kramers-Kronig relation
<φ(ω) = PZ∞
−∞
dx
π=φ(x+)
x−ω+φ(∞) (A3)
Appendix B: Spectral representation -
Electron-electron bubble
The electron-electron bubble has a simpler spectral
representation. It is not needed for the spectral self-
energy, but its spectral representation is useful the de-
termination of the effective interaction Λ at low temper-
atures with a high precision. It has no contribution from
anomalous Green functions. Its imaginary part can be
represented as
=ψ(ω+) = −Z∞
−∞
dx
π[fx)−f(x−ω)]
× =G↑(ω+−x)=G↓(x+) (B1)
Taking into account the induced gap on the dot we can
represent the three contribution from the band and gap
states.
=ψbb(ω+) = −"Zmin(−∆,ω−∆)
−∞
+Z∞
max(∆,ω+∆)#dx
π[f(x)−f(x−ω)] =G↑(ω+−x)=G↓(x+),(B2a)
=ψbg(ω+) = −"Zmax(−∆,ω−∆)
min(−∆,ω−∆)
+Zmax(∆,ω+∆)
min(∆,ω+∆) #dx
π[f(x)−f(x−ω)] =G↑(ω+−x)=G↓(x+),(B2b)
=ψgg (ω+) = −Zmin(∆,ω+∆)
max(−∆,ω−∆)
dx
π[f(x)−f(x−ω)] =G↑(ω+−x)=G↓(x+).(B2c)
Appendix C: Atomic limit - Exact solution
We summarize the basic results of the exact solution
of the atomic limit with infinite superconducting gap.
The atomic Hamiltonian is a matrix
H=


0 0 0 ΓcΦ
0d+h0 0
0 0 d−h0
ΓcΦ0 0 2d+U


.(C1)
One can simply diagonalize the Hamiltonian matrix,
Eq.(C1), by observing that the central 2×2 sub-block is
decoupled from the others. As a result, the 4 eigenstates
are summarized in the following: (i) E−
d=d−hwith the
eigenstates |1,0i; (ii) E+
d=d+hwith the eigenstates
|0,1i; (iii) E−
s=1
2h2d+U−p(2d+U)2+ 4Γ2c2
Φi
with the eigenstates 1
√Γ2c2
Φ+(E−
s)2(ΓcΦ|0,0i+E−
s|1,1i);
and (iv) E+
s=1
2h2d+U+p(2d+U)2+ 4Γ2c2
Φiwith

18
the eigenstates 1
√Γ2c2
Φ+(E+
s)2(ΓcΦ|0,0i+E+
s|1,1i).
The general thermodynamic quantity is
Q=1
ZX
ihEi|ˆ
Q|Eiie−β(Ei−µNi),(C2)
where Eiand |Eiiare the eigenvalues and correspond-
ing eigenstates of the Hamiltonian, β= 1/kBT, and the
partition function is
Z=e−βE−
s+e−βE+
s+e−β(d+h)+e−β(d−h).(C3)
The thermodynamic properties can, alternatively, be
derived from the derivatives of the grand potential F=
−1
βln Z. The charge and spin densities are
n=1
Z"X
σ
e−β(d−σh)+2(E−
s)2
Γ2c2
φ+ (E−
s)2e−βE−
s
+2(E+
s)2
Γ2c2
φ+ (E+
s)2e−βE+
s#,(C4a)
m=1
Z"X
σ
σe−β(d−σh)#.(C4b)
The density of the Cooper pairs is
ν=1
ZΓE−
s
Γ2c2
Φ+ (E−
s)2e−βE−
s
+ΓE+
s
Γ2c2
Φ+ (E+
s)2e−βE+
s.(C5)
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