# Anisotropy-driven topological quantum phase transition in magnetic
impurities

G. G. Blesio  Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
Instituto de Física de Rosario (CONICET) and Facultad de Ciencias Exactas,
Ingeniería y Agrimensura, Universidad Nacional de Rosario, Rosario, Argentina
L. O. Manuel  Instituto de Física de Rosario (CONICET) and Facultad de
Ciencias Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario,
Rosario, Argentina  A. A. Aligia  Instituto de Nanociencia y Nanotecnología
CNEA-CONICET, GAIDI, Centro Atómico Bariloche and Instituto Balseiro, 8400
Bariloche, Argentina<EMAIL_ADDRESS>
###### Abstract

A few years ago, a topological quantum phase transition (TQPT) has been found
in Anderson and Kondo 2-channel spin-1 impurity models that include a hard-
axis anisotropy term $DS_{z}^{2}$ with $D>0$. The most remarkable
manifestation of the TQPT is a jump in the spectral density of localized
electrons, at the Fermi level, from very high to very low values as $D$ is
increased. If the two conduction channels are equivalent, the transition takes
place at the critical anisotropy $D_{c}\sim 2.5\;T_{K}$, where $T_{K}$ is the
Kondo temperature for $D=0$. This jump might be important to develop a
molecular transistor. The jump is due to a corresponding one in the Luttinger
integral, which has a topological non-trivial value $\pi/2$ for $D>D_{c}$.
Here, we review the main results for the spectral density and highlight the
significance of the theory for the interpretation of measurements conducted on
magnetic atoms or molecules on metallic surfaces. In these experiments, where
$D$ is held constant, the energy scale $T_{K}$ is manipulated by some
parameters. The resulting variation gives rise to a differential conductance
$dI/dV$, measured by scanning-tunneling spectroscopy, which is consistent with
a TQPT at an intermediate value of $T_{K}$. We also show that the theory can
be extended to integer spin $S>1$ and two-impurity systems. This is also
probably true for half-integer spin and non-equivalent channels in some cases.

  * December 2023

##  1 Introduction

Systems with individual magnetic atoms [1] or molecules [2, 3, 4, 5, 6] on
metallic surfaces are being studied extensively in the last years due to their
peculiar properties and potential application in spintronics and molecular
electronics. A fundamental component within an integrated circuit is the
transistor, which can be realized in these magnetic impurity systems through
the controlled switching of the electric current by varying some physical
parameters [4].

A realistic analysis of magnetic impurity systems requires taking into account
their potential multiorbital nature and hybridization with more than one
conduction channel. Indeed, a plethora of electronic states, ranging from
Landau-Fermi liquid to singular Fermi liquid and non-Fermi liquid, emerges
depending on the specific value of the spin of the magnetic impurity and the
number of conduction channels [7]. An important, although often overlooked,
physical ingredient of multiorbital systems is the single-ion magnetic
anisotropy $DS_{z}^{2}$ due to the spin-orbit coupling, enhanced in the low-
symmetry arrangements of atoms or molecules on surfaces.

As an example of the relevance of single-ion anisotropy, some years ago, it
has been found that the conductance of a system composed of a Ni impurity in a
Au chain doped with oxygen has a jump as a function of the anisotropy $D$ of
the spin 1 of the Ni atom, suggesting that the system could act as a
transistor [8, 9]. The underlying model is the 2-channel spin-1 Anderson model
with anisotropy (2CS1AMA) or its integer valent limit, the anisotropic two-
channel spin-1 Kondo model (A2CS1KM). Both models are described in Section 2.
With such models, it can be seen that the jump is due to a topological quantum
phase transition (TQPT) between two phases that differ in the value of the so-
called Luttinger integral $I_{L}$, whose zero value had been for decades a
hallmark of a Fermi liquid [10]. For low $D/T_{K}$ where $T_{K}$ is the Kondo
temperature for $D=0$, the system is in the topologically trivial phase with
$I_{L}=0$, characterized by a large spectral density of localized electrons at
the Fermi level and large conductance at low temperatures and bias voltage.
For large $D/T_{K}$ the system is topologically non-trivial with
$I_{L}=\pi/2$, with a pronounced dip in the conductance and spectral density
of localized electrons at low energies. This phase has been called “non-Landau
”Fermi liquid, because it cannot be adiabatically connected to a non-
interacting system for which $I_{L}=0$ [8, 10].

For degenerated channels and zero magnetic field $B=0$, the transition takes
place at the critical anisotropy $D_{c}\sim 2.5\;T_{K}$. However, the TQPT
still exists for $B\neq 0$ or inequivalent channels with $B=0$ [11]. For non-
equivalent channels and $B\neq 0$, the system is a topologically trivial
ordinary Fermi liquid, but a crossover from a dip to a peak can be induced by
modifying a parameter like $B$ [11] and has been actually observed, as
explained in Section 4.1.

In this article, we review the main results concerning the above mentioned
TQPT. While the original model was proposed for a particular system that has
not been realized up to date [12], we show that several systems with magnetic
atoms or molecules on metallic surfaces can be actually described by the
2CS1AMA, or the simpler A2CS1KM, providing a consistent and unified
description of several experiments. Alternatives theories, sometimes
inconsistent between them or physically unjustified, have been proposed to
interpret the outcome of those experiments. This is probably due to the fact
that the theory is rather recent, and that to describe the TQPT, a very
accurate technique like the numerical renormalization group (NRG) is required,
which is computationally expensive for two or more channels. So far, no other
technique has been able to capture the TQPT. We also show that the theory can
be extended to a larger spin and two-impurity systems.

The paper is organized as follows. In Sec. 2 we introduce the two models,
Kondo and Anderson, for the spin-1 impurity coupled with two conduction
channels. In Sec. 3 the NRG predictions for the localized electron spectral
density are presented and discussed in the context of the TQPT. In Sec. 4, the
differencial conductance ($dI/dV$) measurements for five different systems are
discussed and compared with the theoretical calculations: 4.1) FePc on
Au(111), 4.2) MnPc on Au(111), 4.3) nickelocene on Cu(100), 4.4) Fe atoms on
MoS2/Au(111) and 4.5) Fe porphyrin molecules on Au(111). In Sec. 5 the theory
is generalized for $S>1$ and two-impurity systems. Finally, we conclude with a
brief summary in Sec. 6.

##  2 Models

The simplest model to describe the TQPT is the anisotropic two-channel spin-1
Kondo model (A2CS1KM). It takes the form

$H_{K}=\sum_{k\tau\sigma}\varepsilon_{k\tau}c_{k\tau\sigma}^{\dagger}c_{k\tau\sigma}+\sum_{k\tau\sigma\sigma^{\prime}}\frac{J_{K\tau}}{2}c_{k\tau\sigma}^{\dagger}\vec{\sigma}_{\sigma\sigma^{\prime}}c_{k\tau\sigma^{\prime}}\cdot\vec{S}+DS_{z}^{2},$
(1)

where $c_{k\tau\sigma}^{\dagger}$ creates a conduction electron with point-
group symmetry $\tau$ (channel index), spin $\sigma$ and remaining quantum
numbers $k$. The first term describes the substrate conduction bands, the
second term is the Kondo exchange interaction between conduction electrons and
the localized spin $\vec{S}$ with exchange couplings $J_{K\tau}$, and the last
term is the single-ion uniaxial magnetic anisotropy. $\vec{\sigma}$ is the
vector of Pauli matrices. The simplest case is when the two channels are
equivalent: both are degenerate ($\varepsilon_{k\tau}=\varepsilon_{k}$) and
the localized spin is equally coupled to them ($J_{K\tau}=J_{K}$).

When intermediate valence of the magnetic impurity is included, the model is
the 2-channel spin-1 Anderson model with anisotropy (2CS1AMA). It can be
written in the form used first for Ni compounds with holes in the $xz$ and
$yz$ orbitals [9, 13, 14]. Extension to other cases, for example, FePc on
Au(111) [11] are straightforward. Neglecting the pair-hopping term [9], which
is irrelevant in all cases considered so far as the intra-orbital Coulomb $U$
repulsion is considerably larger that the inter-orbital $U^{\prime}$ one, the
Anderson Hamiltonian is

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{k\tau\sigma}\varepsilon_{k}c_{k\tau\sigma}^{\dagger}c_{k\tau\sigma}+\sum_{k\tau\sigma}\left(v_{\tau}{c}_{k\tau\sigma}^{\dagger}{d}_{\tau\sigma}+\mathrm{H.c.}\right)+$
(2) $\displaystyle+$ $\displaystyle\sum_{\tau\sigma}\epsilon
d_{\tau\sigma}^{\dagger}d_{\tau\sigma}+\sum_{\tau}Un_{\tau\uparrow}n_{\tau\downarrow}+U^{\prime}n_{xz}n_{yz}-J_{H}{\vec{S}}_{xz}\cdot{\vec{S}}_{yz}+DS_{z}^{2},$

where $d_{\tau\sigma}^{\dagger}$ ($c^{\dagger}_{k\tau\sigma}$) creates a hole
with energy $\epsilon$ ($\varepsilon_{k}$) in the $d$ orbital $\tau$
(conduction band $\tau$ with momentum $k$), with $\tau=xz,yz$.
$n_{\tau\sigma}=d_{\tau\sigma}^{\dagger}d_{\tau\sigma}$ and
$n_{\tau}=\sum_{\sigma}n_{\tau\sigma}$. $v_{\tau}$ is the tunneling or
hybridization amplitude between impurity and conduction states (assumed to be
independent of $k$), while $J_{H}$ is the strong Hund ferromagnetic exchange
responsible for the spin 1 of the impurity.

For small hybridization and when the two-particle configuration dominates, the
model (2) reduces to the A2CS1KM [9].

##  3 Results

For equivalent channels, it has been found that the TQPT in the A2CS1KM takes
place for anisotropy $D_{c}\sim 2.5\;T_{K}$, where $T_{K}$ is the Kondo
temperature for $D=0$ [9]. The evolution of the spectral density for localized
states $\rho(\omega)$ with $D$ for constant $T_{K}$ is shown in Fig. 11 of
Ref. [9], where an abrupt change at the Fermi level signals the critical value
$D_{c}$

In Fig. 1 we represent $\rho(\omega)$ for different Kondo exchange couplings
$J_{K}$, keeping $D=0.0027\;W$ constant, where $W$ is the half-bandwidth of
the conduction bands, taken as the energy unit. $r\equiv J_{K}/J_{Kc}$, where
$J_{Kc}$ is the critical Kondo coupling for the given $D$ ($J_{K}$ such that
$D\simeq 2.5\;T_{K}$). The numerical calculations were performed with the
Ljubljana code of the NRG [15, 16]. We assume flat conduction bands extending
from $-W$ to $W$ for both symmetries.

Figure 1: (Color online) Spectral density of localized electrons of the
A2CS1KM as a function of energy for several values of $r=J_{K}/J_{Kc}$, where
$J_{Kc}$ is the value of $J_{K}$ at the TQPT. $\rho_{0}$ is given by the
ordinary Friedel sum rule with vanishing Luttinger integral [11]. Vertical
dotted lines are at $\omega=\pm D$. $\omega=0$ corresponds to the Fermi level.

For $r$ significantly larger than 1, the spectral density of localized states
is similar to that of an ordinary Kondo peak. For $r$ slightly larger than 1,
$\rho(\omega)$ has the form of a narrow peak mounted on a broad peak. The
latter, at the Fermi level, has around half the magnitude expected for the
usual compensated Kondo model and therefore, it is similar to the spectral
density expected for the spin-1/2 two-channel Kondo model. However, as long as
$r>1$, the system satisfies the ordinary Friedel-Langreth sum rule (the small
deviation in the figure is due to numerical errors of the NRG) and $\rho(0)$
has its maximum possible value. However, as soon as $r<1$, $\rho(0)$ jumps to
0. This is due to a jump in the Luttinger integral from 0 to $\pi/2$ at $r=1$
[8, 11], as discussed below. If $r$ is decreased further, the shape of the
spectral density tends to be rectangular, with two jumps at $-D$ and $D$,
typical of inelastic scattering when only one channel is present [17, 18, 19,
20]. These steps are overbroadened in Fig. 1 due to technical reasons that
limit the resolution of the NRG at large energies [21].

A detailed description of what happens with the Friedel-Langreth sum rule and
the Luttinger integrals in the more general case, which includes intermediate
valence, different channels, and a magnetic field has been explained in detail
in the Supplemental Material of Ref. [11]. To avoid including many technical
details, we outline the main facts for the simpler case of equivalent
conduction channels and zero magnetic field in the two-orbital Anderson model,
for which the Luttinger integral $I_{L}$ does not depend on the spin and
channel quantum numbers.

Using conservation laws, it can be shown that the spectral function of the
localized states for each orbital and spin, at the Fermi level and $T=0,$ is
given by

$\rho_{\tau\sigma}(\omega=0)=\frac{1}{\pi\Delta}\sin^{2}(\delta_{\tau\sigma}),$
(3)

where $\Delta=\pi v^{2}\rho_{c}$, with $v$ the hybridization and $\rho_{c}$
the density of conduction states assumed independent of energy. The phase
shift suffered by the conduction electrons at the Fermi level due to the
presence of the impurity is

$\delta_{\tau\sigma}=\pi\langle n_{\tau\sigma}\rangle-I_{L}.$ (4)

The Luttinger integral $I_{L}$ (independent of orbital and spin indices in the
simplest case) is defined as

$I_{L}={\rm Im}\int_{-\infty}^{0}d\omega
G^{d}_{\tau\sigma}(\omega)\frac{\partial\Sigma^{d}_{\tau\sigma}(\omega)}{\partial\omega},$
(5)

where $G^{d}_{\tau\sigma}(\omega)$ is the impurity Green function for orbital
$\tau$ and spin $\sigma$, and $\Sigma^{d}_{\tau\sigma}(\omega)$ is the
corresponding self energy.

For a long time, $I_{L}$ has been assumed to vanish for a Fermi liquid, based
on perturbation calculations starting from a non-interacting electronic system
[10, 22, 23]. However, rather recently [24, 25] it has been found that this is
not always the case for local Fermi liquids. A topological interpretation of
$I_{L}$ was provided for extended systems in Ref. [26] and extended to the
impurity case in Ref. [8]. Previously a non-trivial value of $I_{L}$ has been
found in underscreened one-channel spin-1 models, for which the system is a
singular Fermi liquid [27].

An explicit calculation has shown that in the “non-Landau ”phase, for large
$D/T_{K}$, $I_{L}=\pi/2$ [11]. In the Kondo limit, $\langle
n_{\tau\sigma}\rangle=1/2$ and Eq. (3) gives $\rho_{\tau\sigma}(0)=0$ in this
phase, whereas in the ordinary Fermi liquid phase, with $I_{L}=0$, the
spectral density at the Fermi level has its maximum possible value
$\rho_{\tau\sigma}(0)=1/(\pi\Delta)$. This explains the jump observed in Fig.
1 at the TQPT.

##  4 Experimental relevance of the topological quantum phase transition

In this section, we present several systems of atoms or molecules on metallic
surfaces, and we show that scanning-tunneling spectroscopy experiments
performed on them can be explained qualitatively or semiquantitatively by
means of the two-channel $S=1$ Kondo or Anderson models with anisotropy. In
most of these cases, alternative explanations were proposed in the literature
that contradict basic physical principles or are unsatisfactory. As the theory
of the topological quantum phase transition is relatively new and it seems to
be only captured by NRG calculations, while alternative explanations fail, it
is important to show that a consistent explanation using new concepts and
models exists. This is our focus, and not to provide an accurate fit of all
the experimental curves mentioned below, an objective that lies beyond the
scope of this paper.

###  4.1 FePc on Au(111)

The system of iron phthalocyanine (FePc) on the Au(111) surface has attracted
a lot of attention during the last 15 years [11, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38]. According to LDA + U calculations, the basic electronic
structure of the molecule is that shown in Fig. 2, taken from Ref. [36]. The
partially filled orbitals of Fe are those of symmetry $3z^{2}-r^{2}$ with
nearly one electron, and the degenerate so-called $\pi$ orbitals, of symmetry
$xz$ and $yz$ with three electrons, resulting in a spin 1. This is in
agreement with X-ray magnetic circular dichroism experiments [29].

Figure 2: (Color online) Electronic structure of MnPc and FePc. Reprinted with
permission from Ref. [36]. Copyright 2020 American Chemical Society.

The differential conductance $dI/dV$ observed by scanning-tunneling
spectroscopy of a single FePc molecule on Au(111) in the on-top position shows
a narrow dip mounted on a broad peak [31, 34, 37] (corresponding to the curves
at the bottom of Figs. 3 and 4). This is very suggestive of the spectral
density of localized electrons near the topological transition, for anisotropy
$D$ slightly larger than the critical one [8, 9] (See Fig. 1). However, most
experiments were done before the development of the theory of the TQPT and
were interpreted in a different fashion, as a two-stage Kondo effect: the
$3z^{2}-r^{2}$ orbitals hybridize strongly with the conduction electrons with
the same symmetry, giving rise to a first-stage Kondo effect and to the broad
peak in $dI/dV$ around 20 meV. At a lower temperature, the Kondo effect due to
the $\pi$ orbitals sets in giving rise to a dip of half-width $\sim 0.6$ meV
in $dI/dV$ [31, 35]. The fits of the spectrum suggest the following hierarchy
of the different orbitals in decreasing order of hopping amplitude to the tip:
3d${}_{3z^{2}-r^{2}}$ , conduction electrons with $3z^{2}-r^{2}$ symmetry,
conduction electrons with $\pi$ symmetry and 3dπ [35].

However, an experiment that can discern between both scenarios (TQPT or two-
stage Kondo effect) has been made. Raising the FePc molecule from the surface
[34], the hybridization amplitudes are weakened and, with them, the exchange
interactions between localized 3d electrons and conduction electrons. In the
two-stage scenario explained above one expects that both features, the broad
peak and the Kondo dip _narrow_ since the corresponding Kondo temperatures
should decrease. Instead, if the system is a non-Landau Fermi liquid close to
the topological transition, decreasing the exchange interactions with respect
to $D$ should _broaden_ the dip as the system moves away from the TQPT. This
last scenario is what is observed experimentally (see Fig. 3), giving support
to the TQPT picture.

Figure 3: Left: experimental differential conductance of FePc on Au(111) as a
function of voltage, taken from Fig. 2 (b) of Ref. [34]; and right:
theoretical spectral density of 3d $3z^{2}-r^{2}$ electrons as a function of
energy, as the molecule is raised from the surface, taken from Supplementary
Fig. 3 of Ref. [11]. Parameters are $D=0.005$ $J_{z}=0.44f$, $J_{a}=0.22f$ in
units of the band width taken as 1 eV. From top to bottom, the factor $f$ used
is 0.1, 0.2, 0.5, 0.6, 0.7 and 0.8.

To construct the adequate model for FePc, one has to take into account the
splitting of the $\pi$ orbitals as a consequence of the spin-orbit coupling
(SOC) [38], neglected in previous treatments. The states $|\pi\sigma\rangle$
with one hole in the $\pi$ orbitals are (except for an irrelevant phase)

$\displaystyle|a$ $\displaystyle\uparrow$
$\displaystyle\rangle=\frac{|xz\uparrow\rangle+i|yz\uparrow\rangle}{\sqrt{2}},|a\downarrow\rangle=\frac{|xz\downarrow\rangle-i|yz\downarrow\rangle}{\sqrt{2}},$
$\displaystyle|b$ $\displaystyle\uparrow$
$\displaystyle\rangle=\frac{|xz\uparrow\rangle-i|yz\uparrow\rangle}{\sqrt{2}},|b\downarrow\rangle=\frac{|xz\downarrow\rangle+i|yz\downarrow\rangle}{\sqrt{2}}.$
(6)

The $|b\sigma\rangle$ states lie above the $|a\sigma\rangle$ by an energy of
the order of the SOC, estimated in 76 meV for Fe [39]. This also leads to a
significant orbital polarization which was observed [29], and to an anisotropy
$D\sim 5$ meV.

Therefore, an appropriate model to describe the system is the anisotropic two-
channel spin-1 Kondo model (A2CS1KM), one channel with strong exchange
coupling $J_{z}$ for the $3z^{2}-r^{2}$ electrons and another channel of the
$a$ states $J_{a}$. The effect of raising the molecule is incorporated in our
model by reducing both $J_{z}$ and $J_{a}$ by the same factor $f$. In this
way, the experiments can be semiquantitatively explained [11], as shown in
Fig. 3.

In Ref. [37] the dependence of $G=dI/dV$ with temperature and magnetic field
has been measured. The results have been also explained using the A2CS1KM
[11]. As the temperature is raised, the dip is reduced and disappears at $\sim
10$ K. The dependence with the magnetic field shown in Fig. 4 is striking: the
narrow dip is converted into a narrow peak as the magnetic field is increased.
The theoretical results, taken from Ref. [11], reproduce semiquantitatively
the experimental data. To take into account the asymmetry of the shape, we
assume that the STM tip senses mainly the localized electrons of symmetry
$\tau=3z^{2}-r^{2}$ with some admixture of conduction electrons with the same
symmetry weighted by the parameter $q$ [40], which we take as $q=0.4$. The
conductance given by our model $G_{m}$, represented at the right of Fig. 4 is
therefore given by [11]:

$G_{m}(V)=-\left[(1-q^{2}){\rm Im}G_{\tau\sigma}^{d}(\omega)+2q{\rm
Re}G_{\tau\sigma}^{d}(\omega)\right],$ (7)

where $G_{\tau\sigma}^{d}(\omega)$ is the Green function of localized
electrons for symmetry $\tau$ and spin $\sigma$.

A better agreement can probably be obtained by enlarging $J_{z}$, which has
the effect of broadening the broad peak, adjusting $J_{a}$ to broaden the dip
a little bit, and by including the orbital polarization, which increases the
effective coupling with the magnetic field by a factor of 3/2.

Figure 4: Experimental (left) and theoretical (right) differential conductance
of FePc on Au(111) as a function of voltage for several values of the magnetic
field. The experimental curves are taken from Fig. 2 (b) of Ref. [37], while
the theoretical curves are taken from Supplementary Fig. 4 of Ref. [11].
Parameters as in Fig. 3 with $f=1$ and asymmetry parameter $q=0.4$.

###  4.2 MnPc on Au(111)

Curiously, in spite of having one electron less than FePc, the observed
differential conductance in Mn phthalocyanine on Au(111) [41] is qualitatively
very similar to that observed in the Fe system. There is a dip of half-width
about 0.5 mV, mounted on a broad peak. Under the application of a magnetic
field, the dip turns to a peak at $\sim 4$ Tesla, and for a larger magnetic
field the peak splits (see Fig. 5). The latter behavior was not observed in
the FePc system, but is expected if larger fields were applied in that case.

Figure 5: (Color online) Differential conductance $dI/dV$ of MnPc on Au(111)
as a function of magnetic field $B$ and bias voltage $V_{sd}$, taken from Fig.
2(d) of Ref. [41].

It has been suggested that the observed behavior can be explained by a singlet
ground state and a triplet excited state with a small excitation energy [41].
However, on one hand, the Hund coupling in 3d transition-metal elements are of
the order of 0.7 eV, favoring a total spin 3/2. On the other hand, specific
calculations for the singlet-triplet model with a small triplet excitation
energy shows a $dI/dV$ that decreases slightly with increasing voltage $V$ for
small $V$ (see Fig. 7 of Ref. [42]), in constrast to the experimental
observations (see Fig. 5).

The calculated electronic structure for planar MnPc, is different from that in
the gas phase (represented in Fig. 2) and corresponds to the intermediate-spin
quartet ${}^{4}E_{g}$ $[(xy)^{1}(\pi)^{3}(3z^{2}-r^{2})^{1}]$ [43, 44]. This
is in agreement with polarization-dependent N K-edge x-ray absorption spectra
for MnPc on Au [45]. Therefore, the difference with the electronic structure
for FePc on Au(111) is that, in the Mn system, there is a hole in the 3dxy
which is absent in the Fe system, leading to total spin 3/2 in the Mn case.

A realistic model for MnPc on Au(111) involves therefore three channels
[states with symmetry $3z^{2}-r^{2}$, $a$ of Eq. (6) and $xy$] and is almost
intractable with NRG. In order to have a qualitative understanding, we have
studied a spin 3/2, two-channel Kondo model including anisotropy, assuming
$J_{z}\sim 2J_{a}$ and $J_{xy}=0$. From the temperature dependence of the
conductance, we see a first-stage Kondo effect in which the contribution of
the dominant $3z^{2}-r^{2}$ channel is the usual one for a spin 1/2 system,
saturating near $2e^{2}/h$ at low temperatures, without a dip. However, the
contribution of the $a$ channel has a dip, so that the total conductance
presents a narrow dip mounted on a broad peak. The ground state is a singlet.

The results can be qualitatively understood as follows. At temperatures of the
order of the Kondo temperature of the dominant $3z^{2}-r^{2}$ channel (near 20
meV in the Fe system), the spin of that channel is screened and one is left
with a spin 1 screened partially by the exchange in the $a$ channel and with
anisotropy $D$. This model has been studied and, in presence of any $D>0$ [19,
20], the conductance and the spectral density have a dip, whose width
decreases exponentially with $\sqrt{D}$, while the application of a magnetic
field leads to a differential conductance of the same form as that shown in
Fig. 5 of Ref. [20]. If, in the effective model after screening the spin 1/2
of the dominant channel, one includes the exchange of the third channel $xy$,
the model is precisely the A2CS1KM, and one expects a topological quantum
phase transition at finite $D$. The parameters should be renormalized, as
expected from approximate treatments of similar 3-channel models [35].

###  4.3 Nickelocene on Cu(100)

The system of isolated double-decker nickelocene (Nc) molecules on Cu(100)
substrates have been experimentally studied in detail [13, 46, 47, 48, 49,
50]. Density functional theory (DFT) calculations show that the electronic
structure of Ni is basically 3d8, with one hole in each of the nearly
degenerate $\pi$ orbitals ($xz$ and $yz$), with some mixing with the 3d9
configuration [48, 13]. Therefore the appropriate model to describe the system
is actually the 2CS1AMA proposed for Ni impurities in a Au chain doped with
oxygen [8, 9] or its integer-valent limit, the A2CS1KM [9].

Figure 6: (Color online) Differential conductance of several samples of Nc on
Cu(111) (a) [(b)] correspond to spectra observed with frequency 2/3 [1/3],
approximately. Within each panel, the higher a curve is, the shorter the
distance from the molecule to the substrate. Taken from Fig. 5 of Ref. [13].
Copyright 2020 American Physical Society.

The experimentally observed spectra for the differential conductance $dI/dV$
are shown in Fig. 6. Left and right panels correspond to spectra observed with
frequency 2/3 (case A) and 1/3 (cases B), respectively. The curves within each
panel correspond to different positions of the tip. As the STM tip is
approached to the molecule, the hybridization between tip and molecule states
increases and a jump from a dip to a peak near $V=0$ takes place, which is
consistent with the TQPT.

In the theory that was presented alongside the experimental observations, the
ocurrence of a peak or a dip has been tentatively ascribed to a crossover in
the spin of the molecule. This transition is noted as shifting from 1/2 in the
contact regime (STM tip near the molecule) to 1 in the tunneling regime (STM
tip far from the molecule), a deduction based on first-principle calculations
[13, 48]. However, on one hand, these calculations miss relevant dynamical
correlations and, therefore, they do not properly treat the Kondo effect which
tends to screen the spin. On the other hand, the electronic structure does not
change much between the two regimes and, as admitted by the authors, the
change in the molecular charge is actually insufficient to account for the
large change in the spin.

The observed spectra have many similarities with the spectral density of
localized electrons in the A2CS1KM (see Fig. 1) but also important
differences. For case A, there seems to be a first-order transition as the tip
is approached to the molecule, avoiding the transition zone with a very narrow
peak or dip, as shown in Fig. 1. For case B, the transition seems to be
continuous, without a jump from a dip to a peak at zero voltage, as the
hybridization (or exchange) between localized and conduction electrons is
increased.

The first-order transition of case A can be understood as follows: as in FePc
on Au(111), in which the molecule is raised when the STM approaches it [34],
we expect that some variable $\eta$ which determines either the position or
the shape of the molecule, modifies the hybridization $v$ of the 2CS1AMA [Eq.
(2) assuming $v_{\tau}=v$, the same for both channels] [14], leading to a
coupling of $\eta$ with our electronic model. In the absence of this coupling,
one expects that the elastic energy is $E_{e}=K\eta^{2}/2$ (shifting the 0 of
$\eta$ if necessary). It has been shown that the second derivative of the
energy of the electronic model is strong and negative near the TQPT [14]. This
means that for a soft spring (small $K$) the second derivative of the total
energy is also negative at the TQPT, leading to a first-order transition in a
Maxwell construction [14]. This reasoning provides a natural explanation of
the observed behavior for case A.

Figure 7: (Color online) Differential conductance as a function of voltage for
different values of $\Delta$. Other parameters are $U=3.5$, $U^{\prime}=2.5,$
$J_{H}=0.5$, $\epsilon=-3.0$, $T=0.0005$ [see Eq. (2)].

Case B probably corresponds to a hard spring (large $K$) and the first-order
transition does not take place. The reason why an abrupt jump like that in
Fig. 1 is not observed is two-fold: i) the magnitude of the jump decreases
with the degree of intermediate valence [see Eqs. (3) and (4)] and ii) finite
temperature. This is the most relevant parameter. These effects were
investigated using the 2CS1AMA described by Eq. (2) [14]. In Fig. 7 we show
the evolution of the differential conductance for different values of
$\Delta=\pi v^{2}\rho_{c}$ where $\rho_{c}=1/(2W)$ is the density of
conduction electrons assumed constant in the range $-W<\omega<W$. The half-
band width $W=$ 1 eV is taken as the unit of energy. For small $\Delta$,
$dI/dV$ has a dip mounted on a broader peak, as usual. As $\Delta$ increases,
the dip narrows, but in contrast to the case of zero temperature, the minimum
of the dip increases, and a very sharp dip like that of Fig. 1 is absent. For
larger $\Delta$ the dip gradually disappears and the magnitude of $dI/dV$ near
zero voltage increases. The overall behavior reproduces semiquantitatively the
experimental results for case B shown in the right panel of Fig.6.

###  4.4 Fe Atoms on MoS2/Au(111)

Trishin et al. [51] studied experimentally a system consisting of an Fe atom
on top of a monolayer of MoS2 deposited in turn on a Au(111) surface. As
argued below, it is very natural to expect that the system is described by the
A2CS1KM. MoS2 on Au(111) forms a Moiré structure, which implies strong local
variations of the density of conduction electrons $\rho_{c}$. Therefore,
depending on the specific position at which the Fe adatom is located, dramatic
variations of the adimensional parameter $J^{\prime}=\rho_{c}J_{K}$ (which
determines the Kondo temperature) are expected, and one might expect to
observe the TQPT as in Fig. 1.

Figure 8: (Color online) Differential conductance corresponding to the
experimental curves (b) to (f) (full lines, each curve is labeled as
exp(b),etc.) and the results for our model (dashed lines,shifted upward by the
same magnitude as the corresponding experimental curve). Note that (a) is not
included and the top figure corresponds to (b). See the main text for the
meaning of the $r$, $q$ and $f$ fitting parameters. Figure taken from Ref.
[14]. Copyright 2023 American Physical Society

The validity of the A2CS1KM to describe the system can be justified as
follows. DFT calculations of Fe atoms on free-standing MoS2 indicates that the
spin state of the atoms is either $S=1$ [52] or $S=2$ [53]. However, for
$S=2$, one would expect a second jump in $G(V)$ at larger $|V|$ in the regime
of low $J^{\prime}$, which is not observed experimentally [Fig. 3(h) of Ref.
[51]]. On the other hand, experiments and DFT calculations indicate that the
Fe atoms are located in positions with symmetry corresponding to the point
group $C_{3v}$. Therefore, the Fe $3d$ orbitals are split into one $A_{1}$
singlet and two $E$ doublets [54]. Our comparison with the experiment (shown
in Fig. 8), indicates that the spin 1 is formed by occupying the two states of
an $E$ doublet (the agreement worsens when non-equivalent channels are
considered). In this case, it is clear that the spin-orbit coupling originates
a hard axis anisotropy $D(S_{z})^{2}$ with $D>0$ [14]. In addition, each of
the degenerate orbitals of the doublet hybridizes with conduction states of
the same symmetry [8, 9, 14]. This reasoning naturally leads to the 2CS1AMA
with degenerate channels used in Refs. [8, 9, 14] and to the A2CS1KM in the
integer valence limit.

The differential conductance $G(V)=dI/dV$ has been measured on nearly 40
different Fe positions. Six of them [(a) to (f)] are presented in Ref. [51]
and five of them [(b) to (f)] are reproduced in our Fig. 8. In Ref. [51], fits
of the different spectra were done using three different approaches: i)
perturbation theory in the exchange coupling in an $S=1$ anisotropic one-
channel Kondo model for cases (a) to (d), ii) a Frota peak (expected for the
simplest Kondo model) for case (f), and iii) a Lorentz peak plus Frota dip
(without justification) for case (e). Note that the fit of the peak and the
dip requires 3 parameters for each one (determining position, width and
intensity) in addition to a linear background. Therefore the resulting good
fit is not surprising [51], but it lacks a physical justification.

In contrast, as shown in Fig 8, the A2CS1KM can semiquantitatively explain the
data in a unified fashion. A better agreement with the experiment can be
obtained by allowing some degree of intermediate valence for cases (e) and
(f), as explained below, but we wanted to keep the number of free parameters
as minimal as possible.

The numerical calculations were performed with the Ljubljana code of the NRG
[15, 16], and were reported previously in Ref. [55]. We assume flat conduction
bands extending from $-W$ to $W$ for both symmetries. We take $W=1$ eV and
$D=2.7$ meV. The product $J^{\prime}=\rho_{c}J_{K}$ is assumed to vary among
the different cases, due to the Moiré modulation. The TQPT is at
$J^{\prime}_{c}\sim 0.135$. The different theoretical curves in Fig. 8
correspond to different values of the ratio $r=J^{\prime}/J^{\prime}_{c}$.

The structure at low voltage $V$ of the differential conductance $G(V)=dI/dV$
is determined by the localized and conduction electrons of symmetry $\tau$
included in the model. We assume that the STM tip senses mainly the localized
$3d$ states with some admixture of conduction states. Thus, the contribution
of the model to $G(V)$ at zero temperature is given by Eq. 7, where the Green
functions $G_{\tau\sigma}^{d}(\omega)$ depend only on $r$, and $q$ is a
measure of the contribution of the conduction states. In the experiment, there
is also a linear background due to the contribution of other states, and
$G_{m}(V)$ is affected by a factor $f$ which depends on the distance of the
STM tip to the system. Therefore, to fit the experiment, the following
expression is used

$G(V)=fG_{m}(V)+A+BV,$ (8)

which contains five parameters ($r,f,q,A$ and $B$), but the shape of each
curve depends essentially on $r$, while $q$ controls the asymmetry. We have
not included in the comparison with experiments the curve (a), which is
similar to a rectangular dip formed by two step-like functions, because these
steps are overbroadened in our calculations due to the limited resolution of
the NRG at large energies [21].

The agreement between theory and experiment for cases (e) and (f) could be
considerably improved by introducing intermediate valence effects. This is in
fact expected since a larger conduction density of states increases the
parameter $\Delta=\pi v^{2}\rho_{c}$ discussed in Section 4.3, which controls
the degree of intermediate valence. It is well known that a smaller occupancy
of the localized states shifts the Kondo peak to higher energies. Concerning
case (e), the dip in the theory is more pronounced than in the experiment.
However, as shown in Section 4.3, intermediate valence and finite temperature
reduces the magnitude of the dip compared to that predicted by the A2CS1KM.

The comparison with the experiment can also be affected by the assumption of a
constant density of conduction states and the effect of other orbitals not
considered in our model. In any case, the experimental and theoretical
results, including the LDA and NRG ones, strongly suggests that the underlying
physics is that of the 2CS1AMA.

###  4.5 Fe porphyrin molecules on Au(111)

Experiments similar to those for the Nc molecule on Cu(100) (see Section 4.3),
in which the STM tip is approached to the molecule, have been carried out for
iron porphyrin molecules on Au(111) [56]. The LDA calculations indicate that
the Fe spin is 1, with a partial occupancy of two orbitals (of symmetry
$3z^{2}-r^{2}$ and $x^{2}-y^{2}$), that belong to different irreducible
representations. In other words, the two channels are not equivalent. One
expects that the A2CS1KM or the corresponding Anderson model with non-
equivalent channels, should be appropriate for the system. In these
experiments, the dip narrows as the contact regime is approached, but never
turns to a peak, in contrast to Nc on Cu(100). Nevertheless, the results are
consistent with our model assuming a weaker hybridization of the localized
states with the substrate in comparison to Nc/Cu(100).

Figure 9: (Color online) Differential conductance obtained at different
positions (II and III) of a chain of three Fe-porphyrin-based molecules on top
of herringbone-reconstructed Au(111) containing a Br atom. The presence of a
peak (dip) signals a molecule in a “Kondo” (“spin-flip”) position. The solid
curves correspond to phenomenological fits of the data. Reprinted with
permission from [57]. Copyright 2023 American Chemical Society.

.

More recently similar experiments have been carried out using a Br decorated
Au(111) surface [57]. In this case, depending on the particular position of
the molecule with respect to the defects, sometimes a broad dip is observed
and sometimes a narrow peak (see Fig. 9), consistent with the A2CS1KM for
parameters near the TQPT on the non-trivial and trivial topological sectors,
respectively.

##  5 Extension to larger spin and two-impurity systems

For impurities involving transition metal ions, the localized spin $S$ can
vary in the interval $0\leq S\leq 5/2$. Clearly, for $S=0$ there is no Kondo
effect. The number of channels can also vary reaching up to 5 channels. Since
treating more than two channels is computationally too expensive with the NRG,
we restrict the present study to two channels, assuming for the moment
equivalent channels.

The effects of anisotropy when only one channel is present was studied before
[17, 18, 19, 20]. For $S=1$, the transition occurs at $D_{c}=0^{+}$ [19, 20]
and the Luttinger integral takes the value $\pi/2$ at this point [27]. For any
single-channel $S\geq 1$ impurity, there is a competition between the
underscreened Kondo effect [7] and the single-ion anisotropy for $D>0$, giving
rise to a complex low energy behavior that strongly depends on the integer or
half-integer nature of $S$ and the $D=0$ Kondo temperature $T_{K}$ (see Table
1).

For two channels and $S=1/2$, the anisotropy is irrelevant and one has a non-
Fermi liquid behavior corresponding to the spin-1/2 two-channel Kondo model
(2CKM) [58]. The $S=3/2$ case has been studied before with NRG [59, 60], and
it was found that the two-channel Kondo effect also describes its low-energy
physics as in the $S=1/2$ case. On the other hand, for $S=2$ we find a similar
topological transition as for $S=1$. Therefore, the low-temperature behavior
can be divided into two groups depending if the spin is integer or a half-
integer. Table 1 presents a summary of the different cases.

$\\#$ ch. | $S$ | D | Low energy electronic state
---|---|---|---
1 | 1/2 | irrelevant | fully compensated KE, ordinary FL
1 | integer | $D_{c}=0^{+}$ | two-stage KE:
|  |  | effective (anisotropic) S=1/2 KE for $D<T_{K};$
|  |  | quenched impurity spin for $D>T_{K}$ [17]
1 | half-integer | $D_{c}=0^{+}$ | two-stage KE: quenched impurity spin for $D<T_{K};$
|  |  | effective and complex S=1/2 KE for $D>T_{K}$ [17]
2 | 1/2 | irrelevant | two-channel KE: non-Fermi liquid [7]
2 | 1 | $D_{c}>0$ |  $D<D_{c}$: fully compensated KE
|  |  |  $D>D_{c}$: quenched impurity spin, non-Landau FL [8]
2 | 3/2 | $D_{c}=0^{+}$ | effective two-channel (anisotropic) KE [59]
2 | 2 | $D_{c}>0$ |  $D<D_{c}$: fully compensated KE
|  |  |  $D>D_{c}$: quenched impurity spin, non-Landau FL [8]
Table 1: Low energy states for a magnetic impurity of spin $S$ with $D>0$ and
coupled to $\\#ch.$ equivalent conduction channels. KE = Kondo effect; FL =
Fermi liquid. $T_{K}$ is the Kondo temperature for $D=0$. Figure 10: (Color
online) Entropy as a function of temperature for $J_{K}=0.16$ and different
values of $S$ and $D$. $D>0$ ($D<0$) corresponds to $D=0.002$ ($-0.002$). $W$
is taken as the unit of energy.

In Fig. 10 we show the evolution of the entropy with temperature for different
spins and anisotropy. For $D=0$ (orange curves), the two equivalent
hybridizing channels reduce the ground state spin to a residual value
$S^{\prime}=S-1$ and the entropy at zero temperature approaches
ln($2S^{\prime}-1$). For negative $D$ (blue curves) the ground state is two-
fold degenerate. The most interesting case is for positive $D$. In this case,
for half-integer $S$ the low-energy behavior is dominated by the physics of
the spin-1/2 two-channel Kondo model and the zero-temperature entropy is
$\frac{1}{2}\ln(2)$. For integer $S$, the ground state is non-degenerate.

In Fig. 11 we show the conductance $G(T)$ (related to the spectral density) as
a function of temperature for different $S$ and $D$. For $D=0$ (orange
curves), $G(T)$ for $T\rightarrow 0$ tends to the unitary value
$G_{0}=4e^{2}/h$, characteristic of two orbital- and spin-degenerate
conduction channels. The case $D<0$ is complex and is not of interest to the
central objective of this article, because in such a case, there is no quantum
phase transition. For $D>0$ and half-integer $S$, the low-temperature value is
$G_{0}/2$, characteristic of the spin-1/2 two-channel Kondo model. For integer
$S$, $G(0)=G_{0}$ (zero) if $D$ is below (above) the critical anisotropy
$D_{c}$ of the topological transition. In the figure only the case $D>D_{c}$
is shown.

We have also analyzed what happens for the half-integer case when the
couplings of the two channels are different in presence of a small $D>0$. The
ground state becomes a singlet and therefore, the entropy goes to zero for
$T\rightarrow 0$. The contribution to the conductance of the channel with a
larger coupling constant is similar to that of the simplest spin-1/2 1-channel
Kondo model, increasing at $T_{K}$ and reaching $2e^{2}/h$ at zero
temperature. At a lower characteristic temperature the conductance of the
other channel decreases with decreasing temperature, indicating a dip in the
corresponding spectral density. This seems to be the case of MnPc on Au(111).
See Section 4.2.

Figure 11: (Color online) Conductance as a function of temperature for
$J_{K}=0.16$ and different $S$ and $D$. $D>0$ ($D<0$) corresponds to $D=0.002$
($-0.002$). $W$ is taken as the unit of energy.

Finally, we briefly comment that the two-channel spin-1 Anderson model with
single-ion anisotropy [Eq. (2)] can be exactly mapped to a system of two
$S=1/2$ impurities coupled through an anisotropic exchange interaction between
them. The Hamiltonian is

$\displaystyle H_{2I}$ $\displaystyle=$
$\displaystyle\sum_{k\alpha\sigma}\varepsilon_{k}c_{k\alpha\sigma}^{\dagger}c_{k\alpha\sigma}+\sum_{k\alpha\sigma}\left(v{c}_{k\alpha\sigma}^{\dagger}{d}_{\alpha\sigma}+\mathrm{H.c.}\right)+$
(9) $\displaystyle+$ $\displaystyle\sum_{\alpha\sigma}\epsilon^{\rm
2I}d_{\alpha\sigma}^{\dagger}d_{\alpha\sigma}+\sum_{\alpha}U^{\rm
2I}n_{\alpha\uparrow}n_{\alpha\downarrow}+J_{z}s^{z}_{1}s^{z}_{2}+\frac{J_{\perp}}{2}\left(s^{+}_{1}s^{-}_{2}+s^{-}_{1}s^{+}_{2}\right).$

For simplicity we consider two equivalent channels, and $\alpha=1,2$ refers to
both impurities.

Changing the basis of the four impurity states from the individual spin
projections $|s^{z}_{1}s^{z}_{2}\rangle$ to the total spin and projection
$|SM\rangle$ with $S=0,1$, $M=-1,0,1$, and denoting by $E_{SM}$ the energies
of these states in the new basis, one obtains an effective Hund coupling
$J_{H}=E_{00}-E_{10}=-J_{\perp}$ and an exchange anisotropy directly related
with the single-ion anisotropy of the spin-1 model through the relation
$D=E_{11}-E_{10}=\frac{1}{2}(J_{z}-J_{\perp})$. It is interesting to note that
the properties of the system are invariant under a rotation of one of the
impurity spins (for example $s_{2}$) in $\pi$ around the $z$ axis (or
equivalently a change of sign of the states with $s^{z}_{2}=-1/2$). This
transformation changes the sign of $J_{\perp}$ and interchanges the states
$|00\rangle$ and $|10\rangle$. In particular, the isotropic two-impurity model
with $J_{\perp}=J_{z}>0$ is mapped into our A2CS1KM model with
$J_{H}=D=J_{z}$, for which one expects a topological phase for low or moderate
$T_{K}$.

Therefore, the two-impurity system [Eq.( 9)] will undergo a topological
quantum phase transition depending on the exchange anisotropy. In fact, one of
the first systems in which the existence of a non-trivial Fermi liquid was
detected was the two-impurity system in Ref. [24]. This significantly broadens
the range of impurity systems in which the TQPT theory could be useful to
interpret scanning-tunneling spectroscopy measurements.

##  6 Summary and discussion

Since the seminal work of Nozières and Blandin [7], the Kondo effect in
situations involving multiorbital magnetic impurities and/or coupled to more
than one conduction channels has become a highly active area of research
within condensed matter physics. The primary motivation stems from the
prediction of exotic electronic states, such as non-Fermi liquids
(overscreened Kondo effect) and singular Fermi liquids (underscreened Kondo
effect), exhibiting distinctive dynamic and thermal behaviors, that, on one
hand, could shed light on unconventional physics in heavy fermion compounds,
and, on the other hand, they serve as paradigmatic models for quantum many-
body phenomena.

The simplest generalization of the conventional Kondo effect to multiorbital
systems, where a spin $S>1/2$ is fully screened by $n=2S$ channels, has been
relatively underexplored, perhaps because it was believed that no new physics
could be found in this context. However, we have shown in a series of recent
publications that the presence of single-ion magnetic anisotropy drives these
systems through a topological quantum phase transition, that separates two
topologically distinctive local Fermi liquids.

Besides the theoretical relevance of our finding, we have shown that several
systems consisting of isolated magnetic atoms or molecules on noble metal
surfaces can be described by the 2-channel spin-1 Anderson model with
anisotropy or its integer valent limit, the anisotropic two-channel spin-1
Kondo model. Both models exhibits the topological quantum phase transition, as
a consequence of which the spectral density at zero temperature of the
localized electrons has a jump between high values in the topologically
trivial Fermi liquid phase to very low values in the non-trivial ”non-Landau”
Fermi liquid phase. Near the transition, the spectral density is characterized
by a narrow peak or dip (depending on the phase) mounted on a broad peak.

Several experiments with scanning-tunneling spectroscopy in different systems
have in fact identified similar structures in the differential conductance.
Five of these systems are listed in Section 4. They were usually interpreted
using different, more conventional theories, probably due to the fact of the
novelty of the concepts related to the topological quantum phase transition.
We expect that this work contributes to disseminate these rather novel ideas
to the community of condensed-matter and nanoscience researchers.

We have also shown that the concepts can be extended to larger spin and two-
impurity systems.

## Acknowledgments

We thank Rok Žitko and Pablo Roura-Bas for several collaborations in subjects
of this review. AAA acknowledges financial support provided by PICT 2018-01546
and PICT 2020A-03661 of the Agencia I+D+i, Argentina. LOM and GGB acknowledge
financial support provided by PIP 3220 of CONICET, Argentina. GGB is supported
by Slovenian Research Agency (ARRS) under Grant no. P1-0044 and J1-2458.

## ORCID iDs

A A Aligia https://orcid.org/0000-0001-9647-3926
L O Manuel https://orcid.org/0000-0001-7793-1709
G G Blesio https://orcid.org/0000-0002-1962-0969

## References

## References

  * [1] Yang K, Paul W, Phark S H, Willke P, Bae Y, Choi T, Esat T, Ardavan A, Heinrich A J and Lutz C P 2019 Coherent spin manipulation of individual atoms on a surface Science 366 509
  * [2] Aradhya S V and Venkataraman L 2013 Single-molecule junctions beyond electronic transport Nature Nanotechnology 8 399
  * [3] Cuevas J C and Scheer E 2010 Molecular Electronics: An Introduction to Theory and Experiment (Singapore:World Scientific)
  * [4] Mathew P Th and Fang F 2018 Advances in Molecular Electronics: A Brief Review Engineering 4 760
  * [5] Kügel J, Karolak M, Krönlein A, Serrate D, Bode M and Sangiovanni G 2018 Reversible magnetic switching of high-spin molecules on a giant Rashba surface npj Quantum Materials 3 53
  * [6] Evers F, Korytár R, Tewari S and van Ruitenbeek J 2020 Advances and challenges in single-molecule electron transport Rev. Mod. Phys. 92 035001
  * [7] Nozieres Ph and Blandin A 1980 Kondo effect in real metals J. Phys. France 41 193
  * [8] Blesio G G, Manuel L O, Roura-Bas P Aligia A A 2018 Topological quantum phase transition between Fermi liquid phases in an Anderson impurity model Phys. Rev. B 98 195435
  * [9] Blesio G G, Manuel L O, Roura-Bas P and Aligia A A 2019 Fully compensated Kondo effect for a two-channel spin $S=1$ impurity Phys. Rev. B 100 075434
  * [10] Luttinger J M and Ward J C 1960 Ground-state energy of a many-fermion system. II Phys. Rev. 118 1417
  * [11] Žitko R, Blesio G G, Manuel L O and Aligia A A 2021 Iron phthalocyanine on Au(111) is a “non-Landau ”Fermi liquid Nat. Commun. 12 6027
  * [12] Di Napoli S, Barral M A, Roura-Bas P, Manuel L O, Llois A M and Aligia A A 2015 Kondo physics in a Ni impurity embedded in O-doped Au chains Phys. Rev. B 92 085120
  * [13] Mohr M, Gruber M, Weismann A, Jacob D, Abufager P, Lorente N and Berndt R 2020 Spin dependent transmission of nickelocene-Cu contacts probed with shot noise Phys. Rev. B 101 075414
  * [14] Blesio G G, Žitko R, Manuel L O and Aligia A A 2023 Topological quantum phase transition of nickelocene on Cu(100) SciPost Phys. 14 042
  * [15] Žitko R and Pruschke T 2009 Energy resolution and discretization artifacts in the numerical renormalization group Phys. Rev. B 79 085106
  * [16] Žitko R NRG Ljubljana-open source NRG code https://github.com/rokzitko/nrgljubljana and http://nrgljubljana.ijs.si/
  * [17] Žitko R, Peters R and Pruschke T 2008 Properties of anisotropic magnetic impurities on surfaces Phys. Rev. B 78 224404
  * [18] Žitko R and Pruschke T 2010 Many-particle effects in adsorbed magnetic atomswith easy-axis anisotropy: the case of Fe on the CuN/Cu(100) surface New J. Phys. 12 063040
  * [19] Parks J J, Champagne A R, Costi T A, Shum W W, Pasupathy A N, Neuscamman E, Flores-Torres S, Cornaglia P S, Aligia A A, Balseiro C A, Chan C K L, Abruña H D and Ralph D C 2010 Mechanical Control of Spin States in Spin-1 Molecules and the Underscreened Kondo Effect Science 328 1370
  * [20] Cornaglia P S, Roura-Bas P, Aligia A A and Balseiro C A 2011 Quantum transport through a stretched spin-1 molecule Europhys. Lett. 93 47005
  * [21] Bulla R, Costi T and Pruschke T 2008 The numerical renormalization group method for quantum impurity systems Rev. Mod. Phys. 80 395
  * [22] Langreth D C, 1966 Friedel Sum Rule for Anderson’s Model of Localized Impurity States Phys. Rev. 150 516
  * [23] Yoshimori A and Zawadowski A 1982 Restricted Friedel sum rules and Korringa relations as consequences of conservation laws J. Phys. C 15 5241
  * [24] Curtin O J, Nishikawa Y, Hewson A C and Crow D J G 2018 Fermi liquids and the Luttinger theorem J. Phys. Commun. 2 031001
  * [25] Nishikawa Y, Curtin O J, Hewson A C and Crow D J G 2018 Magnetic field induced quantum criticality and the Luttinger sum rule Phys. Rev. B 98 104419
  * [26] Seki K and Yunoki S 2017 Topological interpretation of the Luttinger theorem Phys. Rev. B 96 085124
  * [27] Logan D E, Wright C J and Galpin M R 2009 Correlated electron physics in two-level quantum dots: Phase transitions, transport, and experiment Phys. Rev. B 80 125177
  * [28] Gao L, Ji W, Hu Y B, Cheng Z H, Deng Z T, Liu Q, Jiang N, Lin X, Guo W, Du S X, Hofer W A, Xie X C and Gao H-J 2007 Site-Specific Kondo Effect at Ambient Temperatures in Iron-Based Molecules Phys. Rev. Lett. 99 106402
  * [29] Bartolomé J, Bartolomé, García L M, Filoti G, Gredig T, Colesniuc C N, Schuller I K and Cezar J C 2010 Highly unquenched orbital moment in textured Fe-phthalocyanine thin films Phys. Rev. B 81 195405
  * [30] Tsukahara N, Shiraki S, Itou S, Ohta N, Takagi N and Kawai M 2011 Evolution of Kondo Resonance from a Single Impurity Molecule to the Two-Dimensional Lattice Phys. Rev. Lett. 106 187201
  * [31] Minamitani E, Tsukahara N, Matsunaka D, Kim Y, Takagi N and Kawai M 2012 Symmetry-Driven Novel Kondo Effect in a Molecule Phys. Rev. Lett. 109 086602
  * [32] Lobos A M, Romero M A and Aligia A A 2014 Spectral evolution of the SU(4) Kondo effect from the single impurity to the two-dimensional limit Phys. Rev. B 89 121406(R)
  * [33] Fernández J, Aligia A A and Lobos A M 2015 Valence fluctuations in a lattice of magnetic molecules: Application to iron(II) phthalocyanine molecules on Au(111) Europhys. Lett. 109 37011
  * [34] Hiraoka R, Minamitani E, Arafune R, Tsukahara N, Watanabe S, Kawai M and Takagi N 1027 Single-molecule quantum dot as a Kondo simulator Nat. Commun. 8 16012
  * [35] Fernández J, Roura-Bas P, Camjayi A and Aligia A A 2018 Two-stage three-channel Kondo physics for an FePc molecule on the Au(111) surface J. Phys.: Condens. Matter 30 374003; Corrigendum J. Phys. Condens. Matter 31 029501
  * [36] Aykanat A, Meng Z, Benedetto G and Mirica K A 2020 Molecular Engineering of Multifunctional Metallophthalocyanine Containing Framework Materials Chem. Mater. 32 5372
  * [37] Yang K, Chen H, Pope Th, Hu Y, Liu L, Wang D, Tao L, Xiao W, Fei X, Zhang Y-Y, Luo H-G, Du S, Xiang T, Hofer W A and Gao H-J 2019 Tunable giant magnetoresistance in a single-molecule junction Nat. Commun. 10 1038
  * [38] Aligia A A 2022 Low-energy physics for an iron phthalocyanine molecule on Au(111) Phys. Rev. B 105 205114
  * [39] Fisk C, Valdemoro C and Fraga S 1968 Spin–Orbit Coupling in Some Positive Ions of Cr, Mn, and Fe J. Chem. Phys. 48 2923
  * [40] Žitko R 2011 Kondo resonance lineshape of magnetic adatoms on decoupling layers Phys. Rev.´ B 84 195116
  * [41] Guo X, Zhu Q, Zhou L, Yu W, Lu W and Lian W 2021 Evolution and universality of two-stage Kondo effect in single manganese phthalocyanine molecule transistors Nat. Commun. 12 1566
  * [42] Roura Bas P and Aligia A A 2010 Nonequilibrium dynamics of a singlet-triplet Anderson impurity near the quantum phase transition J. Phys.: Condens. Matt. 22 025602
  * [43] Liao M-S, Watts J D and Huang M J 2005 DFT Study of Unligated and Ligated Manganese II Porphyrins and Phthalocyanines Inorg. Chem. 44 1941
  * [44] Brumboiu I E, Totani R, de Simone M, Coreno M, Grazioli C, Lozzi L, Herper H C, Sanyal B, Eriksson O, Puglia C and Brena B 2014 Elucidating the 3d Electronic Configuration in Manganese Phthalocyanine J. Phys. Chem. A 118 927
  * [45] Petraki F, Peisert H, Hoffmann P, Uihlein J, Knupfer M and Chasse T 2012 Modification of the 3d-Electronic Configuration of Manganese Phthalocyanine at the Interface to Gold J. Phys. Chem. C 116 5121
  * [46] Bachellier N, Ormaza M, Faraggi M, Verlhac B, Vérot M, Le Bahers T, Bocquet M-L and Limot L 2016 Unveiling nickelocene bonding to a noble metal surface Phys. Rev. B 93 195403
  * [47] Ormaza M, Robles R, Bachellier N, Abufager P, Lorente N and Limot L 2016 On-surface engineering of a magnetic organometallic nanowire Nano Lett. 16 588
  * [48] Ormaza M, Abufager P, Verlhac B, Bachellier N, Bocquet M-L, Lorente N and Limot L 2017 Controlled spin switching in a metallocene molecular junction Nat. Commun. 8 1974
  * [49] Ormaza M, Bachellier N, Faraggi M N, Verlhac B, Abufager P, Ohresser P, Joly L, Romeo M, Scheurer F, Bocquet M-L, Lorente N and Limot L 2017 Efficient spin-flip excitation of a nickelocene molecule Nano Lett. 17 1877
  * [50] Verlhac B, Bachellier N, Garnier L, Ormaza M, Abufager P, Robles R, Bocquet M-L, Ternes M, Lorente N and Limot L 2019 Atomic-scale spin sensing with a single molecule at the apex of a scanning tunneling microscope Science 366 623
  * [51] Trishin S, Lotze C, Bogdanoff N, von Oppen F and Franke K J 2021 Moiré Tuning of Spin Excitations: Individual Fe Atoms on MoS2/Au(111) Phys. Rev. Lett. 127 236801
  * [52] Wang Y, Wang B, Huang R, Gao B, Kong F and Zhang Q 2014 First-principles study of transition-metal atoms adsorption on MoS2 monolayer Physica (Amsterdam) 63E 276
  * [53] Chen X, Zhong L, Li X and Qi J 2017 Valley splitting in the transition-metal dichalcogenide monolayer via atom adsorption Nanoscale 9 2188
  * [54] Moro-Lagares M, Fernández J, Roura-Bas P, Ibarra M R, Aligia A A and Serrate D 2018 Quantifying the leading role of the surface state in the Kondo effect of Co/Ag(111) Phys. Rev. B 97 235442
  * [55] Blesio G G and Aligia A A 2023 Topological quantum phase transition in individual Fe Atoms on MoS2/Au(111) Phys. Rev. B 108 045113
  * [56] Meng X, Möller J, Mansouri M, Sánchez-Portal D, Garcia-Lekue A, Weismann A, Li C, Herges R and Berndt R 2023 Controlling the Spin States of FeTBrPP on Au(111) ACS Nano 17 1268
  * [57] Gao Y, Vlaic S, Gorni T, de’ Medici L, Clair S, Roditchev D and Pons S 2023 Manipulation of the magnetic state of a porphyrin-based molecule on gold: From Kondo to quantum nanomagnet via the charge fluctuation regime ACS Nano 17 9082
  * [58] Mitchell A K, Sela E and Logan D E 2012 Two-channel Kondo physics in two-impurity Kondo models Phys. Rev. Lett. 108 086405
  * [59] Di Napoli S, Weichselbaum A, Roura-Bas P, Aligia A A, Mokrousov Y and Blügel S 2013 Non-Fermi-Liquid Behavior in Transport Through Co-Doped Au Chains Phys. Rev. Lett. 110 196402
  * [60] Di Napoli S, Roura-Bas P, Weichselbaum A and Aligia A A 2014 Non-Fermi-liquid behavior in nonequilibrium transport through Co-doped Au chains connected to fourfold symmetric leads Phys. Rev. B 90 125149