# Topological quantum phase transition of nickelocene on Cu(100)

G. G. Blesio  Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
R. Žitko  Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19,
SI-1000 Ljubljana, Slovenia  L. O. Manuel  Instituto de Física Rosario
(CONICET) and Facultad de Ciencias Exactas, Ingeniería y Agrimensura,
Universidad Nacional de Rosario, 2000 Rosario, Argentina  A. A. Aligia
Instituto de Nanociencia y Nanotecnología CNEA-CONICET, Centro Atómico
Bariloche and Instituto Balseiro, 8400 Bariloche, Argentina

###### Abstract

Local quantum phase transitions driven by Kondo correlations have been
theoretically proposed in several magnetic nanosystems; however, clear
experimental signatures are scant. Modeling a nickelocene molecule on a
Cu(100) substrate as a two-orbital Anderson impurity with single-ion easy-
plane anisotropy coupled to two conduction bands, we find that recent scanning
tunneling spectra measured at different microscope tip heights reveal the
existence of a topological quantum phase transition from the usual local Fermi
liquid with high zero-bias conductance to a non-Landau Fermi liquid,
characterized by a non-trivial quantized Luttinger integral, with a small
conductance. The effects of intermediate valence, finite temperature, and
structural relaxation of the molecule position allow us to explain the
different observed behaviors.

###### pacs:

73.23.-b, 71.10.Hf, 75.20.Hr

Quantum phase transitions (QPT) where the Kondo effect is destroyed by
competing interactions have been theoretically found in several magnetic
nanosystems vojta06 . They were also invoked to explain the unconventional
quantum criticality in some heavy fermion compounds si01 . However,
experimental realizations of such QPT bork11  are elusive, as they require
fine control of the parameter that triggers the transition. Due to their
variety and tunability, magnetic single molecules in contact with metal
electrodes, probed and manipulated with a scanning tunneling microscope (STM),
are distinguished candidate systems. They are being extensively studied
because of their novel properties and also their potential use in new
spintronic devices naturefocus ; cuevas ; evers20 .

The double-decker nickelocene (Nc) molecules on the Cu(100) substrates have
been experimentally studied in several articles bache ; orma1 ; ormaza17 ;
orma3 ; verlhac ; mohr . As the STM tip is approached to the molecule, the
tunneling regime is followed by the contact regime ormaza17 . While in the
latter the differential conductance $\mathrm{d}I/\mathrm{d}V$ for small bias
voltage $V$ displays the characteristic Kondo peak due to screening of the
molecular magnetic moment by conduction electrons liang02 ; parks07 ; roch08 ;
parks10 , in the tunneling regime $\mathrm{d}I/\mathrm{d}V$ shows a low-bias
dip and finite-$V$ rise characteristic of inelastic spin-flips due to single-
ion anisotropy for a spin $S>1/2$ zitko08 ; zitko10 . The different behaviors
have been tentatively ascribed to a crossover in the spin of the molecule from
1/2 in the contact regime to 1 in the tunneling regime based on first-
principle calculations ormaza17 ; mohr  that, however, neglect dynamical
correlations and therefore do not properly treat the Kondo effect. Moreover,
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 magnetization. This calls
for an alternative interpretation without such discrepancy.

The minimal Hamiltonian that captures the many-body physics of this system is
an impurity Anderson model in which the dominant configuration has two holes
in the Ni 3d shell occupying the nearly degenerate orbitals with $xz$ and $yz$
symmetry, and forming a spin $S=1$ due to the atomic Hund coupling. These
triplet states are split by the single-ion easy plane anisotropy $D$. Both
localized orbitals are equally hybridized with conduction electrons with the
same symmetry mohr . A recent numerical renormalization group (NRG) study of
this class of Hamiltonians blesio18 ; blesio19  demonstrated that with the
increasing ratio $D/T^{0}_{K}$, where $T^{0}_{K}$ is the $D=0$ Kondo
temperature, the system undergoes a topological quantum phase transition
(TQPT) from a phase with high impurity spectral density $\rho(\omega)$ at zero
frequency $\omega$ (proportional to $\mathrm{d}I/\mathrm{d}V$ for $V=0$) to a
topologically distinct phase with a low $\rho(0)$.

In Fig. 1 we show the evolution of the spectral density for the anisotropic
two-channel spin-1 Kondo model (A2CS1KM) blesio19  (the integer valence limit
of the above Anderson model) for fixed $D$ as the Kondo exchange $J$ is
increased. For large $J$, $\rho(\omega)$ has the usual shape expected for a
Kondo resonance. As $J$ get closer to the critical value $J_{c}\sim 0.4095$
from above, the shape evolves to a very narrow peak mounted on a much broader
one. For $J=J_{c}$ the narrow peak transforms suddenly into a sharp dip, which
broadens with decreasing $J$. For small $J$, $\rho(\omega)$ tends to have two
well defined steps at $\omega=\pm D$. A similar behavior has been found in
other strongly correlated impurity models leo ; curtin18 ; nishi ; zitko21 .
The phase with large $D/T^{0}_{K}$ is a Fermi liquid characterized by a non-
zero Luttinger integral $I_{L}=\pi/2$ for each orbital and spin blesio18 ;
zitko21 . Since for a non-interacting system $I_{L}=0$, this phase breaks
Landau adiabatic hypothesis. Thus for $D/T^{0}_{K}>(D/T_{K}^{0})_{c}$ the
system is a non-Landau Fermi liquid (NLFL). Recently, we have shown zitko21
that this concept can explain in a unified and consistent fashion several
experiments in iron phthalocyanine (FePc) on Au(111) mina ; hira ; yang  and
we have predicted the possibility of driving that system directly through the
TQPT, although such measurement remains to be performed.

Figure 1: (Color online) Normalized spectral density of the A2CS1KM as a
function of frequency for several values of the Kondo coupling $J$, given in
units of the half band with $W$ of the conduction channels.

The contrasting $\mathrm{d}I/\mathrm{d}V$ spectra reported in Ref. ormaza17,
for Nc on Cu(100) in tunneling and contact regimes coincide with those of the
two distinct phases of the model, with the tunneling regime corresponding to
the NLFL phase. Moreover, assuming a half band with $W=$ 1 eV, the critical
anisotropy $D_{c}\sim 3.8$ meV blesio18  is of the order of the reported
anisotropy $D=4.2$ meV of the system. Then, it is natural to assume that Nc on
Cu(100) is a physical realization of the TQPT. This would be the first
experimental realization of a TQPT to a NLFL phase. Previously, we have
identified FePc on Au(111), with a clear dip in $\mathrm{d}I/\mathrm{d}V$ at
$V=0,$ as a NLFL zitko21 , using a similar model as the present one with two
non-equivalent channels note . Nevertheless, an abrupt jump to the
conventional Fermi liquid phase has not been reported yet in this system note2
.

A recent detailed study of Nc/Cu(100) consisting of a careful tip-molecule
distance variation mohr  revealed more complex behavior that does not quite
correspond to that observed in Ref. ormaza17,  and shown in Fig. 1. Two
different types of evolution of $\mathrm{d}I/\mathrm{d}V$, probably depending
on the exact adsorption geometry, were observed. In case A, the evolution from
tunneling to contact regime is discontinuous and hysteretic, with a jump
between low and high $\mathrm{d}I/\mathrm{d}V$, seemingly skipping the regime
where a narrow peak/dip for $J\sim J_{c}$ is expected (as in Fig. 1). In case
B, observed experimentally in 1/3 of the cases, the evolution is instead
continuous, but it also lacks the narrow peak/dip structures.

To explain these observations, we calculate the phase diagram of the Anderson
impurity model [see Eq. (1)] and trace the TQPT as a function of hybridization
and energy level. We find that the behavior observed in case A can be
explained by a first-order transition in the position of the molecule that
avoids parameter values near the TQPT for which the system is less stable or
unstable. For this purpose we extend the model by including the effects of the
structural relaxation of the molecule, observed in similar systems hira ;
karan  and expected to play a role also in Nc/Cu(100). For a soft “spring”
related to the molecular displacement, we find that the system is unstable
near the quantum critical point (QCP) and a first-order TQPT takes place in
which the critical point is avoided. The continuous transition experimentally
observed in case B is due to two effects: i) the magnitude of the jump in the
spectral density at zero frequency $\rho(0)$ decreases as the degree of
intermediate valence increases and, more importantly, ii) finite temperatures
blur the narrow peaks and dips in $\rho(\omega)$ near the TQPT.

The Hamiltonian that describes the system can be written in the form mohr ;
blesio19

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{k\tau\sigma}\varepsilon_{k}c_{k\tau\sigma}^{\dagger}c_{k\tau\sigma}+\sum_{\tau\sigma}\epsilon
d_{\tau\sigma}^{\dagger}d_{\tau\sigma}+\sum_{\tau}Un_{\tau\uparrow}n_{\tau\downarrow}+$
(1)
$\displaystyle+U^{\prime}n_{xz}n_{yz}-J_{H}{\vec{S}}_{xz}\cdot{\vec{S}}_{yz}+DS_{z}^{2}+$
$\displaystyle+\sum_{k\tau\sigma}\left(v{c}_{k\tau\sigma}^{\dagger}{d}_{\tau\sigma}+\mathrm{H.c.}\right),$

where $d_{\tau\sigma}^{\dagger}$ ($c^{\dagger}_{k\tau\sigma}$) creates a hole
with energy $\epsilon$ ($\varepsilon_{k}$) in the Ni $d$ orbital $\tau$ (Cu
band $\tau$ with momentum $k$), with $\tau=xz,yz$.
$n_{\tau\sigma}=d_{\tau\sigma}^{\dagger}d_{\tau\sigma}$ and
$n_{\tau}=\sum_{\sigma}n_{\tau\sigma}$. The actual localized orbitals might
extend in the Nc molecule beyond the Ni orbitals, but for simplicity we refer
to them as the Ni $d$ orbitals. We also assume degenerate conduction bands,
hybridizations $v$ independent of energy and channel, and the same energy
$\epsilon$ for both orbitals notaSOC . For small hybridization and when the
two-particle configuration dominates, the model (1) reduces to the A2CS1KM
blesio19 .

For the discussion of topological properties, we remind the reader that the
impurity spectral function per orbital and spin, at the Fermi level
$(\omega=0)$ at zero temperature, is related to the quasiparticle scattering
phase shift $\delta_{\tau\sigma}$ by taylor ; friedel ; lutti ; langer ; lang
; shiba ; yoshi

$\rho_{\tau\sigma}(0)=-\frac{1}{\pi}\mathrm{Im}G_{\tau\sigma}^{d}(0)=\frac{1}{\pi\Gamma}\sin^{2}\delta_{\tau\sigma},$
(2)

where $G_{\tau\sigma}^{d}(\omega)=\left\langle\left\langle
d_{\tau\sigma};d_{\tau\sigma}^{\dagger}\right\rangle\right\rangle$ is the
impurity Green’s function and
$\Gamma=\pi\sum_{k}|V|^{2}\delta(\omega-\varepsilon_{k})$ is the hybridization
strength, assumed independent of energy. In turn, the phase shift is related
to the number of displaced electrons by the impurity for each channel and spin
which, in the case of wide flat conduction density of states, coincides with
the expectation value of the Ni occupancy for the corresponding spin and
orbital. Therefore, in this limit, the generalized Friedel sum rule reads
blesio18 ; zitko21

$\delta_{\tau\sigma}=\pi\left\langle
n_{\tau\sigma}\right\rangle+I_{\tau\sigma},\text{
}I_{\tau\sigma}=\mathrm{Im}\int_{-\infty}^{0}d\omega\,G_{\tau\sigma}^{d}(\omega)\frac{\partial\Sigma_{\tau\sigma}^{d}(\omega)}{\partial\omega},$
(3)

where $\Sigma_{\tau\sigma}^{d}(\omega)$ is the impurity self energy. In our
case, by symmetry, the four integrals are equal, $I_{\tau\sigma}=I_{L}$. Until
recently, $I_{L}=0$ was generally assumed as a hallmark of a Fermi liquid, but
several local Fermi liquids were found in which $I_{L}=\pm\pi/2$ blesio18 ;
blesio19 ; curtin18 ; nishi ; zitko21 .

Since $I_{L}=(\pi/2)\theta(D-D_{c})$ blesio18 ; blesio19 ; zitko21 , Eqs. (2)
and (3) imply that for a total Ni occupancy of $\left\langle
n\right\rangle=4\left\langle n_{\tau\sigma}\right\rangle=2$, the spectral
densities $\rho_{\tau\sigma}(0)$ jump from $\rho_{0}=1/\pi\Gamma$ for
$D<D_{c}$ to 0 for $D>D_{c}$. For fixed $D$ the TQPT can be obtained by
decreasing $\Gamma$ or the exchange constant $J$, as displayed in Fig. 1. For
other occupancies, the jump in the spectral densities at the TQPT is reduced
as $\Delta\rho_{\tau\sigma}(0)=-\frac{1}{\pi\Gamma}\cos(\pi\left\langle
n\right\rangle/2)$.

Figure 2: Phase diagram of the Anderson model in the $\epsilon,\Gamma$ plane
for $U=3.5$, $U^{\prime}=2.5$ and $J_{H}=0.5$. Below (above) the curve the
phase is a NLFL (usual Fermi liquid). Inset: occupancy of the Ni at the TQPT.

Clearly this relation predicts that the jump disappears as the Ni occupancy is
reduced to one particle. More fundamentally, the formation of the $S=1$ state
requires a strong contribution of the two-particle configuration in the
ground-state wave function. This naturally leads to the question of the very
existence of the TQPT away from half-filling. To elucidate this, we have
calculated the position of the TQPT in the $\epsilon,\Gamma$ plane. We take
similar parameters for the model as those reported previously mohr  with the
half band width $W=1$ eV as the unit of energy and a flat density of states
for the conduction electrons. The result is displayed in Fig. 2. For
$\epsilon=-(U+U^{\prime})/2=-4.25$, $\left\langle n\right\rangle=2$ and
$\Gamma_{c}$ reaches its maximum value $\sim 350$ meV. As $\epsilon$ increases
one expects that the Kondo temperature increases for constant $\Gamma$, and
then to keep the same ratio $D/T_{K}\sim 2.6$ at the TQPT blesio18 ,
$\Gamma_{c}$ should decrease. This decrease is more abrupt near
$\epsilon\sim-2.5$ until for $\epsilon=\epsilon_{0}=-2.375$,
$\Gamma_{c}\rightarrow 0$ at the TQPT. For larger $\epsilon$, the system is
always in the topologically trivial ($I_{L}=0$) phase.

Note that the end point of the TQPT $(\epsilon,\Gamma)=(\epsilon_{0},0)$ is
the point of degeneracy of the ground state configurations of the isolated
molecule ($\Gamma=0$) between the two-particle state with $S_{z}=0$ (with
energy $2\epsilon+U^{\prime}-J_{H}/4$) and the four one-particle states (with
energy $\epsilon$). A TQPT at this point might be expected on general physical
grounds. If the one-particle configuration is that of lowest energy, the
splitting of the orbitals is physically relevant and cannot be neglected for a
realistic description (as for FePc note ). Neglecting the excited doublet, the
model mixes a doublet and a triplet. For $D=0$, the model has been solved
exactly bethe  and the ground state is a triplet. For arbitrary $D$ and small
$\Gamma$, a Schrieffer-Wolf transformation like that performed in Ref. bethe,
leads to the following Kondo interaction

$H_{I}=\left(\frac{-2v^{2}}{\Delta E+D}+\frac{v^{2}}{\Delta
E}\right)S_{d}^{z}S_{c}^{z}-\frac{v^{2}}{\Delta
E}\left(S_{d}^{x}S_{c}^{x}+S_{d}^{y}S_{c}^{y}\right),$ (4)

where $\vec{S}_{d}$ and $\vec{S}_{c}$ are the total spin of the localized and
conduction electrons respectively, and $\Delta E$ is the excitation energy
from the one-particle GS configuration to the $S_{z}=0$ triplet excited state.
According to poor man’s scaling ander , for $D>0$, the ground state is a
singlet and corresponds to an ordinary Fermi liquid as we found. If instead,
the two-particle configuration dominates, the model is equivalent to the
A2CS1KM blesio19  with exchange $J\rightarrow 0$ which is a NLFL.

In the inset of Fig. 2 we show the Ni occupancy $\left\langle n\right\rangle$
along the TQPT line. It lies always above $\sim 1.5$ . For $\left\langle
n\right\rangle=1.5$, the jump in the spectral density is reduced by a factor
0.71. Therefore, while the TQPT is present with some degree of intermediate
valence, it seems that the two-particle triplet Ni configuration should be the
dominant one for the TQPT to be observed.

We now discuss the evolution of $\mathrm{d}I/\mathrm{d}V$ for type B mohr ,
where a jump the spectral density at the TQPT is not observed, but there is a
rather continuous evolution as the STM tip is pressed against the surface,
increasing $\Gamma$. As discussed above, the jump can be reduced up to 30% by
the effects of intermediate valence. Such interpretation is supported by the
measured conductances that saturate well below the unitary limit (see Fig. 4
in Ref. mohr, ). Nevertheless, an even more important effect is that due to
the finite temperature of the experiment ($T=4.5\,\mathrm{K}$): the narrow
dip/peak near the TQPT (see Fig. 1) is blurred by temperature and,
furthermore, at finite temperature $\mathrm{d}I/\mathrm{d}V$ is not
proportional to $\rho_{\tau\sigma}(\omega)$ but to its convolution with the
derivative of the Fermi function (see for example Ref. zitko21 ) that further
broadens sharp features near the TQPT.

Figure 3: (Color online) Differential conductance as a function of voltage for
different values of $\Gamma$. Other parameters are $U=3.5$, $U^{\prime}=2.5,$
$J_{H}=0.5$, $\epsilon=-3.0$, $T=0.0005$.

In Fig. 3 we show the evolution of the differential conductance as $\Gamma$ is
increased at a finite temperature. For small $\Gamma$,
$\mathrm{d}I/\mathrm{d}V$ has a dip mounted on a broader peak. As $\Gamma$
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 $\Gamma$ the dip disappears and the magnitude of
$\mathrm{d}I/\mathrm{d}V$ near zero voltage increases. Therefore, the main
features of Fig. 5 (b) of Ref. mohr  are reproduced.

To explain the evolution of $\mathrm{d}I/\mathrm{d}V$ for the experimental
situation of type A mohr , we need to extend the model to permit the
structural relaxation of the molecule. Clearly $\Gamma$ increases as the STM
tip approaches to the Cu(100) surface, and (as in other systems hira ; karan )
the Nc molecule accommodates itself to the tip position, mainly tilting its
axis relative to the surface. This shift in the position of the molecule,
which we denote as $\eta$, also affects $\Gamma$. On general physical grounds
one expects, to leading order in $\eta$,

$\Gamma=\Gamma_{0}+a\eta,$ (5)

where $\Gamma_{0}$, which depends on the position of the STM tip, is the value
of $\Gamma$ for $\eta=0$ and $a$ is a constant. The on-site energy $\epsilon$
might depend slightly on $\eta$ but this does not affect the main argument
below. The total Hamiltonian $H_{t}$ should also include the elastic energy
due to the displacement $\eta$,

$H_{t}=H+\frac{1}{2}K\eta^{2},$ (6)

where $K>0$ is another constant. The optimum value of $\eta$ for each
$\Gamma_{0}$ is obtained by minimizing the total energy $E_{t}=\left\langle
H_{t}\right\rangle=E(\Gamma)+b\eta^{2}$, where $E=\left\langle
H\right\rangle$. Differentiating this equation one has

$\displaystyle\frac{dE_{t}}{d\eta}=\frac{\partial
E}{\partial\Gamma}a+K\eta,~{}~{}\frac{d^{2}E_{t}}{d\eta^{2}}=\frac{\partial^{2}E}{\partial\Gamma^{2}}a^{2}+K.$
(7)

In order to have a locally stable minimum at the value $\eta=\eta_{1}$ for
which $dE_{t}/d\eta=0$, one needs that $d^{2}E_{t}/d\eta^{2}>0$ for
$\eta=\eta_{1}$. If, however, for some value $\Gamma=\Gamma_{u}$,
$\partial^{2}E/\partial\Gamma^{2}<-K/a^{2}$, the position $\eta=\eta_{1}$ of
the molecule is unstable and, as $\Gamma_{0}$ is varied, the system has a
first-order transition between two states, one with $\Gamma<$ $\Gamma_{u}$ and
another one with $\Gamma>$ $\Gamma_{u}$, avoiding the value
$\Gamma=\Gamma_{u}$.

To analyze this possibility in our system, we have calculated the ground state
energy $E(\Gamma)$ and numerically differentiated it, see Fig. 4. Remarkably,
$\partial^{2}E/\partial\Gamma^{2}$ is negative and large in magnitude at the
value $\Gamma=\Gamma_{c}$ that corresponds to the TQPT. Therefore, one expects
that for a soft enough spring constant
($K<a^{2}|\partial^{2}E/\partial\Gamma^{2}|$), as the STM tip is pressed into
the molecule, there is a first-order transition from the NLFL phase to the
usual Fermi liquid phase, avoiding the values $\Gamma\sim\Gamma_{c}$. This
agrees with the experimental observations reported in Ref. ormaza17  and Fig.
5 (a) of Ref. mohr  for systems of type A. For the cases of type B, it is
likely that relaxation of the molecule is inhibited by a larger spring
constant $K$. In the contact regime, the Kondo resonance is much broader for
the type B situation than for the type A, suggesting an upright molecular
configuration in case B and a tilted one in case A mohr .

Figure 4: Ground-state energy and its first and second derivatives as a
function of $\Gamma$ for $U=3.5$, $U^{\prime}=2.5$ and $J_{H}=0.5$, and
$\epsilon=-3.8$.

In summary, we have uncovered a topological quantum phase transition (TQPT) in
published tunneling spectra of nickelocene molecules on Cu(100). The TQPT is
controlled by the distance between the STM tip and the surface. We have
modeled the system as a spin-1 two-channel Anderson model with easy-plane
magnetic anisotropy, which has a TQPT between a non-trivial topological non-
Landau Fermi liquid phase in the tunneling regime to a topologically trivial
Fermi liquid phase in the contact regime. The transition extends to the zero-
hybridization limit at the degeneracy point between the two-particle
configuration and the one-/three-particle configuration. We find that the
different behaviors of the differential conductance $\mathrm{d}I/\mathrm{d}V$
as a function of voltage can be explained in terms of this TQPT. The
transition is characterized by a jump in $\mathrm{d}I/\mathrm{d}V$ at $V=0$ at
zero temperature. However, in most experimental observations (case A) there is
a hysteretic jump in $\mathrm{d}I/\mathrm{d}V$ also at finite voltage which
can be explained by the structural relaxation of the molecule. In the absence
of this relaxation and at finite temperature, the change in
$\mathrm{d}I/\mathrm{d}V$ is continuous as the tip of the STM is approached to
the molecule (case B).

RŽ acknowledges the support of the Slovenian Research Agency (ARRS) under
P1-0044 and P1-0416. GGB is also support by the Slovenian Research Agency
(ARRS) under P1-0044 and J1-2458. GGB and LOM are supported by PIP No. 3220 of
CONICET, Argentina. AAA is supported by PICT 2017-2726 and PICT 2018-01546 of
the ANPCyT, Argentina.

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