# Long-lived magnetization in an atomic spin chain tuned to a diabolic point

R.J.G. Elbertse1, D. Borodin2, J. Oh2,3, T. Ahn2,3, J. Hwang2,3, J.C.
Rietveld1,
A.J. Heinrich2,3, F. Delgado4∗, S. Otte1∗, Y. Bae2,3∗†
1Department of Quantum Nanoscience, Kavli Institute of Nanoscience,
Delft University of Technology, Delft, The Netherlands
2Center for Quantum Nanoscience, Institute for Basic Science (IBS), Seoul,
South Korea
3Department of Physics, EWHA Womans University, Seoul, South Korea
4Instituto Universitario de Estudios Avanzados IUDEA, Departamento de Física,
Universidad de La Laguna
La Laguna, Tenerife, Spain
†Current address: Empa, Swiss Federal Laboratories for Materials Science and
Technology,
nanotech@surfaces Laboratory, Switzerland

###### Abstract

Scaling magnets down to where quantum size effects become prominent triggers
quantum tunneling of magnetization (QTM), profoundly influencing magnetization
dynamics. Measuring magnetization switching in an Fe atomic chain under a
carefully tuned transverse magnetic field, we observe a non-monotonic
variation of magnetization lifetimes around a level crossing, known as the
diabolic point (DP). Near DPs, local environment effects causing QTM are
efficiently suppressed, enhancing lifetimes by three orders of magnitude.
Adjusting interatomic interactions further facilitates multiple DPs. Our study
provides a deeper understanding of quantum dynamics near DPs and enhances our
ability to engineer a quantum magnet.

In quantum mechanical systems, unusual dynamic processes occur when energy
levels approach and mix with each other. In a two-parameter space, the
degeneracy between orthogonal states creates a level crossing of energy
surfaces (Fig. 1a), the shape of which reminds of the toy, diabolo, and thus
is dubbed a diabolic point (DP) [1]. This DP has attracted significant
attention in quantum magnets [2], which are characterized by two metastable
magnetization states separated by an energy barrier [3, 4]. In the vicinity of
the DP, quantum tunneling of magnetization (QTM) between these states is
suppressed due to destructive interference among separate tunneling paths [5,
6, 7]. While the importance of DPs has been shown from ensembles of molecular
magnets [2, 8, 9], precise control of local environments as a control knob of
DPs has remained elusive.

Manipulation of magnetic atoms with a scanning tunneling microscope (STM)
allows for the assembly of prototypical quantum magnets with controllable
energy barriers ranging from 100 $\mu$eV to 100 meV [10, 11, 12]. The lifetime
of magnetization states in these magnets can be determined by monitoring the
spin polarized current through one of the magnet’s atoms over time [13]. When
the magnetic anisotropy barrier exceeds the thermal energy, the lifetime is
dominated by through-the-barrier transitions, i.e. QTM, resulting from
hybridization between quantum states on either side of the barrier. While
various systems have been studied with different degree of spin state
hybridization [13] and spin-spin interactions [13, 14], it remains challenging
to vary individual parameters due to the discrete nature of binding sites on
surfaces. Instead, a more effective control knob for QTM may be achieved by
exploiting the physics of a DP, where the hybridization of the quantum states
is expected to quench, allowing, at least in principle, arbitrarily long
lifetimes.

In this work, we demonstrate the manifestation of DPs through the spin
dynamics of nanomagnets by assembling Fe atoms into chains on Cu2N/Cu(100).
Precisely adjusting chain length and interatomic spacing enables us to tailor
the spin-spin interactions and thereby to engineer DPs in a controlled manner.
When tuning the direction and strength of the external magnetic field near a
DP, we observed a significant increase of magnetic lifetimes, with enhancement
of up to three orders of magnitude. We provide a comprehensive picture of the
quantum state composition near the DP, offering a rational strategy to control
the spin dynamics of quantum magnets.

To model a chain of $N$ Fe atoms on Cu-sites of Cu2N, we consider a spin
Hamiltonian that includes the Zeeman energy, the uniaxial and transverse
magnetic anisotropy terms for each atom, as well as the Heisenberg exchange
interaction between neighboring atoms [13, 15, 16, 17]:

$\split
H=&\sum_{i}^{N}\left[g_{i}\mu_{\mathrm{B}}\bm{B}^{\mathrm{tot}}_{i}\cdot\bm{S}_{i}+D_{i}S_{i,z}^{2}+E_{i}(S_{i,x}^{2}-S_{i,y}^{2})\right]\\\
+\sum_{i}^{N-1}J_{i}\bm{S}_{i}\cdot\bm{S}_{i+1}.$ (1)

For an Fe atom on site $i$,
$\bm{B}^{\mathrm{tot}}_{i}=\bm{B}+\bm{B}^{\mathrm{tip}}_{i}$ is the total
magnetic field composed of external and tip fields, $\mu_{\mathrm{B}}$ the
Bohr magneton, and $\bm{S}_{i}$ the spin operator with a magnitude of
$S_{i}=2$. Note that we allow for subtle variations in the values of the
g-factor $g_{i}$, anisotropy parameters $D_{i}$ and $E_{i}$, and exchange
interaction strength $J_{i}$ between atoms in the chain, since these
parameters might vary due to subtle changes in the local strain in the
underlying Cu2N layer, as evidenced by variations of Hamiltonian parameters
throughout the literature (see Table S1). The easy-axis $z$ is oriented along
the in-plane Cu-N bonds [10]; we define the $x$ and $y$ axes as the remaining
in-plane direction and the out-of-plane direction, respectively (Fig. 1b).
Since the chain is not perfectly aligned with the external magnetic field in
the experiment, the angle $\alpha$ is used to decompose the external field
into both transverse ($B_{x}$) and longitudinal ($B_{z}$) components.

We investigate the DPs as a function of the transverse magnetic field,
$B_{x}$. Solving the spin Hamiltonian (Eq. 1) for a single Fe atom ($N=1$)
gives the analytical solution for the $B_{x,n}$ fields where the DPs appear
[18, 19]:

$B_{x,n}=\frac{n\sqrt{2E(E-D)}}{g\mu_{\mathrm{B}}}.$ (2)

Here, $n$ is the diabolic point index that ranges from $2S-1$ to $1-2S$ in
double-integer steps. To estimate the location of DPs, we use Hamiltonian
parameters which have been obtained from previous work, see Supplementary Note
1. For a single Fe atom at the Cu site, the lowest positive magnetic field DP
($n=1$) is expected at $B_{x,1}\approx 9.5$ T. A second DP ($n=3$) can be
reached at even larger values of $B_{x}$. As depicted in Fig. 1a, when
$B_{x}=B_{x,1}$ and $B_{z}=0$, an energy level crossing occurs between the two
lowest-lying eigenstates, $\psi_{0}$ and $\psi_{1}$. When a small $B_{z}$ is
applied, sweeping $B_{x}$ results in an avoided level crossing, as shown by
the surface cut in Fig. 1a.

Diabolic points in atomic chains of length $N>1$ with $|J/D|\lesssim 1$ can be
understood in a similar fashion, although there are now $N$ diabolic points
for each diabolic point index, leading to a total of $2NS$ diabolic points.
Now, the spin-spin interaction between the atoms can be used as a control knob
for the location of DPs as a function of $B_{x}$. By adjusting the number of
atoms in the chain and the interatomic distance, we are able to precisely
determine the magnetic field values at which DPs are expected to occur (see
Supplementary Note 2).

Figure 1: Diabolic points in Fe atomic chains on Cu2N. (a) Energy levels for
the two lowest states of a single Fe atom (orange for the ground state
$\psi_{0}$ and pink for the first-excited state $\psi_{1}$) with a diabolic
point at the crossing. Thick lines show a cut of the energy surfaces,
indicating the corresponding energies of the two states at a finite $B_{z}$
outlined by the dashed rectangle. (b) Constant-current STM image of a Fe5
chain ($V_{\mathrm{DC}}=100$ mV, $I=10$ pA, $B_{x}=2$ T). Atoms are labeled in
Roman numerals (I–V). Schematics shows Fe atoms (white circles) on top of Cu2N
lattice. The intersections of grid lines correspond to the nitrogen atom
position in Cu2N. Magnetic field directions are tilted by the angle $\alpha$
with respect to the crystal axis. (c) Schematic overview of state composition
for $\psi_{0}$ and $\psi_{1}$ of Fe5 along a similar cut as in panel a. (d)
Current traces taken for atom III of antiferomagnetically coupled Fe5 and
different magnetic fields around the DP, located at $4.2$ T, indicated in the
schematics above. Unless stated otherwise, all traces are obtained at 3 mV and
approximately 10 pA, where magnetization switching occurs due to QTM and over-
the-barrier excitation can be neglected, see Supplementary Note 8 for further
details.

The magnetic fields needed to identify the DP in a single Fe atom are beyond
available transverse magnetic field ($6$ T) of our instrument [10, 19].
However, for longer chains, the DP eventually becomes accessible within the
range of our experimental capabilities. Our initial calculations using Eq. 1
predicted a DP for an antiferromagnetically coupled Fe5 chain (Fig. 1b) at
$B_{x,1}\sim 4$ T (see Supplementary Note 2), which guided our spin lifetime
measurements. We investigate the influence of DPs on the magnetization
bistability of the Fe atomic chains with spin-polarized STM.

Figure 1c schematically shows the energy levels of the two lowest-lying states
($\psi_{0}$ and $\psi_{1}$) in the antiferomagnetic Fe5 chain as a function of
transverse magnetic fields. For $B_{x}\ll B_{x,1}$, these two states are
mainly composed of Néel states, denoted as
$\mathrm{N_{A}}=\\{-2,+2,-2,+2,-2\\}$ and
$\mathrm{N_{B}}=\\{+2,-2,+2,-2,+2\\}$ (expressed in the $S_{z}$-basis), with
subtle contributions from other spin states. Owing to the presence of a finite
longitudinal component of the field and $N$ being an odd number, the ground
state $\psi_{0}$ has a larger contribution from $\mathrm{N_{A}}$ ($>98\%$)
compared to $\mathrm{N_{B}}$, while the opposite holds for $\psi_{1}$. The
finite contributions of both $\mathrm{N_{A}}$ and $\mathrm{N_{B}}$ in these
two states demonstrate that the Néel states are hybridized, enabling QTM
between them.

In $\psi_{0}$, the contributions of $\mathrm{N_{A}}$ and $\mathrm{N_{B}}$ are
symmetric, whereas in $\psi_{1}$ they are antisymmetric. Around
$B_{x}=B_{x,1}$, the two states undergo an avoided level crossing, beyond
which their symmetry is inverted. At the DP, the contribution of the minority
Néel state vanishes, significantly enhancing the purity of $\psi_{0}$ and
$\psi_{1}$ as mainly composed of $\mathrm{N_{A}}$ and $\mathrm{N_{B}}$,
respectively, thereby suppressing QTM.

Using spin-polarized STM, we are able to capture the time-dependent
magnetization switching of the Fe chain at different $B_{x}$. By positioning
the tip above one of the Fe atoms in the chain, we observe telegraph noise in
the current signals, arising from the magnetization switching between the two
lowest-lying states of the chain (Fig. 1d). The specific spin polarization of
the tip, and which atom in the chain is being probed, determines the current
value characteristic for $\mathrm{N_{A}}$ and $\mathrm{N_{B}}$. For $B_{x}\ll
B_{x,1}$, we detect rapid, yet clearly distinguishable switching events
between two distinct current values. As $B_{x}$ approaches the DP, the
switching rate markedly decreases, to gradually increase again beyond the DP.

To quantitatively investigate the evolution of spin dynamics around the DPs,
we extract the values of spin lifetimes from a current trace as demonstrated
in Fig. 2a. Shown are the lifetimes $\tau_{\mathrm{A}}$ and
$\tau_{\mathrm{B}}$, representing the duration between consecutive switches in
eigenstates dominated by $\mathrm{N_{A}}$ and $\mathrm{N_{B}}$, respectively.
By collecting sufficiently long traces, we obtain histograms for
$\tau_{\mathrm{A}}$ (Fig. 2b) and $\tau_{\mathrm{B}}$ (Fig. 2c), enabling us
to extract characteristic lifetimes $T_{\mathrm{A}}$ and $T_{\mathrm{B}}$,
respectively [13]. Finally, we define $T_{\rm
avg}=\left(T_{\mathrm{A}}^{-1}+T_{\mathrm{B}}^{-1}\right)^{-1}$ as the average
lifetime of the magnetization states. The extracted $T_{\rm avg}$ for atom
I–III of the antiferromagnetic Fe5 chain are shown as a function of the
transverse magnetic field in Fig. 2d, showing a pronounced peak at $B_{x}=4.1$
T spanning two orders of magnitude. As we will demonstrate below, we can
associate this field value to the first DP of the chain $B_{x,1}$. This DP
coincides with a minimum in the scattering amplitude, shown by the black solid
line in Fig. 2d and defined as
$\sum_{a=x,y,z}|\langle\psi_{0}|S_{a,i}|\psi_{1}\rangle|^{2}$, which is an
indication of the hybridization between $\psi_{0}$ and $\psi_{1}$.

Note that the presence of a small longitudinal field induces an avoided level
crossing instead of a level crossing, see Supplementary Notes 3 and 6 for
further details. This provides the peak in the lifetime with a finite width,
enabling us to measure it. The observed trend happens for all atoms of the
chain, and is robust to many experimental parameters (see Supplementary Note
4).

Figure 2: Lifetime of magnetization states. (a) A current trace obtained at a
constant tip height on atom III of the Fe5 chain near the diabolic point
($B_{x}=4$ T). A spin-polarized tip reveals two magnetization states of the
chain: $\mathrm{N_{A}}$ (red) and $\mathrm{N_{B}}$ (blue). The data was
shifted by about 8 pA to center the middle values around zero, mitigating
background fluctuations caused by drift. (b,c) Histograms of lifetimes
$\tau_{\mathrm{A}}$ and $\tau_{\mathrm{B}}$, as defined in a, with a fit to an
exponential function to determine lifetimes $T_{\mathrm{A}}$ and
$T_{\mathrm{B}}$, respectively. (d) Average lifetimes $T_{\rm avg}$ measured
for atoms I, II and III of an antiferromagnetic Fe5 chain at different
$B_{x}$. The corresponding longitudinal component $B_{z}$ is approximately
$0.35\%$ of the transverse magnetic field, due to a small angle
($\alpha\approx 0.2^{\circ}$) between the external magnetic field and the
crystal axes. Error bars are the standard deviation $\sigma$. The lifetime and
the scattering intensities calculated using master rate equations are given in
the purple dashed and black solid lines, respectively. Background color and
diagrams indicate the quanta of $S_{x}$ in the ground state: zero before the
diabolic point and one after the diabolic point, with the diabolic point at
$B_{x,1}\approx 4.1$ T. Different colors of data points correspond to
measurements performed on different atoms along the chain, see diagram. (e)
Lifetime obtained similar to d, but for a magnetic field applied along the
longitudinal axis.

The dashed line is a simulation of the lifetime measurements of the chain
based on master rate equations (see Supplementary Note 2). Neither the
simulations nor the measurements show a peak reaching infinity, owing to
minute contributions of states other than $\mathrm{N_{A}}$ and
$\mathrm{N_{B}}$. The experimental data shows slightly lower lifetimes than
the simulation, especially close to the diabolic point. We attribute this
deviation to accidental high-energy electrons caused by voltage noise, leading
to over-the-barrier excitations.

Until now we have described the situation in terms of the $S_{z}$-basis.
However, for interpretation purposes, it is insightful to consider the
situation in terms of the $S_{x}$-basis. For $B_{x}<4.1$ T, the expectation
value of $S_{x}$ in the ground state approaches zero, indicated by the white
background on the left-side of Fig. 2d and the diagram on the left hand-side.
The first excited state contains a single quantum of $S_{x}$ (i.e. $|\langle
S_{x}\rangle|=1$). Past the DP, i.e. the avoided level crossing, the states
are inverted, transferring the finite $S_{x}$ magnetization to the ground
state, as indicated by the green color of the right-side of Fig. 2d and the
diagram on the right hand-side. Note that the ground state now consists of a
superposition of five spin states, each having the quantum of $S_{x}$ on a
different atom.

We also demonstrate that the increase of lifetime through the DP emerges only
for specific orientation of the magnetic field with respect to the
quantization axis. When the magnetic field is swept along the easy axis of the
antiferromagnetically coupled Fe5 chain, no peak in the lifetime is observed,
see Fig. 2e. The lifetime slightly increases as a result of decreasing
scattering intensity. Within the range of available magnetic fields of $B_{z}$
(up to $\sim 6$ T), there is no energy level crossing. Thus, this behaviour is
strictly monotonic and results from an increasing imbalance in the Néel state
contributions in each eigenstate.

The finite energy difference between the two lowest energy states at the
avoided level crossing is responsible for the width of the observed peak in
Fig. 2d and therefore limits the efficiency of increasing spin lifetimes. The
energy difference at the DP emerges as a consequence of Zeeman energy,
resulting from a small angle $\alpha$ between the quantization axis of the Fe
atoms with respect to the applied magnetic field. This suggests that in order
to achieve a sharper peak, one must achieve a null magnetic field along
$B_{z}$. This is practically impossible, as $\alpha$ will inevitably be
nonzero in our experimental setup. An alternative approach would be to make
use of even-length antiferromagnetic chains, where both Néel states have equal
energy, irrespective of $B_{z}$. In line with this idea, Fig. 3a shows spin
lifetimes measured on a Fe6 chain, where a sharper peak than on the Fe5 chain
is observed.

Despite the expected absence of Zeeman splitting between the two states, the
observed peak shows a finite width, which implies that there must be factors
breaking the symmetry between the Néel states. Our analysis (see Supplementary
Note 5) suggests that the primary cause of the asymmetry is variations in the
$g$-factors of the atoms in the chain. The asymmetry introduced from the tip
field has a minor importance. This difference in $g$-factors results in an
energy discrepancy of approximately 50 $\mu$eV at $B_{1}=6$ T, favoring one
Néel state at higher magnetic fields.

The lifetime of the Fe6 chain increases by nearly three orders of magnitude at
the peak, which is shifted to a lower magnetic field compared to Fe5. The
overall lifetime has also increased, as larger chains are naturally more
stable [13]. It is noteworthy that this considerable enhancement of lifetime
near the DP is only apparent when the magnetization lifetime is primarily
determined by the QTM. Once the over-the-barrier transitions become frequent,
the magnetization anomaly diminishes, and we only observe a subtle change in
lifetime, as shown in the lifetime curves measured at 5 mV (see also
Supplementary Notes 4, 8 and 9). While dehybridization also occurs among
higher energy states, as shown in Supplementary Note 2, it appears not to
affect the rate of over-the-barrier excitations, for voltages close to the
over-the-barrier threshold.

Figure 3: Tuning diabolic points. (a) Lifetime of an antiferromagnetic Fe6
chain. Data points given in filled and empty circles were obtained at $3$ mV
and $5$ mV, respectively, for atoms I-III in the chain. Dashed lines are from
a simulation of the experiment. Background color and diagram indicate the
quanta of $S_{x}$ in the ground state, which increases from zero to one upon
passing the diabolic point. The solid black line indicates the scattering
amplitude from the simulation. In this measurement,at $10$ pA, $1.3$ K,
$\alpha\approx 5^{\circ}$. (b) Similar to panel a, but for a ferromagnetic Fe5
chain, and $\alpha\approx 0.2^{\circ}$. Background color and diagrams indicate
a second quantum of $S_{x}$ entering the ground state as the second diabolic
point at around $5$ T is passed. Note that the easy axis $z$ and hard axis $x$
are defined along the crystallographic directions, see inset. Error bars
indicate 2$\sigma$.

Antiferromagnetic Fe atom chains showed one DP in the transverse magnetic
field ranging from 0 T to 6 T. However, adjusting magnetic interactions
between atoms in the chain through atom manipulation gives us the possibility
to change the location and spacing of DPs. Our simulations indicate that a
ferromagnetic Fe5 chain ($J<0$) is the most likely candidate where multiple
diabolic points can be observed, given the magnetic field range of our
experimental setup. In this case, the chain tends to behave as a macrospin
with a large total spin. When the coupling is sufficiently strong, this leads
to $NS$ positive DPs all equally spaced across the transverse magnetic field.
The highest-field DP corresponds to the $B_{x,3}$ value of a single Fe, while
the lowest-field DP occurs at $B_{x,3}/(2NS-1)$. Consequently, the DPs for
ferromagnetic chains occur at lower magnetic field values.

We built a ferromagnetic Fe5 chain by placing the 5 atoms diagonally with
respect to the easy axis, see inset of Figure 3b, leading to $J=-0.7$ meV
[14]. Figure 3b shows that this chain exhibits two distinct peaks in lifetimes
as a function of $B_{x}$ within our operation range; one at $2.5$ T and one at
$5.0$ T. The second DP is associated with delocalized states where two
different atoms in the chain gain a quantum of $S_{x}$ (i.e. $|\langle
S_{x}\rangle|=1\rightarrow 2$). Hence, this may be associated with two-magnon
states [20, 21].

The simulation results, see the dashed line in Fig. 3b, qualitatively
reproduce the experimental observations but fail to reproduce the exact
positions of the DPs. While the scattering intensity does still contain two
clearly distinguishable peaks, both the measured and simulated lifetime curves
show comparably shallow peaks. We believe that this is due to the overlapping
of the two lifetime peaks as well as a larger Zeeman energy associated with
the ferromagnetic coupling. The larger energy not only broadens the peaks but
also decreases the energy difference between $\psi_{1}$ and higher energy
states, resulting in more over-the-barrier transitions.

Our work presents a comprehensive approach, combining experimental and
theoretical methods, to elucidate the physics of magnetization stability of
individual atomic spin chains near a diabolic point. We have demonstrated that
the switching rate between the two lowest energy levels can be precisely
controlled through tailored transverse magnetic fields, which suppresses
quantum tunneling of magnetization. This suppression emerges as a consequence
of dehybritization of the lowest lying spin states near an avoided level
crossing, which, in the case of Fe chains on Cu2N, results in strong
enhancement of lifetimes up to three orders of magnitude.

While effects of diabolic points have been observed previously in single
molecule magnets [2, 8, 9] through ensemble measurements, we showed that
diabolic points in quantum magnets can be manipulated and rationally designed
by tailoring the interaction of individual atomic spins on surfaces. The
composite nature of atomic spin chains provides the possibility to understand
the topology of diabolic points, with different predicted behaviour for
ferromagnetic and antiferromagnetic chains, and a crucial role of parity
predicted in the latter case.

The dramatic enhancement of the lifetime provides an interesting avenue into
spintronics [17, 22] and applications of coherent spin dynamics [23, 24]. The
extreme sensitivity of spin lifetimes of a quantum magnet near a diabolic
point could be exploited for precise sensing of local and external magnetic
fields at the atomic scale. This sensitivity near diabolic points can also be
used to determine parameters of spin Hamiltonians, such a magnetic anisotropy
and g-factors, with exquisite precision, leading to further understanding of
magnetic material in various applications [25, 26, 27].

R.J.G.E., J.C.R. and S.O. acknowledge support from the Netherlands
Organisation for Scientific Research (NWO) and from the European Research
Council (ERC Starting Grant 676895 “SPINCAD”). D.B., J.O., T.A., J.H., A.J.H.,
and Y.B. acknowledge support from the Institute for Basic Science
(IBS-R027-D1). Y.B. acknowledges support from Asian Office of Aerospace
Research and Development (FA2386-20-1-4052). D.B. acknowledges support from
the Alexander von Humboldt Foundation for financial support through a Feodor-
Lynen Research Fellowship. F.D. acknowledges support from
MCIN/AEI/10.13039/501100011033, and “FEDER, a way to make Europe”, by the
European Union (PID2022-138269NB-I00).

All simulations, raw data, code to process the data, figures and data points
on the figures in the main text and the Supplementary Information [28] are
available from the Open Data folder accessible through the digital object
identifier (”DOI”) 10.5281/zenodo.10906000.

The authors declare no competing interests.

∗Corresponding authors: F.D. (fdelgadoa@ull.edu.es), S.O.
(a.f.otte@tudelft.nl), Y.B<EMAIL_ADDRESS>

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## Supplementary Note 1: Parameters used in the simulations

In the Hamiltonian given in the main text (Eq. 1), the parameters ($D_{i}$,
$E_{i}$, $g_{i}$, and $J_{ij}$) specified for Fe on Cu2N have been
experimentally investigated. The values found in the literature are summarized
in Table 1, showing significant variations among adatoms. These discrepancies
can be attributed to variations of local environments influenced by factors
such as strains in the underlying Cu2N layer, nearby defects, and adjacent
atoms. While previous studies derived parameters from bias spectroscopy, our
approach involved extracting values only based on lifetime curves, but
referred to the literature values in Table 1 to facilitate optimizing the
parameters for simulations. The simulations are based on the Master Rate
Equations [29, 30, 31].

$N$ |  $D$ (meV) |  $E$ (meV) |  $J$ (meV) | $g$ | Source | Year
---|---|---|---|---|---|---
$1$ | $-1.55\pm 0.01$ | $0.31\pm 0.01$ |  | $2.11\pm 0.05$ | [10] | 2007
$1$ | $-1.55\pm 0.02$ | $0.31\pm 0.01$ |  | $2.11$ | [32] | 2010
$3$ (sim.) | $-1.53$ | $0.37$ | $>25$ |  | [33] | 2011
$\geq 3$ |  |  | $1.3\pm 0.1$ |  | [13] | 2012
$2$ | $-1.87$ | $0.31$ | $0.7$ |  | [34] | 2013
$2$ (FM) | $-1.37$ | $0.31$ | $-0.69$ |  | [34] | 2013
$6$ (FM) | $-1.29$ | $0.31$ | $-0.73$ |  | [14] | 2014
$1$ | $-1.59\pm 0.03$ | $0.31\pm 0.01$ |  | $2.07\pm 0.19$ | [35] | 2015
$3$ | $\\{-2.1,-3.6,-2.1\\}$ | $0.31$ | $1.15\pm 0.1$ | $2.1$ | [16] | 2015
$3$ | $\\{-2.1,-3.6,-2.1\\}$ | $0.3$ | $1.15$ | $2$ | [36] | 2017
$3$ | $\\{-1.88,-2.42,-1.87\\}$ | $\\{0.39,0.2,0.31\\}$ | $\\{0.71,0.66\\}$ |  | [37] | 2017
$1$ (sim.) | $-1.53$ | $<0$ |  | $2.14\pm 0.12$ | [38] | 2019
$3,5$ | $-1.87$ | $0.31$ | $0.7$ | $2.11$ | [17] | 2020
Table 1: Overview of literature values for parameters in the Hamiltonian. The
first column shows the number of atoms $N$. The values given in the table are
mostly obtained from experiments conducted on antiferromagnetically coupled Fe
atomic chains, except for instances labeled as ”(sim.)” and ”(FM)”, denoting
studies based on simulation results and ferromagnetic chains, respectively.
The second column describes the uniaxial magnetic anisotropy term $D$, either
considered equal for all atoms or uniquely defined for each atom (three cases
of trimers). The third column describes the transverse magnetic anisotropy
parameter $E$ and has one study with a unique definition for each individual
atom of a trimer. Ref. [38] found this to be negative, contrary to our
definition. The fourth column describes the Heisenberg exchange coupling
strength $J$, showing both ferromagnetic ($J<0$) and antiferromagnetic ($J>0$)
couplings. A study [33] reporting $J$ larger than $25$ meV is inconsistent
with our observations. Ref. [37] describes different coupling strengths
between different neighbours in a trimer. The fifth column describes the
g-factor. The sixth and seventh columns provide the reference and publication
year of these studies, respectively.

Table 2 provides a summary of the parameters used in the simulations of each
figure. The simulation result used for Fig. 1a was processed by subtracting a
linear slope along $B_{x}$ for clear visualization of the diabolic point. In
the other simulations, we considered angles, $\alpha$ and $\beta$,
representing the directions of the applied magnetic field with respect to the
crystal axes (further details in Supplementary Note 3). In addition to the
external magnetic field, the tip magnetic field induced by the magnetic
interaction between the magnetic tip and the atom beneath it was considered.
Note that we considered the $B_{z}$ component only, given the negligibly small
tip field along $B_{x}$ compared to the external field. The coupling with the
substrate is given as $G_{\rm SS}=1$ $\mu$S, following the definition given in
[37].

Noting that the simulation shows lifetimes larger than the experimental
values, especially closer to the peak center, we believe the measured lifetime
is limited by accidental voltage spikes that may cause over-the-barrier
transitions. This is corroborated by longer lifetimes measured upon better
grounding of thermometry lines close to the current line. While we did not
find a proper way to include such spurious voltages in the model, the
simulations were performed with elevated temperatures to account for the on-
average higher electron energy. We believe there has been a small voltage
offset during the measurements of Fig. 3a, resulting in voltages slightly
below the intended values. For Fig. 2 and 3b, we introduced a relatively
strong tip polarization $\eta$ (defined between $-1$ and $+1$), which gives
better match between the simulation and experimental results. To expedite the
processing time, only the lowest 250 energy eigenstates were considered, with
only the 500 largest contributions of spin eigenstates within these energy
eigenstates.

Paramater | Figure 1a | Figure 2d | Figure 2e | Figure 3a | Figure 3b
---|---|---|---|---|---
$N$ | 1 | 5 | 5 | 6 | 5
$D$ (meV) | -1.87 | $\\{-2.05,-2.35,-2.85,...\\}$ | $\\{-2.1,-2.3,-2.85,...\\}$ | $\\{-1.95,-2.25,-2.80,...\\}$ |  $-1.70$ (all)
$E$ (meV) | 0.31 | $\\{0.33,0.32,0.32,...\\}$ |  $0.32$ (all) |  $0.3$ (all) |  $0.4$ (all)
$J$ (meV) |  | $\\{0.85,1,...\\}$ |  $0.83$ (all) | $\\{0.85,0.95,1.35...\\}$ |  $-0.95$ (all)
$g$ | 2.11 | $\\{2.5,2.11,2.11,2.11,2.11\\}$ | $\\{2.5,1.9,2.5,1.9,1.8\\}$ | $\\{2.4,2.11,2.11,2.11,2.11,2.11\\}$ | $\\{2.5,2.11,2.11,2.11,2.11\\}$
$B_{\rm tip}$ (mT) |  | 50 on atom i | 100 on atom i | -110 on probed atom | -60 on atom i
$\alpha(^{\circ})$ |  | 0.2 | 0.8 | 4.1 | 0.2
$T$ (K) |  | 3 | 1.3 | 2.3 | 1.3
$V$ (mV) |  | 3 | 3 | 2.75 (purple), 4.75 (grey) | 3
$G_{\rm SS}$ ($\mu$S) |  | 1 | 4 | 4 | 4
$\eta$ |  | 0.5 | 0.5 | 0 | 0.5

Table 2: Overview of parameters used in the simulations presented in the main
text. $D$, $E$, $J$, and $g$ are the same parameters as given in Table 1.
$B_{\rm tip}$ represents the magnetic field induced on a single atom due to
the tip field along $B_{z}$, $\alpha$ the angle between the external magnetic
field and the crystal lattice, $V$ the voltage applied for measurements,
$G_{\rm SS}$ the electron coupling with the bath, and $\eta$ the polarization
of the tip. For $D$, $E$, $J$, and $g$, a list of values is given,
corresponding to the values on or between the various atoms, starting with
atom I. Here an ellipsis (”…”) indicates that the remaining values are
symmetric with respect to the previous values (following $a,b,c,b,a$ for
$N=5$, and $a,b,c,c,b,a$ for $N=6$).

## Supplementary Note 2: Simulations of lifetime around a diabolic point

The simulations to describe the evolution of lifetime as a function of the
external magnetic fields are based on rate equations [29, 30, 39], which
include transitions between various states following spin-selection and energy
conservation rules. These transitions can be derived by including the exchange
coupling of the local spins with the itinerant electrons [40]. In a
semiclassical description, under an applied longitudinal field $B_{z}$, a spin
$S$ with an easy-axis anisotropy exhibits the ground state $\psi_{0}$ with
$m_{s}=-S$ along the easy axis and the first excited state $\psi_{1}$ with
$m_{s}=+S$. Thus transitions between these states while exchanging spin
angular momentum with tunneling electrons necessitate passing through
intermediate states, adhering to the spin conservation rule ($\Delta m_{S}=\pm
1$). These intermediate states are higher in energy, which requires tunneling
electrons with sufficient energy to overcome the effective energy barrier. In
a quantum mechanical framework, states $\psi_{0}$ and $\psi_{1}$ experience
slight hybridization, leading to direct transitions between them without
exchanging the spin angular momentum ($\Delta m_{s}=0$). For a single Fe atom
on Cu2N, the transverse magnetic anisotropy induces this hybridization,
resulting in the ground state comprising a linear combination of spin states
$m_{s}=-2$ and $m_{s}=+2$. This hybridization facilitates Quantum Tunneling of
Magnetization (QTM), leading to direct transitions between both states.

We can thus differentiate between two types of transitions. First, there is
the so-called flip-flop transition where the magnetic angular momentum of the
tunneling electron is exchanged with the one of the atom, leading to $\Delta
m_{s}=\pm 1$, which is the primary cause of over-the-barrier transitions and
is expressed as $\langle\psi_{0}|S_{\pm,i}|\psi_{1}\rangle$. Secondly, there
is a so-called direct transition, where $\Delta m_{s}=0$, which is the primary
cause of QTM. By suppressing the hybridization, the direct transitions, whose
intensities depend on $\langle\psi_{0}|S_{z,i}|\psi_{1}\rangle$, are
suppressed. This quenching reduces the switching rates due to QTM,
consequently leading to longer lifetimes in the absence of over-the-barrier
excitations. The transition likelihood between $\psi_{0}$ and $\psi_{1}$ for a
single tunneling electron can be encapsulated by the scattering intensity
$P_{0,1}=\sum_{a=x,y,z}|\langle\psi_{0}|S_{a,i}|\psi_{1}\rangle|^{2}$, which
is plotted in Figs. 2d, 2e, 3a and 3b and is a measure for how likely QTM is.

More specifically, in the absence of an external magnetic field, hybridization
occurs from the transverse magnetic anisotropy term
$E(S_{x}^{2}-S_{y}^{2})=E(S_{+}^{2}+S_{-}^{2})/2$. For a single Fe atom with
$S=2$, the magnetic anisotropy terms, $D=-1.87$ meV and $E=0.31$ meV, result
in the ground state of $\psi_{0}=(0.7,0,-0.139,0,0.7)$ as a symmetric
superposition of $m_{s}=\pm 2$. The first excited state
$\psi_{1}=(0.707,0,0,0,-0.707)$ represents an antisymmetric superposition.
Figure 4a and b show how these states evolve when sweeping $B_{x}$ as given by
the $S_{z}$ and $S_{x}$ basis, respectively.

Figure 4: State decomposition of the single Fe atom as a function of $B_{x}$.
(a) Decomposition of $\psi_{0}$ and $\psi_{1}$ (top) in the $S_{z}$-basis, and
$\psi_{2}$ through $\psi_{4}$ (bottom) as a function of $B_{x}$ divided by
$B_{x,1}$, the field value of the first diabolic point. (b) Same but in the
$S_{x}$-basis. Red curves represent $m_{s}=\pm 2$, blue curves correspond to
$m_{s}=\pm 1$, and a green curve is for $m_{s}=0$ in the respective basis.
Dashed lines are for $m_{s}>0$. (c) (top) Energies of the five states of the
single Fe atom over the same magnetic field range as panels a and b. The
abrupt changes in the decomposition curves around $B_{x,1}$ and $3B_{x,1}$
correspond to avoided level crossings in the energy spectrum. (bottom) Energy
difference between $\psi_{0}$ and $\psi_{1}$ from the top panel. (d) Energies
of the five states when the magnetic field is applied along $B_{z}$ instead.
In all cases, there is $10$ mT in the other in-plane direction to induce
smooth avoided level crossings.

Around $B_{x,1}$ and $3B_{x,1}=B_{x,3}$, there are avoided level crossings
between $\psi_{0}$ and $\psi_{1}$ (Fig. 4c), where the two states effectively
swap their spin composition. In terms of $S_{z}$-basis, this results in a
switching of the symmetry, i.e., for the first avoided level crossing, a
transition from $|\psi_{0}\rangle\sim(|-2\rangle_{z}+|+2\rangle_{z})/\sqrt{2}$
to $|\psi_{0}\rangle\sim(|-2\rangle_{z}-|+2\rangle_{z})/\sqrt{2}$. In the
process, the $\langle\psi_{0}|+2\rangle_{z}$ contribution goes through zero
while the $\langle\psi_{0}|-2\rangle_{z}$ one increases in contribution. A
similar effect happens at the second avoided level crossing. In both cases, at
the exact crossing points, $\psi_{0}$ has zero contribution of $m_{z}=+2$ and
$\psi_{1}$ has zero contribution of $m_{z}=-2$. This significantly suppresses
the $\Delta m_{z}=0$ transition probability between the two states, compared
to field values far from these crossings. The width of the lifetime peak vs.
$B_{x}$, or, alternatively, the width of the avoided level crossing in the
energy spectrum (see Fig. 4c) is ultimately determined by the $B_{z}$-field,
being strictly zero when $B_{z}=0$. For a finite $B_{z}$, the hybridization
between the $|\pm 2\rangle_{z}$ states is reduced, and the quenching of the
QTM near a DP survives in a larger $B_{x}$-window. This quenching also happens
between $\psi_{2}$ and $\psi_{3}$, although the effect is much sharper as the
$B_{x}$ field has a much more drastic effect on the energy splitting between
the $\psi_{2}$ and $\psi_{3}$ states.

While the $S_{z}$-basis is convenient for explaining the critical behaviour of
lifetimes near the diabolic point, a better insight into the underlying
mechanism can be gained using the basis of spin states along the applied
field, i.e., the $S_{x}$-basis (Fig. 4b). For small $B_{x}$, $\psi_{0}$ is
dominated by $m_{x}=0$ with minor contribution in $m_{x}=\pm 2$ due to the
transverse magnetic anisotropy. At the first diabolic point, $\psi_{0}$ and
$\psi_{1}$ swap their spin compositions. Thus, $\psi_{0}$ converts from a
state dominated by $m_{x}=0$ to one dominated by $m_{x}=-1$. Then, at the
second diabolic point, $\psi_{0}$ converts into a state dominated by
$m_{x}=-2$. Hence, at each diabolic point, a quantum of $S_{x}$ enters the
ground state. As shown in Fig. 4, abrupt changes in both energy spectra and
states’ composition only appears near the DPs.

In summary, we can understand each crossing between $\psi_{0}$ and $\psi_{1}$
as the change of a quantum of $S_{x}$ in the ground state. Thus, as $B_{x}$ is
swept, the ground state changes from the majority state of $m_{x}=0$ to
$m_{x}=-1$ after the first crossing and to $m_{x}=-2$ after the second
crossing. In terms of the $S_{z}$ state composition, an additional unit of
$S_{x}$ results in alternating symmetry in the linear combinations of
$m_{z}=\pm m$ for any $m$. This swapping of symmetry results in values of
$B_{x}$ where the hybridization is minimized, ultimately leading to maxima in
the lifetimes.

Compared to the single Fe atom, the behaviour of spin chains is more intricate
but fundamentally based on the same physics for weak coupling between the
atoms. The number of diabolic points grows with the total spin of the system
and, consequently, with the number of atoms $N$. This is illustrated in Fig. 5
where, using the same values of $D$, $E$, and $g$, we calculated all positive
diabolical points $B^{N,j}_{x}$ for the $j\in\\{1,...,NS\\}$ values of chains
with $N=2-6$ spins for different coupling strengths of $J$ including both
ferromagnetic ($J<0$, Fig. 5a) and antiferromagnetic ($J>0$, Fig. 5b) cases.
The $N=5$ case is shown in thick black lines for clarity. The value $j=1$
corresponds to the lowest DP. In Fig. 5c, we narrow our focus to a smaller
window aligned with experimental conditions.

Figure 5: Magnetic field values of diabolic points for Fe chains with
different lengths $N=\\{2,3,4,5,6\\}$ and different coupling strengths
$J/|D|$. (a) Ferromagnetically coupled atomic chains ($J/|D|\leq 0$) in a
linear scale. (b) Antiferromagnetically coupled case ($J/|D|>0$) in a
logarithmic scale to facilitate comparison under different regimes. (c) Zoom-
in for $|J/D|\leq 1$ and $B_{x}\leq 0.4B_{x,3}$. The shaded region contains
values of $B_{x}$ and $J/|D|$ relevant to the experiments performed throughout
this work. The horizontal shaded region indicates our external magnetic field
range. All plots were constructed by finding degeneracies between $\psi_{0}$
and $\psi_{1}$ as a function of $B_{x}$ (here, $B_{z}=0$). The legend
presented in panel b applies to all panels. To facilitate comparison, solid
lines (dashed lines) indicate odd (even) number chains.

For $J=0$, the $N$ atoms are decoupled, resulting in the same diabolic points
at $B_{x,1}$ and $B_{x,3}$. As $|J|$ increases, the degeneracy between atoms
is lifted, leading to the dispersion of diabolic points into multiple
branches. The number of these branches corresponds to the number of atoms in
the chain. Consequently, by increasing $j$, the quanta of $S_{x}$ in the
ground state increase upon passing the $j$th-DP, as happened twice for Fig.
3b.

There is a major difference between the ferromagnetic and antiferromagnetic
chains. The ferromagnetic case (Fig. 5a) is certainly more intuitive: as
$-J/|D|$ increases, the diabolic points tend towards the DPs of a macrospin of
dimension $NS$. This can be observed in the spacing between each subsequent
diabolic point, which follows the same spacing as described in Eq. 2 of the
main text. Surprisingly, irrespective of the length of the chain, the highest
diabolic point is always equal to $B_{x,3}$. This implies that the lowest
diabolic point can be found at $B_{x,3}/(2SN-1)$, and the next diabolic point
can be found at three times this value. The ferromagnetic chain in the main
text has a ratio $B^{5,2}_{x}/B^{5,1}_{x}\sim 2$, suggesting that $|J|$ is too
small to consider the chain as a single macrospin, consistent with our
simulations.

For the antiferromagnetic ($J>0$) chain (Fig. 5b), when $J/|D|\gg 1$, the odd-
numbered chains present $S$ DPs at $B_{x}\lesssim B_{x,2S-1}$. This
corresponds to the uncompensated spin in such chains. This may also open the
possibility of exploring diabolic points in MnN-chains as the ones explored by
Hirjibehedin et al. [15], where $J/|D|\approx 160$ [10] and $B_{x,1}\approx
0.5$ T. Simulations show lifetime peaks at around $0.75$ T and $1.5$ T for Mn3
and at around $1$ T and $2$ T for Mn5.

In agreement with our experimental observations, Fig. 5c shows that as $N$
increases for values $|J|/|D|\approx 0.5$, the lowest diabolic point
$B^{N,1}_{x,1}$ is found at smaller values of $B_{x}$. For much larger $J$,
all DPs of even-numbered antiferromangetic chains diverge to extreme values.
It also shows that in our operable window (up to about $0.21B_{x,3}$), the
easiest way to measure multiple DPs is by going to ferromagnetic chains. We
had considered antiferromagnetic chains of longer length, like Fe8. However,
the time needed to measure statistically relevant data at the DP exceeded the
scope of this work.

The only noteworthy effect of local changes in the $g$-factors is a small
splitting of the $B^{N,j}_{x}$ diabolic points when $|J/D|\rightarrow 0$, not
shown in the figure.

## Supplementary Note 3: Sample orientation

Figure 6: Alignments of the sample with magnetic fields. (a) The substrate is
given as a square plate, on which five red spheres represent the Fe5 atom
chain. The substrate’s orientation can be adjusted along three axes through
angles $\alpha$, $\beta$, and $\gamma$. An external magnetic field,
facilitated by a two-axis vector magnet, can be applied along $\bm{B}_{1}$ and
$\bm{B}_{2}$. Due to misalignment between the crystal axes and the magnetic
field directions, the applied fields can be decomposed into $\bm{B}_{x}$,
$\bm{B}_{y}$ and $\bm{B}_{z}$ for the chains in terms of crystal axes. (b)
Tait-Bryan rotation along the z-axis with an angle $\beta$. A negative value
of $\beta$ is shown to represent the actual angle $\sim-8^{\circ}$. (c) Tait-
Bryan rotation along the x’-axis with an angle $\gamma$. A small value
$\gamma$ is shown to represent the near negligible actual value. (d) Tait-
Bryan rotation along the y”-axis with an angle $\alpha$. A large value of
$\alpha$ is shown to represent the actual large variability of $\alpha$ in the
work presented here.

Since the shape of the lifetime curves strongly depends on the exact
orientation and magnitude of the applied magnetic field, particularly the
$B_{z}$ component, the precise characterization of magnetic field components
with respect to the crystal axes is crucial. Figure 6 shows the Fe5 chain on a
Cu(100) crystal, illustrating the directions of the external magnetic field
and the crystal axes. In our experiment, we found that the crystal axes are
not fully aligned with the external magnetic field directions due to several
reasons:

  1. 1.

The sample holder, designed for a certain angle ($\sim 8^{\circ}$) for the
cold deposition [41], contributes significantly to the misalignment between
$\bm{B}_{1}$ and $\bm{B}_{x}$, resulting in the angle $\beta=\sim-8^{\circ}$.

  2. 2.

Mounting the Cu(100) crystal to the sample holder introduces a slight in-plane
rotation, which gives a finite $\alpha$. This can be estimated from the atomic
resolution images and adjusted by remounting the crystal (Fig. 9b,d,f).

  3. 3.

There is a subtle misalignment between the STM stage and the magnetic field
axes, which affects all three angles. From the experiment, we found the
effects on $\gamma$ and $\beta$ are negligibly small while giving $\sim
3^{\circ}$ deviation for $\alpha$, denoted as $\alpha_{\rm tilt}$.

The resulting combination of rotations can be expressed in a sequence of Tait-
Bryan rotations, as indicated in Fig. 6b-d. Applying the rotations in the
given order results in the following decomposition of the external field:

$\aligned
B_{x}=&B_{1}(\cos\alpha\cos\beta-\sin\alpha\sin\beta\sin\gamma)+B_{2}(\cos\alpha\sin\beta+\sin\alpha\cos\beta\sin\gamma)\\\
B_{y}=B_{1}(-\sin\beta\cos\gamma)+B_{2}(\cos\beta\cos\gamma)\\\
B_{z}=B_{1}(\sin\alpha\cos\beta+\cos\alpha\sin\beta\sin\gamma)+B_{2}(\sin\alpha\sin\beta-\cos\alpha\cos\beta\sin\gamma)\\\
$ (3)

In the limit where $\gamma=0$, $B_{2}=0$, $|\beta|\ll\pi/2$ and
$|\alpha|\ll\pi/2$, these equations can be simplified to:

$\aligned B_{x}=&B_{1}(\cos\alpha\cos\beta)\\\ B_{y}=B_{1}(-\sin\beta)\\\
B_{z}=B_{1}(\sin\alpha\cos\beta)\\\ $ (4)

## Supplementary Note 4: Variation of lifetime curves at different conditions

Figure 7: Lifetime curves for antiferromagnetic Fe5 under various experimental
conditions. (a) Lifetimes measured for each Fe atom in the chain at different
out-of-plane fields, $\bm{B}_{y}$. The variation in $\bm{B}_{y}$ is primarily
achieved by adjusting the $\bm{B}_{2}$ field. For the green curves,
$\bm{B}_{2}=\sin(8^{\circ})B_{1}$, while for the red curves, $\bm{B}_{2}=0$.
(b) Lifetimes measured for atom III at different $\alpha$ values obtained by
rotating the crystal. The labels indicate the actual angle. The data in the
main text (Fig 2d) is included in blue. (c) Lifetimes for each Fe atom in a
chain measured at different bias voltages. (d) Lifetimes measured under the
same conditions on the same chain but using two different tips. (e) Up-to-up
switching time, defined as $1/(1/\tau_{A}+1/\tau_{B})$ (gray), while sweeping
the external magnetic field from $4.75$ T to $3.25$ T. During the sweep, the
tip remained in constant-current mode over the atom III to compensate for
drift. The time between switches was long enough such that the tip height
followed telegraphic noise-like behaviour. A rolling average of 50 points is
given in black. All magnetic field values, $B_{x}$ and $B_{z}$, were
determined by converting the applied external magnetic fields considering the
crystal axes. All data (except for the blue curve in (b) and the green curve
in (c)) were obtained with $V_{\rm bias}=5$ mV, $I=10$ pA, $T=1.3$ K (blue
data obtained at $T=1.4$ K and $V=3$ mV). All data (except the red and blue
curves in (b)) were obtained with $\alpha=5^{\circ}$. Error bars represent 2
standard deviations. Connecting lines between data points are guides to the
eye. Solid lines correspond to atom III, dashed lines to atoms II and IV, and
dotted lines to atoms I and V.

To clarify major effects of $B_{y}$ (Fig. 7a), $\alpha$ (Fig. 7b),
$V_{\mathrm{bias}}$ (Fig. 7c) and tip apex (Fig. 7d) on the shape of the
lifetime curves, we present further data sets obtained under various
conditions on an antiferromagnetic Fe5 chain. Further analyses of voltage
dependence, as well as current dependence, are explored in Supplementary Notes
7 and 8, respectively.

As explained in the previous section, although the external magnetic fields
$\bm{B}_{1}$ and $\bm{B}_{2}$ are closely aligned along the in-plane and out-
of-plane directions, respectively, simultaneous adjustments of $\bm{B}_{1}$
and $\bm{B}_{2}$ are necessary to selectively apply a field along one of
$\hat{B_{x}}$, $\hat{B_{y}}$ and $\hat{B_{z}}$. For instance, applying
$\bm{B}_{1}$ to change the in-plane field induces a finite out-of-plane field,
given by $B_{y}=B_{1}\sin(8^{\circ})$. To understand the effect of the out-of-
plane magnetic field on the lifetime, we varied $\bm{B}_{2}$ in a way to
double the total $B_{y}$ experienced by the chain, i.e.,
$B_{2}=B_{1}\sin(8^{\circ})$. As shown in Fig. 7a, the overall lifetimes show
a slight reduction, and the peak near the diabolic point is slightly
attenuated with increased $B_{y}$, consistent with simulations. Although a
more thorough analysis is required, we expect that fully compensating $B_{y}$
could yield slightly more pronounced peaks near the diabolic point. However,
since the influence of $B_{y}$ is not substantial and a small $B_{y}$ would
not change the underlying physics behind our studies, we kept $B_{2}=0$ for
the rest of our experiments.

Adjusting the angle $\alpha$ is correlated with changing the ratio between two
in-plane fields, $B_{x}$ and $B_{z}$. A larger $\alpha$ corresponds to a
higher $B_{z}$ at a certain $B_{1}$. The increase in $B_{z}$ is associated
with an increasing contribution of the main Néel states for $\psi_{0}$ and
$\psi_{1}$, at the expense of the minor contribution. In other words,
$\psi_{0}\rightarrow N_{A}$ and $\psi_{1}\rightarrow N_{B}$, leading to a
reduction in hybridization. Thus, at a larger $\alpha$, we observed an overall
increase in the lifetime compared to a smaller $\alpha$. Secondly, the Zeeman
splitting between these two lowest-lying states was increased, which results
in the smearing out of the peak near the diabolic point (Fig. 7b). Note that
the lifetime curves were measured with different samples and different tips
since we needed to remount the crystal to rotate it and thus to change
$\alpha$, which may introduce additional effects of local environments on the
chain, such as local defects, strain in the underlying Cu2N layer, diverse tip
apexes, and so on. The influence of different tip apexes is shown in Fig. 7d.
As measured on the same Fe5 chain but with different spin-polarized tips, the
lifetime varies not only in the magnitudes but also in the magnetic field
value for the lifetime peak. This indicates variations of the tip magnetic
fields along both $B_{x}$ and $B_{z}$.

Figure 7c shows the effect of the applied bias voltages. Increasing the
voltage results in more over-the-barrier transitions and, thus, larger
variations in the lifetime among the atoms in the chain (consistent with the
data shown in Fig. 3a of the main text). See also Supplementary Notes 8 and 9
for further discussions.

Figure 7e shows the up-to-up switching times during a magnetic field sweep.
The raw data is given in gray, while the black line shows a rolling average
using SciPy’s 1D Gaussian filter (50 data points). Even for this rough
measurement, we can clearly see the switching rate is significantly reduced
near 4.15 T, corresponding to the diabolic point. Considering that obtaining a
dataset for lifetime curves takes at least one day, this 45-minute measurement
provides a quick way to verify the location of the diabolic point.

## Supplementary Note 5: Symmetry breaker of Fe6

In the absence of external magnetic fields, the two lowest-lying eigenstates,
$\psi_{0}$ and $\psi_{1}$, in an Fe chain are composed of the symmetric and
antisymmetric superposition of the Néel states, respectively. The energy
difference between these states arises from transverse magnetic anisotropy
($E(S_{x}^{2}-S_{y}^{2})$). This energy difference is effectively mitigated by
applying a transverse magnetic field $B_{x}$. At the diabolic point, the
$B_{x}$ fully compensates for the energy difference, leading to an energy
level crossing. In an ideal situation, where all atoms on a surface are
identical and devoid of any variations in local environments, the diabolic
point is expected to manifest in an exceedingly sharp window of $B_{x}$ (a
singular point value) with nearly infinite lifetimes. This sharp transition
can be smeared out by converting the level crossing into an avoided-level
crossing, where the broadening is proportional to the energy difference
between the two states.

For a ferromagnetic chain or an odd-numbered antiferromagnetic chain, this
avoided-level crossing can be easily induced by applying a longitudinal
magnetic field $B_{z}$, which yields the energy difference between two Néel
states by the Zeeman energy. In contrast, achieving an avoided-level crossing
for an even-numbered antiferromagnetic chain is nontrivial, as the two Néel
states possess identical energies across all $B_{z}$ fields. This conflicts
with our observations from the Fe6 chain in Fig. 3a, which suggests a symmetry
break between the two Néel states, ultimately making one state more favored
than the other under a finite Zeeman energy. We attribute this symmetry break
to inhomogeneity in the chain. In reality, there are subtle variations in
local environments among atoms on surfaces, which results in different
$g$-values and magnetic interactions between them. In addition, the presence
of a magnetic tip located over one atom in the chain provides a tip-induced
local magnetic field.

In our simulations, the Zeeman energy induced by variations in $g$-values is
included by assigning a different $g$-value to atom I in the chain rather than
introducing varied $g$-values for all atoms. Consequently, the total Zeeman
splitting is proportional to an applied field along the $\hat{z}$-axis and the
$g$-factor mismatch between the atom I and the rest of the atoms
$g_{1}-g_{\mathrm{rest}}$.

Figure 8: Lifetime ratios defined as $T_{\mathrm{H}}/T_{\mathrm{L}}$ for odd-
and even-numbered chains. (a) Lifetime ratios of a Fe5 chain obtained at
$\alpha=5^{\circ}$. In the chain, even-numbered atoms II and IV exhibit a
lifetime ratio of about 3, while odd-numbered atoms I, III, and V show a
lifetime ratio of about $1/3$ across the given magnetic field range. (b)
Lifetime ratios of Fe6 for the data presented in the main text. Blue and green
colors indicate even- and odd-numbered atoms, respectively. A split between
even- and odd-numbered atoms starts to emerge around $2$ T, diminishes around
the diabolic point, and increases for higher magnetic fields. Lines are guides
to the eye. Error bars represent two standard deviations. Data in panel a
corresponds to the $10$ pA data presented in Fig. 12.

To confirm the symmetry break, we compare the lifetime ratios between atoms in
the Fe5 and Fe6 chains. While our primary focus in the main text is on
$T_{\mathrm{A}}$ and $T_{\mathrm{B}}$ for the average lifetimes associated
mostly with Néel states $N_{\mathrm{A}}$ and $N_{\mathrm{B}}$, respectively,
we shift our attention here to the average lifetimes of $T_{\mathrm{H}}$ and
$T_{\mathrm{L}}$ related to high and low spin-polarized current states,
respectively. This keeps the analysis more general and highlights the
alternating pattern of antiferromagnetic chains better. At the diabolic point,
in the absence of Zeeman energy, the two lowest-lying states are degenerate,
resulting in equal lifetimes for $T_{\mathrm{H}}$ and $T_{\mathrm{L}}$ (i.e.
$T_{\mathrm{H}}/T_{\mathrm{L}}\approx 1$). Importantly, this means that a
larger imbalance between $T_{\mathrm{H}}$ and $T_{\mathrm{L}}$ indicates a
greater energy difference between the two states. Thus, we use the lifetime
ratio $T_{\mathrm{H}}/T_{\mathrm{L}}$ as a proxy for the ratio of state
ensemble occupation, which, according to the Boltzmann distribution, provides
insights into the energy difference between the two states.

Figures 8a and b present the lifetime ratios ($T_{\mathrm{H}}/T_{\mathrm{L}}$)
for the Fe5 and Fe6 chains, respectively. In the Fe5 chain, the lifetime ratio
for atoms I, III and V is about $1/3$, while for atoms II and IV, it is about
$3$. This simple inversion in the $T_{\mathrm{H}}$ and $T_{\mathrm{L}}$ ratios
indicates that $\psi_{0}$ and $\psi_{1}$ predominantly consist of specific
Néel states, namely $N_{\mathrm{A}}$ and $N_{\mathrm{B}}$, respectively.
Interpreted as population ratios, the lifetime ratios given in Fig. 8a
correspond to an energy difference between $\psi_{0}$ and $\psi_{1}$ of about
$140$ $\mu$eV at around 4T. Note that this dataset was obtained with the
sample’s angle of $\alpha\approx 5^{\circ}$, which is larger than the one in
Fig. 2.

The Fe6 chain (Fig. 8b) shows relatively small variation in the lifetime
ratios among the atoms. As mentioned before, in an ideal situation, the even-
numbered AFM chain should yield $T_{\mathrm{H}}/T_{\mathrm{L}}=1$. However, as
shown in Fig. 8b, especially at larger $B_{x}$ values, a finite splitting
occurs. We attribute this splitting to slight variations in the $g$-factors of
the atoms arising from local imperfections near the atoms, such as defects or
edges of the Cu2N island (indicated by white arrows in STM images of Fig.9).
For $B_{x}<2$ T, the lifetime ratios measured on each atom are approximately
the same, albeit slightly larger than unity. We interpret this as the result
of a longitudinal tip field of about $30$ mT causing a similar imbalance on
each atom. Around the diabolic point ($3.5$ T), the lifetime ratios approach
unity. Meanwhile, at $B_{x}=6$ T, the lifetime ratios are about $1.6$ and
$0.70$ for even- and odd-numbered atoms in the chain, respectively. This
corresponds to an energy difference of about $50$ $\mu$eV, which clearly
indicates the Zeeman energy between two ground states induced due to the
inhomogeneity among atoms in the chain.

## Supplementary Note 6: Investigating $\alpha_{\mathrm{tilt}}$

While the crystal orientation with respect to the STM stage ($\alpha_{\rm
atomic}$) can be estimated by scanning the sample surface at atomic
resolution, the alignment between the STM stage and the magnets need to be
thoroughly investigated to identify the magnetic fields with respect to the
crystal axes. When applying $B_{1}$, the $B_{z}$ field expressed in the
crystal axes is given by $B_{z}\approx B_{1}\sin(\alpha)$. Due to a slight
misalignment (”tilt”) between the magnetic field axes and the STM stage, the
angle $\alpha_{\rm atomic}$, derived from atomic resolution topographic
images, needs to be corrected by a value $\alpha_{\rm tilt}$ such that the
angle $\alpha=\alpha_{\rm atomic}-\alpha_{\rm tilt}$. Note that $\alpha_{\rm
tilt}$ remains constant throughout our experiments, set by the installation of
the STM stage with respect to the magnets, while $\alpha_{\rm atomic}$ varies
depending on how the substrate is mounted on the sample holder. In this
section, we demonstrate that $\alpha_{\rm tilt}=3^{\circ}$, estimated from the
analysis of lifetime ratios and peak widths of lifetime curves.

The peak width of a lifetime curve and the lifetime ratios primarily depend on
the energy difference between $\psi_{0}$ and $\psi_{1}$, dominated by the
Zeeman energy. Accurate determination of the Zeeman energy necessitates the
determination of the angle $\alpha$, considering $\alpha_{\mathrm{atomic}}$
and $\alpha_{\mathrm{tilt}}$. In our experiments, crystal orientations can be
adjusted by rotating the sample with respect to the sample holder. Figures
9a–f show lifetime ratios and corresponding STM images for three different
chains, each with a different $\alpha_{\rm atomic}$. From STM images, we can
extract the angle $\alpha_{\rm atomic}$ for each of the chains:
$\alpha_{\mathrm{atomic}}=-2^{\circ}$ for Fig. 9b, $+3^{\circ}$ for Fig. 9d,
and $+12^{\circ}$ for Fig. 9f. Depending on $\alpha_{\mathrm{atomic}}$, we
observe clear variations in the lifetime ratios and in their deviations
between even- and odd-numbered atoms of the chains due to different Zeeman
energies. For $\alpha_{\mathrm{atomic}}=3^{\circ}$ (Fig. 9c), we observe a
negligibly small difference in lifetime ratios between even- and odd-numbered
atoms, which already implies the minimal Zeeman energy ($\alpha\approx 0$).
Based on the extracted angle $\alpha_{\rm atomic}$, we deduce $B_{z}$ values
relative to applied magnetic fields $B_{1}$ and, thus, calculate Zeeman
energies for each chain. For a single unpaired spin with $S=2$ for the Fe5
chain, the Zeeman energy is given by $\Delta E\approx
4g\mu_{\mathrm{B}}B_{z}\approx 4g\mu_{\mathrm{B}}B_{1}\sin(\alpha)$. We
subsequently calculate lifetime ratios based on the populations of $\psi_{0}$
and $\psi_{1}$ through the Boltzmann distribution at the Zeeman energy and
$T=1.8$ K, slightly exceeding the measurement temperature of $T=1.3\sim 1.4$ K
(see also Table 2).

Assuming $\alpha_{\rm tilt}=0$ (thus, $\alpha=\alpha_{\mathrm{atomic}}$), the
calculated Zeeman energies and corresponding lifetime ratios are depicted by
yellow shading in Fig. 9a,c,e. We found substantial deviations between our
measured values and the calculated Zeeman energies for $\alpha_{\rm tilt}=0$,
which suggests the influence of a nonzero $\alpha_{\rm tilt}$ and variations
in the $g$-factors among the chain’s atoms. As discussed in Supplementary Note
5, we consider the inhomogeneity of $g$-values by assigning a distinct
$g$-factor to atom I, differing from the rest ($g_{\rm rest}=2.11$ following
literature [10]). To optimize $g_{1}$ and $\alpha_{\rm tilt}$, we initially
assume $\alpha_{\rm tilt}=0$ and adjust $g_{1}$ to scale the calculated Zeeman
energies, aligning them with our results. Subsequently, we fine-tune
$\alpha_{\mathrm{tilt}}$ to compensate the scaling by $g_{1}$ to keep $g_{1}$
within a reasonable range. To characterize both odd- and even-numbered chains
uniformly, we introduce the concept of “unpaired spins”. For an ideal Fe5
chain ($g_{1}=g_{\rm rest}$), there would be one unpaired spin with
uncompensated Zeeman energy. Conversely, in an ideal Fe6 chain, no unpaired
spins would be present. However, considering inhomogeneity ($g_{1}\neq g_{\rm
rest}$), the unpaired spin for antiferromagnetically coupled even-numbered
chains is defined as ($g_{1}/g_{\rm rest}-1$).

To match the calculated Zeeman energies with our experimental results, we
adjust the values of unpaired spins (i.e., $g_{1}$). We found optimal
agreement when we set the unpaired spins as 4.25, 0.1, and 0.65 for Fig. 9a,c,
and e, respectively, while keeping $\alpha_{\rm tilt}=0$. This is shown in
orange shading. Next, we try to compensate for an offset of the angle,
$\alpha_{\rm tilt}$, to bring the number of unpaired spins in all measured odd
chains as close to unity as possible. This adjustment is carried out as
follows: Fig. 9g shows the determined number of unpaired spins with
$\alpha_{\mathrm{tilt}}=0$ (bottom) and with
$\alpha_{\mathrm{tilt}}=3^{\circ}$ (top). The overall values of unpaired spins
are much closer to 1 for $\alpha_{\mathrm{tilt}}=3^{\circ}$. Note that the
angles ($\alpha$) used for the simulations in the main text all lie within an
error margin of 1 degree compared to the measured angles, after accounting for
the tilt ($\alpha_{\mathrm{tilt}}-\alpha_{\mathrm{atomic}}$). Note also that
outside of this Supplementary Note $\alpha$ will always be presented as
$|\alpha|$.

Figure 9: Determining crystal axes. (a) Lifetime ratios
$T_{\mathrm{H}}/T_{\mathrm{L}}$ and (b) STM image for the Fe5 chain. The
corresponding lifetime curve for this chain is given in red in Fig. 7a. The
atomic resolution image presents $\alpha_{\rm atomic}=-2^{\circ}$. Considering
the Zeeman energy of one “unpaired spin” for the Fe5 chain, the calculated
lifetime ratios at this angle are represented by the yellow shading, where
$\alpha_{\rm tilt}=0^{\circ}$ and $g$-factors are homogeneous between atoms.
However, our measurement shows 4.25 times larger Zeeman splittings, as
indicated by the orange shading. (c,d) Same as (a,b), but $\alpha_{\rm
atomic}=+3^{\circ}$. The corresponding lifetime data are shown in Fig. 2d. For
$\alpha_{\rm atomic}=+3^{\circ}$, the measured Zeeman energy (orange shading)
is now 0.1 times smaller than the calculated Zeeman energy (yellow shading).
(e,f) Same as (a,b), but $\alpha_{\rm atomic}=+12^{\circ}$. The corresponding
lifetime curve is shown in a red curve in Fig. 7b. The lifetime ratios given
from the measurement show the Zeeman energy is 0.65 times smaller than the
calculated value. Arrows given in each STM image indicate nearby defects that
may cause symmetry breaking. The angle $\alpha=\alpha_{\rm tilt}-\alpha_{\rm
atomic}$ is given for $\alpha_{\rm tilt}=+3^{\circ}$. (g) Overview of all
determined unpaired spins throughout this work before (”$\alpha_{\rm
tilt}=0^{\circ}$”) and after (”$\alpha_{\rm tilt}=3^{\circ}$”) adjusting the
angle $\alpha_{\rm tilt}$. (h) Overview of unpaired spins and peak width for
the various data sets throughout this work. The peak width for atom III is
defined as half the range of $B_{x}$ values over which the lifetime is at most
one order of magnitude smaller than the peak lifetime. For panels g and h:
Green circles indicate data taken at $V_{\rm bias}=3$ mV, red circles taken at
$5$ mV. Inserted labels indicate the correspondent dataset and the angle
$\alpha$.

Lastly, using the adjusted tilt angle $\alpha_{\mathrm{tilt}}=3^{\circ}$ and
the corresponding calculated Zeeman energies in terms of unpaired spins, we
show the peak width for several different Fe chains as a function of unpaired
spins in Fig. 9h. The green and red symbols indicate data taken at $3$ mV and
$5$ mV, respectively. For the latter, we need to consider over-the-barrier
transitions, thereby leading to a slight reduction in peak height and an
increase in peak width. This effect is clearly observed in the data for Fe6
and the data presented in Fig. S4c. Furthermore, an increase in Zeeman energy
(through an increase in unpaired spins or angle) is also associated with a
larger peak width, as expected. Note that the data in Fig. 2d was obtained at
a much smaller angle than the rest of the dataset in this overview, resulting
in a smaller Zeeman energy.

Despite the Fe5 chain having an estimated Zeeman energy of around $30$
$\mu$eV, it is surprising that the peak width in Fig. 2d is somewhat larger
than that of the Fe6 chain with a Zeeman energy of about $50$ $\mu$eV. Given
the very small energy difference between $\psi_{0}$ and $\psi_{1}$ for the
data in Fig. 2d, we did not find any convincing indicator to associate the low
current in Fig. 2a with Nèel state A. Without loss of generality we picked
this based on Fig. 2b and c, where, for the specific case of $B_{x}=4$ T, the
average lifetime of the low current state is longer than the average lifetime
of the high current state. Code for processing all the lifetime ratios is
available in the Open Data folder.

## Supplementary Note 7: Lifetime of the Fe3 chain

Figure 10: Pump and probe measurements for the Fe3 chain at different $B_{x}$
fields. Pump-probe signals measured on the center atom of an Fe3 chain at
different $B_{x}$ magnetic fields. Considering the crystal axes with respect
to the magnets’ axes, increasing the $B_{1}$ results in increasing both
$B_{x}$ and $B_{z}$ in a way of $B_{z}\approx 0.09\cdot B_{x}$. With
increasing $B_{x}$, we found the lifetimes monotonically increase (see inset).
For the Fe3 chain, our simulation shows the diabolic point (almost) exceeding
our magnetic field range. Setpoint: $V_{\rm DC}=10$ mV, $I=200$ pA,
$V_{\mathrm{pump}}=35$ mV, $V_{\mathrm{probe}}=5$ mV, and $T=1.3$ K. Each pump
and probe measurement is offset for clarity.

## Supplementary Note 8: Voltage dependence of lifetimes

To identify the regimes for the quantum tunneling of magnetization (QTM) and
over-the-barrier transitions, we measured the spin lifetimes of Fe5 chains as
a function of bias voltages. At bias voltages corresponding to electron
energies below the magnetic anisotropy barrier (i.e. less than the energy of
$\psi_{2}$), to first approximation, we expect the lifetimes to be constant,
while above the barrier the lifetimes are expected to decrease gradually.
Figure 11a shows the average spin lifetimes of an antiferromagnetically
coupled Fe5 chain. Below $4$ mV the spin lifetime is constant with bias,
indicating the QTM regime. Above $4$ mV, spin lifetimes of the outer atom
decrease as bias increases. This threshold voltage appears higher for the
inner atoms. Due to this variation of threshold voltages between the atoms in
the chain, the lifetime curves given in Fig. 3a show strong dependence on the
atoms when measured at 5 mV. This observation can be understood by larger
values of $J_{i}$ and $D_{i}$ further towards the center of the chain, see
also Supplementary Note 1. Additionally, owing to the nodal structure of
excitation modes, the next lowest excited energy states (e.g. $\psi_{2}$) are
expected to be primarily localized on the outer atoms [14].

In Fig. 11b, we show analogous data for the ferromagnetically coupled Fe5
chain. The blue and black circles present the results measured at the same
conditions as in Fig. 11a for the central and outer atoms, respectively. Here
we clearly observe a maximum lifetime at around $3\,\mathrm{mV}$. This feature
likely emerges as a consequence of increased tip-chain exchange interactions
at lower bias voltages, which decreases the spin lifetimes. To keep the tip-
chain interaction constant, we fixed the conductance to $\sim
6.6\,\mathrm{nS}$ and repeated the lifetime measurements at different voltages
as given by faint blue and black squares in Fig. 11b. In this constant height
measurement, we found a plateau of the lifetimes below $\sim
3.5\,\mathrm{mV}$. Unlike the antiferromagnetic chain, the threshold voltages
for the over-the-barrier transitions are similar between the atoms in the
chain. Thus, for both antiferromagnetic and ferromagnetic chains, we chose
$3\,\mathrm{mV}$ to characterize the spin lifetimes due to the quantum
tunneling of magnetization near the diabolic point.

Figure 11: Voltage threshold for over-the-barrier transitions. (a) Bias
dependence of spin lifetimes for the antiferromagnetically coupled Fe5 chain
presented in Fig. 2d of the main text measured under constant-current mode at
$10\,\mathrm{pA}$. The measurements are shown for three atoms (I–III) in the
chain. (b) Bias dependence of spin lifetimes for the ferromagnetically coupled
Fe5 chain presented in Fig. 3b. The blue and black circles show the data
measured at a constant current of $10\,\mathrm{pA}$. Faint blue and black
squares represent the data obtained at a constant tip height (6.6 nS). Lines
connecting data points are guides to the eye. The error bars ($2\sigma$) are
smaller than the marker size. All data was obtained at $T=1.4$ K,
$B_{x}=4.125$ T near the diabolic point for (a) and $B_{x}=4.75$ T near the
second diabolic point for (b).

## Supplementary Note 9: Current dependence

In this section, we consider the effects of the current ($I$) on the lifetime
of the antiferromagnetic Fe5 chain. Figure 12 shows the lifetimes measured on
each atom in the chain at different currents. Note that the bias voltage was
set to $5$ mV, at which lifetime reductions due to over-the-barrier
transitions become apparent, most notably on the outer atoms. We thus consider
two separate transition rates: $R_{\mathrm{T}}$ for transitions due to
tunneling of magnetization and $R_{\mathrm{O}}$ for over-the-barrier
transitions. With increasing current, the lifetime generally decreases, due to
both $R_{\mathrm{T}}$ and $R_{\mathrm{O}}$ increasing linearly with $I$. The
lifetimes can be expressed in terms of transition rates:
$(T_{\mathrm{avg}})^{-1}\propto
R_{\mathrm{O}}+R_{\mathrm{T}}=Ir_{\mathrm{O}}+(I_{0}+I)r_{\mathrm{T}}=I(r_{\mathrm{O}}+r_{\rm
T})+I_{0}r_{\rm T}$, where the coefficients $r_{\mathrm{O}}$ and
$r_{\mathrm{T}}$ are based on the amplitudes of the scattering paths and
$I_{0}$ is the current from the bath interacting with the chain, which has too
little energy to induce over-the-barrier excitations. Note that
$r_{\mathrm{O}}$ and $r_{\mathrm{T}}$ are independent of $I$ but may be
dependent on $V_{\mathrm{bias}}$, $T$, atom location in the chain ($i$), and
magnetic fields. We found that $r_{\mathrm{O}}$ does not depend on the
transverse magnetic field, but $r_{\mathrm{T}}$ does.

Figure 12: Current dependence of spin lifetimes. (a) Lifetimes measured on
each atom at 3 pA, (b) 100 pA, and (c) 500 pA. All data was taken at $5$ mV
and $T=1.3$ K. Associated $B_{\mathrm{z}}$ is shown on the top axes. Lines are
guide to the eye. Error bars are $2\sigma$. (d) Lifetimes of each atom as a
function of current at $B_{x}=3.5$ T (dotted) and at the first DP
$B^{5,1}_{x}$ (solid). Dashed glowing lines represent lifetimes calculated
considering transition rates. (e) Lifetimes of atom III as a function of
magnetic fields at different currents. The blue dashed line indicates
$B^{5,1}_{x}$ used in (d) corresponding to the magnetic fields of the maximum
of lifetimes.

Figures 12a–c show the lifetimes measured for each atom in the Fe5 chain at a
current of 3 pA, 100 pA and 500 pA, respectively. When increasing the current,
the outer atoms show no signature of a diabolic point, as $R_{\rm O}$ becomes
too large. In contrast, the diabolic point remains visible for atom III but
appears at higher magnetic fields with overall reduced lifetimes (Fig. 12e).
The dashed blue line in Fig. 12e indicates the shift of the DP as a result of
the tip field. Increasing the current leads to smaller tip-sample distances
and larger tip fields, resulting in the DP at larger external magnetic fields.
This indicates that the tip field is counteracting the external magnetic
field. Note that this corresponds to tip 2 of Fig. 7d. Importantly, atom III,
not limited by over-the-barrier transitions, shows that the tip field does not
destroy the appearance of the diabolic point, despite the field being applied
to only one atom of the chain. This further supports the claim that this
effect is robust to many variations in the parameters of the Hamiltonian.

This current dependence is further analyzed in Fig. 12d by plotting the
lifetimes as a function of current at each magnetic field of $B_{x}=3.5$ T
(dotted lines) and $B_{x}=B^{5,1}_{x}$ (solid lines). We found four different
effects of the current on the lifetimes:

  1. 1.

As indicated by the orange lines in Fig. 12d, for the inner atoms, the
lifetimes are nearly constant for smaller $I$, where the plateau value and the
threshold current depend on $B_{x}$. At the plateaus, the lifetimes are mostly
determined by $I_{0}r_{\rm T}$, which depends on the scattering intensity and,
thus, on $B_{x}$. Consistent with the rest of this work, in the absence of
over-the-barrier transitions, a longer lifetime is observed at $B_{x}$ closer
to the DP.

  2. 2.

Excitation over the barrier due to an applied bias and finite temperature
causes $R_{\mathrm{O}}$ to limit the lifetime for all atoms [42]. At
$V_{\mathrm{bias}}=5$ mV and $T=1.3$ K, this effect is less significant
towards the center of the chain (see Supplementary Note 8). This effect does
not depend on field and should therefore result in constant lifetimes
throughout the full range of $B_{x}$. This is the case for atoms I/V in Figs.
12b,c and atoms II/IV in Fig. 12c. This contribution scales linearly with
current, as depicted by the red and blue glowing straight lines in Fig. 12d.

  3. 3.

Additional electrons might cause through-the-barrier transitions if the
electrons do not have enough energy to cause over-the-barrier transitions. We
find that for the experimental conditions for Fig. 12, only atom III shows
this behavior, as the shape of the lifetime curves in Figs. 12a–c does not
really flatten with increasing current. These transition events also depend on
the scattering amplitude between the lowest energy eigenstates. In Fig. 12d,
this is highlighted with the green glowing dashed lines, which intersects with
the orange lines at around 200 pA. This suggests the rate of bath electrons
interacting with the system is of a similar order.

  4. 4.

In Fig. 12c, the lifetimes of atoms II/IV appear higher than atom III for
$B<3.75$ T, which can be attributed to the atomic exchange bias as the
magnetic tip approaches close to the chain at higher current. This exchange
bias can be modeled as an increased $B_{z}$ for atoms II/IV and a decreased
$B_{z}$ for atoms I/III/V in their ground state [16]. For $B>3.75$ T, the
lifetimes of atoms II/IV are limited by over-the-barrier transitions, while
the atom III is free from this and, thus, shows longer lifetime.

## Supplementary Note 10: Methods and Data Acquisition

### Sample Preparation

We used a home-built STM system [41], operating at $1.3$ K and $B=0-6$ T in
the plane of the sample, mainly perpendicular to the axis of the chain. The
Cu2N/Cu(100) sample was prepared as described in [15]. The tip was prepared as
in [13]. Fe atoms on the Cu2N were picked up by applying voltage pulses of
$\sim 1$ V (setpoint of $100$ pA, $20$ mV, then moved $-300$ pm), and dropped
at $-600$ $\mu$V with the tip gradually approaching the surface until an
abrupt change in current was observed. The Fe was subsequently hopped into
place with a pulse of $\sim 1$ V at ($100$ pA, $20$ mV). Preferred hopping
directions were determined by straining of the Cu2N lattice in line with
previous works [43], and utilized for efficient construction of the chains.

### Data Acquisition

The spin-polarized STM tip was prepared by attaching several Fe atoms to the
Cu-coated tip apex. Using this spin-polarized tip, the magnetization switching
was measured in either a constant-current or constant-height mode. To measure
the switching in the order of a millisecond or below, a DAQ (NI 782258-01) was
used to record an incoming current stream of up to 10 seconds with the
feedback turned off with a sampling rate of $10$ kHz. For slightly longer
lifetimes, the data was recorded directly through the internal DAQ of Nanonis
electronics with a sampling rate of $2$ kHz. For lifetimes longer than $100$
ms, the feedback was turned on with an extremely long time constant in the
feedback loop. This allows the feedback to account for drift, but keep the
switching signals in the current data stream. For very long lifetimes ($\geq
1$ s), a constant-current mode was used such that the variation of tip heights
was used to determine the magnetization switches. The tip’s feedback was set
such that the response time is much faster than the average switch time. The
gradual drift of the tip height was subtracted from the data. The angle
$\alpha$ was derived from the angle determined by atomic resolution STM
images, and adjusted to include a tilt of $3^{\circ}$, see Supplementary Note
6.


the N1 (top) and N2 (bottom) cavity. The vertical dashed lines indicates the
expected central velocity of the line which remains undetected in both
cavities.

##  Appendix G The combination of a 70 $\mu$m bright cavity and 70 $\mu$m
quiet core in the radiative transfer simulations

We combine the gathered information on both the emission properties of the
nebulosity and that of the core to check whether the radiative transfer
simulations with nebulosities also correspond to the best core runs. To do
this we take all the radiative transfer calculations for which a nebulosity
has formed, and plot the corresponding 70 $\mu$m nebulosity fluxes (normalised
to the observed nebulosity flux) as a function of their associated
$\chi^{2}_{\rm{core}}$ (see Fig. 21). First, we find that only a small number
of runs with nebulosity have a $\chi^{2}_{\rm{core}}<3$. Second, those good
core runs are also the worst at reproducing the 70 $\mu$m nebulosity flux. In
fact, these two conditions (i.e. a quiet mid-infrared core and a bright mid-
infrared cavity) are conflicting, with the best models for each sitting at
both ends of the point distribution in Fig. 21. This important discrepancy and
the fact that it appears impossible to even create radiative transfer
nebulosities with more than 15% of the observed 70 $\mu$m flux in G345.88-1.10
strongly suggests that radiative heating from an embedded object is not
responsible for the the bright cavities in G345.88-1.10. This also justifies
our approach not to apply a model optimizer and explore the parameter space by
hand as it will search for a non existing solution.

As already pointed out in Sect. 7.4.2, radiative transfer calculations with a
central source with $L\geq$ 4000 $L_{\odot}$ are not compatible with the
observations of H1. Investigating the relation between the 70 $\mu$m flux of
the core and the density of the models (see Fig. 22), it is also found that a
high density is necessary to reproduce a weak 70 $\mu$m core emission. At 500
L⊙, the central density $n_{c}$ of the Plummer profile, or the characteristic
density $n_{0}$ of the passive flared disc, needs to be larger than
n${}_{H_{2}}$ = 5$\times$106 cm-3 to sufficiently suppress the 70 $\mu$m
emission towards the core. For an embedded object with a luminosity of 2000 L⊙
and more, the required density increases to values n${}_{H_{2}}$ $\geq$ 107
cm-3. However, as pointed out in Sec. 7.4.2, at 2000 L⊙ there is still the
problem that emission at longer wavelengths is not in agreement with what is
observed towards the H1 fragment.

Figure 21: $\chi^{2}_{core}$ for the radiative transfer models that have
nebulosities as a function of the 70 $\mu$m flux for the simulated
nebulosities (normalised to the observed nebulosities). This shows that none
of the radiative transfer simulations with 70 $\mu$m nebulosities fit with the
core at longer wavelengths. Figure 22: The 70 $\mu$m flux predicted by the
radiative transfer calculations as a function of the central (nc) and
characteristic (n0) density for all models with a luminosity of 500 L⊙. The
horizontal line indicates the upper limit for the 70 $\mu$m flux towards the
H1 fragment.

##  Appendix H Calculating the cooling timescale for the H1 fragment and the
cavities

The cooling time scale is given by

$\tau_{cool}=\frac{\rho\text{u}}{\lambda_{cool}}$ (15)

with $\rho$ the gas density, u the specific internal energy and
$\lambda_{cool}$ the cooling rate per unit volume. The specific internal
energy is given by

$\text{u}=\frac{3}{2}\frac{\text{k}_{b}\text{T}_{gas}}{\mu\text{m}_{p}}$ (16)

with kb the Boltzmann constant, Tgas the kinetic gas temperature, $\mu$ the
mean molecular mass (=2.33) and mp the mass of a proton. The last term that
has to be constrained is the cooling rate per unit volume ($\lambda_{cool}$).
If the gas is dense enough, typically n${}_{H_{2}}>$ 105 cm-3 (Glover & Clark
2012), then it can be assumed that the cooling is driven by dust through gas-
grain energy transfer. Given that the estimated average density in the H1
fragment is n${}_{H_{2}}$ = 3.2$\times$105 cm-3, we can use this condition
(Glover & Clark 2012). When the cooling is dominated by gas-grain energy
transfer the cooling rate per unit volume is given by

$\lambda_{cool}=4\pi\rho\text{D}\int_{0}^{\infty}\kappa_{\nu}\text{B}_{\nu}\text{(T)}\text{d}\nu$
(17)

where $\kappa_{\nu}$ is the dust opacity, D the dust to gas mass fraction and
Bν the Planck function. Using Ossenkopf & Henning (1994) this results in

$\lambda_{cool}=4.68\times 10^{-31}\text{n}\text{T}_{dust}^{6}\text{( in erg
s}^{-1}\text{ cm}^{-3}\text{)}$ (18)

with Tdust the dust temperature and n the number density of the gas for which
we assume it is fully molecular in the H1 fragment.
We considered several input parameters for the heated gas in the fragment
ranging from 20 K to 100 K and for Tdust we considered that the heated dust
could reach 20 K, 30 K and 40 K. The resulting cooling timescales are given in
Tab. 7 and are typically of the order of a few days to a few years for these
conditions. Assuming even higher dust temperature values would only further
decrease the cooling timescale.
For the cavities we get typical densities n${}_{H_{2}}$ = 1.0-2.5$\times$104
cm-3 based on the Herschel column density and cavity size. From Goldsmith &
Langer (1978) we find that for a typical density n${}_{H_{2}}$ = 104 cm-3,
$\lambda_{cool}$ is given by

$\lambda_{cool}=1.5\times 10^{-26}\text{T}^{2.7}\text{( in erg s}^{-1}\text{
cm}^{-3}\text{)}$ (19)

where T is the kinetic temperature in the region. Using the same average
kinetic temperatures of 20-100 K in the cavities, we obtain $\lambda_{cool}$
between 4.9$\times$10-23 \- 3.8$\times$10-21 erg cm-3 s-1 which is consistent
with cooling rates found in the simulations by Glover & Clark (2012) for these
densities. Using these values, we obtain estimated cooling timescales between
1.7$\times$103 and 2.7$\times$104 yr for the cavities. Note that the lower end
of these estimated cooling timescales is of the same order of magnitude as the
propagation timescale for the jet through the cavity. This could thus be
consistent with the observation that the full cavity is bright and not a
specific subregion, Because of the lower densities, the cavities thus have
significantly longer cooling timescales than the dense fragment.

Table 7: The cooling timescales for the H1 fragment for a grid of dust and gas temperatures. | Tgas = 20 K | Tgas = 40 K | Tgas = 60 K | Tgas = 80 K | Tgas = 100 K
---|---|---|---|---|---
Tdust = 20 K | 4.4 yr | 8.8 yr | 13 yr | 18 yr | 22 yr
Tdust = 30 K | 3.9$\times$10-1 yr | 7.7$\times$10-1 yr | 1.2 yr | 1.5 yr | 1.9 yr
Tdust = 40 K | 6.9$\times$10-2 yr | 1.4$\times$10-1 yr | 2.1$\times$10-1 yr | 2.7$\times$10-1 yr | 3.4$\times$10-1 yr