Tracking a spin-polarized superconducting bound state across a quantum phase transition

The magnetic exchange coupling between magnetic impurities and a superconductor induce so-called Yu-Shiba-Rusinov (YSR) states which undergo a quantum phase transition (QPT) upon increasing the exchange interaction beyond a critical value. While the evolution through the QPT is readily observable, in particular if the YSR state features an electron-hole asymmetry, the concomitant change in the ground state is more difficult to identify. We use ultralow temperature scanning tunneling microscopy to demonstrate how the change in the YSR ground state across the QPT can be directly observed for a spin-1/2 impurity in a magnetic field. The excitation spectrum changes from featuring two peaks in the doublet (free spin) state to four peaks in the singlet (screened spin) ground state. We also identify a transition regime, where the YSR excitation energy is smaller than the Zeeman energy. We thus demonstrate a straightforward way for unambiguously identifying the ground state of a spin-1/2 YSR state.

Unpaired spins in impurities coupled to a superconductor induce discrete sub-gap excitations, the Yu-Shiba-Rusinov (YSR) states [1][2][3][4], through an exchange interaction produced locally via impurity-superconductor coupling.If the exchange coupling increases beyond a critical value the YSR states undergo a quantum phase transition (QPT) such that the initially free spin becomes screened [5,6].The transition through a QPT has been attributed to a reversal in the asymmetry of the spectral weight of electron and hole excitation components, which are readily observed in a scanning tunneling microscope (STM) [7][8][9][10][11][12][13].This reversal in spectral weight holds, however, only in the simplest approximation that all higher order e ects are ignored.The spectral weight does not re ect the particle-hole asymmetry if, for example, the system is already in the resonant Andreev re ection regime [14] or tunneling paths are interfering [15].Most crucially, it is a priori not possible with the STM to identify to which side of the quantum phase transition the system belongs.
A straightforward albeit indirect and not entirely unambiguous way to manipulate the ground state of an atomic scale YSR resonance is to change the impuritysubstrate coupling if the YSR impurity is susceptible to the atomic forces acting between tip and sample in the STM tunnel junction [7,11,[16][17][18].The ambiguity arises because it is not a priori clear whether the impuritysubstrate coupling increases or decreases upon reducing the tip-sample distance.This calls for an unambiguous manifestation going beyond auxiliary measurements [18] to distinguish the ground state of the YSR excitation.
An independent observation identifying the ground state of the system across the QPT can be made by placing the YSR state in a Josephson junction (0- transition) [19].Also, the zero-eld splitting of YSR excitations due to effective anisotropic interactions in high-spin systems has been used to assign the ground state of di erent molecules on either side of the QPT [20].While di erent YSR states have been studied with the STM in the presence of a mag-netic eld [21][22][23][24], a continuous evolution of the YSR state across the QPT in a magnetic eld has not been observed like in mesoscopic systems [25].The challenge in observing a sizeable Zeeman splitting in a YSR state lies with the typically rather small critical magnetic eld that quenches superconductivity.Here, we circumvent this problem by placing the YSR state at the tip apex [19,26], where the superconductor is dimensionally con ned, such that the critical eld is considerably enhanced (Meservey-Tedrow-Fulde (MTF) e ect) [27][28][29].We use an ultralow temperature STM at 10 mK to reduce the thermal energy much below the Zeeman energy and trace the spectral signatures associated with the changes in the YSR ground state across the QPT by continuously changing the impuritysubstrate coupling (see Fig. 1

(a)).
A typical spectrum measured with a YSR functionalized superconducting vanadium tip on a superconducting V(100) sample at 10 mK is shown in Fig. 1(b).The electron and hole parts of the YSR state with energy  appear at a bias voltage  = ±( +  s ) as prominent peaks.Due to the superconducting sample, the YSR peaks shift by the sample gap  = ± s away from zero bias voltage.The coherence peaks at  = ±(  +   ), which is the sum of the tip and sample gaps, are small indicating a dominant transport channel through the YSR state.We change the impurity-substrate coupling by varying the tip-sample distance, which modi es the atomic force acting on the impurity [30,31].This concomitantly changes the exchange coupling  causing an evolution of the YSR energy  as shown in Fig. 1(c).At a critical exchange coupling  T , when the YSR energy is at zero, the system moves across a QPT such that the free impurity-spin ( <  T ) becomes screened ( >  T ) bringing about a change in the fermionic parity of the ground state [32].This scenario is schematically depicted in the insets of Fig. 1(c), where a doublet ( = 1 /2) transforms into a singlet ( = 0) leading to the screening of the impurity spin.junction transmission  =   / 0 (  : normal state conductance;  0 = 2 2 /ℎ: conductance quantum with  being the elementary charge and ℎ Planck's constant).The YSR peaks evolve continuously reaching the bias voltage closest to zero at the QPT.Because of the shift of the YSR state by the superconducting gap  s of the other electrode (the substrate in this case) in the conductance spectrum, the zero crossing at the QPT is not observed directly.An inversion of the asymmetry in the YSR peak intensities is clearly visible, when the electron and hole excitation components switch sides across the QPT.However, it is not possible to judge from the tunneling spectra alone, on which side of the QPT the system is.
Turning on a magnetic eld, the  = 1 /2-state splits into two levels.In the free spin regime, the spin down state (see Fig. 3(a)) turns into the non-degenerate ground state.Its higher lying spin-ipped partner is thermally not populated due to the extremely low temperature of 10 mK.Only the screened  = 0-state appears as a transport channel lying energectically above the doublet.Since it does not change in the magentic eld, it induces only one spectral feature on either side of the Fermi level.In contrast, beyond the QPT, the  = 0-state becomes the ground state and charge transfer is possible through the spin-doublet (details see below).This can be seen in Fig. 2(a), which shows two representative di erential conductance spectra on either side of the QPT at a magnetic eld of 750 mT.The orange spectrum shows two features (one on either side of the Fermi level), which indicates that the system is in the free spin regime.The sample is already normal conducting at 750 mT, such that there is no shift of the YSR peak by  s .The YSR tip is still superconducting due to the MTF e ect.In the screened spin regime, ground state and excited state are interchanged, such that now two transitions into the upper and lower Zeeman split  = 1 /2 levels are possible from the single ground state  = 0 level.As a result, the spectrum measured in the screened spin regime (the blue curve) shows four spectral features (two on either side of the Fermi level).This distinction is only possible, if the Zeeman energy is much larger than the thermal energy.If this is not the case, two spectral features will be visible on either side of the QPT and a more detailed analysis of the spectral weight has to be done to distinguish the ground states [20].
As has been demonstrated before [7-13, 18, 19, 26], we exploit the changing atomic forces in the tunnel junction when reducing the tip-sample distance to change the impurity-superconductor coupling thereby shifting the YSR state energy.We note that depending on the particular system, the impurity-substrate coupling can increase or decrease during tip approach.The evolution of the YSR state through the QPT for two di erent magnetic elds are shown in Figs.2(b) and (c) as function of the tunnel junction transmission  (i.e.junction conductance).Here, we can see directly that the screened spin regime featuring four spectral peaks is at higher transmissions and the free spin regime featuring only two spectral peak is at lower transmissions.This actually implies that the impuritysuperconductor coupling increases with increasing transmission, which is veri ed by an additional analysis of the Kondo e ect at higher magnetic elds below.The data in Fig. 2(b) was taken at 750 mT, which results in a stronger Zeeman splitting than the data in Fig. 2(c), which was taken at 500 mT having less Zeeman splitting.Still, both data sets show qualitatively the same behavior across the QPT as expected from the discussion above.
In addition to these two regimes, we found a crossover regime, where the two outer spectral features extend into the free spin regime, which is seen for both magnetic eld values in Fig. 2(b) and (c).Due to the higher magnetic eld in Fig. 2(b) than in (c), the crossover regime is also wider.The crossover regime marks a small region, where the excitation energy of the YSR state is smaller than the Zeeman splitting ( <  Z ).The outer spectral feature (marked by the arrow in Fig. 2(b)) in the crossover regime is a combination of quasiparticle tunneling from the thermally excited YSR state, which becomes exponentially suppressed as the YSR energy increases, and two-electron tunneling processes, i.e. resonant Andreev processes (see below and Supplementary Information [33]).), because the Zeeman split state | cannot be thermally excited at 10 mK.We can reproduce the experimental ndings theoretically by calculating a tunneling current from a master equation involving both single electron and two electron processes (for details see the Supplementary Information [33]).The calculation in Fig. 3(b) has been done for a magnetic eld of 750 mT comparable to the experimental data in Fig. 2(b).All the features that we observed experimentally are reproduced in the calculations.
In order to independently verify the evolution of the YSR state through the QPT, we take a closer look at the YSR peak height and the resulting Kondo e ect in the normal conducting state.The evolution of the YSR peak height is plotted in Fig. 2(d) in blue for the left and right peak as function of junction transmission.In the same graph the YSR peak energy is shown in orange.At the QPT (vertical dashed line), the YSR energy is zero and the peak height reverses indicating the QPT.This reversal is observable so clearly because resonant Andreev processes have not yet become signi cant.Further, we increase the magnetic eld to 2.75 T such that both tip and sample become normal conducting and a Kondo peak appears [35][36][37][38].This is shown in Fig. 4(a), where the Kondo peak around zero bias voltage is displayed as a function of the junction transmission .We already see that the splitting of the Kondo peak in the magnetic eld decreases as the transmission increases, which indicates that the Kondo temperature increases with increasing transmission.The higher Kondo temperature implies a stronger screening, which means that the Kondo peak starts splitting at a higher critical magnetic eld  c .We have tted the Kondo spectra using numerical renormalization group (NRG) theory [39].This allows us to directly determine the Kondo temperature  K from the microscopic parameters extracted from the t.The extracted Kondo temperature is shown in Fig. 4 Furthermore, scaling the Kondo temperature  K and the YSR energy  to the superconducting gap , we compare the evolution across the QPT to the universal behavior predicted by NRG theory [44][45][46].The blue data points in Fig. 4(c) show the evolution of the YSR state across the QPT as function of the scaled Kondo temperature, which follows the predicted universal scaling (dashed line) with a slight o set.This deviation of the data from the universal curve is presumably due to subtle changes in the impurity-substrate coupling as a result of modi cations in the atomic forces acting in the junction with and without the applied magnetic eld.We, therefore, nd a consistent picture for the behavior of the YSR state in a magnetic eld across the quantum phase transition.
The evolution of the YSR state splitting across the QPT clearly demonstrates the change in the YSR ground state.For a spin-1 ⁄ 2 system, the nature of the ground state can be straightforwardly identi ed simply by the number of peaks in the spectrum.For higher order spins, the situation remains simple as long as the system can be assumed to be magnetically isotropic [24].If the system experiences a magnetic anisotropy, the analysis of the YSR states becomes more cumbersome [21,47].Still, the evolution in a magnetic eld as well as with changing impurity-superconductor coupling (if susceptible to the atomic forces of the tip) greatly facilitates the identi cation of the ground state maybe even the spin state itself.In summary, we present the evolution of a spin-1 ⁄ 2 impurity derived YSR state in a magnetic eld across the QPT.Due to the extremely low temperature of the STM, the change from a single feature spectrum (free spin regime) to a double feature spectrum (screened spin regime) is clearly visible.This allows for an unambiguous determination of the ground state of the YSR state.

METHODS
The V(100) single crystal was sputtered (with Ar + ), annealed to about 925 K, and cooled to ambient temperature repeatedly in ultra-high vacuum, ensuring an atomically at sample surface.Typical surface reconstructions form with oxygen di used from the bulk [26,48,49].A small fraction of these defects exhibit YSR states [26].Similarly, we produce YSR states at the vanadium tip apex by repeatedly dipping the tip in situ into the substrate [19,26], which is veri ed in the conductance spectrum.This gives us full control to reproducibly design and de ne the junctions under investigation.We choose to use YSR functionalized tips for our experiments as they o ered the exibility to single out those ful lling the required response to tip approach.Moreover, the YSR tips feature a range of YSR state energies and show a better junction stability at higher conductance than YSR states in the sample.
The experiments were performed in a low-temperature scanning tunneling microscope operating at 10 mK.Differential tunneling conductance ( / ) spectra were recorded using an open feedback loop with a standard lock-in technique (10 μV rms , 727.8 Hz).In Fig. 4(a) a modulation amplitude of 50 μV rms was used.The tunneling current was measured through the tip with the voltage bias applied to the sample.
The calculations for the current based on the master equation are detailed in the Supplementary Information [33].For the Kondo spectra, we used the numerical renormalization group (NRG) theory in the framework of the single impurity Anderson model (SIAM) as implemented in "NRG Ljubljana" code [34] to model the Kondo e ect in a magnetic eld.We xed the Hubbard term  = 10 to be much larger than the half bandwidth  = 1 and modeled the asymmetry of the Kondo spectra using the intrinsic asymmetry parameter  =  +  /2 where  is the impurity level [39].The best agreement with the experiment corresponds to  = −2.The only free parameter left for tting the Kondo spectra is the impurity-substrate coupling  .The Kondo temperature was extracted from the t through its de nition with respect to the SIAM parameters [45,50,51]

Supplementary Material THEORY: INTRODUCTION
A single spin-1 ⁄ 2 impurity gives rise to an in-gap Yu-Shiba-Rusinov state (YSR).Based on the occupation of the YSR state, the system wavefunction can be a spin doublet, when the YSR state is unoccupied, or a singlet, in the opposite case [1][2][3][4].
The presence of a magnetic eld splits the doublet states.We denote by |0 ↑ and |0 ↓ the spin doublet states, that correspond to the free impurity spin.The states indicate that the spin-1 ⁄ 2 impurity is aligned, respectively antialigned, with the external magnetic eld.The spin singlet, that corresponds to the impurity spin, is denoted by |1 .The notation adopted here di ers from the notation in the main text, by adding an emphasis on the occupation of the YSR state, that we believe is more intuitive for transport calculations.The notation in the main text for the doublet states | and | , indicating the total spin state, is equivalent to the notation here |0 ↑ and |0 ↓ , indicating an empty YSR state (free impurity spin) and the total spin.The notation in the main text for the singlet state |0 , indicating the total spin  = 0, is equivalent here to the state |1 , indicating the occupation of the YSR state.
The energies of the three states  0↑ ,  0↓ , and  1 depend on the exchange coupling strength and give rise to three regimes, as shown in Fig. 3 of the main text.The free spin regime  1 >  0↑ ,  0↓ , a crossover regime  0↑ >  1 >  0↓ , and the screened spin regime  0↑ ,  0↓ >  1 .We have assumed that  0↑ >  0↓ , with the energy di erence determined by the Zeeman splitting   of the doublet state  0↑ −  0↓ =   (as illustrated in Fig. S1).
For transport calculations, it is useful to work with fermionic excitation energies.We de ne the energy required to add a quasiparticle with spin ↑ to the YSR state as  ↑ =  1 −  0↓ .We note that the state involved is |0 ↓ , such that the impurity spin ↓ is screened by the added quasiparticle ↑.Similarly, we denote by  ↓ =  1 −  0↑ , the energy required to add a quasiparticle with spin ↑ to the YSR state.Consistent with our assumptions above, we have  ↑ −  ↓ =   , therefore  ↑ >  ↓ (as shown in Fig. S1).
We note that it is possible to remove either a spin ↑, or a spin ↓ quasiparticle from the YSR state, when it is in the singlet state |1 .The implication therefore is that transport processes can result in the ip of the impurity spin, e.g.|0 ↓ Crucially, also the transport between the impurity and its host superconducting substrate can lead to such spin ip, as pointed out in Ref. [5].

Rate equations
The simplest theoretical framework that captures the transport properties of the Zeeman split states takes the form of a rate equation for the probabilities to be in one of three states,  0↑ ,  0↓ , and  1 .We denote by  1↑ ( 1↓ ) the rate to remove a quasiparticle with spin ↑ (↓) from the YSR state, and by  2↑ ( 2↓ ) the rate to add a quasiparticle with spin ↑ (↓) to the YSR state.
The rates represent a sum of all the contributing processes, intrinsic processes and tunneling processes, and depend on the bias voltage and the YSR state energy, that we parameterize by the exchange coupling strength  (see Fig. S1).
The probabilities characterizing the three states obey the following rate equations, ) with the normalization condition The steady state probabilities are given by Where we have used the notation In the following, we discuss the rates, which consist of intrinsic rates ( () 1 ,  () 2 ) and tunneling rates ( ( ) 1 ,  ( ) 2 ), with  ∈ {↑, ↓}, such that

1𝜎
and

Intrinsic rates
The occupation of the YSR can change due to intrinsic processes involving the quasiparticle population above the superconducting gap.We stress that these processes do not involve tunneling between the tip and substrate.The intrinsic process that describes the addition of a quasiparticle to the YSR state, changing the state from |0 , with  ∈ {↑, ↓}, to the singlet state |1 , requires that the quasiparticle has spin σ, opposite to .We denote the corresponding rate by  () 2 σ .Similarly, the rate to emit a quasiparticle with spin σ into the continuum, is denoted by  ()  1 σ .The latter process transforms the initial state |1 into the doublet state |0 .
In absence of tunneling between tip and substrate, the intrinsic processes in the tip and are responsible for the equilibrium value of the YSR state occupation.We apply detailed balance to determine the relations between the intrinsic rates.The equilibrium population in each state in absence of tunneling  (eq) 0↑ ,  (eq) 0↓ , and  (eq) 1 , are obtained from Eqs. S4 by setting the tunneling rates to zero, such that  1 =  ()  1 and  2 =  () 2 .We require that the equilibrium populations are related by the Fermi-Dirac distribution . Furthermore, we will assume that the rate to emit a quasiparticle with spin  into the continuum, transition from |0 σ to state |1 , is independent of the orientation of the impurity spin σ.Therefore, we have This assumption is not necessary, but convenient to reduce the number of parameters of the model.The physical mechanism behind such intrinsic processes, as well as the origin of the quasiparticle population above the gap at mK temperatures, remain unknown.The two relations obtained by applying detailed balance, together with our assumption, express the intrinsic rates in terms of a single free parameter, which we denote   , an intrinsic rate of relaxation.We parameterize the intrinsic rates in terms of   , as follows The intrinsic rates are shown in Fig. S2.The parametrization chose ensures that the intrinsic rates  () and  ()

2↑
are bound by   .The intrinsic rate  () 2↓ becomes much larger than   in the regime  ↑ < 0, indicating that the higher excited state |0 ↑ relaxes to the ground state |1 in this regime at a rate much faster than the relaxation of the lower excited state |0 ↓ .

Tunneling rates
We assume that the tunneling process is spinconserving and characterized by a spin-and momentumindependent tunneling amplitude .The density of states of the substrate is denoted   (), that may di er for the two spin species  = {↑, ↓}.The chemical potentials for di erent spin species align and are denoted by   for the substrate and   for the tip, respectively.The occupation of the electronic states of the substrate is given by the Fermi-Dirac distribution   ().
In the experiment, the magnetic eld is su ciently large such that the substrate is in the normal-conducting state.This provides a simpli cation, since the density of states is approximately at at the scale of the Zeeman splitting   , and therefore becomes spin-independent.However, we will provide expressions for the rate equations that can account for a future experimental situation where the substrate density of states could be potentially spin-dependent.The rates describing tunneling processes contribute to the total rates  1 and  2 , as follows.When the initial state is either one of the doublets, |0 , a tunneling process will add a quasiparticle with spin σ, opposite , resulting in the singlet.We distinguish two possibilities: either i. an electron with spin σ tunnels into the YSR state, with rate denoted by  ( ) 2, σ ; or ii. a hole with spin  tunnels into the YSR state, with rate denoted by  ( )  2,ℎ also Fig. S3).Since both processes create a quasiparticle excitation with spin σ in the YSR state, they add up to give the contribution to  2 σ due to tunneling, denoted  ( ) 2 σ .
the expressions above, we have introduced the coherence factors of the YSR state,  and .We denoted by | | 2 the probability to add an electron to the YSR state, while | | 2 represents the probability to add a hole, respectively.Furthermore, we have used the convention that  =   −   , where   and   are the chemical potentials of the substrate and tip, respectively.We have also introduced the common notation n () = 1 −   () to denote the occupation for holes.
Similarly, we obtain the rates of tunneling processes that remove a quasiparticle from the YSR state, leading to the transition from state |1 to one of the dublet states |0 .A quasiparticle can be removed either by removing from the YSR state an electron with spin σ,  ( )  1, σ , or a hole with spin ,  ( )  1,ℎ .We nd in analogy to the results for adding a quasiparticle above, . Two types of elementary transport processes.Left: illustration of an elementary transport process carrying charge .In the rst step, a tunneling event leads to the transition |0 ↓ to |1 .In a second step, the state |1 relaxes back to state |0 ↓ by an intrinsic process without charge transport.Right: illustration of an elementary transport process carrying charge 2.The rst step is identical to the left illustration.In the second step, a hole tunnels out of the YSR state, restoring the state |0 ↓ and leading to a total 2 charge transfer.The hole tunneling can be seen as an electron tunneling in the opposite direction, forming a Cooper pair with the electron that tunneled in the rst step.

Steady state current
The electrical current is expressed in terms of tunneling rates and the steady state probabilities.The latter are given by Eq. (S4), with the total rates  1 =  ()  1 +  ( ) 1 and  2 =  ()  2 +  ( ) 2 , given by sum of intrinsic and tunneling rates.
The total steady state current is given by the expression, The expression accounts for all charge transfer processes across the tip-substrate junction.The rst term accounts for the possibility to add a quasiparticle to the YSR state |0 either by transporting an electron with spin σ, or a hole with spin .The second term, similarly, accounts for the possibility to remove a quasiparticle with spin  =↑ or ↓ from the YSR state |1 , by transporting an electron with spin σ, or a hole with spin .The total current can be further understood in terms of elementary transport processes.An elementary transport process consists of two transitions: the rst changes the occupation of the YSR state, and the second restores the original occupation, thereby completing the transport cycle.We distinguish two types of elementary transport processes: i. when one transition occurs due to a tunneling process, while the other transition is intrinsic, such that a total charge  is transported; and ii.when both transitions occur due to a tunneling process, such that a total charge 2 is transported by sequential charge  tunneling events.Fig. S4 illustrates an example of the two types of processes.
The total steady state current is the sum of currents  Charge  elementary transport process For this transport process, charge is transported in a single tunneling event.We must account both for forward transport and for backward transport, as follows.(S9) The rst term, proportional to  0 , describes the process when an electron, or a hole, tunnels into the YSR state changing the occupation from |0 to |1 , as shown in Fig. S3.The second step of the elementary process occurs therefore via an intrinsic process, restoring an unoccupied YSR state |0 , without charge transport.The total process is depicted in the left side of Fig. S4.Alternatively, the transition from |0 to |1 could occur by an intrinsic process, while the second step, from |1 to |0 , could occur by tunneling.Note that the information about the energy of the states and lling factors of the substrate are all encoded in the rates and indirectly, in the steady state probabilities.Therefore, the expressions apply for all regimes, free spin, intermediate, as well as the screened spin regime.

Charge 2𝑒 elementary transport process
When both steps of the elementary transport process involve a tunneling event, such as the process depicted in the right side of Fig. S4, the total transported charge is 2.The transport of 2 charge is reminiscent of Andreev re ection.Indeed, the two charged particles involved in transport change the number of Cooper pairs in the tip condensate by one.However, there is also an important di erence, the process described here is a sequential process consisting of two single particle transport events.
Elementary transport processes involving two sequen-

Figure 1 (
Figure 1(d) shows how the YSR peaks evolve with the

Figure 1 .
Figure 1.YSR state in the vicinity of a QPT (a) Schematic of the tunnel junction incorporating a magnetic impurity at the tip apex.(b) Di erential conductance spectrum at zero eld showing the impurity-induced YSR states at  = ±( +  s ).(c) The YSR excitation energy  vs. magnetic exchange coupling  .At the crossing of the YSR energies, the system undergoes a quantum phase transition (QPT) from a free spin doublet into a screened spin singlet state (see inset).(d) Normalized di erential conductance spectra as function of junction transmission .The QPT occurs, when the YSR peaks are closest to zero.The YSR peak crossing is not directly visible because both tip and sample are superconducting shifting the YSR peaks by the sample gap ±  .(b) and (d) The coherence peaks are visible at the sum of the tip and sample gap  = ±(  +   ).

Figure 2 .
Figure 2. Magnetic eld dependence of YSR states across QPT.(a) Di erential conductance spectra at a magnetic eld of  = 750 mT at two junction transmissions, one below and one above the QPT marked by arrows in panel (b).(b) Di erential conductance map at 750 mT as function of junction transmission revealing the dispersion of the YSR states across the QPT.The sample is normal conducting, so that the spectral features cross at the Fermi level.(c) Same as (b) for 500 mT with a correspondingly reduced Zeeman splitting.(d) YSR peak heights (blue) at zero eld of the left and right peak.The height inverts across the QPT, where the YSR energies  (orange) are zero.
(b) as the blue line.It monotonously increases with increasing junction transmission , which corroborates the previous nding that the exchange coupling increases with increasing transmission (cf.Fig. 2(b) and (c)) [18, 40].We also extracted the critical eld  c , where the Kondo peaks starts splitting, as a function of transmission.The values for the critical eld  c are plotted in Fig. 4(b) as a red line.The critical eld increases with increasing transmission and follows the Kondo temperature very well.This corroborates very well the increase in impurity-substrate coupling for an increasing junction transmission.We further nd a relation of  B  K =   B  c between the Kondo temperature and the critical eld with  = 1.6, which compares well with what has been found in the literature [41-43].

Figure 3 .
Figure 3. Theoretical modeling of the Zeeman splitting across the QPT.(a) The sketches illustrate the level structures of a YSR state at nite magnetic eld in di erent regimes across the QPT.The Zeeman e ect lifts the spin degeneracy of the doublet state leading to two possible transitions in the screened spin regime (shaded blue).The YSR states are split by the Zeeman energy   =   , where  is the -factor and  B , is the Bohr magneton.The crossover regime (shaded yellow) in the proximity of the QPT still allows for a second transition involving thermal excitation of the YSR state and two-electron tunneling processes.The free spin regime (shaded red) allows only for one transition.(b) Calculation of the di erential conductance spectra across the QPT based on a master equation including single electron and two electron processes.The experimental data in Fig. 2(b) is well reproduced.

Figure 4 .
Figure 4. Kondo e ect and numerical renormalization group (NRG) analysis (a) Di erential conductance spectra featuring a Kondo peak at di erent junction transmissions .Dashed gray lines represent ts of the data to NRG calculations [34].(b) The Kondo temperature extracted from the t in (a) as function of transmission  in blue along with the critical eld  c , above which the Kondo peak starts splitting.(c) YSR state energy vs. Kondo temperature both scaled to the superconducting gap .The dashed black curve represents the universal scaling predicted from the NRG model.

Figure S1 .
Figure S1.Energy of YSR states.Above: energies as a function of exchange coupling.The critical exchange coupling   corresponds to the phase transition at zero magnetic eld.The Zeeman energy is chosen as   = 0.1.Below: excitation energies as a function of exchange coupling.Both   and −  are shown, for  =↑, ↓.The dashed lines indicate the excitation energy involves two excited states, while the solid lines indicate excitations from the ground state.The background color indicates the free spin, crossover and screened spin regions, as in Fig. 3 of the main text.

Figure S2 .
FigureS2.Intrinsic rates as a function of exchange coupling.The critical exchange coupling   corresponds to the phase transition at zero magnetic eld.

Figure S3 .
Figure S3.Illustration of two transport processes that add a spin quasiparticle to the YSR state, one realized by transporting a spin ↑ electron, the other realized by transporting a spin ↓ hole.The latter is equivalent to an electron with spin ↓ resulting from a Cooper pair splitting event, traveling in the opposite direction of the hole.

Figure S5 .
Figure S5.Current-voltage curves.Left: Density plot of the current as a function of voltage  and exchange coupling  .Right: cuts showing the I-V curves before and after the phase transition.The blue and gold curves show the contributions   and  2 , respectively, while the green curves show the total current.
Figure S6.Di erential conductance curves.Left: Density plot of the differential conductance as a function of voltage  and exchange coupling  .Right: cuts showing the di erential conductance curves before and after the phase transition.The blue and gold curves show the contributions   / and  2 / , respectively, while the green curves show the di erential conductance.