SI_ArXiv.pdf

# Observation of a Kondo impurity state and universal screening using a charge
pseudospin

C. Piquard  Université Paris-Saclay, CNRS, Centre de Nanosciences et de
Nanotechnologies, 91120 Palaiseau, France  P. Glidic  Université Paris-Saclay,
CNRS, Centre de Nanosciences et de Nanotechnologies, 91120 Palaiseau, France
C. Han  Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv
University, Tel Aviv 69978, Israel  A. Aassime  Université Paris-Saclay, CNRS,
Centre de Nanosciences et de Nanotechnologies, 91120 Palaiseau, France  A.
Cavanna  Université Paris-Saclay, CNRS, Centre de Nanosciences et de
Nanotechnologies, 91120 Palaiseau, France  U. Gennser  Université Paris-
Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120 Palaiseau,
France  Y. Meir  Department of Physics, Ben-Gurion University of the Negev,
Beer-Sheva, 84105 Israel  E. Sela  Raymond and Beverly Sackler School of
Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel  A. Anthore
Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies,
91120 Palaiseau, France  Université Paris Cité, CNRS, Centre de Nanosciences
et de Nanotechnologies, F-91120 Palaiseau, France  F. Pierre  Université
Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120
Palaiseau, France

###### Abstract

The Kondo effect, deriving from a local magnetic impurity mediating electron-
electron interactions, constitutes a flourishing basis for understanding a
large variety of intricate many-body problems. Its experimental implementation
in tunable circuits has made possible important advances through well-
controlled investigations. However, these have mostly concerned transport
properties, whereas thermodynamic observations - notably the fundamental
measurement of the spin of the Kondo impurity - remain elusive in test-bed
circuits. Here, with a novel combination of a ‘charge’ Kondo circuit with a
charge sensor, we directly observe the state of the impurity and its
progressive screening. We establish the universal renormalization flow from a
single free spin to a screened singlet, the associated reduction in the
magnetization, and the relationship between scaling Kondo temperature and
microscopic parameters. In our device, a Kondo pseudospin is realized by two
degenerate charge states of a metallic island, which we measure with a non-
invasive, capacitively coupled charge sensor. Such pseudospin probe of an
engineered Kondo system opens the way to the thermodynamic investigation of
many exotic quantum states, including the clear observation of Majorana zero
modes through their fractional entropy.

The Kondo model has proved to be an essential framework for the understanding
and engineering of unconventional behaviors that develop in strongly
correlated systems1, 2, 3, 4. It underpins current insights into a variety of
promising phenomena from the emergence of exotic non-abelian particles5, 6, 7
to heavy fermions1, 4 and high-$T_{\mathrm{c}}$ superconductivity. The central
element of this model is a local, energy degenerate ‘Kondo’ spin that
effectively mediates interactions between itinerant electrons: As the
temperature goes down, an initially weak antiferromagnetic coupling of the
Kondo impurity with the spin of the electrons progressively grows, thereby
giving rise to strong electron-electron correlations. The Kondo model and its
variants are the continuing focus of huge body of theoretical8, 9, numerical10
and experimental works11, 12, 13, 14, 15, 16, 17. While the complexity of bulk
materials impedes the data-theory comparison18, a major step forward was made
in 1998 with the experimental implementation in nano-circuits of tunable Kondo
impurities, from two degenerate spin states of a quantum dot11, 12. However,
experimental investigations of such Kondo circuits have essentially relied on
their transport properties, which are inherently non-equilibrium quantities,
whereas thermodynamic properties of primary interest remain elusive. In
particular, due to the difficulty in measuring a single elementary magnetic
moment, the spin of a Kondo impurity has not been possible to probe so far, in
spite of its central role.

The present work overcomes this obstacle, demonstrates the universal screening
of a Kondo impurity and provides a thermodynamic window into the underlying
many-body physics with a ‘charge’ pseudospin implementation19, 20, 21. As the
role of the Kondo spin is here played by two charge states, it can be
sensitively measured with a capacitively coupled detector22, 23, 24, 25. Such
thermodynamic charge probe permits us to investigate the primary (1-channel)
Kondo effect with ‘charge’ Kondo circuits. In contrast, with multiple contacts
required for transport characterizations, a different physics emerges in these
circuits 20, 21, 26. One noteworthy challenge to achieve a full picture is the
logarithmic spread of the Kondo crossover, which extends over many orders of
magnitude in $T/T_{\mathrm{K}}$ with $T_{\mathrm{K}}$ the scaling Kondo
temperature. Here, it is addressed through the particularly broad field-effect
tunability of charge Kondo circuits, allowing for large variations of
$T_{\mathrm{K}}$. With this approach, we observe the crossover experienced by
a Kondo impurity as the temperature $T$ is lowered, from an asymptotically
free (charge pseudo-) spin-$1/2$ to a screened singlet. The full Kondo
screening is first evidenced from the saturation of the charge pseudospin
susceptibility at low temperature. The complete universal crossover predicted
by theory is then confronted with the temperature evolution of the Curie
constant, which is considered to provide a measure of the effective spin27.
This comparison also informs on the relation between $T_{\mathrm{K}}$ and
device parameters. Finally, we determine the screened Kondo (charge pseudo-)
spin by measuring its fully polarized value, whose universal evolution is
controlled by the ratio between Zeeman (charge states) energy splitting and
Kondo temperature $T_{\mathrm{K}}$.

Figure 1:  Metal-semiconductor charge Kondo device with a charge sensor. a,
Colored e-beam micrograph of the measured device. A micron-scale metallic
island (brighter) is connected through a tunable quantum point contact (QPC)
formed by field effect using split gates (red, top-right) in a two-dimensional
electron gas (2DEG, darker gray areas delimited by bright edges). The island’s
charge $Q$ is controlled with the plunger gate voltage $V_{\mathrm{pl}}$
(orange, bottom-right), and measured with a capacitively coupled sensor
separated by a barring gate (blue, left). The sensor consists of a
constriction formed near the tip of the lateral gate (uncolored). The current
propagates along the edges of the 2DEG set in the integer quantum Hall regime
($B\simeq 5.3$ T, filling factor $\nu=2$), as depicted by orange lines with
arrows (reflected inner channel not shown). A schematic (bottom-left)
illustrates the charge Kondo mapping. b, Sensor calibration with a weakly
connected island, implementing a single-electron box. The change $\delta
G_{\mathrm{sens}}^{\mathrm{1e}}$ in sensor conductance $G_{\mathrm{sens}}$ for
an additional $1e$ charge is given by the periodic jumps when sweeping
$V_{\mathrm{pl}}$, which are associated with discrete increments in the mean
number $\left<N_{\mathrm{isl}}\right>$ of electrons in the island.

The ‘charge’ Kondo mapping proposed by Matveev28, 29 is implemented in the
device shown in Fig. 1a. In this mapping, the local magnetic impurity of the
original Kondo model is replaced by a Kondo pseudospin of $1/2$
($S=\\{\Downarrow,\Uparrow\\}$) made of the two quantum charge states of
lowest energy of a metallic island (bright central part), which can be tuned
to degeneracy with a plunger gate voltage ($V_{\mathrm{pl}}$). Such a
reduction of the charge degree of freedom necessitates low temperatures and
voltages with respect to the charging energy $E_{\mathrm{C}}=e^{2}/2C$ ($e$
the elementary electron charge, $C$ the geometrical capacitance of the
island), such that the other charge states are effectively frozen out. In
practice, $E_{\mathrm{C}}\simeq 39\,$µeV$\simeq k_{\mathrm{B}}\times 450\,$mK
thus sets a high energy cutoff for Kondo physics. The important role of the
magnetic field in the spin Kondo model is played by the detuning $\delta
V_{\mathrm{pl}}$ from charge degeneracy. It induces an energy difference
$2E_{\mathrm{C}}\delta V_{\mathrm{pl}}/\Delta$ ($\Delta$ the plunger gate
voltage period) between the pseudospin charge states, which mimics the Zeeman
splitting of a Kondo magnetic impurity. Whereas the Kondo model involves a
spin-exchange interaction, the island’s charge (the impurity pseudospin) is
not coupled to the real spin of electrons. Matveev instead introduced an
electron localization pseudospin of $1/2$ ($s=\\{\uparrow,\downarrow\\}$) that
labels wave functions according to their position, either within
($\downarrow$) or outside ($\uparrow$) of the island. This localization
pseudospin description requires an effectively continuous electronic density
of states in the metallic island (in contrast with small quantum dots with
discrete energy levels). Failing that, an electron state outside of the island
($\uparrow$) could end up without a matching state of identical energy inside
it, and thus no associated pseudospin ($\downarrow$). In this representation,
each time an electron enters the island, both the island’s charge pseudospin
$\vec{S}$ and the electron’s localization pseudospin $\vec{s}$ flip. This
spin-exchange process coincides with the Kondo model (with an anisotropic
coupling, as the irrelevant $S_{\mathrm{z}}s_{\mathrm{z}}$ coupling is
absent28). The strength of the Kondo coupling is adjusted with a tunable
quantum point contact (QPC, colored red) controlling the connection to the
island, and characterized by the electron transmission probability $\tau$
across the QPC. Tuning the QPC to a larger $\tau$ increases the scaling Kondo
temperature, thereby opening access to lower $T/T_{\mathrm{K}}$. Note however
that non-universal behaviors involving the high energy cutoff $E_{\mathrm{C}}$
might develop at high $\tau$, upon reaching $k_{\mathrm{B}}T_{\mathrm{K}}\sim
E_{\mathrm{C}}$ (ultimately, at precisely $\tau=1$, the charge quantization
completely vanishes29, 30 and there is no Kondo effect). The primary
(1-channel) Kondo effect of present interest is realized with a single
electronic quantum channel (non-degenerate in real spin) connected to the
island through the QPC.

The sample is nanofabricated on a Ga(Al)As heterojunction hosting a two-
dimensional electron gas (2DEG) buried 90 nm below the darker surface
delimited by etch-defined edges appearing bright in Fig. 1a. It is installed
in a dilution refrigerator with heavily filtered and thermalized measurement
lines31, cooled down to an electronic temperature $T\simeq 9\,$mK obtained
from on-chip noise thermometry (Methods). It is also immersed in a strong
perpendicular magnetic field ($B\simeq 5.3$ T) corresponding to the integer
quantum Hall effect at filling factor $\nu=2$. In this regime the current
propagates along two quantum Hall edge channels. The outer one is
schematically shown as orange lines with arrows indicating the propagation
direction, while the irrelevant inner one, reflected at all QPCs, is not
shown. This edge channel is in essentially perfect electrical connection with
the thermally annealed AuGeNi metallic island, such that the Kondo coupling
strength is entirely determined by the transmission probability $\tau$ across
the single connected QPC (upper-right in Fig. 1a). Note that measuring $\tau$
requires us to connect additional channels to the island, as well as the
application of a large dc voltage bias ($\sim
50\,\mu\mathrm{V}>E_{\mathrm{C}}/e$) to minimize Coulomb effects (Methods).
The two other QPCs controlled by uncolored metallic split gates are only used
for characterization purposes and to establish the generic (QPC independent)
character of our observations (Methods). Indeed, in contrast with the small
quantum dot implementation of the Kondo model32, 33, multiple connected
channels would compete to screen the charge Kondo pseudospin 28, 29, 20, 21, a
phenomenon with profound consequences described by another model referred to
as multi-channel Kondo 34, 1, 2, 3. For the data shown in the main manuscript,
the path across each of these two other QPCs is disconnected.

The island’s charge sensor consists in an additional QPC whose conductance
$G_{\mathrm{sens}}$ changes by capacitive coupling with the electrical
potential of the island. The sensor QPC is located along the barring gate
colored blue in Fig. 1a, near the tip of the uncolored gate. The barring gate
is negatively voltage biased to create a narrow depleted region galvanically
isolating the sensor from the nearby edge channel emanated and reabsorbed by
the island (carrying the same electrical potential). The conversion factor
between a change in $G_{\mathrm{sens}}$ and a change in the island’s charge
can be straightforwardly calibrated with the connected QPC set to a weak,
tunnel contact ($\tau\simeq 0.04\ll 1$, the island then implements a single-
electron box). In this tunnel regime and at low temperatures $T\ll
E_{\mathrm{C}}/k_{\mathrm{B}}$, the mean number of electrons in the island
$\langle N_{\mathrm{isl}}\rangle$ is quantized and periodically increases, one
electron at a time, when raising $V_{\mathrm{pl}}$. The amplitude of the
corresponding periodic jumps observed in $G_{\mathrm{sens}}(\delta
V_{\mathrm{pl}})$ (see Fig. 1b) hence provides the change $\delta
G_{\mathrm{sens}}^{1e}$ (vertical arrow) associated with the addition of one
electron. Note that the charge probed by $G_{\mathrm{sens}}$ includes the
linear electrostatic contribution of $V_{\mathrm{pl}}$ mediated by the island
(a much smaller direct cross-talk contribution is separately measured and
compensated for, Methods), which results in the usually observed saw-tooth
shape of $G_{\mathrm{sens}}(\delta V_{\mathrm{pl}})$ instead of a Coulomb
staircase (see e.g. Ref. 35). The difference $\delta N_{\mathrm{isl}}$ in the
mean number of electrons in the island with respect to the charge degeneracy
point ($\delta V_{\mathrm{pl}}=0$) is obtained from

$\delta N_{\mathrm{isl}}\equiv\langle N_{\mathrm{isl}}\rangle(\delta
V_{\mathrm{pl}})-\langle N_{\mathrm{isl}}\rangle(0)=\frac{\delta
G_{\mathrm{sens}}}{\delta G_{\mathrm{sens}}^{1e}}+\frac{\delta
V_{\mathrm{pl}}}{\Delta},$ (1)

with $\delta G_{\mathrm{sens}}=G_{\mathrm{sens}}(\delta
V_{\mathrm{pl}})-G_{\mathrm{sens}}(0)$ and $\Delta$ the period in
$V_{\mathrm{pl}}$. At $|\delta V_{\mathrm{pl}}|<\Delta/2$, the mean value of
the charge Kondo pseudospin is simply given by $\langle
S_{\mathrm{z}}\rangle=\delta N_{\mathrm{isl}}$. Interestingly, despite the
large geometrical difference with small quantum dots where this charge
detection strategy was previously implemented22, 23, 24, 25, we obtain a
comparable conductance sensitivity per electron of $|\delta
G_{\mathrm{sens}}^{1e}|\simeq 0.04e^{2}/h$ (see Methods for checks with the
sensor tuned to a larger sensitivity $|\delta G_{\mathrm{sens}}^{1e}|\simeq
0.09e^{2}/h$). In principle, for strong enough measurements, the charge sensor
could interfere with the probed Kondo pseudospin, by projecting it. In
practice, we avoid any discernible back-action notably by driving the sensor
with a rather small ac voltage bias $V_{\mathrm{sens}}^{\mathrm{rms}}\lesssim
3k_{\mathrm{B}}T/e$ (see Methods for specific tests and discussions).

Figure 2:  Charge sensing in tunnel and near ballistic limits. a, b, Sensor
conductance signal plotted in $1e$ step units versus plunger gate voltage
difference to charge degeneracy $\delta V_{\mathrm{pl}}$ in units of one
period $\Delta$. Each set of identical symbols correspond to data points at a
distinct temperature $T$ for a connected QPC set either in the tunnel
($\tau\simeq 0.04\ll 1$, a) or near-ballistic ($1-\tau\simeq 0.015\ll 1$, b)
limit. Measurements are plotted alongside the corresponding quantitative
prediction (nearest black dashed line). Inset of a shows the circuit’s
schematic.

In a preliminary step, we ascertain the charge detection procedure and
validate that the device is described by Matveev’s model where charge Kondo
physics is predicted to develop. To this aim, we directly confront in Fig. 2
the charge measurements performed over a full plunger gate voltage period
$\Delta\simeq 1.14$ mV (symbols) with the quantitative analytical predictions
(dashed lines) available for a connected QPC in the tunnel ($\tau\simeq
0.04\ll 1$, panel a) and near ballistic ($1-\tau\simeq 0.015\ll 1$, panel b)
opposite limits.
First, in the top panel (a), the displayed predictions correspond to the
straightforward expression $\delta
N_{\mathrm{isl}}=\tanh{\left(E_{\mathrm{Z}}/2k_{\mathrm{B}}T\right)}/2$ for
the statistical population of a pseudospin of $1/2$ in the presence of the
effective ‘Zeeman’ splitting $E_{\mathrm{Z}}=2E_{\mathrm{C}}\delta
V_{\mathrm{pl}}/\Delta$, to which is added the linear plunger gate
contribution $-\delta V_{\mathrm{pl}}/\Delta$ (see Eq. 1). Note that the
charging energy $E_{\mathrm{C}}\simeq k_{\mathrm{B}}\times 450$ mK and the
electronic temperature $T\in\left\\{9.4,21.6,30.4,68.7\right\\}$ mK are
separately characterized (Methods), leaving no free parameters in the
comparison. The match between data and theory shows, at our instrumental
accuracy, that the sensor solely probes the island’s charge, that additional
charge states of the island can here be considered as frozen, and that the
detector back-action is negligible even in this most sensitive tunnel
configuration.
Second, in the bottom panel (b) addressing the near ballistic regime, the
displayed theoretical prediction is a novel result obtained using Matveev’s
model of our device 29, with a second-order perturbation treatment of the
back-scattering amplitude $\sqrt{1-\tau}$ valid at arbitrary
$k_{\mathrm{B}}T/E_{\mathrm{C}}$ (see Supplementary Information for the
derivation and full expression, see Eq. 4 in Methods for a $O(\pi
k_{\mathrm{B}}T/E_{\mathrm{C}})^{2}$ analytical prediction very accurate
except at $T\simeq 69$ mK). A remarkable, parameter-free agreement is observed
at $\tau\simeq 0.985$ (corresponding to a back-scattering amplitude of
$0.12$). This validates the non-invasive island charge sensing in the present
regime of weak charge modulations, more sensitive to nearby charged defects
and small artifacts. This agreement also attests of the adequation between
device and theoretical model at high $\tau$. On these firm grounds the Kondo
screening of the charge pseudospin is now explored.

Figure 3:  Screened charge Kondo pseudospin at strong Kondo coupling. The
impurity charge pseudospin susceptibility measured at degeneracy ($\delta
V_{\mathrm{pl}}\simeq 0$) is plotted versus $T^{2}$ at $T<40$ mK well-below
the high energy cutoff $\sim E_{\mathrm{C}}\simeq k_{\mathrm{B}}\times 450$
mK. The continuous lines display the strong-coupling/near-ballistic
quantitative predictions ($\tau$-dependent, see Methods), converging as
$\propto T^{2}$ toward a finite zero-temperature value. A non-diverging
susceptibility as $T\rightarrow 0$ signals a fully screened pseudospin. The
dashed line shows the weak-coupling/tunnel quantitative prediction $\propto
1/T$ for an asymptotically free impurity.

A central feature of the 1-channel Kondo physics is that a spin-$1/2$ Kondo
impurity becomes fully screened at low enough temperatures $T\ll
T_{\mathrm{K}}$. This can be demonstrated from the low-temperature behavior of
the magnetic susceptibility $\chi$ of the impurity, since a finite (non-
diverging) $\chi(T\rightarrow 0)$ implies a fully screened singlet ground
state 1, 27. We observe here such a screening signature of the charge Kondo
pseudospin. In the present ‘charge’ implementation, the zero-field
susceptibility $\chi/(g\mu_{\mathrm{B}})^{2}$ corresponds to $\left(\partial
N_{\mathrm{isl}}/\partial V_{\mathrm{pl}}\right)\Delta/2E_{\mathrm{C}}$ at
$\delta V_{\mathrm{pl}}\simeq 0$. Measurements of this charge Kondo
susceptibility are displayed in Fig. 3 versus $T^{2}$, where different symbols
are associated with different Kondo couplings ($\tau$). In the weak Kondo
coupling limit of a tunnel QPC ($\tau\simeq 0.04$, open squares), the
susceptibility increases like $1/T$ when $T$ is reduced (dashed line), as
expected in the corresponding asymptotic freedom regime of an unscreened Kondo
impurity ($T\gg T_{\mathrm{K}}$, see Eq. 8 in Methods). In contrast, for the
stronger Kondo couplings implemented through at higher QPC transmission
($\tau\gtrsim 0.9$), the susceptibility does not diverge but instead
approaches a finite low temperature limit. This mere observation establishes
the screening of the Kondo impurity predicted in the corresponding limit $T\ll
T_{\mathrm{K}}$. Quantitatively, the high transmission data closely match the
parameter-free predictions derived within Matveev’s model of our device
(straight continuous lines, Eq. 5 in Methods). The quadratic temperature
scaling also corroborates the expected universal Kondo behavior at $T\ll
T_{\mathrm{K}}$ 1. However, the universal character does not extend to the
numerical value of the $T^{2}$ coefficient, which could be attributed 36 to
$k_{\mathrm{B}}T_{\mathrm{K}}$ being insufficiently small compared to the
high-energy cutoff $E_{\mathrm{C}}$ for large $\tau$ (see Eqs. 5,7 and related
discussion in Methods).

Figure 4:  Universal temperature renormalization flow from free to screened
Kondo spin, observed on the zero-field Curie constant (spin susceptibility
coefficient) $T\chi$. Identical symbols (including in Figs. 3 and 5a)
represent data points obtained for the same QPC setting (see b for
correspondence symbol-$\tau$) at different temperatures. Lines are universal
predictions within the standard spin Kondo model (continuous: numerical
calculations, dashed: asymptotic limits). a, Measurements of
$k_{\mathrm{B}}T\left(\partial N_{\mathrm{isl}}/\partial
V_{\mathrm{pl}}\right)\Delta/2E_{\mathrm{C}}\equiv
k_{\mathrm{B}}T\chi/(g\mu_{\mathrm{B}})^{2}$ at $\delta V_{\mathrm{pl}}\propto
B=0$ are plotted vs $T$ in a linear-log scale. The (near) logarithmic behavior
is indicative of the Kondo effect. b, The same $T\chi$ data points are plotted
vs $T/T_{\mathrm{K}}$, with $T_{\mathrm{K}}(\tau)$ (see inset) obtained by
matching with theory the $T\simeq 9$ mK measurements. c, Parameter-free data-
theory comparison on $\partial(T\chi)/\partial\log{T}$ vs $T\chi$,
corresponding to the $\beta$-function characterizing the underlying Kondo
renormalization flow.

Beyond the screened and free Kondo impurity limits, we now extend our
investigation to the full, universal renormalization flow as
$T/T_{\mathrm{K}}$ is reduced. To this aim, we focus in Fig. 4 on the Curie
constant, i.e. the susceptibility coefficient $T\chi$ (from dimensional
scaling, $\chi$ without a prefactor $T$ or $T_{\mathrm{K}}$ is not a universal
function of $T/T_{\mathrm{K}}$). For a free impurity, the Curie constant is
directly related to its spin:
$k_{\mathrm{B}}T\chi/(g\mu_{\mathrm{B}})^{2}=S(S+1)/3$ ($0.25$ for $S=1/2$).
More generally, the Curie constant is considered to provide a measure of the
effective spin and, in particular, of the screened spin of the Kondo
impurity27. In the left panel (a), a representative selection of ‘charge’
$T\chi$ data, spanning a complete range of $\tau$ (see panel b for the
distribution) is displayed versus $T<40$ mK in a linear-log scale.
First, a log-like temperature dependence characteristic of the Kondo effect is
evidenced from the near-linearity of the data, except at the lowest and
highest $T\chi$. However, as each tuning of $\tau$ corresponds to a different
value of $T_{\mathrm{K}}$, each data set seems unrelated to the others when
plotted versus $T$.
Second, the central panel (b) shows a comparison of the same data, now plotted
vs $T/T_{\mathrm{K}}$, with the universal curve for the standard spin Kondo
model (1-channel, isotropic, $S=1/2$) obtained combining a numerical
calculation (continuous line, extracted from Ref. 8) and asymptotic
predictions (dashed lines, see Eqs. 9 and 10 in Methods). To perform this
comparison, the a priori unknown value of $T_{\mathrm{K}}$ must be fixed for
each $\tau$. In practice, we determine $T_{\mathrm{K}}(\tau)$ such that, by
construction, the data point at the lowest temperature ($T\simeq 9$ mK) lies
on the predicted universal curve. The comparison in the main panel of Fig. 4b
is then in the evolution as $T$ is increased for each set of identical
symbols. At our experimental accuracy, we observe a good match with the
universal theoretical prediction, on an explored range extending over several
orders of magnitude in $T/T_{\mathrm{K}}$. Although $T_{\mathrm{K}}$ is a free
parameter in the main panel of Fig. 4b, the underlying Kondo physics is
stringently tested by further confronting this experimental
$T_{\mathrm{K}}(\tau)$ with predictions, see inset of b. The standard
theoretical prediction for small couplings1, 28
$T_{\mathrm{K}}/E_{\mathrm{C}}\propto\tau^{1/4}\exp{\left(-\pi^{2}/2\tau^{1/2}\right)}$
is shown as a dashed blue line, and the prediction obtained expanding upon
Matveev’s model for high transmissions is displayed as a dashed red line (Eq.
6 in Methods). The data-theory agreement in the inset strengthens the
comparison in the main panel, as the only fit parameter in the latter is seen
to obey Kondo predictions in the former. Note that at large $\tau$, the Kondo
temperature increases up to $k_{\mathrm{B}}T_{\mathrm{K}}/E_{\mathrm{C}}\simeq
0.12$. Whereas non-universal deviations could develop, as pointed out in the
previous paragraph, we find that they remain relatively small.
Third, for a direct, $T_{\mathrm{K}}$-free comparison, the so-called
$\beta$-function $\partial(T\chi)/\partial\log T$ vs $T\chi$, characterizing
the universal renormalization flow of $T\chi$ is shown in Fig. 4c (continuous
line) together with its analytical asymptotes (dashed lines, see expressions
in Methods). Although the experimental uncertainty is stronger in the
$\beta$-function representation because of the discrete temperature
differentiation of the measurements, the similitude between the data (symbols)
and the numerical Kondo prediction can be directly appreciated.

Figure 5:  Kondo impurity magnetization. a,
$\left<S_{\mathrm{z}}\right>\equiv\delta N_{\mathrm{isl}}$ is plotted versus
the ‘Zeeman’ energy splitting $E_{\mathrm{Z}}\equiv 2E_{\mathrm{C}}|\delta
V_{\mathrm{pl}}|/\Delta$ in units of the Kondo energy scale
$k_{\mathrm{B}}T_{\mathrm{K}}$. The Kondo prediction
$S_{\mathrm{thy}}^{E_{\mathrm{Z}}\gg
T}(E_{\mathrm{Z}}/k_{\mathrm{B}}T_{\mathrm{K}})$, in the universal limit
$E_{\mathrm{Z}}\gg k_{\mathrm{B}}T$ where it shows the transition from free to
screened spin, is represented by a thick continuous line. Measurements at
$T\simeq 9$ and $14$ mK are shown as dashed and dotted lines, respectively,
with each color corresponding to a different $\tau$ (see inset in Fig. 4b for
the separately obtained $T_{\mathrm{K}}(\tau)$). Symbols (full for 9 mK, open
for 14 mK) highlight data points for which a good agreement with the universal
prediction is expected ($5k_{\mathrm{B}}T<E_{\mathrm{Z}}<E_{\mathrm{C}}/5$).
b, Progressive polarization of
$\left<S_{\mathrm{z}}\right>/S_{\mathrm{thy}}^{E_{\mathrm{Z}}\gg T}$ with
increasing $E_{\mathrm{Z}}/k_{\mathrm{B}}T$ at fixed
$E_{\mathrm{Z}}/k_{\mathrm{B}}T_{\mathrm{K}}$. The thermal crossover shifts
and broadens with increasing $E_{\mathrm{Z}}/k_{\mathrm{B}}T_{\mathrm{K}}$.
Gathered data points restricted to $E_{\mathrm{Z}}<E_{\mathrm{C}}/5$ are shown
as symbols. Kondo predictions are shown as continuous lines. The free-spin
prediction $\tanh(E_{\mathrm{Z}}/2k_{\mathrm{B}}T)$ is shown as a dashed line.

In contrast to the above susceptibility studies performed in the absence of a
Zeeman-like energy splitting of the Kondo impurity ($E_{\mathrm{Z}}\ll
k_{\mathrm{B}}T$), we explore here the pseudospin polarization
$\left<S_{\mathrm{z}}\right>$ as the degeneracy between the two charge states
is lifted (see Fig. 5). At large energy splitting with respect to the
temperature ($E_{\mathrm{Z}}\gg k_{\mathrm{B}}T$),
$\left<S_{\mathrm{z}}\right>$ corresponds to the partially screened spin of
the impurity along a Kondo renormalization flow controlled by
$E_{\mathrm{Z}}/k_{\mathrm{B}}T_{\mathrm{K}}$. The universal Kondo prediction
for $S_{\mathrm{thy}}^{E_{\mathrm{Z}}\gg
T}(E_{\mathrm{Z}}/k_{\mathrm{B}}T_{\mathrm{K}})$ is shown as a continuous line
in Fig. 5a 27. Representative measurement sweeps of $\delta
N_{\mathrm{isl}}\equiv\left<S_{\mathrm{z}}\right>$, each performed for a
specific setting of $\tau$ and a fixed $T$, are plotted in Fig. 5a as a
function of $E_{\mathrm{Z}}/k_{\mathrm{B}}T_{\mathrm{K}}$ (thin dashed and
dotted lines indicate, respectively, $T\simeq 9$ and 14 mK). Here
$T_{\mathrm{K}}(\tau)$ is not a free parameter, but the value separately
obtained from the previous data-theory comparison on the Curie constant (see
inset in Fig. 4b). Only a reduced $E_{\mathrm{Z}}$ interval of these
measurement sweeps, highlighted with symbols, should be compared to the
universal Kondo prediction. The highlighted $E_{\mathrm{Z}}$ interval is
limited on the low side by the requirement of a sufficiently large
$E_{\mathrm{Z}}/k_{\mathrm{B}}T$ ratio; in practice, we set the minimum value
to $5$. On the high side, the comparison to the universal Kondo prediction is
limited to $E_{\mathrm{Z}}\ll E_{\mathrm{C}}$ (equivalently, $\delta
V_{\mathrm{pl}}\ll\Delta/2$), since other charge states are otherwise
relevant, which breaks the Kondo mapping. In practice, we highlight
$E_{\mathrm{Z}}<E_{\mathrm{C}}/5$ ($|\delta V_{\mathrm{pl}}|<0.1\Delta$). In
this interval, as shown in Fig. 5a, a data-theory difference smaller than
$0.03$ is observed over the wide explored parameter range, for both $T\simeq
9$ and 14 mK, hence establishing the Kondo impurity magnetization vs
$E_{\mathrm{Z}}/k_{\mathrm{B}}T_{\mathrm{K}}$. (Note that for experimental
temperatures above $14\,$mK, there are no data points fulfilling
$5k_{\mathrm{B}}T<E_{\mathrm{Z}}<E_{\mathrm{C}}/5$.)
In a second step, we investigate the crossover from
$\left<S_{\mathrm{z}}\right>=0$ to the above universal regime as
$E_{\mathrm{Z}}/k_{\mathrm{B}}T$ increases, for a fixed
$E_{\mathrm{Z}}/k_{\mathrm{B}}T_{\mathrm{K}}$, which is expected to exhibit
specific Kondo signatures markedly different from the polarization of a free
spin8. For this purpose, we display in Fig. 5b the Kondo impurity
magnetization $\left<S_{\mathrm{z}}\right>$ normalized by the predicted
universal value $S_{\mathrm{thy}}^{E_{\mathrm{Z}}\gg
T}(E_{\mathrm{Z}}/k_{\mathrm{B}}T_{\mathrm{K}})$. Whereas for a free spin the
crossover follows $\tanh\left(E_{\mathrm{Z}}/2k_{\mathrm{B}}T\right)$ (dashed
line), as $E_{\mathrm{Z}}/k_{\mathrm{B}}T_{\mathrm{K}}$ is reduced a
logarithmic broadening is predicted to develop with the Kondo effect
(continuous lines) 8. Data points shown as identical symbols and plotted in
linear-log scale versus $E_{\mathrm{Z}}/k_{\mathrm{B}}T$ are measurements
gathered from different settings of $\tau$ (and thus different
$T_{\mathrm{K}}$) and different Zeeman splittings whose ratio matches the same
value of
$E_{\mathrm{Z}}/k_{\mathrm{B}}T_{\mathrm{K}}\in\\{0.476,0.952,4.76\\}$. As in
panel a, the collected data are limited on the high Zeeman splitting side to
$E_{\mathrm{Z}}<E_{\mathrm{C}}/5$; however $E_{\mathrm{Z}}/k_{\mathrm{B}}T$
can be arbitrarily low. The present observation hence establishes, without any
free parameter, the unconventional Kondo crossover to a fully polarized,
partially screened impurity spin.

By implementing a single Kondo impurity as two charge states measured by a
capacitively coupled sensor, we have directly observed the central Kondo spin
as it progressively hybridizes with the conduction electrons. With this
approach, we demonstrated the screening of the Kondo impurity and observed the
universal crossover from asymptotic freedom to the strong coupling limit on a
broadly tunable and fully characterized device. The present strategy can be
applied to the large and continuously expanding family of Kondo-type models,
hence providing a platform to engineer and convincingly observe exotic
states5, 2, 3, 6, 26. In particular, such thermodynamic measurement of the
charge Kondo impurity gives access to the associated entropy25, 37, 38, whose
predicted fractional value of $k_{\mathrm{B}}\log 2/2$ for the two-channel
Kondo model would provide a clear signature for the highly debated emergence
of a free Majorana mode.

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Acknowledgments. This work was supported by the European Research Council
(ERC-2020-SyG-951451), the French National Research Agency
(ANR-18-CE47-0014-01) and the French RENATECH network. We thank K. Ensslin, J.
Folk and A. Mitchell for discussions.

Author Contributions. C.P. performed the experiment with inputs from A.Aa.,
A.An., F.P. and P.G.; C.P., A.An. and F.P. analyzed the data; C.P., A.Aa and
F.P. fabricated the sample; A.C. and U.G. grew the 2DEG; C.H., E.S. and Y.M.
developed the theory; C.P. and F.P. wrote the paper with inputs from all
authors; A.An. and F.P. led the project.

Author Information. The authors declare no competing financial interests.
Correspondence and requests for materials should be addressed to A.An.
<EMAIL_ADDRESS>and F.P. (frederic.pierre@cnrs.fr).

METHODS

Nanofabrication. The sample is patterned by standard e-beam lithography on a
GaAlAs heterostructure forming an electron gas 90 nm below the surface, with a
density of $2.6\times 10^{11}\,\mathrm{cm}^{-2}$ and a mobility of $0.5\times
10^{6}\,\mathrm{cm}^{2}V^{-1}\mathrm{s}^{-1}$. The 2DEG mesa is delimited by a
wet etching approximately 100 nm deep with a
$\mathrm{H}_{3}\mathrm{PO}_{4}/\mathrm{H}_{2}\mathrm{O}_{2}/\mathrm{H}_{2}\mathrm{O}$
solution. The ohmic contacts are realized with a metallic multilayer
deposition of Ni(10 nm)-Au(10 nm)-Ge(90 nm)-Ni(20 nm)-Au(170 nm)-Ni(40 nm)
followed by an annealing at 440°C where the metal penetrates the GaAlAs. Note
that the active outer quantum Hall edge channel is found to be perfectly
connected to the small metallic island at experimental accuracy (with a
reflection probability below $0.1\%$). The gates forming the QPCs by field
effect are made of aluminum deposited directly on the surface of the GaAlAs
heterostructure.

Experimental setup. The measurements are performed in a cryo-free dilution
refrigerator with extensive measurement lines filtering and thermalization.
Details on the fridge wiring are provided in Ref. 39. Measurements of the
conductance across the sample (for the device characterization) and across the
charge sensor QPC are carried out with standard lock-in techniques at low
frequencies, below 150 Hz, with ac excitations of rms amplitude below
$k_{\mathrm{B}}T/e$. Noise measurements for the thermometry described below
are performed near 1 MHz with home-made cryogenic amplifiers40.

Electronic temperature. The electronic temperature $T$ is obtained from on-
chip thermal noise measured on an ohmic contact. The conversion factor is
calibrated from the linear slope of thermal noise vs temperature of the mixing
chamber, at sufficiently high temperature where the difference between
electron and mixing chamber temperatures is negligible. In practice, the
calibration is performed above 40 mK where the high linearity of noise vs
temperature attests of the good thermal anchoring of the electrons (here,
differences between in-situ electronic temperature $T$ and mixing chamber
temperature develop essentially below 20 mK). The data displayed in the main
manuscript are obtained at $T=\\{9.4\pm 0.3,14.4\pm 0.1,21.6\pm 0.2,30.4\pm
0.1,46.2\pm 0.1,68.7\pm 0.5\\}$ mK, where the indicated uncertainties
correspond to the temperature drifts occurring during the measurements.

Extended Data Figure 1:  QPC characterization. Intrinsic transmission
probability $\tau$ across the connected QPC vs applied voltage to the split
gate. It is obtained from $\tau=1/(e^{2}/hG-1/2)$ with $G$ the differential
conductance of the device in the configuration shown in the schematic. Symbols
represent the specific settings in the main manuscript.

QPC characterization. The Kondo coupling strength is controlled by the bare
(unrenormalized) transmission probability $\tau$ across the connected QPC. The
value of $\tau$ is obtained by setting the two other QPCs (uncolored in Fig.
1a and normally closed) well within their broad transmission plateau and by
applying a dc bias of 50 $\mu$V ($>E_{\mathrm{C}}$ to suppress most of the
dynamical Coulomb reduction of the conductance). The device resistance is then
equal to the sum of the QPC resistance $h/\tau e^{2}$ and the well defined
$h/2e^{2}$ resistance. Note that changing the tuning of the two other,
normally closed QPCs impacts significantly, through capacitive cross-talk, the
QPC of present interest. This effect can be calibrated and is mostly corrected
for, as detailed below.

Capacitive cross-talk. The capacitive cross-talk that underpins the charge
sensing mechanism also results in cross-correlations between the tunings of
the different QPCs. Thanks to the distance of a few microns between
constrictions, the influence of remote gates remains at a relatively small
level of a few percent. Accordingly, the corresponding cross-talk can be
mostly compensated for by relatively small, linear corrections calibrated
separately, one pair of gates at a time. These corrections are employed for
the determination of $\tau$ (see above in Methods), and also to maintain a
charge sensor as stable as possible with respect to changes in the tuning of
the QPC, as well as during sweeps in plunger gate voltage $V_{\mathrm{pl}}$.

Extended Data Figure 2:  Charging energy extracted with the island connected
by two QPCs, in the single electron transistor (SET) configuration
schematically shown in b. Both QPCs are set to a small transmission
probability. a, The differential conductance $G_{\mathrm{SET}}$ across the
device is displayed on a color-map versus plunger gate voltage ($\delta
V_{\mathrm{pl}}$) and dc voltage ($V_{\mathrm{dc}})$. The
$E_{\mathrm{C}}=39\,$µeV prediction for the Coulomb diamonds edges is shown as
red straight lines. c, The normalized conductance peak at zero dc bias
(symbols, each color for a different $T$) can be fitted with the asymptotic
tunnel theory (lines). The resulting $E_{\mathrm{C}}^{\mathrm{fit}}$ are shown
as symbols in d (open and full for, respectively, lowest and slightly higher
QPCs transmissions). Lines in d represent the corresponding predictions for a
fixed $E_{\mathrm{C}}=39\,$µeV, also including the small Kondo renormalization
effectively narrowing the peak (reducing $E_{\mathrm{C}}^{\mathrm{fit}}$).

Charging energy. The experimental procedure to obtain the central charging
energy $E_{\mathrm{C}}\equiv e^{2}/2C\simeq 39\,$µeV $\simeq
k_{\mathrm{B}}\times 450$ mK combines two methods, performed with the device
set to have two QPCs weakly connected to the island (see Extended Data Fig.
2a).

In this single-electron transistor regime, we first measure the differential
conductance across the device $G_{\mathrm{SET}}$ as a function of plunger gate
voltage $\delta V_{\mathrm{pl}}$ and dc voltage bias $V_{\mathrm{dc}}$. The
height of the resulting Coulomb diamonds displayed in Extended Data Fig. 2a
corresponds to $2E_{\mathrm{C}}$. The red continuous lines shows the diamonds
boundaries for $E_{\mathrm{C}}=39\,$µeV. Although very straightforward, the
accuracy of this approach is mostly limited by the observed broadening of the
diamond edges as $V_{\mathrm{dc}}$ is increased.

Second, the resolution on this central quantity is refined from the width of
the $G_{\mathrm{SET}}$ peak at zero dc bias. In the asymptotic tunnel limit,
where Kondo renormalization is ignored, theory predicts at
$V_{\mathrm{dc}}=0$:

$G_{\mathrm{SET}}(\delta
V_{\mathrm{pl}}/\Delta)=G_{\mathrm{SET}}(0)\frac{E_{\mathrm{Z}}/k_{\mathrm{B}}T}{\sinh(E_{\mathrm{Z}}/k_{\mathrm{B}}T)},$
(2)

with $E_{\mathrm{Z}}=2E_{\mathrm{C}}\delta V_{\mathrm{pl}}/\Delta$. As
illustrated in Extended Data Fig. 2c for the setting $\tau\simeq 0.04$ and
$T\in\\{9,22,46\\}$ mK, a very good match can be achieved between data
(symbols) and Eq. 2. For each separately calibrated temperature a separate
fitted value $E_{\mathrm{C}}^{\mathrm{fit}}$ is extracted, shown in the panel
d as open and full symbols for $\tau\simeq 0.04$ and $0.08$, respectively.
These fitted values provide lower bounds for $E_{\mathrm{C}}$ since the
residual Kondo renormalization slightly narrows the peak (by increasing the
maximum conductance at degeneracy). The theoretical predictions for the
effective, Kondo renormalized $E_{\mathrm{C}}^{*}$ that is to be compared with
$E_{\mathrm{C}}^{\mathrm{fit}}$ is given, in the limit of tunnel contacts,
by41, 35:

$\frac{E_{\mathrm{C}}^{*}}{E_{\mathrm{C}}}=1-\frac{\textstyle\sum\tau}{2\pi^{2}}\left(5.154+\ln\frac{E_{\mathrm{C}}}{\pi
k_{\mathrm{B}}T}\right)+O\left[\left(\frac{\textstyle\sum\tau}{4\pi^{2}}\right)^{2},\left(\frac{k_{\mathrm{B}}T}{E_{\mathrm{C}}}\right)^{2}\right],$
(3)

with $\textstyle\sum\tau$ the sum of the transmission probabilities of all the
channels connected to the metallic island (see Ref. 35 for a quantitative
experimental observation). Grey and black lines in Extended Data Fig. 2d show
the theoretical predictions of Eq. 3 for $E_{\mathrm{C}}^{*}$ in the presence
of two connected channels with the same transmission
($\textstyle\sum\tau=2\tau$) of, respectively, $\tau=0.04$ and $0.08$, and
using an unrenormalized charging energy of $E_{\mathrm{C}}=39\,$µeV. We
estimate our experimental uncertainty on the charging energy to be of about
$\pm 1\,$µeV.

Charge data acquisition. The charge data are obtained from multiple
repetitions of sweeps extending over several plunger gate periods. The
displayed charge data within $|\delta V_{\mathrm{pl}}|<\Delta/2$ are hence
obtained from an ensemble of at least 30 measurements. As anomalies such as
nearby charge jumps can affect the integrity of the measurements, several
procedures are used to automatically discard suspicious data points or sweeps
(in practice, less than 10%). 1) We first check the integrity of each one
period sweep. For this purpose, we integrate with $V_{\mathrm{pl}}$ both the
charge sensor conductance, as well as the absolute value of the difference
between period sweep and reversed period sweep around the degeneracy point.
Then we automatically discard each period sweep with a statistically anomalous
integral value, defined in practice as being away from the mean by more than
three times the standard deviation. 2) Second, we perform a separate
statistical analysis for each individual value of $\delta V_{\mathrm{pl}}$ on
the ensemble of corresponding data points from the remaining, preserved, one
period sweeps. Specific data points away from the mean by more than three
times the standard deviation are automatically dropped out before the final
averaging is performed.

Extended Data Figure 3:  Reproducibility. The independence of the charge
measurements on the connected QPC and on the charge sensor setting attests of
the robustness and reproducibility of our results. a, Reproducibility with
different connected QPCs. Measurements at $T\simeq 9.4\,$mK of the sensor
conductance signal in units of the $1e$ sensibility are plotted as symbols
over one plunger gate voltage period. The different settings of $\tau$ (see
color code) are each implemented on three different QPCs (different symbols)
connected one at a time to the metallic island. b, Reproducibility with
different charge sensor tunings. The charge measurements shown as symbols were
repeated with two sensor settings, corresponding to different voltages
$V_{\mathrm{B}}$ applied to the blue barring gate (see inset for
$G_{\mathrm{sens}}(V_{\mathrm{B}})$, the dashed line for
$G_{\mathrm{sens}}\lesssim 0.1$ indicates a less reliable measurement).
Squares correspond to the sensor setting used for the data in the main text,
for which $\delta G_{\mathrm{sens}}^{1e}\simeq 0.0375\,e^{2}/h$. Triangles
correspond to a more sensitive setting $\delta G_{\mathrm{sens}}^{1e}\simeq
0.0937\,e^{2}/h$. The normalized sensor signal at $9.4$ mK is plotted versus
$\delta V_{\mathrm{pl}}/\Delta$ along with three different QPC transmission
$\tau$ to the metallic island (see color code).

Reproducibility. The generic character of the results shown in this article is
ascertained by checking their independence on (i) which specific QPC is
connected to the metallic island, and (ii) the tuning of the charge sensor.

(i) We confront in Extended Data Fig. 3a, the charge measurements performed at
$T\simeq 9$ mK using individually each of the three physical QPCs to connect
the metallic island (distinct symbols). For a direct comparison, the three
QPCs are (one at a time) tuned to the same quantitative value of $\tau$. Then
we match the sensor conductance normalized by $\delta G_{\mathrm{sens}}^{1e}$.
For a more thorough test, the comparison is performed over a full $\delta
V_{\mathrm{pl}}$ period and for five representative values of $\tau$ spanning
the full range from tunnel to ballistic (see scale bar for color code). The
different symbols of the same color fall on each other, essentially within the
symbols’ size, showing the QPC-independent character of our results. Note that
small systematic differences most likely result from the uncertainty on the
experimental values of $\tau$.

(ii) The reproducibility of the charge measurement versus charge sensor tuning
is specifically checked in Extended Data Fig. 3b. For this purpose, we
confront the sensor conductance normalized by $\delta G_{\mathrm{sens}}^{1e}$
obtained either with the main sensor tuning used for the data shown in the
main text (open squares, $\delta G_{\mathrm{sens}}^{1e}\simeq
0.0375\,e^{2}/h$) or with a different, more sensitive tuning (open triangles,
$\delta G_{\mathrm{sens}}^{1e}\simeq 0.0937\,e^{2}/h$) achieved with a more
negative voltage applied to the barring gate (see inset). The comparison is
then repeated for three different settings of $\tau$ across the QPC connected
to the island, over a full $\delta V_{\mathrm{pl}}$ period. The corresponding
data points are nearly indistinguishable.

Extended Data Figure 4:  Charge sensor back-action vs dc bias. The conductance
through the island in a SET configuration, with two connected QPCs in the
tunnel regime (see schematic), is displayed as symbols versus $\delta
V_{\mathrm{pl}}/\Delta$ for different dc bias voltage $V_{\mathrm{dc}}$
applied to the charge sensor constriction, at $T\simeq 9.4\,$mK. The peak
width starts to broaden for a relatively large $V_{\mathrm{dc}}\gtrsim
30\,$µV. The displayed effective temperature $T_{\mathrm{eff}}$ are fits using
the temperature as a free parameter.

Charge sensor back-action. The charge sensor can impact the probed charge
Kondo physics through the back-action induced by the charge measurement. This
phenomenon is more important for stronger measurements, and therefore for
larger voltages applied to determine $G_{\mathrm{sens}}$, as well as for
larger tunings of the sensitivity $\delta G_{\mathrm{sens}}^{1e}$. In
practice, we probe the sensor conductance using a small ac voltage of rms
amplitude smaller than $3k_{\mathrm{B}}T/e$. Note however that, even at
thermal equilibrium, the mere presence of a coupling to the charge sensor
and/or the thermal fluctuations could be sufficient to perturb the probed
system. In this section, we establish that the back-action can be neglected
from (i) the negligible effect of increasing $\delta G_{\mathrm{sens}}^{1e}$,
(ii) the large bias voltage margin before the $1e$ charge step is broadened,
(iii) the good fit of the charge step with the tunnel and near ballistic
models ignoring the coupling to the sensor.

(i) We find that the charge data are indistinguishable when increasing by a
factor of $2.5$ the sensor $1e$ sensitivity, including in the most sensitive
regime of a tunnel coupled island (see Extended Data Fig. 3b). If the back-
action was not negligible and controlled by the charge detector sensitivity, a
difference should have developed.

(ii) We added a dc bias voltage to the small ac signal used to measure the
sensor conductance to determine the back-action threshold. In practice, we
chose to probe not the charge (the charge sensor is impacted by the
application of large biases) but the width of the peak displayed by the
conductance $G_{\mathrm{SET}}$ across the island connected by two QPCs both in
the tunnel limit (see schematic in inset). In this SET regime, the peak width
of $G_{\mathrm{SET}}(\delta V_{\mathrm{pl}})$ is proportional to the
temperature if the sensor can be ignored (and in the absence of other
artifacts such as nearby charge fluctuators). We find at the most sensitive,
lowest temperature $T\simeq 9\,$mK that a dc voltage of nearly $30\,$µV
applied to the sensor is required to induce a visible change in
$G_{\mathrm{SET}}$ peak width (see $T_{\mathrm{eff}}$ values for the
corresponding effective temperature). This is a much larger sensor drive than
both the applied ac excitation amplitude of $2\,$µVrms and the thermal
fluctuations $k_{\mathrm{B}}T/e\simeq 0.8\,$µV.

(iii) The quantitative, parameter-less match with theory ignoring the sensor
back-action further establish that it is negligible. Such comparisons are
shown in Fig. 2 for the measured charge, and in the good match between blue
symbols and blue line in Extended Data Fig. 4 for $G_{\mathrm{SET}}$.

Charge Kondo circuit predictions at large QPC transmission. In this section we
recapitulate the theoretical predictions at $1-\tau\ll 1$ used to compare with
the data. Their derivation is provided in the Supplementary Information.

The central new prediction, directly used in Fig. 2b, is the island charge in
electron charge units $\delta Q/e\equiv\delta N_{\mathrm{isl}}$ for arbitrary
$\delta V_{\mathrm{pl}}$, obtained by a second order expansion in
$\pi^{2}k_{\mathrm{B}}T/E_{\mathrm{C}}$ and in the reflection amplitude
$\sqrt{1-\tau}$ of the connected QPC:

$\begin{split}\delta N_{\mathrm{isl}}&=\frac{\delta
V_{\mathrm{pl}}}{\Delta}+\frac{\gamma\sqrt{1-\tau}}{\pi}\left(1-\frac{(\pi^{2}k_{\mathrm{B}}T)^{2}}{3E_{\mathrm{C}}^{2}}\right)\sin{\frac{2\pi\delta
V_{\mathrm{pl}}}{\Delta}}\\\
&+\frac{2\gamma^{2}(1-\tau)}{\pi^{2}}\left(1-\frac{(\pi^{2}k_{\mathrm{B}}T)^{2}}{E_{\mathrm{C}}^{2}}\right)\sin{\frac{4\pi\delta
V_{\mathrm{pl}}}{\Delta}},\end{split}$ (4)

where $\gamma=\exp(C_{\mathrm{E}})\simeq 1.78107$ with $C_{\mathrm{E}}\simeq
0.577216$ the Euler constant. The dashed lines in Fig. 2b represent $\delta
N_{\mathrm{isl}}-\delta V_{\mathrm{pl}}/\Delta$, with $\delta
N_{\mathrm{isl}}$ given by Eq. 4, using the separately characterized values of
$T$, $\tau=0.985$ and $E_{\mathrm{C}}=39\,$µeV.

From Eq. 4, one straightforwardly obtains for
$\pi^{2}k_{\mathrm{B}}T/E_{\mathrm{C}},\sqrt{1-\tau}\ll 1$ the charge analog
of the zero-field magnetic susceptibility:

$\begin{split}\frac{\chi}{(g\mu_{\mathrm{B}})^{2}}\equiv&\frac{\Delta}{2E_{\mathrm{C}}}\frac{\partial
N_{\mathrm{isl}}}{\partial V_{\mathrm{pl}}}(\delta V_{\mathrm{pl}}=0)\\\
\simeq&\frac{1}{2E_{\mathrm{C}}}\left[1+2\gamma\sqrt{1-\tau}+\frac{8}{\pi}\gamma^{2}(1-\tau)\right.\\\
&\left.-\frac{(\pi^{2}k_{\mathrm{B}}T)^{2}}{E_{\mathrm{C}}^{2}}\left(\frac{2}{3}\gamma\sqrt{1-\tau}+\frac{8}{\pi}\gamma^{2}(1-\tau)\right)\right].\end{split}$
(5)

This expression together with the separately characterized $T$, $\tau$ and
$E_{\mathrm{C}}$ gives the predictions displayed as continuous lines in Fig.
3. The $T^{2}$ approach to a low temperature fixed value of $\chi$ corresponds
to the standard Kondo prediction1 (see Eq. 7).

The theoretical expression of the Kondo temperature can be obtained by
comparing Eq. 5 with the asymptotic spin Kondo prediction in the low
temperature limit $\chi(T\ll T_{\mathrm{K}})\simeq
n_{\mathrm{w}}/4k_{\mathrm{B}}T_{\mathrm{K}}$, with $n_{\mathrm{w}}\simeq
0.4107$ the Wilson number (see Eq. 7). This gives in the present
$\sqrt{1-\tau}\ll 1$ limit:

$k_{\mathrm{B}}T_{\mathrm{K}}\simeq\frac{E_{\mathrm{C}}n_{\mathrm{w}}/2}{1+2\gamma\sqrt{1-\tau}+8/\pi\gamma^{2}(1-\tau)}.$
(6)

This expression for the Kondo temperature is represented as a red line in the
inset of Fig. 4b. Note that it is not possible based on Eq. 5 to write
$k_{\mathrm{B}}T\chi$ as a function of $T/T_{\mathrm{K}}$ beyond the linear
term. This could be explained by the fact that, in the corresponding
$\sqrt{1-\tau}\ll 1$ limit, the value of $k_{\mathrm{B}}T_{\mathrm{K}}$ is
comparable to the high energy cutoff $E_{\mathrm{C}}$ thus giving rise to non-
universal contributions beyond lowest order in temperature.

Universal Kondo predictions. In this section, we recapitulate some expressions
used in the main text for the Kondo behavior of the susceptibility $\chi$, the
renormalization flow of the effective spin $k_{\mathrm{B}}T\chi$ and its
$\beta$-function, and the renormalization flow of the Kondo impurity
magnetization vs $E_{\mathrm{Z}}/k_{\mathrm{B}}T_{\mathrm{K}}$.

The magnetic susceptibility
$\chi/(g\mu_{\mathrm{B}})^{2}\equiv\partial\left<S_{\mathrm{z}}\right>/\partial(g\mu_{\mathrm{B}}B)$
of a spin-$1/2$ Kondo impurity is predicted to asymptotically approach at low
temperatures (see e.g. Ref. 1, Eqs. 4.58, 6.31, 6.79):

$\frac{\chi(T\ll
T_{\mathrm{K}})}{(g\mu_{\mathrm{B}})^{2}}=\frac{n_{\mathrm{w}}}{4k_{\mathrm{B}}T_{\mathrm{K}}}\left(1-\frac{\sqrt{3}}{4}\pi^{3}n_{\mathrm{w}}^{2}\left(\frac{T}{T_{\mathrm{K}}}\right)^{2}+O\left(\frac{T}{T_{\mathrm{K}}}\right)^{4}\right),$
(7)

with $n_{\mathrm{w}}=\exp(C_{\mathrm{E}}+1/4)/\pi^{3/2}\simeq 0.4107$ the
Wilson number. In the opposite limit of high temperatures with respect to
$T_{\mathrm{K}}$, the susceptibility reads (see e.g. Ref. 1, Eq. 3.53):

$\begin{split}\frac{\chi(T\gg
T_{\mathrm{K}})}{(g\mu_{\mathrm{B}})^{2}}=\frac{1}{4k_{\mathrm{B}}T}&\bigg{(}1-\frac{1}{\log(T/T_{\mathrm{K}})}-\frac{\log(\log(T/T_{\mathrm{K}}))}{2\log^{2}(T/T_{\mathrm{K}})}\\\
&+O\left(\log^{-2}(T/T_{\mathrm{K}})\right)\bigg{)}.\end{split}$ (8)

The magnetic susceptibility coefficient (Curie constant) displayed in Fig.
4a,b is defined as $k_{\mathrm{B}}T\chi/(g\mu_{\mathrm{B}})^{2}$. The
theoretical asymptotic expression at low $T/T_{\mathrm{K}}$ displayed in Fig.
4b reads:

$\frac{k_{\mathrm{B}}T\chi(T\ll
T_{\mathrm{K}})}{(g\mu_{\mathrm{B}})^{2}}\simeq\frac{n_{\mathrm{w}}}{4}\frac{T}{T_{\mathrm{K}}}.$
(9)

The high temperature asymptotic prediction for the magnetic susceptibility
coefficient (Curie constant) displayed in Fig. 4b is calculated by solving
(see e.g. Ref. 1, Eqs. 4.51 and 4.52):

$\Phi(2k_{\mathrm{B}}T\chi/(g\mu_{\mathrm{B}})^{2}-1/2)=\log(T/T_{\mathrm{K}}),$
(10)

with the function $\Phi(x)$ given by:

$\Phi(x)=1/2x-\log|2x|/2+3.1648x+O(x^{2}).$ (11)

Without the linear term in $x$ in the above expression of $\Phi$, solving Eq.
10 leads to the expression of $\chi$ given in Eq. 8. Including the linear term
increases the range of validity in $T/T_{\mathrm{K}}$ of the asymptotic
solution, allowing us to make contact with the numerical calculation available
up to $T/T_{\mathrm{K}}\approx 10^{3}$.

The theoretical $\beta$-function characterizing the renormalization flow of
the magnetic susceptibility coefficient (Curie constant) is defined as
$\beta\equiv\partial\left(k_{\mathrm{B}}T\chi/(g\mu_{\mathrm{B}})^{2}\right)/\partial\log
T$ as a function of $k_{\mathrm{B}}T\chi/(g\mu_{\mathrm{B}})^{2}$. From Eq. 9,
which is linear in $T$, the low $T\chi$ asymptote displayed as a dashed
straight line in Fig. 4c is simply:

$\beta(x\ll 1)\simeq x,$ (12)

with $x\equiv k_{\mathrm{B}}T\chi/(g\mu_{\mathrm{B}})^{2}$. From Eq. 8,
keeping only the first two terms in the parenthesis, the asymptote near
$k_{\mathrm{B}}T\chi/(g\mu_{\mathrm{B}})^{2}\approx 1/4$ displayed as a dashed
line in Fig. 4c reads:

$\beta(1/4-x\ll 1)\simeq(1-4x)^{2}/4.$ (13)

The full prediction shown as a continuous line in Fig. 4c is obtained by
discrete differentiation of the numerical calculation (continuous line in Fig.
4b), averaged by Fourier filtering and completed by the asymptotic analytical
predictions.

The magnetization of a Kondo impurity in the limit of low temperatures with
respect to the Zeeman splitting ($k_{\mathrm{B}}T\ll E_{\mathrm{Z}}$, both
small compared to the high energy cutoff) is predicted to follow a universal
renormalization flow controlled by the parameter
$E_{\mathrm{Z}}/k_{\mathrm{B}}T_{\mathrm{K}}$, which is shown as a continuous
line in Fig. 5a. This prediction can be expressed analytically, with a
different equation depending on whether
$x\equiv\frac{E_{\mathrm{Z}}}{k_{\mathrm{B}}T_{\mathrm{K}}}\frac{\exp{(C_{\mathrm{E}}+3/4})}{2\pi\sqrt{2}}$
is larger or smaller than 1 (see Eqs. 4.29c in Ref. 27, and Ref. 1 or the
comparison with Eq. 7 at $x\ll 1$ for the numerical factor between $x$ and
$E_{\mathrm{Z}}/k_{\mathrm{B}}T_{\mathrm{K}}$):

$\begin{split}S_{\mathrm{thy}}^{E_{\mathrm{Z}}\gg
T}(x>1)=&\frac{1}{2}-\frac{1}{2\pi^{3/2}}\int_{0}^{\infty}dt\frac{\sin{(\pi
t)}}{t}e^{-t\ln{t}+t}x^{-2t}\Gamma(t+\frac{1}{2}),\\\
S_{\mathrm{thy}}^{E_{\mathrm{Z}}\gg
T}(x<1)=&\frac{1}{2\sqrt{\pi}}\sum_{k=0}^{\infty}(-1)^{k}\frac{(k+\frac{1}{2})^{k-1/2}}{k!}e^{-k-1/2}x^{2k+1}.\end{split}$
(14)

in 1,…,0 See pages \x, of SI_ArXiv.pdf