Exotic quantum critical point in a two-site charge Kondo circuit
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Exotic quantum critical point in a two-site charge Kondo circuit
Winston Pouse1,2,3† , Lucas Peeters3,4† , Connie L. Hsueh1,2,3 , Ulf Gennser5 , Antonella Cavanna5 ,
Marc A. Kastner4,6 , Andrew K. Mitchell7,8∗ , and David Goldhaber-Gordon2,4∗
1
Department of Applied Physics, Stanford University, Stanford, CA 94305, USA
2
Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
3
Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA
arXiv:2108.12691v1 [cond-mat.mes-hall] 28 Aug 2021
4
Department of Physics, Stanford University, Stanford, CA 94305, USA
5
Centre de Nanosciences et de Nanotechnologies (C2N), CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91120 Palaiseau, France
6
Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
7
School of Physics, University College Dublin, Belfield, Dublin 4, Ireland
8
Centre for Quantum Engineering, Science, and Technology, University College Dublin, Belfield, Dublin 4, Ireland
† These authors contributed equally to this work.
∗
To whom correspondence should be addressed;
E-mail: andrew.mitchell@ucd.ie, goldhaber-gordon@stanford.edu.Abstract—The physical properties of a material tuned to electrons, and magnetic ordering of the local moments driven
the cusp between two distinct ground states can be quite by a through-lattice RKKY exchange interaction. A similar
exotic, and unlike those in either of the neighboring phases competition can be found in the two-impurity Kondo (2IK)
[1, 2]. The prospect of capturing such behavior in a simple
model is tantalizing; for example, the interplay between heavy model [3, 29]–[34], which consists of two ‘impurity’ quantum
fermion physics and magnetic ordering in certain materials is spins- 21 coupled to each other and also to their own leads. The
often rationalized in terms of the quantum phase transition in Hamiltonian reads
the two-impurity Kondo model [3, 4]. However, this model is
oversimplified for the purpose: its quantum critical point does not H2IK = Helec + JL S~L · ~sL + JR S~R · ~sR + JC S~L · S~R , (1)
reflect the distinctive properties of a magnetic lattice surrounded
by mobile electrons [5, 6]. In this work, we study a tunable where Helec describes conduction electrons in distinct left
nanoelectronic circuit comprising two coupled charge-Kondo and right leads with local spin densities ~sL,R , and S~L,R are
quantum islands, realizing a new model which captures the quantum spin- 12 operators for the local moments. Although
essence of competition between local and collective screening of real f -electron heavy fermion materials contain a lattice
magnetic moments. This may have relevance for materials in of many such local moments immersed in a common
which collective many-body effects drive lattice coherence [7]–[9].
We tune our device to a novel quantum critical point, and show reservoir of mobile electrons, the 2IK considers just two
experimentally that deviations as we tune away from this point local moments, and two distinct electron reservoirs. The
match non-trivial predictions from the model. This work on the quantum phase transition resulting from competition between
crucial role of inter-island interactions is a necessary first step JL,R and JC has been intensively studied theoretically, but
in scaling up such circuits from individual sites to networks or has not been observed experimentally. This is because in
lattices.
real double-quantum-dot systems charge fluctuations destroy
the QCP, replacing it with a smooth crossover [35, 36].
I NTRODUCTION
However, recent work in which charge fluctuations correspond
The rich behavior seen in bulk materials emerges from the to pseudospin flips giving rise to Kondo interactions has been
microscopic quantum interactions between many constituent successful in observing multi-channel Kondo effects [24, 25].
atoms. Even simple individual interactions can yield complex Extending this approach to two-impurity systems may allow
collective behavior. When competing interactions favor accessing a QCP associated with coupling two local moments.
different collective quantum states, one can often tune In addition, these charge-Kondo systems hold the promise
from one quantum state to another by applying pressure, of mirroring the effects of conduction electron-mediated
electromagnetic fields, or chemical doping; in principle this magnetic exchange between local sites in real materials: the
can even happen at the absolute zero of temperature: a source of coherence within the lattice of sites [6]. To model
quantum phase transition [1, 2]. Of course, experimental a circuit involving two occupied charge-Kondo islands, one
systems cannot be studied at zero temperature. Remarkably, has to take into account the competition between local Kondo
the behavior is controlled by the zero-temperature quantum screening for each of the two local moments and a many-body
critical point (QCP) over a widening regime as the temperature collective screening that couples the two local moments. In
is increased, so that signatures of criticality are experimentally this article, we develop such a model and identify a novel
accessible. Further, seemingly very different systems can share QCP with exotic properties. Furthermore, we implement the
the same type of critical point: each system behaves the same circuit and are able to directly probe its QCP via transport
‘universal’ way near such a critical point. measurements, taking a next step towards building multi-site
A fundamental microscopic description of the range and clusters and ultimately lattices.
character of different phases, as well as the transitions The double charge-Kondo (DCK) model reads,
between them, is often hindered by the sheer chemical
complexity of real bulk materials. For this reason, simplified HDCK =Helec + JL ŜL+ ŝ− + −
L + JR ŜR ŝR + H.c.
(2)
models are constructed to capture the essential physics − †
+ JC ŜL+ ŜR cCL cCR + H.c. + Htune .
of interest. One prominent class of such effective models
are so-called ‘quantum impurity models’ that involve a Compared to the 2IK model, Eq. 1, it has two differences.
few interacting quantum degrees of freedom coupled to First, the spin couplings are anisotropic, of the form
non-interacting conduction electrons. These models can Ŝ + Ŝ − instead of S~ · S.
~ This is known not to make a
be realized experimentally in nanoelectronic circuits based fundamental difference [37]. Second, and more interestingly,
on semiconductor quantum dots, offering the opportunity the inter-site spin coupling comes with an additional tunneling
to engineer, manipulate and study quantum effects in a of conduction electrons c†CL cCR . For generality, we included
controlled way [10, 11]. These circuits display a range of a term Htune = I ŜLz ŜR z
+ BL ŜLz + BR ŜRz
corresponding
non-trivial physics such as the Coulomb blockade [12], various to an inter-site Ising interaction I, and local pseudo-Zeeman
Kondo effects [13]–[20], emergent symmetries [21, 22], and fields BL and BR , which give us the control to navigate the
fractionalization [23]–[26]. Quantum phase transitions with phase diagram, and correspond to physical parameters of our
universal properties can also be realized [23]–[25, 27, 28]. experimental system.
A classic paradigm for such physics is the competition The DCK model is realized experimentally in a circuit
between Kondo screening of local moments by conduction consisting of two coupled hybrid metal-semiconductor islands,a
(N+1,M)
C
L
(N,M+1)
(N,M)
R
c ( p )
d 20 mK (NRG) e 2 mK (NRG)
0.2
40 0.12 40 40
0.1
0.1 0.15
20 (N+1,M) 20 0.08 20
(N+1,M+1)
G (e2/h)
UL ( eV)
0.08
0 0 0.06 0 0.1
0.06
0.04
−20 0.04 −20 −20
(N,M) 0.05
(N,M+1) 0.02
0.02
−40 −40 −40
0 0 0
−40 −20 0 20 40 −40 −20 0 20 40 −40 −20 0 20 40
40 40 0.2
40 0.1
0.1
20 0.08 20 0.15
20 0.08
G (e2/h)
UL ( eV)
0.06
0 0.06 0 0 0.1
0.04 0.04
−20 −20 −20 0.05
0.02 0.02
−40 −40 −40
0 0 0
−40 −20 0 20 40 −40 −20 0 20 40 −40 −20 0 20 40
UR (eV) UR (eV) UR (eV)
(eV)LCR(N,M)(N+1,M)(N,M+1)acde
(NRG)UR
mK
(eV)−40−2002040−40−200204000.020.040.060.080.1−40−2002040−40−200204000.020.040.060.080.120
(eV)UL
(eV)UL
(e2/h)−40−2002040−40−200204000.020.040.060.080.1−40−2002040−40−200204000.020.040.060.080.10.12(p)(N,M)(N+1,M)(N,M+1)(N+1,M+1)UR
(eV)G
(NRG)UR
mK
(e2/h)2
−40−2002040−40−200204000.050.10.150.2−40−2002040−40−200204000.050.10.150.2G
(eV)LCR(N,M)(N+1,M)(N,M+1)acde
(NRG)UR
mK
(eV)−40−2002040−40−200204000.020.040.060.080.1−40−2002040−40−200204000.020.040.060.080.120
(eV)UL
(eV)UL
(e2/h)−40−2002040−40−200204000.020.040.060.080.1−40−2002040−40−200204000.020.040.060.080.10.12(p)(N,M)(N+1,M)(N,M+1)(N+1,M+1)UR
(eV)G
(NRG)UR
mK
(e2/h)2
−40−2002040−40−200204000.050.10.150.2−40−2002040−40−200204000.050.10.150.2G
(eV)LCR(N,M)(N+1,M)(N,M+1)acde
(NRG)UR
mK
(eV)−40−2002040−40−200204000.020.040.060.080.1−40−2002040−40−200204000.020.040.060.080.120
(eV)UL
(eV)UL
(e2/h)−40−2002040−40−200204000.020.040.060.080.1−40−2002040−40−200204000.020.040.060.080.10.12(p)(N,M)(N+1,M)(N,M+1)(N+1,M+1)UR
(eV)G
(NRG)UR
mK
(e2/h)2
−40−2002040−40−200204000.050.10.150.2−40−2002040−40−200204000.050.10.150.2G
(eV)LCR(N,M)(N+1,M)(N,M+1)acde
(NRG)UR
mK
(eV)−40−2002040−40−200204000.020.040.060.080.1−40−2002040−40−200204000.020.040.060.080.120
(eV)UL
(eV)UL
(e2/h)−40−2002040−40−200204000.020.040.060.080.1−40−2002040−40−200204000.020.040.060.080.10.12(p)(N,M)(N+1,M)(N,M+1)(N+1,M+1)UR
(eV)G
(NRG)UR
mK
(e2/h)2
−40−2002040−40−200204000.050.10.150.2−40−2002040−40−200204000.050.10.150.2G
Fig. 1: Two island charge-Kondo Device. a, Schematic layout of the device structure, consisting of two metallic islands (yellow-green)
coupled to quantum Hall edges (red lines) in a buried 2DEG (blue) via QPCs (black). Only the top and central QPCs are used throughout this
work. The island levels are controlled via plunger gates (green). b, Three neighbouring charge states are interconverted by direct tunneling of
electrons at each of the three QPCs, characterized by transmissions τL , τR , τC . Distinct Kondo effects arise along each two-state degeneracy
line. At the triple point connecting them, the three different Kondo interactions cannot simultaneously be satisfied, leading to frustration
and a quantum critical point. c, d, Experimentally measured series conductance and NRG calculations at 20 mK for τL = τR ≡ τ = 0.38
(JL = JR ≡ J = 0.35) as the island potentials UL , UR are varied via plunger gate voltages PL and PR . The top row corresponds to
τC = 0.9 (JC = 0.5) and the bottom row τC = 0.7 (JC = 0.3). The bright conductance spots in the top row correspond to the triple points.
In the bottom row, the triple points are closer and somewhat merged. e, NRG calculated stability diagram at 2 mK for the same settings as
in d. While experimentally inaccessible, we see clear peaks at the triple points, with suppressed conductance elsewhere.
each also coupled to its own lead, as illustrated schematically inter-island capacitive interaction). Lithographically patterned
in Fig. 1a. Even though each island is small enough to have metallic top gates form quantum point contacts (QPCs, black
substantial charging energy, because of the metal component in Fig. 1a). The transmissions τL and τR control the left
each effectively has a continuum of single particle states, in and right island-lead tunnel couplings, while τC controls the
contrast to the situation for purely semiconductor quantum coupling between the islands. Each coupling is through the
dots. Our circuit is based on a GaAs/AlGaAs heterostructure outermost quantum Hall edge state; QPC voltages are set so
which hosts a buried two-dimensional electron gas (2DEG). the second, inner channel, is completely reflected. Throughout
Mesas are lithographically patterned (blue regions in Fig. 1a, the experiment we fix τL = τR ≡ τ and keep all other QPCs
outside of which the 2DEG is etched away). Metallic islands closed. Finally, plunger gates (green) control the electrostatic
are deposited bridging the various mesas, then are electrically potential, and hence electronic occupancy, on each island. We
connected to the 2DEG by thermal annealing. The device measure the conductance G from left lead to right lead through
is operated in a magnetic field of 4.3 T, corresponding to both islands in series, as a function of the left and right plunger
a quantum Hall filling factor of 2 in the 2DEG bulk. The gate voltages PL and PR . See Methods for further details of
left and right islands are designed to behave identically: the device and measurement setup.
the spacing of single-particle states on each island is far
below kT at our base temperature of 20 mK; and their Following the charge-Kondo mapping for a single island
L R introduced theoretically by Matveev [38], and validated
charging energies EC ≈ EC ≈ 25 µeV are equal to
within our experimental resolution (V ≈ 10 µeV is the experimentally by Iftikhar [24, 25], the effective model
describing this device is the DCK model, Eq. 2 (see Methods).The mapping associates effective pseudospin degrees of the series conductance from left to right leads through the
freedom to the macroscopic island charge states. The double island structure is suppressed by this effect, since the
raising/lowering operator ŜL± converts a state (N, M ) with N conductive pathway involves virtual polarization of the Kondo
electrons on the left island and M on the right, into the state singlet through the state (N, M + 1), which is an excited
±
(N ± 1, M ); while ŜR yields (N, M ± 1). These processes state when the charge dynamics of the right island are frozen.
naturally arise due to single electron tunneling through the This is supported by NRG calculations at T = 2 mK (Fig. 1e)
various QPCs. In the limit of large island charging energies which show this incipient ‘Kondo blockade’ [41] in the series
L,R ±
EC , the operators ŜL,R become pseudospin- 12 operators. The conductance. A similar effect is seen along the degeneracy line
inter-island capacitive interaction then generates the term I of (N, M )/(N, M + 1), which corresponds to a Kondo effect
Htune in Eq. 2, while the gate voltages PL,R are related to of the right island with the right lead. Along the degeneracy
the pseudo-Zeeman fields BL,R . line (N + 1, M )/(N, M + 1), tunneling with the leads is not
A crucial feature of the charge-Kondo implementation is involved. Instead we may regard (N + 1, M ) and (N, M + 1)
that the pseudospin exchange interactions JL , JR and JC as two components ⇐ and ⇒ of a collective pseudospin state
in the DCK model Eq. 2 are related directly to the QPC of the double island structure, which is flipped by electronic
transmissions of the device τL , τR and τC , and can be large. tunneling at the central QPC. This gives rise to a new kind
By tuning these couplings, one can realize various Kondo of inter-island Kondo effect. The resulting Kondo singlet
effects, and indeed a quantum critical point, at relatively high is disrupted by tunneling at the leads, and hence the low
temperatures. This contrasts with the more familiar coupling temperature conductance is again suppressed.
of spins between two semiconductor quantum dots, where the
effective exchange interactions are perturbatively small, being Triple point
derived from an underlying Anderson model. Furthermore, The triple point (TP), where (N, M )/(N, M + 1)/(N +
relevant perturbations present in the Anderson model destroy 1, M ) are all degenerate, is a special point in the phase
the quantum phase transition of the oversimplified 2IK diagram. Here the three Kondo effects described above are
model, replacing it with a smooth cross-over. The two island all competing, see Fig. 1b. At JL = JR = JC in Eq. 2, the
charge-Kondo system therefore presents a unique opportunity resulting frustration gives rise to a QCP, which will be the
to observe a two-impurity QCP at experimentally relevant main focus of this work.
temperatures. At the TP, the high-temperature series conductance is
enhanced because an electron can tunnel from left lead to
R ESULTS
right lead through the islands without leaving the ground state
Phase diagram and Kondo competition charge configurations. Neglecting interactions and treating the
L,R
The island charging energies EC and inter-island QPCs as three resistors in series, the maximum conductance
capacitive interaction V are finite in the physical device, G = e2 /3h ≡ G∗ occurs when the tunneling rates at each
so multiple island charge states play a role. This gives rise constriction are equal, JL = JR = JC , since then there is no
to a periodic hexagonal structure of the charge stability bottleneck in the flow of electrons through the structure. This
diagram as a function of the left and right plunger gate situation approximately pertains when the QPCs are opened
0
voltages PL,R . We convert these to energies UL,R = UL,R + up, whence G ' e2 /3h is indeed observed in experiment.
αPL,R using the experimentally measured capacitive lever arm Importantly, we also find G = G∗ holds for any JL = JR =
0
α = 50 µeV/mV, relative to an arbitrary reference UL,R . UL,R JC ≡ J at a TP in the full model including interactions, at
are related to the pseudo-Zeeman fields BL,R of Htune in Eq. 2 low temperatures T /TK 1, where TK ∼ EC exp[−1/2νJ]
via B~ = ᾱU~ , which accounts for cross-capacitive gate effects. is the Kondo temperature at the QCP (with ν the electronic
The experimental stability diagram in Fig. 1c allows us density of states at the Fermi energy). In this case, we find
to identify regimes with particular charge states on the two from NRG that the conductance increases with decreasing T ,
islands. In particular, we see distinct charge degeneracy approaching the critical value as G∗ − G(T ) ∼ (T /TK )2/3 .
lines (N, M )/(N, M + 1), (N, M )/(N + 1, M ) and (N + Based on√this and the unusual residual T = 0 entropy
1, M )/(N, M + 1), each of which is associated with single ∆S = ln( 3) seen in NRG calculations, we conclude that the
electron tunneling at one of the three QPCs (see Fig. 1b). This QCP is a non-Fermi liquid with exotic fractional excitations.
structure is reproduced very well by numerical renormalization
group (NRG) [39, 40] calculations of the multiple charge-state Conductance Line Cuts
model by fitting JL,R,C for a given set of experimental We first focus on the behavior of conductance near the TPs.
transmissions τL,R,C , as shown in Fig. 1d for the same Specifically, in Fig. 2a we take cuts along the line between
temperature T = 20 mK (see Methods and Supplementary TPs, UL = UR ≡ U , for different τC at fixed τL = τR ≡ τ .
Info). U = 0 is chosen to be the high-symmetry point between TPs.
Along the degeneracy line (N, M )/(N + 1, M ) the left Experimental data are compared with the corresponding NRG
island charge pseudospin is freely flipped by tunneling at simulations of the device in Fig. 2b.
the QPC between the left island and lead, giving rise to Experiment and theory are seen to match very well, notably
a Kondo effect due to the first term of Eq. 2. However, with regard to the conductance values at all U for most τCa TPs.
U U We also see clear non-monotonicity in the experimental
conductance as a function of τC , reflecting the competition
0.14
between the different Kondo interactions at the TP. Taking the
0.12 τC critical point with completely frustrated interactions to be at
0.98 τC∗ (a monotonic function of τ , and not necessarily τC∗ = τ ),
0.1
0.9 we expect lower conductance for both τC < τC∗ where
G (e2/h)
0.08 0.8 the island-lead Kondo effects dominate, and τC > τC∗ for
0.7 which the inter-island Kondo effect dominates (recall Fig. 1b).
0.06
0.6
Conductance is higher for τC ∼ τC∗ due to incipient critical
0.04 0.5
charge fluctuations.
0.4
0.02 0.3
This latter feature of the experiment is in fact not
0.2 well-captured by NRG calculations at these temperatures, nor
0
−15 −10 −5 0 5 10 15
would it be expected to be. At large τC , many charge states
U (eV) on the islands contribute to transport, but within NRG the
(eV)JCab
(e2/h)U
(eV)τC−15−10−505101500.020.040.060.080.10.120.140.60.50.40.30.250.20.150.10.05UUG
(e2/h)U
−15−10−505101500.020.040.060.080.10.120.140.980.90.80.70.60.50.40.30.2UUG
b number of states included must be restricted (Supplementary
U U Info.), even when using a supercomputer.
Despite some uncertainty in the precise TP location, the
0.14 very existence of a QCP implies an underlying universality, in
0.12
terms of which conductance signatures in its vicinity can be
JC
quantitatively analyzed.
0.6
0.1
0.5 Universal Scaling
G (e2/h)
0.08 0.4
0.3 We now turn to the behavior near the QCP, resulting
0.06 from frustrated island-lead and inter-island Kondo effects. We
0.25
0.04 0.2 focus on parameter regimes with large τ and τC , such that
0.15 the corresponding Kondo temperatures are large. This allows
0.02 0.1 experimental access to the universal regime T /TK . 1. At
0 0.05 the QCP with τC ' τC∗ , our theoretical analysis predicts
−15 −10 −5 0 5 10 15 G ' G∗ = e2 /3h. Moreover, non-trivial behavior is observed
U (eV)
in the vicinity of this singular point, where perturbations drive
Fig. 2: Conductance Line Cuts Between Triple Points. the system away from the QCP and towards a regular Fermi
Experimentally-measured (a) and NRG-calculated (b) line cuts for liquid state. The associated conductance signatures are entirely
τ = 0.38 (J = 0.35 in the model) along the line UL = UR ≡ U for characteristic of the quantum phase transition in this system.
different τC (JC ). Insets show representative 2D PL , PR sweeps from Since the low-T physics near a QCP is universal and
which line cuts are extracted. In a we average over many PL , PR therefore insensitive to microscopic details, we use a minimal
sweeps and utilize the inversion symmetry G(U ) → 21 [G(U ) +
G(−U )] to reduce noise. The model parameters in b are optimized model, with only (N, M )/(N + 1, M )/(N, M + 1) states
to fit the experiment (see Supplementary Info.) retained in Eq. 2. The QCP is destabilized by either detuning
the couplings, corresponding to the perturbation ∆τC = τC −
τC∗ , or moving away from the TP via ∆U = U − UTP (where
(JC ), the width of the peak for τC ≤ 0.7 (JC ≤ 0.4), and the UTP is the putative TP position). Remarkably, we find from
positions of the split peaks at the largest τC (JC ). This level NRG that any combination of ∆τC and ∆U can be captured
of agreement validates the use of the DCK model to describe by a single scale T ∗ characterizing the flow away from the
the physical device. QCP, provided the magnitude of the perturbations is small:
We note that the TP positions in the space of (UL , UR )
T ∗ = a|∆τC |3/2 + b|∆U |3/2 (3)
depend on τC . At large τC , the peaks are rather well separated
and can be easily distinguished, but due to temperature with a, b constants. The resulting universal conductance curve
broadening the peaks are found to merge at low τC . Within as a function of T ∗ /T is shown as the solid line in Fig. 3b.
NRG, this effect is reproduced on decreasing JC . In practice, this can be reconstructed by fixing a small
Although the TP positions at low JC can still be identified T /TK 1 and varying ∆U , say, while keeping ∆τC = 0,
in NRG by going to lower temperatures where the peaks as shown in Fig. 3a (left vertical solid black line, T /TK =
sharpen up (compare Figs. 1d and 1e), even at the experimental 2.5 × 10−4 ). Repeating the procedure at a higher temperature
base electron temperature of 20 mK, thermal broadening (right vertical dashed black line, T /TK = 2.5 × 10−2 ), where
complicates the experimental analysis of the TP behavior. Thus T /TK 1 is not well satisfied, leads to the approximately
care must be taken to estimate the TP positions from the full universal curve shown as the dashed line in Fig. 3b. This
stability diagram and to disentangle the influence of adjacent curve depends both on T ∗ /T as well as T /TK ; the lattera b c U
0.35 0.35
Universal (NRG)
Approx. universal (NRG)
0.3 0.3 0.3
0.25 0.25
0.28
G (e2/h)
G (e2/h)
G (e2/h)
0.2 0.2
0.26
0.15 0.15
0.1 0.1 0.24
0.05 0.05
0.22
0 0
-4
10-8 10 -6
10 10 -2
10 0
10 2
10-2 10 -1
10 0
10 1
10 2
30 20 10 0 10 20 30
T/TK T*/T U (eV)
d e
Approx. universal (NRG) Universal (NRG)
τC : 0.3 20 mK
0.3 0.999
26 mK
0.9985
46 mK
0.998 0.25
75 mK
0.9975
0.25
0.2
G (e2/h)
G (e2/h)
0.2 0.15
0.1
0.15
0.05
0.1 0
0 5 10 15 20 25 80 60 40 20 0 0.5 1 1.5 2 2.5 3 3.5
T* (mK) T* (mK) T*/T
(e2/h)T/TK
00.050.10.150.20.250.30.3510-810-610-410-2100102G
:
(mK)τC
(e2/h)T*
(NRG)0.9990.99850.9980.9975G
universal
(e2/h)10-210-110010110205101520250.10.150.20.250.3Approx.
(NRG)T*/TG
universal
(NRG)Approx.
(e2/h)00.050.10.150.20.250.30.35Universal
(eV)G
(e2/h)0Uabcde30201001020300.220.240.260.280.3U
(mK)T*/TG
mK8060402000.050.10.150.20.250.3T*
mK75
mK46
mK26
(NRG)20
0.511.522.533.5Universal
(e2/h)T/TK
00.050.10.150.20.250.30.3510-810-610-410-2100102G
:
(mK)τC
(e2/h)T*
(NRG)0.9990.99850.9980.9975G
universal
(e2/h)10-210-110010110205101520250.10.150.20.250.3Approx.
(NRG)T*/TG
universal
(NRG)Approx.
(e2/h)00.050.10.150.20.250.30.35Universal
(eV)G
(e2/h)0Uabcde30201001020300.220.240.260.280.3U
(mK)T*/TG
mK8060402000.050.10.150.20.250.3T*
mK75
mK46
mK26
(NRG)20
0.511.522.533.5Universal
(e2/h)T/TK
00.050.10.150.20.250.30.3510-810-610-410-2100102G
:
(mK)τC
(e2/h)T*
(NRG)0.9990.99850.9980.9975G
universal
(e2/h)10-210-110010110205101520250.10.150.20.250.3Approx.
(NRG)T*/TG
universal
(NRG)Approx.
(e2/h)00.050.10.150.20.250.30.35Universal
(eV)G
(e2/h)0Uabcde30201001020300.220.240.260.280.3U
(mK)T*/TG
mK8060402000.050.10.150.20.250.3T*
mK75
mK46
mK26
(NRG)20
0.511.522.533.5Universal
(e2/h)T/TK
00.050.10.150.20.250.30.3510-810-610-410-2100102G
:
(mK)τC
(e2/h)T*
(NRG)0.9990.99850.9980.9975G
universal
(e2/h)10-210-110010110205101520250.10.150.20.250.3Approx.
(NRG)T*/TG
universal
(NRG)Approx.
(e2/h)00.050.10.150.20.250.30.35Universal
(eV)G
(e2/h)0Uabcde30201001020300.220.240.260.280.3U
(mK)T*/TG
mK8060402000.050.10.150.20.250.3T*
mK75
mK46
mK26
(NRG)20
0.511.522.533.5Universal
Fig. 3: Universal conductance scaling near triple point. a. Theoretically calculated conductance as a function of T /TK for different
T ∗ (increasing from the left, blue colored curve, to the right, yellow curve) using a minimal model valid near the TP. The intersection
with the solid (at T /TK = 2.5 × 10−4 ) and dashed (at T /TK = 2.5 × 10−2 ) lines represent ways to reconstruct the universal and
approximately universal curves of b. b. Theoretical universal curve (solid line) as a function of T ∗ /T in the limit T /TK 1, compared
with an approximately universal curve obtained for T /TK . 1 (dashed line). c. Measured line cut between triple points similar to that in
Fig. 2, except here τ = 0.95 and τC ≈ 0.9985. T ∗ is a function of gate-induced detuning ∆U away from a TP (Eq. 3), which we define
relative to UTP estimated at the peak. d. We plot the truncated line cut of c as well as similar line cuts (symbols) for different τC as a
function of T ∗ . These fall on top of each other and the approximately universal curve of b. The uptick in conductance at the tail of each
cut results from influence of a neighboring TP not included in the minimal model. e. Measured truncated line cuts (symbols) at different
temperatures are plotted in the left panel. From lowest to highest temperatures, τ = {0.78, 0.78, 0.81, 0.82} and τC = 0.9 (Methods). Due
to a detuning ∆τC , Eq. 3 implies T ∗ > 0 even at ∆U = 0. This shift is estimated in fitting the tails to the universal curve in the right side
when plotted as a function of T ∗ /T , whence we observe universal scaling collapse.
deduced from experimental data from the saturation value of different temperatures. In the left panel of Fig. 3e, we plot
the conductance. line cuts at different temperatures for slightly lower fixed
We use these NRG results to interpret the experimental data. values of τ, τC than in Fig. 3d, as a function of U . At each
To do this we must identify the TP position, and hence ∆U temperature, τ, τC are adjusted such that the conductance at
from the line cut data. This is relatively straightforward when ∆U = 0 is roughly the same. We do not expect to satisfy
the TPs are well separated, as happens at large τC , and we τC = τC∗ for each temperature. Remarkably, the additive form
simply take UTP as the peak position, Fig. 3c. of the contributions to T ∗ in Eq. 3 means that it is unnecessary
Since the limit T /TK 1 is not perfectly satisfied in to be exactly at the critical value of τC : the detuning in τC
experiment, we use the approximately universal curve to simply generates a finite T ∗ even when ∆U = 0, which we
demonstrate the scaling collapse of data for different τC using can account for by a simple shift when plotting the data in
Eq. 3 in Fig. 3d (Methods). The small T ∗ behavior falls onto terms of T ∗ (Methods).
the theory curve for all τC considered, revealing the non-trivial The right panel of Fig. 3e shows the same data scaled
3/2 power law scaling in the data. Deviations from the line now as T ∗ /T , and compared with the fully universal NRG
are due to influence of the neighboring TP. curve from Fig. 3b. We cannot make the same comparison
Finally, we demonstrate that T ∗ /T is indeed the universal to an approximately universal curve as T /TK would change
scaling parameter by measuring and rescaling line cuts at for each temperature. Instead, we understand the origin of thedeviations at low T ∗ /T and expect agreement at larger T ∗ /T . R EFERENCES
Indeed, the collapse and strong quantitative agreement with [1] Sachdev, S. Quantum Phase Transitions (Cambridge University Press,
the non-trivial universal conductance curve is both striking 2011), 2 edn.
and a direct signature of the novel critical point. Significantly, [2] Paschen, S. & Si, Q. Quantum phases driven by strong correlations. Nat
Rev Phys 3, 9–26 (2021).
the collapse is over the entire range of T ∗ /T for each line [3] Jones, B. A., Varma, C. M. & Wilkins, J. W. Low-Temperature
cut, with limitations at small T ∗ /T due to the finite T /TK Properties of the Two-Impurity Kondo Hamiltonian. Phys. Rev. Lett.
mentioned previously and at large T ∗ /T due to an increasing 61, 125–128 (1988).
[4] Craig, N. J. et al. Tunable Nonlocal Spin Control in a Coupled-Quantum
overlap with other TPs. The observed scaling collapse for Dot System. Science 304, 565–567 (2004).
different temperatures is a result of the universal form of the [5] Mitchell, A. K., Derry, P. G. & Logan, D. E. Multiple magnetic
conductance as a function of T ∗ /T as well as the 3/2 power impurities on surfaces: Scattering and quasiparticle interference. Phys.
Rev. B 91, 235127 (2015).
law scaling of T ∗ with ∆U that arises near the QCP. [6] Burdin, S., Georges, A. & Grempel, D. R. Coherence Scale of the
Kondo Lattice. Phys. Rev. Lett. 85, 1048–1051 (2000).
[7] Yang, Y.-f., Fisk, Z., Lee, H.-O., Thompson, J. D. & Pines, D. Scaling
D ISCUSSION the Kondo lattice. Nature 454, 611–3 (2008).
[8] Si, Q. & Steglich, F. Heavy fermions and quantum phase transitions.
Science 329, 1161–1166 (2010).
In this work, we have presented strong evidence for a [9] Coleman, P. Heavy Fermions and the Kondo Lattice: A 21st Century
completely novel quantum phase transition in a two-site Perspective. In Pavarini, E., Koch, E. & Coleman, P. (eds.) Many-Body
circuit. By exploiting the charge-Kondo paradigm, our device Physics: From Kondo to Hubbard, vol. 5, chap. 1 (Forschungszentrum
Jülich, 2015).
maps to a variant of the celebrated two-impurity Kondo model, [10] Hensgens, T. et al. Quantum simulation of a Fermi-Hubbard model
here featuring a new phase in which the local moments using a semiconductor quantum dot array. Nature 548, 70–73 (2017).
on the two islands are screened collectively by many-body [11] Dehollain, J. P. et al. Nagaoka ferromagnetism observed in a quantum
dot plaquette. Nature 579, 528–533 (2020).
effects driven by conduction electron scattering. This may have [12] Meirav, U., Kastner, M. A. & Wind, S. J. Single-electron charging and
relevance for the development of lattice coherence in Kondo periodic conductance resonances in GaAs nanostructures. Phys. Rev.
lattice systems. Lett. 65, 771–774 (1990).
[13] Goldhaber-Gordon, D. et al. Kondo effect in a single-electron transistor.
We formulate a new model to describe the two island Nature 391, 156–159 (1998).
charge-Kondo device, and demonstrate quantitative agreement [14] Cronenwett, S. M., Oosterkamp, T. H. & Kouwenhoven, L. A Tunable
between NRG calculations and experimentally measured Kondo Effect in Quantum Dots. Science 281, 540–544 (1998).
[15] Sasaki, S. et al. Kondo effect in an integer-spin quantum dot. Nature
conductance, including in the universal regime of the exotic 405, 764–767 (2000).
quantum critical point. [16] Jeong, H., Chang, A. M. & Melloch, M. R. The Kondo effect in an
Our work on the crucial role of the inter-island interaction artificial quantum dot molecule. Science 293, 2221–2223 (2001).
[17] Oreg, Y. & Goldhaber-Gordon, D. Two-Channel Kondo Effect in a
paves the way for a host of other studies. Opening each of Modified Single Electron Transistor. Phys. Rev. Lett. 90, 136602 (2003).
the islands to a second lead (already present but not used in [18] Potok, R. M., Rau, I. G., Shtrikman, H., Oreg, Y. & Goldhaber-Gordon,
the existing device) would produce two sites each hosting a D. Observation of the two-channel Kondo effect. Nature 446, 167–171
(2007).
two-channel Kondo (2CK) state, and would allow studying the [19] Buizert, C., Oiwa, A., Shibata, K., Hirakawa, K. & Tarucha, S. Kondo
effect of coupling the two. Conceivably, the local Majorana Universal Scaling for a Quantum Dot Coupled to Superconducting
zero mode associated with a 2CK state at low temperatures Leads. Phys. Rev. Lett. 99, 136806 (2007).
[20] Takada, S. et al. Transmission Phase in the Kondo Regime Revealed in
on one island [25, 42, 43] could be shifted to the other island a Two-Path Interferometer. Phys. Rev. Lett. 113, 126601 (2014).
by gate voltage tuning (presumably through an as yet unknown [21] Keller, A. J. et al. Emergent SU(4) Kondo physics in a
intermediate collective state). spin–charge-entangled double quantum dot. Nature Phys 10, 145–150
(2013).
This work can be extended to more complex clusters [22] Mitchell, A. K., Liberman, A., Sela, E. & Affleck, I. SO(5) Non-Fermi
of coupled charge-Kondo islands, and ultimately lattices. Liquid in a Coulomb Box Device. Phys. Rev. Lett. 126, 147702 (2021).
[23] Keller, A. J. et al. Universal Fermi liquid crossover and quantum
This provides a way of examining with unprecedented criticality in a mesoscopic system. Nature 526, 237–40 (2015).
control some of the most subtle collective dynamics of [24] Iftikhar, Z. et al. Two-channel Kondo effect and renormalization flow
real correlated materials, and introducing more complicated with macroscopic quantum charge states. Nature 526, 233–236 (2015).
effective interactions. Such charge-Kondo clusters would act as [25] Iftikhar, Z. et al. Tunable quantum criticality and super-ballistic transport
in a “charge” Kondo circuit. Science 360, 1315–1320 (2018).
analog quantum simulators with capabilities beyond classical [26] Landau, L. A., Cornfeld, E. & Sela, E. Charge Fractionalization in the
computation: three coupled islands is already out of reach for Two-Channel Kondo Effect. Phys. Rev. Lett. 120, 186801 (2018).
NRG, while stochastic algorithms such as Quantum Monte [27] Mitchell, A. K., Landau, L. A., Fritz, L. & Sela, E. Universality and
Scaling in a Charge Two-Channel Kondo Device. Phys. Rev. Lett. 116,
Carlo (QMC) [44] may not be able to access the universal 157202 (2016).
low-temperature dynamics of these systems. Indeed, tunable [28] Zhang, G. et al. Nonequilibrium quantum critical steady state: Transport
analog quantum simulators of this type may eventually form through a dissipative resonant level. Phys. Rev. Research 3, 013136
(2021).
the basis for calculations requiring solutions of complex [29] Jayaprakash, C., Krishna-murthy, H. R. & Wilkins, J. W. Two-Impurity
cluster models that are difficult to obtain using NRG or even Kondo Problem. Phys. Rev. Lett. 47, 737–740 (1981).
QMC, as arise for example in extensions of dynamical mean [30] Georges, A. & Sengupta, A. M. Solution of the Two-Impurity,
Two-Channel Kondo Model. Phys. Rev. Lett. 74, 2808–2811 (1995).
field theory (DMFT) [45, 46] for correlated materials such as [31] Affleck, I. & Ludwig, A. W. W. Exact critical theory of the two-impurity
the high-temperature superconductors. Kondo model. Phys. Rev. Lett. 68, 1046–1049 (1992).[32] Simon, P., López, R. & Oreg, Y. Ruderman-Kittel-Kasuya-Yosida and and basic measurements was supported by the National
Magnetic-Field Interactions in Coupled Kondo Quantum Dots. Phys. Science Foundation (NSF) under award no. 1608962. W.P.
Rev. Lett. 94, 086602 (2005).
[33] Mitchell, A. K. & Sela, E. Universal low-temperature crossover in acknowledges support from the Fletcher Jones Fellowship.
two-channel Kondo models. Phys. Rev. B 85, 235127 (2012). C.L.H. acknowledges support from the Gabilan Fellowship.
[34] Mitchell, A. K., Sela, E. & Logan, D. E. Two-Channel Kondo Physics L.P. acknowledges support of the Albion Walter Hewlett
in Two-Impurity Kondo Models. Phys. Rev. Lett. 108, 086405 (2012).
[35] Zaránd, G., Chung, C.-H., Simon, P. & Vojta, M. Quantum Criticality Fellowship.
in a Double-Quantum-Dot System. Phys. Rev. Lett. 97, 166802 (2006).
[36] Jayatilaka, F. W., Galpin, M. R. & Logan, D. E. Two-channel Kondo AUTHOR C ONTRIBUTIONS
physics in tunnel-coupled double quantum dots. Phys. Rev. B 84, 115111 W.P. and L.P. performed the measurements. L.P. fabricated
(2011).
[37] Matveev, K. A. Quantum fluctuations of the charge of a metal particle the device. A.K.M. developed the theory and did NRG
under the Coulomb blockade conditions. Sov. Phys. JETP 72, 892–899 calculations. W.P., L.P., C.L.H., M.A.K., A.K.M., and D.G.-G.
(1991). analyzed the data. A.C. and U.G. grew the heterostructure that
[38] Matveev, K. A. Coulomb blockade at almost perfect transmission. Phys.
Rev. B 51, 1743–1751 (1995). hosts the 2DEG on which these samples are built. D.G.-G.
[39] Bulla, R., Costi, T. A. & Pruschke, T. Numerical renormalization group supervised the project.
method for quantum impurity systems. Rev. Mod. Phys. 80, 395–450
(2008). C OMPETING F INANCIAL I NTERESTS
[40] Mitchell, A. K., Galpin, M. R., Wilson-Fletcher, S., Logan, D. E. &
Bulla, R. Generalized Wilson chain for solving multichannel quantum The authors declare no competing financial interests.
impurity problems. Phys. Rev. B 89, 121105 (2014).
[41] Mitchell, A. K., Pedersen, K. G., Hedegård, P. & Paaske, J. Kondo
blockade due to quantum interference in single-molecule junctions. Nat
Commun. 8, 1–10 (2017).
[42] Emery, V. J. & Kivelson, S. Mapping of the two-channel Kondo problem
to a resonant-level model. Phys. Rev. B 46, 10812–10817 (1992).
[43] Coleman, P., Ioffe, L. B. & Tsvelik, A. M. Simple formulation of the
two-channel Kondo model. Phys. Rev. B 52, 6611–6627 (1995).
[44] Gull, E. et al. Continuous-time Monte Carlo methods for quantum
impurity models. Rev. Mod. Phys. 83, 349–404 (2011).
[45] Maier, T. The Dynamical Cluster Approximation and its DCA+
Extension . In Pavarini, E., Koch, E. & Coleman, P. (eds.) Many-Body
Physics: From Kondo to Hubbard, vol. 5, chap. 14 (Forschungszentrum
Jülich, 2015).
[46] Sénéchal, D. Quantum Cluster Methods: CPT and CDMFT. In Pavarini,
E., Koch, E. & Coleman, P. (eds.) Many-Body Physics: From Kondo to
Hubbard, vol. 5, chap. 13 (Forschungszentrum Jülich, 2015).
DATA AVAILABILITY
All data used in this work are available in the Stanford Digital
Repository at https://doi.org/10.25740/mx151nn9365.
ACKNOWLEDGEMENTS
We thank F. Pierre, I. Safi, G. Zarand, C.P. Moca, I.
Weymann, P. Sriram, E. Sela, Y. Oreg, Q. Si, and C.
Varma for their scientific insights and suggestions. To make
this project work, before coupling two islands we had to
start by reproducing F. Pierre’s tour de force experiments
on single islands of the same type. F. Pierre helped
with comments on our fabrication process, measurement
procedure, and analysis. We acknowledge G. Zarand, C.P.
Moca, and I. Weymann for early discussions of the
Hamiltonian and its implications. Measurement and analysis
were supported by the U.S. Department of Energy (DOE),
Office of Science, Basic Energy Sciences (BES), under
Contract DE-AC02-76SF00515. Growth and characterization
of heterostructures was supported by the French Renatech
network. Theory and computation (A.K.M.) were supported
by the Irish Research Council Laureate Awards 2017/2018
through Grant No. IRCLA/2017/169. Part of this work was
performed at the Stanford Nano Shared Facilities (SNSF),
supported by the National Science Foundation under award
ECCS-2026822. Early research that established how to meet
the demanding technical conditions for sample fabricationM ETHODS This full model is solved numerically-exactly using
Theoretical Modeling of the Device interleaved-NRG (iNRG) [40, 47] to obtain the conductance
plots in Figs. 1d, 1e and 2b. Further details are given in the
Left and right islands are coupled to their respective leads Supplementary Info.
and to each other by QPCs. We define independent electronic
systems around each QPC corresponding to the blue regions Relation to two-impurity Kondo model
in Fig. 1a, L,R
In the limit of large charging energies EC , we may retain
αγk c†αγk cαγk ,
X
Helec = (4) just two charge states around the charge degeneracy point of
±
α,γ,k each island. In this case ŜL,R denote spin- 12 raising/lowering
where α = L, C, R labels the left, center, or right QPC operators, and the model reduces to Eq. 2. This effective model
respectively; γ = 1, 2 labels the electron localization on either is a variant of the celebrated 2IK model Eq. 1, in which
side of the QPC; and αγk is the dispersion of an electron two spin- 12 impurities are each coupled to their own lead and
with momentum k, with corresponding fermionic creation exchange-coupled together. The quantum phase transition of
(annihilation) operators c†αγk (cαγk ). Tunneling at each QPC the standard 2IK model is characterized by the competition
is described by, between separate Kondo effects for each impurity and its
X † connected lead, and an inter-impurity RKKY spin singlet state.
HQPC = Jα cα1 cα2 + H.c. , (5) Here, we have a spin-anisotropic version of the 2IK model, but
α with the dynamics of an additional conduction electron bath in
the central region correlated to the inter-impurity interaction
P
where cαγ = k aαγk cαγk are local orbitals at the QPC.
Finally, electronic interactions are embodied by, – a kind of two-impurity, three channel model. Separate
island-lead Kondo effects now compete with an inter-island
L 2 R
Hint = EC N̂ + EC M̂ 2 + V N̂ M̂ , (6) Kondo effect, giving rise to an entirely new quantum phase
L R transition.
where EC and EC are the local charging energies of the
This reduced model is solved with iNRG to obtain results
left and right islands, while V is the inter-island capacitive
in the universal regime of the TP presented in Figs. 3a and
interaction. Here N̂ = k (c†L2k cL2k + c†C1k cC1k ) and M̂ =
P
P † † 3b.
k (cR2k cR2k + cC2k cC2k ) are operators for the total number
of electrons on the left or right islands. The occupancy of Device
the islands can be tuned by applying plunger gate voltages
0 The device was fabricated on a GaAs/AlGaAs
PL,R , which we convert to energies UL,R = UL,R + αPL,R
heterostructure with a 2DEG approximately 95nm
using the experimentally measurable capacitive lever arm α =
0 deep, density of 2.6 × 1011 cm−2 and mobility
50 µeV/mV, relative to the reference UL,R . This corresponds
2.0 × 106 cm2 V−1 s−1 . An SEM micrograph of an equivalent
to a term in the Hamiltonian Ĥgate = BL N̂ + BR M̂ where device is shown in Extended Data Fig. 1. High quality of
B~ = ᾱU~ accounts for cross-capacitive gate effects through the
the small ohmic contacts is crucial, so we take special steps
dimensionless 2x2 matrix ᾱ. The total Hamiltonian describing to ensure cleanliness. Before any fabrication is done on the
the device is Ĥ = Ĥelec + ĤQPC + Ĥint + Ĥgate . heterostructure, we dip in HCl 3.7% to remove any oxide
Following a similar analysis by Matveev [38] for the layer that has built up. After writing the ohmic layer pattern
single-island case, we transform the model into a generalized using e-beam lithography and developing the PMMA resist,
quantum impurity model involving macroscopic charge we use a light oxygen plasma etch to remove residual PMMA
pseudospins on the two islands. P We define island charge scum. Next, before evaporating the ohmic stack, we use the
+ +
pseudospin operators ŜL = N |N + 1ihN | and ŜR = following chemical treatment procedure: dip in TMAH 2.5%
P − + †
M |M + 1ihM |, with ŜL,R = (ŜL,R ) . We label lead for 20 seconds, 5 seconds in water, 5 seconds in HCl 37%,
electrons αγ = L1 and R1 as ‘up’ spin and the island 5 seconds in water (separate from the first cup of water).
electrons L2 and R2 as ‘down’ spin, such that c†L2 cL1 ≡ ŝ− L Afterwards, we quickly move the chip into a KJL Lab 18
and c†R2 cR1 ≡ ŝ− + − †
R , with ŝL,R = (ŝL,R ) . We also relabel e-beam evaporator, and pump down to vacuum. Reducing the
C1 → CL and C2 → CR . Then Eq. 5 becomes, time in air is important to prevent substantial oxide layer
growth. Finally, we run an in-situ Argon etch for 20 seconds.
HQPC = JL ŜL+ ŝ− + −
L + JR ŜR ŝR + H.c.
(7) Only after this do we evaporate the ohmic stack (107.2 nm
− † Au, 52.8 nm Ge, and 40 nm Ni, in order of deposition).
+ JC ŜL+ ŜR cCL cCR + H.c. .
The first line of Eq. 7 corresponds to the transverse Experimental Setup
components of exchange interactions between each island’s The device was cooled down with a +300 mV bias on all
charge pseudospin and an effective spinful channel of the gates to reduce charge instability by limiting the range
conduction electrons. The second line is the novel ingredient of voltages we need to apply [48]. To reduce thermoelectric
in this system: a correlated tunneling at the central QPC, noise causing unwanted voltage biasing across the device, each
involving a mutual pseudospin flip on the islands. lead has a central ohmic contact (between the source andmeasurement contacts) which are all shorted to each other on recaptured by applying a source drain bias large compared to
chip. The shorted ohmics are connected to a single line and a relevant charging energy (Extended Data Fig. 2a).
grounded at room temperature. Measurements are made at low Alternatively, a measurement pathway which does not
frequencies (< 100 Hz) using an SR830 lock-in amplifier. The go through the metallic island (for example, measuring τR
14 mV output of the SR830 is current biased using a 100MΩ through the plunger gate PR ) effectively shorts the circuit to
resistor, and the current is then converted to a voltage on chip ground, so the ‘intrinsic’ transmission is recovered even at zero
by the quantum Hall resistance (h/2e2 ). A measurement of source drain bias [52, 61]. The measured QPC transmission
either the series transmitted voltage or the reflected voltage is should then be the same as the measurement through the island
amplified by an NF SA-240F5 voltage amplifier. This voltage at high bias. Empirically, this is not the case in our device.
simply converts to the series conductance. For most reported Repeating the measurement through PR at high bias shows
measurements, we source at S2 and measure at M2 (see exact agreement with the high bias measurement through the
Fig. 1a). The series conductance is then related to the reflected island. Bias dependent measurements through PR are entirely
voltage, V2 by Eq. 8. consistent with DCB suppression, contrary to expectation
τ =0 (Extended Data Fig. 2c).
e2 V2 − V2 R,L,C We suspect that this residual DCB effect is due to
G= (8)
3h V2 R,L,C =1 − V2τR,L,C =0
τ
impedances in our measurement setup, external to the device.
Measurements in this way eliminate the need for precise We connect our measurement lines which ground the device
knowledge of many settings in a given setup – excitation through highly resistive coaxial lines and discrete filters
amplitude, amplifier gain, line resistances, etc. For arbitrary located right above the connection to the sample through large
sourcing and measurement configurations, the relations ohmic contacts. This is in contrast to earlier work ([24, 25]) on
between measured voltages and the desired conductances this type of system in which a cold ground is used, effectively
can be calculated straightforwardly through Landauer–Büttiker creating a very low impedance path to ground.
formalism. The important question is which measurement transmission
is relevant for the Kondo interactions. While previous
Electron Temperature work ([24, 25]) found the difference between the ’intrinsic’
The electron temperature is determined from dynamical transmission and the zero bias transmission through an open
Coulomb blockade measurements outlined in Iftikhar [49]. The plunger gate to be small, and thus both equally valid, we find
zero bias suppression of the conductance across a QPC when that the measurement through PR at zero bias empirically
series coupled to another QPC of low resistance can be fit works best in our device. Without any residual DCB, we
to a known theoretical from which directly depends on the believe this should be the exact same as the ‘intrinsic’
electron temperature [50]. In our device we measure through transmission. However, the residual DCB appears to not only
two QPCs across a single island (τC = 0), with one QPC suppress the measured QPC transmissions, but all measured
partially transmitting and the other set to fully transmit a single conductances of our system in any configuration, at zero
edge mode. bias. In particular, when measuring through either one or
both islands, the conductance when using the ‘intrinsic’
Calibrating QPC Transmissions transmission at charge degeneracy points, where Kondo
A standard procedure to measure the transmission of each interactions are most important, is consistently lower than
QPC is to measure the series conductance while varying the both expectations and previous results seen in [24]. With the
applied gate voltage of the QPC, with all other QPCs set single island, we see a suppression consistent with DCB – the
to fully transmit n edge states, acting as an h/ne2 resistor. conductances away from 0 and .5 are suppressed the most,
For our experiments of central interest, we must then adjust and the suppression is reduced at higher temperatures, where
each QPC to a desired transmission. We cannot naively DCB is weaker.
use the gate voltages that produced that transmission with If instead we use not the ‘intrinsic’ transmission,
all other QPCs open, because changing the voltage applied but the zero bias transmissions through PR , we see
to any gate capacitively affects all other QPCs. However, remarkably consistent agreement with previous results
we can calibrate this capacitive effect by measuring how on two-channel Kondo for different transmissions
much each QPC’s transmission curve shifts as we vary each and temperatures. This is shown and described in the
other QPC’s gate voltage by a known amount. This is done Supplementary Info. Our interpretation is that while both
for each QPC we use in the experiment, and with this Kondo and DCB-renormalizations occur, we can work
information we can systematically determine the appropriate in a ‘DCB-renormalized space’ by explicitly setting the
gate voltages to set. However, even this procedure fails in DCB-renormalized transmissions. These DCB-renormalized
our device. When transmission is measured in series through transmissions then act as the transmissions that are important
a QPC and an additional resistance on order h/e2 , dynamical for the Kondo interactions. Importantly the transmissions
Coulomb blockade (DCB) suppresses the conductance relative must be set in the same space that DCB is renormalizing
to that expected from ohmic addition of the ‘intrinsic’ the measurements of interest. This means using the zero bias
transmissions [50]–[60]. The ‘intrinsic’ transmissions can be measurement through PR , where DCB is caused by only theexternal impedances, and not through the island, where there the 75 mK data plotted in Fig. 3e, where the conductance
is additional suppression due to the resistance of the second falloff with T ∗ /T is everywhere slower than in the universal
QPC in series. conductance curve.
On the left island, the QPC we use does not have an In Extended Data Fig. 3a, we show the τC values used in
adjacent pathway through PL , but due to an equivalence of the scaling collapse data of Fig. 3d in the main text. In order
the island-lead QPCs, a consistent mapping can be made to resolve the TP peaks we need a sufficiently large τC , of
from the ‘intrinsic’ transmission to the DCB-renormalized which the value needed grows with τ . To reach a conductance
one that would be measured through the plunger gate. close to e2 /3h and have split triple points, τC must be made
However, we are unable to make this mapping for extremely close to 1. While this may indicate the measured
the inter-island QPC as we observe differences in the line cuts are essentially identical, we see that in the full shape
DCB-renormalization when measured through the islands (Extended Data Fig. 3b) there are large changes to the line
(Extended Data Fig. 2b). Likewise, we are unable to measure cuts. We can compare this to Fig. 2a of the main text where
the extraneous DCB-renormalization of the inter-island QPC large changes in τC are needed for comparable changes in the
due to the device geometry. In our results in the main text, we line cuts.
therefore report ‘intrinsic’ values for τC . This may be why the In Fig. 3e, the set transmissions at each temperature are not
maximum conductance does not appear at τ = τC , and in any the same in order to roughly match the conductance at ∆U =
case it means that the τC relevant for Kondo physics grows 0. If the transmissions are held constant, the conductance
with increasing temperature. would decrease with increasing temperature due to a larger
Regardless of whether the transmission settings in the T /TK . By increasing τ, τC with temperature, T /TK is roughly
main text are reported as the ‘intrinsic’ values or the constant. In our measurement, τ = {0.78, 0.78, 0.81, 0.82}
DCB-renormalized values, the conclusions of the main text at T = {20, 26, 46, 75} mK while τC = 0.9 for all T .
hold. None of our results rely on precise quantitative As described in the Calibrating QPC Transmissions section
knowledge of the transmissions settings, since T ∗ depends of Methods, we know that this corresponds to a slightly
only on detuning of one transmission relative to another. This increasing τC with temperature, although the exact value is
would not be the case in any future work exploring universal unknown. That τC varies between temperatures does not affect
scaling as a function of T /TK , since TK depends directly on the conclusion, since the scaling as a function of ∆U is
the quantitative transmissions. unaffected.
Finally, as stated in the main text, the TPs are not easily
Fitting universal curves identifiable, depending on τ and τC . While in Fig. 3 we
To convert |∆U |3/2 to T ∗ requires the unknown scaling choose line cuts in which the TPs are relatively well separated,
prefactor b of Eq. 3. In Fig. 3d, we do this by least-squares there is inevitably still some error in using the conductance
fitting |∆U |3/2 of the experimental data to T ∗ of the peak location as the TP location. Secondly, for the higher
approximately universal curve, with b as a free parameter. Each temperature data, it is not possible to have extremely well
τC line cut is independently fit, and we average the resulting separated peaks due to the increased broadening. Though an
b values and apply the same rescaling to each curve. We find error in identifying the TP location creates a non-linear error
b = 0.858 mK/µeV3/2 works best. A similar procedure is done when scaling ∆U to T ∗ , we find that small changes in where
in Fig. 3e, except we fit only the 20 mK data to the fully we set ∆U = 0 do not significantly change our results. We
universal curve, obtaining b = 0.769 mK/µeV3/2 . The elevated see this by making small changes in what we use as the TP
temperature data are then rescaled with that same obtained b. location, and observing the resulting collapse.
For every curve, we also fit a constant shift a|∆τC |3/2 to take
R EFERENCES
into account that there is a finite T ∗ even at ∆U = 0 due to
a detuning from the critical couplings. [47] Stadler, K. M., Mitchell, A. K., von Delft, J. & Weichselbaum, A.
Interleaved numerical renormalization group as an efficient multiband
Because of the influence of neighboring TPs, we exclude impurity solver. Phys. Rev. B 93, 235101 (2016).
the data at large T ∗ in our fitting procedure, where there [48] Pioro-Ladrière, M. et al. Origin of switching noise in
is a significant change in the slope. For the temperature GaAs/Alx Ga1−x As lateral gated devices. Phys. Rev. B 72,
115331 (2005).
data specifically, the first couple points at low T ∗ are also [49] Iftikhar, Z. et al. Primary thermometry triad at 6 mK in mesoscopic
excluded, due to the clear deviation from universality which circuits. Nat Commun 7, 12908 (2016).
we understand as a consequence of a finite T /TK . While this [50] Joyez, P. & Esteve, D. Single-electron tunneling at high temperature.
Phys. Rev. B 56, 1848–1853 (1997).
is captured by an approximately universal curve as in Fig. 3d, [51] Souquet, J.-R., Safi, I. & Simon, P. Dynamical Coulomb blockade in an
the comparison fails when exploiting the universal parameter interacting one-dimensional system coupled to an arbitrary environment.
T ∗ /T . At each temperature T /TK also changes, and thus Phys. Rev. B 88, 205419 (2013).
[52] Altimiras, C., Gennser, U., Cavanna, A., Mailly, D. & Pierre, F.
a different approximately universal curve would be needed. Experimental Test of the Dynamical Coulomb Blockade Theory for
A clear limit to how high of temperatures we can observe Short Coherent Conductors. Phys. Rev. Lett. 99, 256805 (2007).
collapse is when the TPs are extremely broadened such that [53] Senkpiel, J. et al. Dynamical Coulomb Blockade as a Local Probe for
Quantum Transport. Phys. Rev. Lett. 124, 156803 (2020).
the conductance is no longer reflecting exclusively the line [54] Jezouin, S. et al. Tomonaga-Luttinger physics in electronic quantum
shape of the TP. In fact, we already see signatures of this with circuits. Nat Commun 4, 1802 (2013).You can also read
