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arXiv:2303.12039 · Nature Communications

Observing the universal screening of a Kondo impurity

C. Piquard, P. Glidic, C. Han, A. Aassime, A. Cavanna, U. Gennser, Yigal Meir, Eran Sela, A. Anthore, F. Pierre

Published Nov 9, 2023• 24 citations
arXivHTMLPDF

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.

DOI
10.1038/s41467-023-42857-4

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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.
Measuring entropy in highly correlated systems extends beyond simply counting accessible microstates in the ground state, offering key insights where traditional observables, such as conductance, may be insufficient. However, traditional entropy measurements, which often rely on bulk properties, lack the sensitivity required for systems with only a few particles.In this study, we investigate the entropy evolution in a non-Fermi liquid by examining aKondo impurity. This impurity mediates strong electron-electron interactions, and reflects theexotic properties of heavy fermions and the non-Fermi liquids behavior that may arise inquantum phase transitions.Specifically, by mapping the Kondo impurity onto the charge degrees of freedom of a metallic island, its state can be measured non-invasively through a capacitively coupledcharge sensor. By linking charge and entropy through a Maxwell relation, we observe a reduction in the impurity's residual entropy. This reduction aligns with Kondo universal renormalization flows, illustrating a transition from a free spin-1/2 state with entropy kᵦ ln(2) to a screened singlet state with zero entropy. Additionally, coupling the impurity to multiple leads, we identify a quantum critical regime where the fractional impurity entropy suggests the emergence of anyons.
The Kondo singlet---a many-body state formed by entanglement between a localized spin and the Fermi sea---has been studied extensively through its transport signatures in quantum dots. Here we report a thermodynamic measurement of the entropy suppression associated with the formation of the Kondo singlet, using temperature-dependent charge sensing and a Maxwell relation to track the suppression of spin entropy as the first electron is added to a strongly-coupled GaAs quantum dot. Plotting dN/dTdN/dTdN/dT against the simultaneously measured occupation NNN reveals an asymmetric lineshape with its peak shifted to N>1/2N>1/2N>1/2---a hallmark of Kondo screening---that weakens with increasing temperature and is qualitatively reproduced by numerical renormalization group (NRG) calculations, with a small but persistent offset to lower occupation relative to the theory. An independent measurement of conductance versus occupation on the same device provides a test of these quantities through the mixed-valence crossover and matches NRG within experimental uncertainty.
The physical properties of a material tuned to the cusp between two distinct ground states can be quite exotic, and unlike those in either of the neighboring phases. The prospect of capturing such behavior in a simple model is tantalizing; for example, the interplay between heavy fermion physics and magnetic ordering in certain materials is often rationalized in terms of the quantum phase transition in the two-impurity Kondo model. However, this model is oversimplified for the purpose: its quantum critical point does not reflect the distinctive properties of a magnetic lattice surrounded by mobile electrons. In this work, we study a tunable nanoelectronic circuit comprising two coupled charge-Kondo quantum islands, realizing a new model which captures the essence of competition between local and collective screening of magnetic moments. This may have relevance for materials in which collective many-body effects drive lattice coherence. We tune our device to a novel quantum critical point, and show experimentally that deviations as we tune away from this point match non-trivial predictions from the model. This work on the crucial role of inter-island interactions is a necessary first step in scaling up such circuits from individual sites to networks or lattices.
Quantum phase transitions (QPTs) are ubiquitous in strongly correlated materials. However, the microscopic complexity of these systems impedes the quantitative understanding of QPTs. We observed and thoroughly analyzed the rich strongly correlated physics in two profoundly dissimilar regimes of quantum criticality. With a circuit implementing a quantum simulator for the three-channel Kondo model, we reveal the universal scalings toward different low-temperature fixed points and along the multiple crossovers from quantum criticality. An unanticipated violation of the maximum conductance for ballistic free electrons is uncovered. The present charge pseudospin implementation of a Kondo impurity opens access to a broad variety of strongly correlated phenomena.
We investigate a model of two Kondo impurities coupled via an Ising interaction. Exploiting the mapping to a generalized single-impurity Anderson model, we establish that the model has a singlet and a (pseudospin) doublet phase separated by a Kosterlitz-Thouless quantum phase transition. Based on a strong-coupling analysis and renormalization-group arguments, we show that at this transition the conductance GGG through the system either displays a zero-bias anomaly, $G\ensuremath{\sim}{\ensuremath{\mid}V\ensuremath{\mid}}^{\ensuremath{-}2(\sqrt{2}\ensuremath{-}1)}$, or takes a universal value, $G=({e}^{2}∕\ensuremath{\pi}\ensuremath{\hbar}){\mathrm{cos}}^{2}(\ensuremath{\pi}∕2\sqrt{2})$, depending on the experimental setup. Close to the Toulouse point of the individual Kondo impurities, the strong-coupling analysis allows us to obtain the location of the phase boundary analytically. For general model parameters, we determine the phase diagram and investigate the thermodynamics using numerical renormalization-group calculations. In the singlet phase close to the quantum phase transition, the entropy is quenched in two steps: first the two Ising-coupled spins form a magnetic minidomain which is, in a second step, screened by a Kondoesque collective resonance in an effective solitonic Fermi sea. In addition, we present a flow-equation analysis which provides a different mapping of the two-impurity model to a generalized single-impurity Anderson model in terms of fully renormalized couplings, which is applicable for the whole range of model parameters.
The prospect of using semiconductor quantum dots as an experimental tool to distinguish Majorana zero modes (MZMs) from other zero-energy excitations such as Kondo resonances has brought up the fundamental question of whether topological superconductivity and the Kondo effect can coexist in these systems. Here, we study the Kondo effect in a quantum dot coupled to a metallic contact and to a pair of MZMs. We consider a situation in which the MZMs are spin polarized in opposite directions. By using numerical renormalization group calculations and scaling analysis of the renormalization group equations, we show that the Kondo effect takes place at low temperatures, regardless of the coupling to the MZMs. Interestingly, we find that the Kondo singlet essentially decouples from the MZMs such that the residual impurity entropy can show local non-Fermi-liquid properties characteristic of the single Majorana excitations. This offers the possibility of tuning between Fermi-liquid and non-Fermi-liquid regimes simply by changing the quantum-dot-MZM couplings.