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.
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.
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.
Entanglement is a defining feature of quantum mechanics, and its relation to quantum criticality is of considerable current interest. Here we show that measurable spin and charge fluctuations provide an entanglement witness of quantum criticality in the two-impurity Kondo model, which has experimental realizations in terms of coupled quantum dots and magnetic impurities added to surfaces. We use density-matrix renormalization group and numerical renormalization group calculations to resolve the non-Fermi-liquid critical point separating two independently Kondo-screened impurities from an inter-impurity singlet and interpret it as a change in the dominant entanglement partner of each local moment: from entanglement of the local moment with an extended set of conduction-electron degrees of freedom to entanglement with the other impurity. This reorganization is accompanied by a singular response of the impurity-bath entanglement and the inter-impurity susceptibility. We show how the same structure is encoded in the quantum Fisher information of collective spin and charge operators at zero and finite temperatures, connecting the entanglement picture to experimentally accessible dynamical response functions. Our results establish impurity systems as controlled settings in which quantum-critical entanglement can be detected through measurable correlations, and provide further insight into the possibility of understanding heavy-fermion physics in terms of entanglement.
Charge Kondo circuits consist of metallic islands connected by single-mode quantum point contacts (QPCs). The island's charging energy makes these circuits tunable quantum simulators of various strongly interacting models. Here we propose a circuit that realizes the Kondo effect with effective Luttinger-liquid interactions, and show that it undergoes a localization transition in which the QPC transmission is fully suppressed below a critical value. Experimental signatures include a diverging charge susceptibility and an entropy step. Our findings open a path toward realizing localization transitions in more exotic settings.
We investigate systems comprising chains and rings of quantum dots, coupled to two metallic leads. Such systems allow to study the competition between orbital and spin degrees of freedom in a nanodevice, and the effect this subtle interplay has on two-channel Kondo (2CK) physics. We demonstrate that a rich range of strongly correlated electron behaviour results, with non-Fermi liquid 2CK phases and non-trivial phase transitions accessible. We employ physical arguments and the numerical renormalization group (NRG) technique to analyse these systems in detail, examining in particular both thermodynamic and dynamical properties. When leads are coupled to either end of a chain of dots, we show that the resulting behaviour on low temperature/energy scales can be understood in terms of simpler paradigmatic quantum `impurity' models. An effective low-energy single-spin 2CK model is derived for all odd-length chains, while the behaviour of even-length chains is related fundamentally to that of the classic `two-impurity Kondo' model. In particular, for small interdot coupling, we show that an effective coupling mediated though incipient single-channel Kondo states drives all odd chains to the 2CK fixed point (FP) on the lowest temperature/energy scales. A theory is also developed to describe a phase transition in even chains. We derive an effective channel-anisotropic 2CK model, which indicates that the critical FP of such models must be the 2CK FP. This physical picture is confirmed using NRG for various chain systems. We also examine the effect of local frustration on 2CK physics in mirror-symmetric ring systems. The importance of geometry and symmetry is demonstrated clearly in the markedly different physical behaviour that arises in systems where two leads are either connected to the same dot, or to neighbouring dots. In the latter case, we show for all odd-membered rings that two distinct 2CK phases, with different ground state parities, arise on tuning the interdot couplings. A frustration-induced phase transition thus occurs, the 2CK phases being separated by a novel critical point for which an effective low-energy model is derived. Precisely at the transition, parity mixing of the quasidegenerate local trimer states acts to destabilise the 2CK FPs, and the critical FP is shown to consist of a free pseudospin together with effective single-channel spin quenching. While connecting both leads to the same dot again results in two parity-distinct phases, a simple level-crossing transition now results due to the symmetry of the setup. The proposed geometry also allows access to a novel ferromagnetically-coupled two-channel local moment phase. Driven by varying the interdot couplings and occurring at the point of inherent magnetic frustration, such transitions in ring structures provide a striking example of the subtle interplay between internal spin and orbital degrees of freedom in coupled quantum dot systems, and the resulting effect on Kondo physics.
Pyrochlore systems (A_{2}B_{2}O_{7}) with A-site rare-earth local moments and B-site 5d conduction electrons offer excellent material platforms for the discovery of exotic quantum many-body ground states. Notable examples include U(1) quantum spin liquid of the local moments and semimetallic non-Fermi liquid of the conduction electrons. Here we investigate emergent quantum phases and their transitions driven by the Kondo lattice coupling between such highly entangled quantum ground states. Using the renormalization group method, it is shown that weak Kondo lattice coupling is irrelevant, leading to a fractionalized semimetal phase with decoupled local moments and conduction electrons. Upon increasing the Kondo lattice coupling, this phase is unstable to the formation of broken symmetry states. Particularly important is the opposing influence of the Kondo lattice coupling and long-range Coulomb interaction. The former prefers to break the particle-hole symmetry while the latter tends to restore it. The characteristic competition leads to possibly multiple phase transitions, first from a fractionalized semimetal phase to a fractionalized Fermi surface state with particle-hole pockets, followed by the second transition to a fractionalized ferromagnetic state. Multiscale quantum critical behaviors appear at nonzero temperatures and with external magnetic field near such quantum phase transitions. We discuss the implication of these results to the experiments on Pr_{2}Ir_{2}O_{7}.
Hybrid metal-semiconductor quantum circuits offer a controlled setting for realizing and probing strongly correlated quantum matter. A largely open question that may be addressed by such simulators is whether coupled critical states can combine into new non-Fermi liquid (NFL) phases. Inspired by the recent realization of a two-site charge-Kondo circuit [W. Pouse \textit{et al.}, \href{https://www.nature.com/articles/s41567-022-01905-4}{Nat. Phys. \textbf{19}, 492 (2023)}], we uncover the fate of coupled Kondo anyons emerging from the two-channel-Kondo critical point at each site. Bosonization and quantum Brownian motion methods yield the full quasiballistic phase diagram and low-temperature transport properties, while numerical renormalization group calculations resolve the complementary weak-coupling limit. Together, these provide a unified description, and reveal robust NFL phases with continuously tunable transport and susceptibility exponents. Stronger inter-site coupling ultimately produces a screened Fermi liquid.