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.
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 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.
Unconventional quantum states defying the ubiquitous Fermi-liquid paradigm can emerge in the presence of strong electronic correlations. Among these, non-Abelian anyons - such as Majorana zero modes and Fibonacci anyons - are of particular interest for topological quantum computing due to their non-integer quantum dimensions d>1, which allows for protected non-local encoding and processing of quantum information. However, despite considerable efforts, the unambiguous characterisation of such anyons via transport measurements has proved challenging. Instead, here we provide experimental evidence for the low-temperature fractional entropy Delta S associated with a single anyon, which directly implies its non-Abelian character through the relation Delta S = kB ln(d). This thermodynamic signature is measured in metal-semiconductor quantum circuits engineered to realize quantum-critical states from frustrated interactions. Using a micrometre-scale metallic island coupled to two or three electronic leads, we tune the system to two-channel and three-channel Kondo critical points. By measuring the island charge and exploiting a thermodynamic Maxwell relation, we estimate the entropy associated with the anyons that emerge in these critical states. Our observations reveal fractional values, exposing non-Abelian anyons. The corresponding scaling dimensions are consistent with theoretical predictions for a Majorana zero mode Delta S = kB ln(sqrt(2)) and a Fibonacci anyon Delta S = kB ln(1 +sqrt(5))/2 for two and three channels. These findings establish entropy measurements as a powerful tool for characterizing exotic quantum states.
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.
The traditional multichannel Kondo effect takes place when several gapless metallic electronic channels interact with a localized spin-S impurity, with the number of channels n exceeding the size of the impurity spin, n>2S, leading to the emergence of non-Fermi liquid impurity behavior at low temperatures. Here, we show that the effect can be realized even when the electronic degrees of freedom are strongly correlated and gapped. The system under consideration consists of a single spin-21 impurity coupled isotropically to n spin singlet superconducting channels realized by one-dimensional leads with quasi-long-range superconducting order. The competition between the Kondo and superconducting fluctuations induces multiple distinct ground states and boundary phases depending on the relative strengths of the bulk and boundary interactions. Using the Bethe Ansatz technique, we identify four regimes: an overscreened Kondo phase, a zero-mode phase, a Yu-Shiba-Rusinov (YSR) phase, and a local-moment phase with an unscreened impurity, each with its own experimental characteristic. We describe the renormalization-group flow, the excitation spectrum, and the full impurity thermodynamics in each phase. Remarkably, even in the presence of a bulk mass gap, the boundary critical behavior in the Kondo phase is governed by the same exponents as in the gapless theory with the low-energy impurity sector flowing to the SU(2)n Wess-Zumino-Witten (WZW) fixed point, and the impurity entropy monotonically decreasing as a function of temperature. In both the overscreened Kondo and zero-mode phases, the residual impurity entropy is Simp(T→0)=ln[2cos(π/(n+2))]. In the YSR and unscreened phases on the other hand the impurity entropy exhibits non-monotonic temperature dependence and is effectively free at low temperatures with Simp(T→0)=ln2.
We study the breakdown of Kondo screening by a local magnetic field from the perspective of a measurement-driven entanglement transition in a monitored quantum system. Here, the Kondo coupling leads to the growth in entanglement of an impurity spin with it's fermionic environment, while the local field plays the role of a continuous observer. Using a non-perturbative Unitary Renormalization Group (URG) approach, we derive coupled renormalization-group flow equations for the Kondo exchange and the local field, and obtain a field-dependent RG phase diagram. The RG flows separate a low-energy Kondo-screened phase, where the impurity is absorbed into the Fermi sea and forms an entangled singlet with the conduction bath, from a polarized local-moment phase in which screening is frustrated and impurity-bath entanglement is suppressed. We identify the fixed-point Hamiltonians governing the two phases and the critical regime, and relate the transition to the emergence of a novel non-Fermi liquid. Various impurity signatures such as the spectral function and thermalisation of impurity observables are used to characterise this entanglement transition. These results offer insight into the interplay of decoherence and measurement in governing the dynamics of a prototypical quantum system.
Quantum impurity models are a fundamental paradigm in the study of many-body quantum matter. They describe metals with actual magnetic impurities, quantum circuits with quantum dots, and are the local effective theories within dynamical mean field theory for correlated materials. They have gained prominence in recent years as a platform for treating non-Markovian open quantum systems exactly in strong coupling, and as a means for accurate simulations of quantum nanoelectronics devices. The work presented in this thesis focuses on the characterization of quantum matter through this lens. Two recurring themes are characterizing systems by their entropy signatures, and through their quench dynamics. Firstly, we investigate the non-equilibrium physics of an interacting quantum dot circuit following a sudden, externally-driven change in the dot potential. We do this by mapping the interacting problem onto a simpler Gaussian theory, which we show is exact for single-particle dynamics in equilibrium. Perhaps surprisingly, we establish that such a mapping is still possible following a quench; although this comes at the cost of numerical optimization of the effective non-interacting auxiliary fermion problem, starting from a tensor network treatment. The method provides a new way of viewing impurity quench dynamics. Secondly, we focus on entropy measurements in quantum dot circuits using a charge sensor. The equilibrium thermodynamic entropy can be extracted from such charge measurements by local Maxwell relations, and we study this connection in detail for a variety of experimentally-relevant quantum nanoelectronic device models, uncovering their entropic phase diagram. The underlying quantum impurity models are solved exactly using numerical renormalization group methods. We extend this to lattice models describing interacting materials, showing that the Maxwell relations provide a route to calculate the lattice entropy and give new physical insights. We go on to consider entropy production in the non-equilibrium setting of a quench, again showing that this can be related to practical charge measurements. In quantum circuit experiments, the dot charge is measured in practice using a secondary, capacitively coupling charge-sensor circuit. We investigate this by explicitly modelling the device together with the charge-sensor in a fully quantum mechanical treatment. This allows us to understand and quantify the measurement backaction. It also allows us to establish when the assumptions underpinning the Maxwell entropy extraction hold. Finally, we consider the strongly correlated electron physics of bilayer Hubbard models within dynamical mean field theory. Such models capture the essence of interface physics in heterostructures. In the basic bilayer, we revisit the phase diagram and study the change in character of the metal-insulator transition from continuous band-insulator gap opening to Mott quasiparticle collapse, as a function of interlayer coupling and local Coulomb interaction strength. For more complex interfaces involving additional correlated sites between layers, many-body quantum interference can be tuned, leading to different types of transition. While a pure impurity model version of this system features a second-order non-Fermi liquid Kondo critical point, full self-consistency on the lattice turns the transition first-order.