We investigate a minimal system consisting of a quantum dot coupled to a Majorana zero mode and a normal lead. We identify the underlying screening process as a novel spin-charge Kondo effect, where the low-energy spin and charge degrees of freedom of the Majorana zero mode-quantum dot subsystem are fully screened by those in the normal lead, resulting in the formation of a spin-charge singlet. An effective low-energy model is derived, with charge fluctuations appropriately accounted for. This spin-charge Kondo effect is found to be consistent with the spin-dependent Andreev/normal boundary conditions induced by the Majorana zero mode. We demonstrate that the anomalous substructure in the spectrum and thermodynamic properties is closely tied to the proportion of the charge component in the screening cloud. The spin-charge screening cloud exhibits scaling behavior analogous to that of traditional Kondo systems, though the sub-leading even-odd effect is subtly modified by the boundary conditions. These findings enhance our understanding of Kondo physics and resolve key debates on quantum dot nanostructures with Majorana zero modes.
Coherent manipulation of Kondo Majoranas in two-channel Kondo setups
University of Cincinnati（辛辛那提大学）；Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superior de Investigaciones Científicas (CSIC)（马德里材料科学研究所，西班牙国家研究委员会）；Universidad Autónoma de Madrid（马德里自治大学）
In this paper, we study a set-up in which quantum dot is tunnel-coupled to Majorana-Kramers pair(MKP) which are the edge modes of time reversal invariant topological superconductor (TRITOPS), which is an experimental realization of time reversal invariant Kitaev chain model. Our motivation is to study the effect of Majorana-Kramers pair on Kondo fixed point and conversely the effect of Kondo interaction on Majorana-Kramers pair. This way we can capture the signatures of Majorana fermions on Kondo effect in quantum dots which can be probed feasibly in experiments. Employing the renormalization …
Non-abelian anyons are highly desired for topological quantum computation purposes, with Majorana fermions providing a promising route, particularly zero modes with non-trivial mutual statistics. Yet realizing Majorana zero modes in matter is a challenge, with various proposals in chiral superconductors, nanowires, and spin liquids, but no clear experimental examples. Heavy fermion materials have long been known to host Majorana fermions at two-channel Kondo impurity sites, however, these impurities cannot be moved adiabatically and generically occur in metals, where the absence of a gap remov…
Electron transport through [normal metal]-[quantum dot]-[topological superconductor] junction is studied and reveals interlacing physics of Kondo correlations with two Majorana fermions bound state residing on the opposite edges of the topological superconductor. When the strength of the Majorana fermion coupling exceeds the temperature $T$, this combination of Kono-Majorana fermion physics can be observed: The usual peak of the temperature dependent zero biased conductance $σ(V=0,T)$ splits and the conductance has a {\it dip} at T=0. The height of the conductance side-peaks decreases with mag…
In an exciting development, SmB6, a Kondo insulator has been shown to exhibit bulk quantum oscillations. We propose that SmB6 is a bulk scalar Majorana Fermi Liquid} (MFL) with a finite gap for charge and spin excitations. In their study of Kondo insulators in 1993, Coleman, Miranda and Tsvelik (CMT) envisaged such a remarkable possibility, using a mean field ansatz. We generalize CMT theory to non-zero magnetic fields and show a counter intuitive result that the scalar Majorana fermi liquid, while remaining electrically insulating, responds to external magnetic field and exhibits Landau diama…
All models of interacting electrons and spins can be reformulated as theories of interacting Majorana fermions. We consider the Kondo lattice model that admits a symmetric representation in terms of Majorana fermions. In the first part of this work we study two variational states, which are natural in the Majorana formulation. At weak coupling a state in which three Majorana fermions tend to propagate together as bound objects is favored, while for strong coupling a better description is obtained by having deconfined Majorana fermions. This way of looking at the Kondo lattice offers an alterna…
We show that a Majorana fermion description of the two channel Kondo model can emerge quite naturally as a representation of the algebra associated with the spin currents in the two channels. Using this representation we derive an exact equivalent Hamiltonian for the two channel model expressed entirely in terms of Majorana fermions. The part of the Hamiltonian that is coupled to the impurity spin corresponds to the vector part of the $σ$-$τ$ model (compactified two channel model). Consequently all the thermodynamic properties associated with the impurity spin can be calculated from the $σ$-$τ…
Quantum dot devices allow one to access the critical point of the two-channel Kondo model. The effective critical theory involves a free Majorana fermion quasiparticle localized on the dot. As a consequence, this critical point shows both the phenomenon of non-Fermi liquid physics and fractionalization. Although a violation of the Wiedemann-Franz law is considered a defining feature of non-Fermi liquid systems, we show by exact calculations that it holds at the critical point, providing a counterexample to this lore. Furthermore, we show that the fractionalized Majorana character of the critic…
It is suggested that networks of Majorana-Cooper pair boxes connected by metallic nanowires can simulate various exotic states of matter. In this simulations Majorana-Cooper boxes play the role of effective spins S=1/2 and the metallic connections generate the Kondo screening and the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. Depending on what prevails - whether it is the Kondo effect or the RKKY exchange, one will have either an effective spin model or a Kondo lattice. The list of exotic stets includes the famous hexagonal Kitaev model, a generalization of this model for a Kondo lattic…
We study the local density of states (LDOS) in systems of Luttinger-liquid nanowires connected to a common mesoscopic superconducting island, in which Majorana bound states give rise to different types of topological Kondo effects. We show that electron interactions enhance the low-energy LDOS in the leads close to the island, with unusual exponents due to Kondo physics that can be probed in tunneling experiments.
Kondo tunneling reveals hidden SO(n) dynamical symmetries of evenly occupied quantum dots. As is exemplified for an experimentally realizable triple quantum dot in parallel geometry, the possible values n=3,4,5,7 can be easily tuned by gate voltages. Following construction of the corresponding o(n) algebras, scaling equations are derived and Kondo temperatures are calculated. The symmetry group for a magnetic field induced anisotropic Kondo tunneling is SU(2) or SO(4).
Formation and Evolution of the Spin-Charge-Entangled Screening Cloud in the Majorana-Kondo System
National Laboratory of Solid State Microstructures and School of Physics, Nanjing University（南京大学固体微结构物理国家重点实验室和物理学院）；Collaborative Innovation Center of Advanced Microstructures, Nanjing University（南京大学先进微结构协同创新中心）；School of Physics and Astronomy, Shanghai Jiao Tong University（上海交通大学物理与天文学院）；College of Physics and Electronic Engineering, Center for Computational Sciences, Sichuan Normal University（四川师范大学物理与电子工程学院计算科学中心）
The Kondo effect is a striking consequence of the coupling of itinerant electrons to a quantum spin with degenerate energy levels. While degeneracies are commonly thought to arise from symmetries or fine-tuning of parameters, the recent emergence of Majorana fermions has brought to the fore an entirely different possibility: a "topological degeneracy" which arises from the nonlocal character of Majorana fermions. Here we show that nonlocal quantum spins formed from these degrees of freedom give rise to a novel "topological Kondo effect". This leads to a robust non-Fermi liquid behavior, known …
We carry out a theoretical analysis of a prototypical Majorana system, which demonstrates the existence of a Majorana-mediated many-body state and an associated intermediate low-energy fixed point. Starting from two Majorana bound states, hosted by a Coulomb-blockaded topological superconductor and each coupled to a separate lead, we derive an effective low-energy Hamiltonian, which displays a Kondo-like character. However, in contrast to the Kondo model which tends to a strong- or weak-coupling limit under renormalization, we show that this effective Hamiltonian scales to an intermediate fixe…
We study entanglement of Kondo clouds in an open triple quantum dots (OTQDs) system based on the dissipaton equation of motion (DEOM) theory. A comprehensive picture of the long-range entanglement of Kondo clouds is sketched by the spectral functions, spin-spin correlation and dot occupancies of OTQDs. We find that for the configuration (N1,N2,N3) = (1, 0, 1), a conduction electrons peak occurs in the spectral function of intermediate QD in Kondo regime. This peak resulting from the overlapping of the two Kondo clouds forming from between the two peripheral QDs and leads, enhances with decreas…
We consider a lateral double-dot system in the Coulomb blockade regime with a single spin-1/2 on each dot, mutually coupled by an anti-ferromagnetic exchange interaction. Each of the two dots is contacted by two leads. We demonstrate that the voltage across one of the dots will have a profound influence on the current passing through the other dot. Using Poor Man's scaling, we find that the Kondo-effect can lead to a strong enhancement of this {\it transconductance}.
The correlations in charge transport in multiple Majorana fermions systems contain much richer physics and carry more information than the conventional transport coefficients. We calculate the auto- and cross-correlations in the spinful topological Kondo model and point out the frequencies dependencies of the correlations that may serve as a proof of the existence of the Majorana fermions in our system, particularly the change of the cross-correlation from positive to negative at higher frequency.
It is shown that the rate limiting time constant for the formation of the Kondo state in a quantum dot can be extracted analytically from known perturbation theoretic results. The prediction obtained is verified via numerical simulations in the noncrossing approximation.
We report anomalous critical transport behavior in a GaAs structure containing a dense array of ErAs quantum dots. The structure displays a voltage (electric field)-controlled insulator-to-metal transition and strong hysteresis in the Kondo-like current-vs-temperature characteristic, with critical temperatures as high as 50 K. We attribute this behavior to a strong distributed Kondo resonance between the quantum dots after the Coulomb blockade of the array is lifted. This is consistent with a high sensitivity of the phase transition to a small external magnetic field that we have observed in t…
We study the transport behavior induced by a small bias voltage through a quantum dot connected to one-channel disordered wires by means of a quantum Monte Carlo method. We model the quantum dot by the Hubbard-Anderson impurity and the wires by the one-dimensional Anderson model with diagonal disorder within a length. We present a complete description of the probability distribution function of the conductance within the Kondo regime.
We predict that an anisotropic charge Kondo effect appears in a triple quantum dot, when the system has two-fold degenerate ground states of (1,1,0) and (0,0,1) charge configurations. Using bosonization and refermionization methods, we find that at low temperature, the system has the two different phases of massive charge fluctuations between the two charge configurations and vanishing fluctuations, which are equivalent with the Kondo-screened and ferromagnetic phases of the anisotropic Kondo model, respectively. The phase transition is identifiable by electron conductance measurement, offerin…
The Kondo effect and the Fano-Kondo effect are important phenomena that have been observed in quantum dots (QDs). We theoretically investigate the transport properties of a coupled QD system in order to study the possibility of detecting a qubit state from the modulation of the conductance peak in the Kondo effect and the dip in the Fano-Kondo effect. We show that the peak and dip of the conductance are both shifted depending on the energy-level of the qubit. In particular, we find that we can estimate the optimal point and tunneling coupling between the |0> and |1> states of the qubit by meas…
The thermopower of few-electron quantum dots with Kondo correlations is investigated via a hierarchial equations of motion approach. The thermopower is determined by the line shape of spectral function within a narrow energy window defined by temperature. Based on calculations and analyses on single-level and two-level Anderson impurity models, the underlying relations between thermopower and various types of electron correlations are elaborated. In particular, an unconventional sign reversal behavior is predicted for quantum dots with a suitable inter-level spacing. The new feature highlights…
We study the AC conductance and equilibrium current fluctuations of a Coulomb blockaded quantum dot. A relation between the equilibrium spectral function and the linear AC conductance is derived which is valid for frequencies well below the charging energy of the quantum dot. Frequency-dependent transport measurements can thus give experimental access to the Kondo peak in the equilibrium spectral function of a quantum dot. We illustrate this in detail for typical experimental parameters using the numerical renormalization group method in combination with the Kubo formalism.
We apply a Gutzwiller-like variational technique to study Josephson conduction across a quantum dot with an odd number of electrons connected to two superconducting leads. We show that, for small values of the superconducting gap, Kondo correlations and superconductivity cooperate to enhance the Josephson current. As the superconducting gap increases, the current changes sign and the system becomes a $π$-junction. The $π$-junction behavior sets in much before antiferromagnetic correlations at the dot can be treated perturbatively.
We investigate theoretically the thermoelectric transport properties of a T-shaped double quantum dot side-coupled to a topological superconducting nanowire hosting Majorana zero-energy modes. The calculations are performed using the numerical renormalization group method focusing on the transport regime, where the system exhibits the two-stage Kondo effect. It is shown that the leakage of Majorana quasiparticles into the double dot system results in a half-suppression of the second stage of the Kondo effect, which is revealed through fractional values of the charge and heat conductances and g…
We examine the transport properties of a double quantum dot system coupled to a topological superconducting nanowire hosting Majorana quasiparticles at its ends, with the central quantum dot attached to the left and right leads. We focus on the behavior of the local density of states and the linear conductance, calculated with the aid of the numerical renormalization group method, to describe the influence of the Majorana coupling on the low-temperature transport properties induced by the Kondo correlations. In particular, we show that the presence of Majorana quasiparticles in the system affe…
The transport behavior of a double quantum dot side-attached to a topological superconducting wire hosting Majorana zero-energy modes is studied theoretically in the strong correlation regime. It is shown that Majorana modes can leak to the whole nanostructure, giving rise to a subtle interplay between the two-stage Kondo screening and the half-fermionic nature of Majorana quasiparticles. In particular, the coupling to the topological wire is found to reduce the effective exchange interaction between the two quantum dots in the absence of normal leads. Interestingly, it also results in an enha…
We consider an interacting nanowire/superconductor heterostructure attached to metallic leads. The device is described by an unusual low-energy model involving spin-1 conduction electrons coupled to a nonlocal spin-1/2 Kondo impurity built from Majorana fermions. The topological origin of the resulting Kondo effect is manifest in distinctive non-Fermi-liquid (NFL) behavior, and the existence of Majorana fermions in the device is demonstrated unambiguously by distinctive conductance lineshapes. We study the physics of the model in detail, using the numerical renormalization group, perturbative …
We study the low energy spectrum of a correlated quantum dot embedded between the normal conducting and superconducting reservoirs and hybridized with the topological superconducting nanowire, hosting the Majorana end-modes. We investigate the leaking Majorana quasiparticle and inspect its interplay with the proximity induced on-dot pairing and correlations. In particular, we focus on the subgap Kondo effect near the quantum phase transition/crossover from the spinfull (doublet) to the spinless (BCS-type singlet) configurations. Treating the correlations perturbatively and within the NRG appro…
We study Kondo screening obtained by coupling Majorana bound states, located on a topological superconducting island, to interacting electronic reservoirs. At the charge degeneracy points of the island, we formulate an exact mapping onto the spin-$1/2$ multi-channel Kondo effect. The coupling to Majorana fermions transforms the tunneling terms into effective fermionic bilinear contributions with a Luttinger parameter $K$ in the leads that is effectively doubled. For strong interaction, $K=1/2$, the intermediate fixed point of the standard multi-channel Kondo model is exactly recovered. It evol…
A one-dimensional time-reversal-invariant topological superconductor hosts a Majorana Kramers pair at each end, where time-reversal symmetry acts as a supersymmetry that flips local fermion parity. We examine the transport anomaly of such a superconductor, floating and tunnel-coupled to normal leads at its two ends. We demonstrate the realization of a topologically-protected, channel-symmetric, two-channel Kondo effect without fine-tuning. Whereas the nonlocal teleportation vanishes, a lead present at one end telecontrols the universal transport through the other end.
We present a powerful and general approach to describe the coupling of Majorana fermions to external leads, of interacting or non-interacting electrons. Our picture has the Klein factors of bosonization appearing as extra Majoranas hybridizing with the physical ones. We demonstrate the power of this approach by solving a highly nontrivial SO(M) Kondo problem arising in topological superconductors with M Majorana-lead couplings, allowing for arbitrary M and for conduction electron interactions. We find that these topological Kondo problems give rise to robust non-Fermi liquid behavior, even for…
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 MBS are spin polarized in opposite directions. By using numerical renormalization-group calculations and scaling analysis of the renormalization group equ…
We analyze the interactions between two Kondo Quantum Dots connected to a Rashba-active Quantum Wire. We find that the Kondo-doublet interaction, at an inter-dot distance of the order of the wire Fermi length, is over an order of magnitude greater than the RKKY interaction. The effects induced on the Kondo-doublet interaction by the wire spin-orbit coupling can be used to control the Quantum Dots spin-spin correlation. These results imply that the widely used assumption that the RKKY is the dominant interaction between Anderson impurities must be revised.
We demonstrate delicate control over the Kondo effect and its interplay with quantum interference in an Aharonov-Bohm interferometer containing one Kondo dot and one noninteracting dot. It is shown that the Kondo resonance undergoes a dramatic evolution as the interdot tunnel coupling progressively increases. A novel triple Kondo splitting occurs from the interference between constant and Lorentzian conduction bands that cooperate in forming the Kondo singlet. The device also manifests a highly controllable Fano-Kondo effect in coherent electronic transport, and can be tuned to a regime where …
We study a Majorana island coupled to a bulk superconductor via a Josephson junction and to multiple external normal leads. In the absence of the Josephson coupling, the system displays a topological Kondo state, which had been largely studied recently. However, we find that this state is unstable even to small Josephson coupling, which instead leads at low temperature $T$ to a new fixed point. Most interesting is the case of three external leads, forming a minimal electronic realization of the long sought two-channel Kondo effect. While the $T=0$ conductance corresponds to simple resonant And…
We investigate a symmetrical double quantum dot system serially attached to the leads. The emphasis is on the numerical analysis of finite inter-dot tunneling in the presence of inter-dot repulsive capacitive coupling. The results reveal the competition between extended Kondo phases and local singlet phases in spin and charge degrees of freedom. The corresponding phase diagram is determined quantitatively.
Double quantum dots offer unique possibilities for the study of many-body correlations. A system containing one Kondo dot and one effectively noninteracting dot maps onto a single-impurity Anderson model with a structured (nonconstant) density of states. Numerical renormalization-group calculations show that while band filtering through the resonant dot splits the Kondo resonance, the singlet ground state is robust. The system can also be continuously tuned to create a pseudogapped density of states and access a quantum critical point separating Kondo and non-Kondo phases.
We have recently shown [Phys. Rev. B {\bf 89}, 165314 (2014)] that a non--interacting quantum dot coupled to a one--dimensional topological superconductor and to normal leads can sustain a Majorana mode even when the dot is expected to be empty, \emph{i.e.}, when the dot energy level is far above the Fermi level of he leads. This is due to the Majorana bound state of the wire leaking into the quantum dot. Here we extend this previous work by investigating the low--temperature quantum transport through an {\it interacting} quantum dot connected to source and drain leads and side--coupled to a t…
We study the Kondo effect in multiple-dot systems for which the inter- as well as intra-dot Coulomb repulsions are strong, and the inter-dot tunneling is small. The application of the Ward-Takahashi identity to the inter-dot dynamical susceptibility enables us to analytically calculate the conductance for a double-dot system by using the Bethe-ansatz exact solution of the SU(4) impurity Anderson model. It is clarified how the inter-dot Kondo effect enhances or suppresses the conductance under the control of the gate voltage and the magnetic field. We then extend our analysis to multiple-dot sy…
We study the gate voltage dependence of the linear conductance through a quantum dot coupled to one-dimensional leads. For interacting dot electrons but noninteracting leads Kondo physics implies broad plateau-like resonances. In the opposite case Luttinger liquid behavior leads to sharp resonances. In the presence of Kondo as well as Luttinger liquid physics and for experimentally relevant parameters, we find a line shape that resembles the one of the Kondo case.
We investigate the dependence of the ground state of a multi-level quantum dot on the coupling to an external fermionic system and on the interactions in the dot. As the coupling to the external system increases, the rearrangement of the effective energy levels in the dot signals the transition from the Kondo regime to a mixed valence (MV) regime. The MV regime in a two level dot is characterized by an intrinsic mixing of the levels in the dot, resulting in non-perturbative sub- and super-tunneling phenomena that strongly influence the Kondo effect.
We investigate entanglement between electrons in serially coupled double quantum dots attached to non interacting leads. In addition to local repulsion we consider the influence of capacitive inter-dot interaction. We show how the competition between extended Kondo and local singlet phases determines the ground state and thereby the entanglement.
We apply a Gutzwiller-like variational technique to study Josephson conduction across a quantum dot with an odd number of electrons connected to two superconducting leads. Our method projects out all states on the dot but the Kondo singlet and is valid when Kondo correlations are dominant and no Andreev bound states localized at the dot are available for Kondo screening. In these conditions superconducting pairing is a competing effect and the junction is $π-$like, to optimize antiferromagnetic correlations on the dot. As the superconducting gap increases, the Josephson current also increases,…
We study the Kondo temperature of a quantum dot embedded into one arm of an Aharonov-Bohm interferometer. The topology of a disordered or chaotic Aharanov-Bohm ring leads to a stochastic term in the scaling equation and in the renormalization procedure. As a result, the Kondo temperature displays significant fluctuations as a function of magnetic flux.
Orbital Kondo effect in a system of two single-level quantum dots attached to external electron reservoirs is considered theoretically. The dots are coupled via direct hoping term and Coulomb interaction. The Kondo temperature is evaluated from the scaling approach and slave boson technique. The later method is also used to calculate linear conductance of the system. Nonlinear conductance, in turn, is calculated in terms of the nonequilibrium Green function formalism.
The Kondo-Heinsberg chain is an interesting model of a strongly correlated system which has a broad superconducting state with pair-density wave (PDW) order. Some of us have recently proposed that this PDW state is a symmetry-protected topological (SPT) state, and the gapped spin sector of the model supports Majorana zero modes. In this work, we reexamine this problem using a combination of numeric and analytic methods. In extensive density matrix renormalization group calculations, we find no evidence of a topological ground state degeneracy or the previously proposed Majorana zero modes in t…
We consider the spin and orbital Kondo effect in a parallel arrangement of two strongly electrostatically coupled quantum dots. Increasing the exchange of electrons between the dots through the attached leads induces the transition between the SU(4) spin- and orbital Kondo effect and SU(2) spin Kondo effect. Being the same for the SU(4) and SU(2) symmetry points, the Kondo temperature drops slightly in the intermediate regime. Experimentally, two kinds of Kondo effects can be discriminated by the sensitivity to the suppression of the spin Kondo effect by Zeeman field. The dependence of Kondo t…
We study theoretically the out-of-equilibrium transport properties of a double quantum dot system in the Kondo regime. We model the system by means of a two-impurity Anderson Hamiltonian. The transport properties are characterized by Kondo effect properties, however, superimposed them, the system possesses novel non-linear bistability behavior.
We show that through an interdot off-site electron correlation in a double quantum-dot (DQD) device, Kondo resonances emerge in the local density of states without the electron spin-degree of freedom. We identify the physical mechanism behind this phenomenon: rather than forming a spin singlet in the device as required in the conventional Kondo physics, we found that exchange of electron position between the two quantum dots, together with the off-site Coulomb interaction, are sufficient to induce the Kondo resonance. Due to peculiar co-tunneling events, the Kondo resonance in one dot may be p…
We consider a N-dot-S junction in the Kondo regime in the limit where the superconducting gap is much smaller than the Kondo temperature. A generalization of the floating of the Kondo resonance is proposed and many body corrections to the average subgap current are calculated. The zero frequency noise is computed and the Fano factor sticks to the value 10/3 for all voltages below the gap. Implications for finite frequency noise are briefly discussed.
We study phase evolution of transmission through a quantum dot with Kondo correlations. By considering a model that includes nonresonant transmission as well as the Anderson impurity, we explain unusually large phase evolution of about $π$ in the Kondo valley observed in recent experiments. We argue that this anomalous phase evolution is a universal property that can be found in the high-temperature Kondo phase in the presence of the time-reversal symmetry.
We consider spin and orbital Kondo effect in a parallel arrangement of two strongly electrostatically coupled quantum dots. Increasing the exchange of electrons between the dots through the attached leads induces a smooth crossover between SU(4) spin- and orbital Kondo effect and SU(2) spin Kondo effect. Being the same for the SU(4) and SU(2) symmetry points, the Kondo temperature drops slightly in the intermediate regime. Experimentally, two kinds of Kondo effect can be discriminated by the sensitivity to the suppression of the spin Kondo effect by Zeeman field. The dependence of the Kondo te…
