A triangular triple quantum dot gives various Kondo effects, such as the emergence of an electric polarization accompanied by a complete compensation of spin degrees of freedom. The interplay of spin and charge of electrons in quantum dots is investigated using an Anderson model with an equilateral triangular spin cluster, in which a single electron state at one site is hybridized with a conduction band in a lead. The numerical renormalization group analysis shows how a nonzero electric polarization develops with local electrons traveling in the loop and how it depends on the hybridization strength as an experimentally controllable parameter in the Kondo effect.
The triangular triple quantum dot is an interesting system which can demonstrate various types of the Kondo effects, such as the one due to the local spin S=1 moment caused by the Nagaoka ferromagnetic mechanism and the SU(4) Kondo effect. We theoretically study the low-temperature properties and the Kondo energy scale of the triangular triple quantum dot, using the Wilson numerical renormalization group. We have explored a wide parameter region of the electron-filling and distortions which break the symmetry of an equilateral structure. Our results give a comprehensive overview of how the Kon…
We study an orbitally degenerate Kondo effect in a triangular triple quantum dot (TTQD), where the three dots are connected vertically with a single metallic lead through electron tunneling. Both spin and orbital degrees of freedom play an important role in the SU(4) Kondo effect. This is demonstrated by an equilateral TTQD Kondo system at half-filling, by Wilson's numerical renormalization group method. We show how an emergent electric polarization of the TTQD is associated with a crossover from SU(4) to SU(2) symmetry in the low-temperature state. A marked sign reversal of the electric polar…
The role of discrete orbital symmetry in mesoscopic physics is manifested in a system consisting of three identical quantum dots forming an equilateral triangle. Under a perpendicular magnetic field, this system demonstrates a unique combination of Kondo and Aharonov-Bohm features due to an interplay between continuous [spin-rotation SU(2)] and discrete (permutation C3v) symmetries, as well as U(1) gauge invariance. The conductance as a function of magnetic flux displays sharp enhancement or complete suppression depending on contact setups.
We study the local antisymmetric spin-orbit (ASO) coupling effect on spin, orbital, and charge degrees of freedom for the Kondo effect in a triangular triple quantum dot (TTQD). Here, one of the three QDs is coupled to a metallic lead through electron tunneling, and a local electric polarization is induced by the Kondo effect. The ASO interaction is introduced in the other two coupled QDs on the opposite side of the lead. Generally, the ASO coupling effect is very weak and not easily detectable, but it essentially causes spin and charge reconfigurations in the TTQD through the Kondo effect. Us…
How many magnetic moments periodically arranged on a metallic surface are needed to generate a coherent Kondo lattice behavior? We investigate this fundamental issue within the particle-hole symmetric Kondo lattice model using quantum Monte Carlo simulations. Extra magnetic atoms forming closed shells around the initial impurity induce a fast splitting of the Kondo resonance at the inner shells which signals the formation of composite heavy-fermion bands. The onset of the hybridization gap matches well the enhancement of antiferromagnetic spin correlations in the plane perpendicular to the app…
We study the conductance through a triangular triple quantum dot, which are connected to two noninteracting leads, using the numerical renormalization group (NRG). It is found that the system shows a variety of Kondo effects depending on the filling of the triangle. The SU(4) Kondo effect occurs at half-filling, and a sharp conductance dip due to a phase lapse appears in the gate-voltage dependence. Furthermore, when four electrons occupy the three sites on average, a local S=1 moment, which is caused by the Nagaoka mechanism, is induced along the triangle. The temperature dependence of the en…
We study strongly correlated states in triangular artificial atoms. Symmetry-driven orbital degeneracy of the single particle states can give rise to an SU(4) Kondo state with entangled orbital and spin degrees of freedom, and a characteristic phase shift $δ= π/4$. Upon application of a Zeeman field, a purely orbital Kondo state is formed with somewhat smaller Kondo temperature and a fully polarized current through the device. The Kondo temperatures are in the measurable range. The triangular atom also provides a tool to systematically study the singlet-triplet transitions observed in recent e…
Transport properties of an interacting triple quantum dot system coupled to three leads in a triangular geometry has been studied in the Kondo regime. Applying mean-field finite-U slave boson and embedded cluster approximations to the calculation of transport properties unveils a set of rich features associated to the high symmetry of this system. Results using both calculation techniques yield excellent overall agreement and provide additional insights into the physical behavior of this interesting geometry. In the case when just two current leads are connected to the three-dot system, interf…
Chiral Magnetism and Quantum Anomalous Hall Effect in a Low-energy Kondo Model on the Triangular Lattice
University of California Santa Barbara（加州大学圣塔芭芭拉分校）；Kavli Institute for Theoretical Physics, University of California, Santa Barbara（加州大学圣塔芭芭拉分校卡弗里理论物理研究所）；French American Center for Theoretical Science, CNRS, KITP, Santa Barbara, California（法国美国理论科学中心）；Canadian Institute for Advanced Research（加拿大高级研究院）
We study the low-energy properties of a triangular triple quantum dot connected to two non-interacting leads in a wide parameter range, using the numerical renormalization group (NRG). Various kinds of Kondo effects take place in this system depending on the electron filling N_{tot}, or the level position $ε_d$ of the triple dot. The SU(4) Kondo behavior is seen in the half-filled case N_{tot} = 3.0 at the dip of the series conductance, and it causes a charge redistribution between the even and odd orbitals in the triangle. We show generally that the quasi-particle excitations from a local Fer…
We study transport through a triangle triple quantum dot connected to two noninteracting leads using the numerical renormalization group (NRG). The triangle has a high-spin ground state of S=1 caused by a Nagaoka ferromagnetism, when it is isolated and has one extra electron introduced into a half-filling. The results show that the conduction electrons screen the local moment via two separate stages with different energy scales. The half of the S=1 is screened first by one of the channel degrees, and then at very low temperature the remaining half is fully screened to form a Kondo singlet. The…
We examine the low energy behavior of a double quantum dot in a regime where spin and pseudospin excitations are degenerate. The individual quantum dots are described by Anderson impurity models with an on-site interaction $U$ which are capacitively coupled by an interdot interaction $U_{12}<U$. The low energy response functions are expressed in terms of renormalized parameters, which can be deduced from an analysis of the fixed point in a numerical renormalization group calculation. At the point where the spin and pseudospin degrees of freedom become degenerate, the free quasiparticle excitat…
We study triangular clusters of three spin-1/2 Kondo or Anderson impurities that are coupled to two conduction leads. In the case of Kondo impurities, the model takes the form of an antiferromagnetic Heisenberg ring with Kondo-like exchange coupling to continuum electrons. We show that this model exhibits many types of the behavior found in various simpler one and two-impurity models, thereby enabling the study of crossovers between a number of Fermi-liquid (FL) and non-Fermi-liquid (NFL) fixed points. In particular, we explore a direct crossover between the two-impurity Kondo-model NFL fixed …
A leading candidate for experimental confirmation of the non-local quantum dynamics of Majorana fermions is the topological Kondo effect, predicted for mesoscopic superconducting islands connected to metallic leads. We identify an anisotropic, Toulouse-like, limit of the topological Kondo problem where the full nonequilibrium conductance and shot noise can be calculated exactly. Near the Kondo fixed point, we find novel asymptotic features including a universal conductance scaling function, and fractional charge quantisation observable via the Fano factor. In the universal regime, our results …
We study the Kondo Lattice and Hubbard models on a triangular lattice for band filling factor 3/4. We show that a simple non-coplanar chiral spin ordering (scalar spin chirality) is naturally realized in both models due to perfect nesting of the Fermi surface. The resulting triple-Q magnetic ordering is a natural counterpart of the collinear Neel ordering of the half-filled square lattice Hubbard model. We show that the obtained chiral phase exhibits a spontaneous quantum Hall-effect with sigma_{xy}= e^2/h.
We show that the Bose-Fermi Kondo model (BFKM), which may find applicability both to certain dissipative mesoscopic qubit devices and to heavy fermion systems described by the Kondo lattice model, can be mapped exactly onto the Caldeira-Leggett model. This mapping requires an ohmic bosonic bath and an Ising-type coupling between the latter and the impurity spin. This allows us to conclude unambiguously that there is an emergent Kosterlitz-Thouless quantum phase transition in the BFKM with an ohmic bosonic bath. By applying a bosonic numerical renormalization group approach, we thoroughly probe…
Emergent Gauge Flux and Spin Ordering in Magnetized Triangular Spin Liquids: Applications to Hofstadter-Hubbard Model
International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料中心）；Beijing Key Laboratory of Quantum Devices, Peking University（北京大学北京量子器件重点实验室）；Collaborative Innovation Center of Quantum Matter（量子物质协同创新中心）
We propose a realization of Kondo-lattice physics in moiré superlattices at the interface between a WX$_2$ homobilayer and MoX$_2$ monolayer (where X=S,Se). Under appropriate gating conditions, the interface-WX$_2$-layer forms a triangular lattice of local moments that couple to itinerant electrons in the other WX$_2$-layer via a gate-tunable Kondo exchange interaction. Using a parton mean-field approach we identify a range of twist-angles which support a gate-tuned quantum phase transition between a heavy-fermion liquid with large anomalous Hall conductance and a fractionalized chiral spin-li…
An idea is proposed for realizing a fully spin-polarized Dirac semimetal in frustrated itinerant magnets. We show that itinerant electrons on a triangular lattice exhibit the Dirac cone dispersion with half-metallic behavior in the presence of a three-sublattice ferrimagnetic order. The Dirac nodes have the same structure as those of graphene. By variational calculation and Monte Carlo simulation, we demonstrate that the ferrimagnetic order with the Dirac node spontaneously emerges in a simple Kondo lattice model with Ising anisotropy. The realization will be beneficial for spintronics as a ca…
We analyze theoretically the phase diagram of a triangular triple quantum dot with strong onsite repulsion coupled to ferromagnetic leads. This model includes the competition of magnetic ordering of local or itinerant magnetic moments, geometric frustration and Kondo screening. We identify all the phases resulting from this competition. We find that three Kondo phases -- the conventional one, the two-stage underscreened one, and the one resulting from the ferromagnetic Kondo effect -- can be realized at zero temperature, and all are very susceptible to the proximity of ferromagnetic leads. In …
We investigate theoretically the thermoelectric transport through a circuit implementation of the three-channel "charge" Kondo model quantum simulator [Z. Iftikhar et al., Science 360, 1315 (2018)]. The universal temperature scaling law of the Seebeck coefficient is computed perturbatively approaching the non-Fermi liquid strong coupling fixed point using abelian bosonization technique. The predicted $T^{1/3}\log T$ scaling behavior of the thermoelectric power sheds a light on the properties of $Z_3$ emerging parafermions and gives an access to exploring pre-fractionalized zero modes in the qu…
A quantum point contact (QPC) is a very basic nano-electronic device: a short and narrow transport channel between two electron reservoirs. In clean channels electron transport is ballistic and the conductance $G$ is then quantised as a function of channel width with plateaus at integer multiples of $2e^2/h$ ($e$ is the electron charge and $h$ Planck's constant). This can be understood in a picture where the electron states are propagating waves, without need to account for electron-electron interactions. Quantised conductance could thus be the signature of ultimate control over nanoscale elec…
We consider a quantum dot with ${\cal K}{\geq} 2$ orbital levels occupied by two electrons connected to two electric terminals. The generic model is given by a multi-level Anderson Hamiltonian. The weak-coupling theory at the particle-hole symmetric point is governed by a two-channel $S{=}1$ Kondo model characterized by intrinsic channels asymmetry. Based on a conformal field theory approach we derived an effective Hamiltonian at a strong-coupling fixed point. The Hamiltonian capturing the low-energy physics of a two-stage Kondo screening represents the quantum impurity by a two-color local Fe…
A Coulomb blockaded $M$-Majorana island coupled to normal metal leads realizes a novel type of Kondo effect where the effective impurity "spin" transforms under the orthogonal group $SO(M)$. The impurity spin stems from the non-local topological ground state degeneracy of the island and thus the effect is known as the topological Kondo effect. We introduce a physically motivated $N$-channel generalization of the topological Kondo model. Starting from the simplest case $N=2$, we conjecture a stable intermediate coupling fixed point and evaluate the resulting low-temperature impurity entropy. Th…
We show that the spin-density wave state on the partially filled honeycomb and triangular lattices is preempted by a paramagnetic phase that breaks an emergent Z_4 symmetry of the system, associated with the four inequivalent arrangements of spins in the quadrupled unit cell. Unlike other emergent paramagnetic phases in itinerant and localized-spin systems, this state preserves the rotational symmetry of the lattice but breaks its translational symmetry, giving rise to a super-lattice structure that can be detected by scanning tunneling microscopy. This emergent phase also has distinctive sign…
We investigate a well studied system of a quantum dot coupled to a Coulomb box and leads, realizing a spin-flavor Kondo model. It exhibits a recently discovered non-Fermi liquid (NFL) behavior with emergent SO(5) symmetry. Here, through a detailed bosonization and refermionization solution, we push forward our previous work and provide a consistent and complete description of the various exotic properties and phase diagram. A unique NFL phase emerges from the presence of an uncoupled Majorana fermion from the flavor sector, whereas FL-like susceptibilities result from the gapping out of a pair…
We analyze the single-channel Kondo model using the recently developed unitary renormalization group (URG) method, and obtain a comprehensive understanding of the Kondo screening cloud. The fixed-point low-energy Hamiltonian enables the computation of a plethora of thermodynamic quantities (specific heat, susceptibility, Wilson ratio, etc.) as well as spectral functions, all of which are found to be in excellent agreement with known results. By integrating out the impurity, we obtain an effective Hamiltonian for the excitations of the electrons comprising the Kondo cloud. This is found to cont…
We study ground-state properties of a triangular triple quantum dot connected to two superconducting (SC) leads. In this system orbital motion along the triangular configuration causes various types of quantum phases, such as a Kondo effect with a four-fold degenerate state and the Nagaoka ferromagnetic mechanism, depending on the electron filling. The ground state also evolves as the Cooper pairs penetrate from the SC leads. We describe the phase diagram in a wide range of the parameter space, varying the gate voltage, the couplings between the dots and leads, and also the Josephson phase bet…
The Kondo insulator compound SmB6 has emerged as a strong candidate for the realization of a topologically nontrivial state in a strongly correlated system, a topological Kondo insulator, which can be a novel platform for investigating the interplay between nontrivial topology and emergent correlation driven phenomena in solid state systems. Electronic transport measurements on this material, however, so far showed only the robust surface dominated charge conduction at low temperatures, lacking evidence of its connection to the topological nature by showing, for example, spin polarization due …
This work considers a two-dimensional artificial triangular anti-dot lattice (TAL); a semiconductor based artificial crystal hosting Dirac cones, flat bands and Fermi surface nesting. All such single particle features have dramatic implications for the emergent correlated phases. This work predominantly focuses on the existence of a robust flatband and enumerates the possible correlated phases that follow. We find that the flatband is generated, in the single-particle theory, when charges align themselves along a kagome lattice with the same period as the TAL. The correlated phases are studied…
Molecular electronic devices currently serve as a platform for studying a variety of physical phenomena only accessible at the nanometer scale. One such phenomenon is the highly correlated electronic state responsible for the Kondo effect, manifested here as a "Kondo resonance" in the conductance. Because the Kondo effect results from strong electron-electron interactions, it is not captured by the usual quantum chemistry approaches traditionally applied to understand chemical electron transfer. In this review we will discuss the origins and phenomenology of Kondo resonances observed in single…
We study nonequilibrium generation of spin-dependent transport through a single-level quantum dot embedded in a ring with the Rashba spin-orbit coupling. We consider nonmagnetic systems, involving no magnetic field nor ferromagnetic leads. It is theoretically predicted that large spin-dependent current occurs as a combined effect of the Rashba spin-orbit interaction, the Kondo effect, and nonequilibrium effect, without using magnetic field or material. The phenomenon is viewed as a new nonequilibrium correlation effect that disappears when either interaction or finite bias is absent. We show h…
Topology and strong electron correlations are crucial ingredients in emerging quantum materials, yet their intersection in experimental systems has been relatively limited to date. Strongly correlated Weyl semimetals, particularly when magnetism is incorporated, offer a unique and fertile platform to explore emergent phenomena in novel topological matter and topological spintronics. The antiferromagnetic Weyl semimetal Mn3Sn exhibits many exotic physical properties such as a large spontaneous Hall effect and has recently attracted intense interest. In this work, we report synthesis of epitaxia…
We study one-dimensional Kondo Lattices (KL) which consist of itinerant electrons interacting with Kondo impurities (KI) - localized quantum magnetic moments. We focus on KL with isotropic exchange interaction between electrons and KI and with a high KI density. The latter determines the principal róle of the indirect interaction between KI for the low energy physics. Namely, the Kondo physics becomes suppressed and all properties are governed by spin ordering. We present a first-ever comprehensive analytical theory of such KL at an arbitrary doping and predict a variety of regimes with differ…
Graphene can develop large magnetic moments in custom crafted open-shell nanostructures such as triangulene, a triangular piece of graphene with zigzag edges. Current methods of engineering graphene nano-systems on surfaces succeeded in producing atomically precise open-shell structures, but demonstration of their net spin remains elusive to date. Here, we fabricate triangulene-like graphene systems and demonstrate that they possess a spin $S=1$ ground state. Scanning tunnelling spectroscopy identifies the fingerprint of an underscreened $S=1$ Kondo state on \rev{these} flakes at low temperatu…
