We investigate the properties of the Kitaev-Kondo lattice model defined on a bilayer honeycomb lattice by means of the SO(3) Majorana representation for spin-$1/2$ moments. We first consider the pairing of neighboring sites for the parent Kitaev spin liquid (KSL) Hamiltonian to render the Majorana and the spin-$1/2$ Hilbert spaces perfectly equivalent to each other. As a consequence, we demonstrate that this decoupling of the Kitaev interaction in terms of the SO(3) Majorana fermions reproduces exactly the spectrum of the KSL model alone. Then, by considering the effect of a local Kondo coupling $J_K$ in the model and decoupling it in terms of an order parameter that physically must have a finite staggering phase, we obtain that the system undergoes a quantum phase transition from a fractionalized Fermi liquid to a nematic triplet superconducting (SC) phase as $J_K$ is increased. Depending on the model parameters, this SC phase can exhibit either Dirac points, Bogoliubov-Fermi lines, or Bogoliubov-Fermi surfaces as nodal bulk manifolds. The surface states in this latter case are also characterized by topologically protected antichiral edge modes. The SC phase breaks time-reversal symmetry and exhibits a coexistence of a dominant odd-frequency pairing with a small even-frequency component for electronic excitations localized on sites of the same sublattice of the system. Finally, we show that this SC phase is in fact a pair-density-wave state, with Cooper pairs possessing a finite center-of-mass momentum in zero magnetic field.
Geometrically frustrated quantum impurities coupled to metallic leads have been shown to exhibit rich behavior with a quantum phase transition separating Kondo screened and local moment phases. Frustration in the quantum impurity can alternatively be introduced via Kitaev-couplings between different spins of the impurity cluster. We use the Numerical Renormalization Group (NRG) to study a range of systems where the quantum impurity comprising a Kitaev cluster is coupled to a bath of non-interacting fermions. The models exhibits a competition between Kitaev and Kondo dominated physics depending…
We investigate possible topological superconductivity in the Kondo-Kitaev model on the honeycomb lattice, where the Kitaev spin liquid is coupled to conduction electrons via the Kondo coupling. We use the self-consistent Abrikosov-fermion mean-field theory to map out the phase diagram. Upon increasing the Kondo coupling, a first order transition occurs from the decoupled phase of spin liquid and conduction electrons to a ferromagnetic topological superconductor of Class D with a single chiral Majorana edge mode. This is followed by a second order transition into a paramagnetic topological supe…
Paramagnetic impurities in a quantum spin-liquid can result in Kondo effects with highly unusual properties. We have studied the effect of locally exchange-coupling a paramagnetic impurity with the spin-1/2 honeycomb Kitaev model in its gapless spin-liquid phase. The (impurity) scaling equations are found to be insensitive to the sign of the coupling. The weak and strong coupling fixed points are stable, with the latter corresponding to a noninteracting vacancy and an interacting, spin-1 defect for the antiferromagnetic and ferromagnetic cases respectively. The ground state in the strong coupl…
Kitaev's honeycomb-lattice compass model describes a spin liquid with emergent fractionalized excitations. Here we study the physics of isolated magnetic impurities coupled to the Kitaev spin-liquid host. We reformulate this Kondo-type problem in terms of a many-state quantum impurity coupled to a multichannel bath of Majorana fermions and present the numerically exact solution using Wilson's numerical renormalization group technique. Quantum phase transitions occur as a function of Kondo coupling and locally applied field. At zero field, the impurity moment is partially screened only when it …
Fractionalized Fermi liquids (FL$^\ast$) have been introduced as non-Fermi-liquid metallic phases, characterized by coexisting electron-like charge carriers and local moments which itself form a fractionalized spin liquid. Here we investigate a Kondo lattice model on the honeycomb lattice with compass interactions among the local moments, a concrete model hosting FL$^\ast$ phases based on Kitaev's $\mathbb{Z}_2$ spin liquid. We characterize the FL$^\ast$ phases via perturbation theory, and we employ a Majorana-fermion mean-field theory to map out the full phase diagram. Most remarkably we find…
We describe a mechanism for order fractionalization in a two-dimensional Kondo lattice model, in which electrons interact with a gapless spin liquid of Majorana fermions described by the Yao-Lee (YL) model. When the Kondo coupling to the conduction electrons exceeds a critical value, the model develops a superconducting instability into a state where the spinor order parameter carries charge $e$ and spin $S=1/2$. The broken symmetry state develops a gapless Majorana Dirac cone in the bulk. By including an appropriate gauge string, we can show that the charge $e$, spinorial order develops off-d…
A spin-$\frac{1}{2}$ Kitaev sublattice interacting with a subsystem of spinless fermions is studied on a honeycomb lattice when the fermion band is half filled. The model Hamiltonian describes a topological Kondo lattice with the Kitaev interaction, it is solved exactly by reduction to free Majorana fermions in a static $\mathbb{Z}_2$ gauge field. An yet unsolved problem of a hybridization of fermions and local moments in the Kondo lattice at low temperatures is solved in the framework of the model proposed. The Kondo hybridization gap is opened and the system is fixed in insulator and spin in…
Renormalization group analysis for bosonization coefficients in half-odd-integer Kitaev spin chains
School of Physics, Nankai University（南开大学物理学院）；Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences（中国科学院大学 Kavli理论科学研究所）
We investigate a modified Anderson model at the large-$N$ limit, where Coulomb interaction is replaced by Sachdev-Ye-Kitaev random interaction. The resistivity of conduction electron $ρ_{c}$ has a minimum value around temperature $\widetilde{T}_{K}$, which is similar to the Kondo system, but the impurity electron's density of state $A_{d}(ω)$ elucidates no sharp-peak like Kondo resonance around the Fermi surface. The impurity electron's entropy $S_{d}$ and specific heat capacity $C_{\textrm{v}}$ illustrate a crossover from Fermi liquid to the non-Fermi liquid. The system is a non-Fermi liquid …
Cotunneling into Kondo systems, where an electron enters a $f$-electron material via a cotunneling process through the local-moment orbital, has been proposed to explain the characteristic lineshapes observed in scanning-tunneling-spectroscopy (STS) experiments. Here we extend the theory of electron cotunneling to Kondo-lattice systems where the bulk hybridization between conduction and $f$ electrons is odd under inversion, being particularly relevant to Kondo insulators. Compared to the case of even hybridization, we show that the interference between normal tunneling and cotunneling processe…
We show that a single-fermion quantum dot acquires odd-frequency Gor'kov anomalous averages in proximity to strongly correlated Majorana zero modes, described by the Sachdev-Ye-Kitaev (SYK) model. Despite the presence of finite anomalous pairing, the superconducting gap vanishes for the intermediate coupling strength between the quantum dot and Majoranas. The increase of the coupling leads to smooth suppression of the original quasiparticles. This effect might be used as a characterization tool for recently proposed tabletop realizations of the SYK model.
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 odd-membered chains of spin-(1/2) impurities, with each end connected to its own metallic lead. For antiferromagnetic exchange coupling, universal two-channel Kondo (2CK) physics is shown to arise at low energies. Two overscreening mechanisms are found to occur depending on coupling strength, with distinct signatures in physical properties. For strong inter-impurity coupling, a residual chain spin-(1/2) moment experiences a renormalized effective coupling to the leads; while in the weak-coupling regime, Kondo coupling is mediated via incipient single-channel Kondo singlet formation. W…
The Kondo lattice model plays a key role in our understanding of quantum materials, but a lack of small parameters has posed a long-standing problem. We present a 3 dimensional S= 1/2 Kondo lattice model describing a spin liquid within an electron sea. Strong correlations in the spin liquid are treated exactly, enabling a controlled analytical approach. Like a Peierls or BCS phase, a logarithmically divergent susceptibility leads to an instability into a new phase at arbitrarily small Kondo coupling. Our solution captures a plethora of emergent phenomena, including odd-frequency pairing, pair …
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 …
A formulation of the Kondo lattice Hamiltonian in terms of bond particles is derived and solved in two different approximations. The bond particles correspond to the eigenstates of a single unit cell and are bosons for states with even electron number and fermions for states with an odd electron number. A s a check various physical quantities are calculated for the 1D Kondo insulator a nd good agreement with numerical results is obtained for J/t>1.
We discuss the odd-frequency pairing correlations discovered by Zachar, Kivelson and Emery (ZKE) in a one dimensional Kondo lattice. A lattice model that realizes the continuum theory of ZKE is introduced and the correlations it gives rise to are identified as odd-frequency singlet pairing. The excitation spectrum is found to contain a spin gap, and a much lower energy band of spinless excitations. We discuss how the power-law correlations realized in the ZKE model evolve into true long-range order when Kondo chains are weakly coupled together and tentatively suggest a way in which the higher …
I construct non-Fermi liquid (NFL) quantum impurity models that are similar to the overscreened multi-channel Kondo models with the difference that an odd number of electron species screen the impurity. The simplest of them, named sesqui-channel (i.e. one-and-a-half-channel) Kondo (1.5CK) model, has less degrees of freedom and is simpler than the two-channel Kondo model, and yet it exhibits NFL physics. Using representation theory I derive the 1.5CK model for a spin-half impurity surrounded with electrons in cubic crystal field and solve it with the numerical renormalization group.
The Kitaev model is an exactly-soluble quantum spin model, whose ground state provides a canonical example of a quantum spin liquid. Spin excitations from the ground state are fractionalized into emergent matter fermions and $Z_2$ fluxes. The $Z_2$ flux excitation is pointlike in two dimensions, while it comprises a closed loop in three dimensions because of the local constraint for each closed volume. In addition, the fluxes obey global constraints involving (semi)macroscopic number of fluxes. We here investigate such global constraints in the Kitaev model on a three-dimensional lattice compo…
We study the thermodynamic properties of modified spin-$S$ Kitaev models introduced by Baskaran, Sen and Shankar (Phys. Rev. B 78, 115116 (2008)). These models have the property that for half-odd-integer spins their eigenstates map on to those of spin-1/2 Kitaev models, with well-known highly entangled quantum spin-liquid states and Majorana fermions. For integer spins, the Hamiltonian is made out of commuting local operators. Thus, the eigenstates can be chosen to be completely unentangled between different sites, though with a significant degeneracy for each eigenstate. For half-odd-integer …
We present a one dimensional model of electrons coupled to localized moments of spin $S\ge 1$ in which magnetism and superconductivity interplay in a nontrivial manner. This model has a non-Fermi liquid ground state of the chiral spin liquid type. A non-conventional odd-frequency pairing is shown to be the dominant instability of the system, together with antiferromagnetism of the local moments. We argue that this model captures the physics of the Kondo-Heisenberg spin S=1 chain, in the limit of strong Kondo coupling. Finally, we discuss briefly the effect of interchain coupling.
We study the behavior of the entropy of the pseudogap Bose-Fermi Kondo model within a dynamical large-$N$ limit, where $N$ is related to the symmetry group of the model. This model is a general quantum impurity model that describes a localized level coupled to a fermionic bath having a density of states that vanishes in a powerlaw fashion near the Fermi energy and to a bosonic bath possessing a powerlaw spectral density below a cutoff energy. As a function of the couplings to the baths various quantum phase transitions can occur. We study how the impurity entropy changes across these zero-temp…
We study the effect of coupling magnetic impurities to the honeycomb lattice spin-1/2 Kitaev model in its spin liquid phase. We show that a spin-S impurity coupled to the Kitaev model is associated with an unusual Kondo effect with an intermediate coupling unstable fixed point K_c J/S separating topologically distinct sectors of the Kitaev model. We also show that the massless spinons in the spin liquid mediate an interaction of the form S_{iα}^{2}S_{jβ}^{2}/R_{ij}^{3} between distant impurities unlike the usual dipolar RKKY interaction S_{iα}S_{jα}/R_{ij}^{3} noted in various 2D impurity prob…
We have studied the two-dimensional two-impurity Anderson model with additional Rashba spin-orbit interaction by means of the modified perturbation theory. The impurity Green's functions we have constructed exactly reproduce the first four spectral moments. We discuss the height and the width of the even/odd Kondo peaks as functions of the inter-impurity distance and the Rashba energy $E_R$ (the strength of the Rashba spin-orbit interaction). For small impurity separations the Kondo temperature shows a non-monotonic dependence on $E_R$ being different in the even and the odd channel. We predic…
We analyze the Kondo effect of a magnetic impurity attached to an ultrasmall metallic wire using the density matrix renormalization group. The spatial spin correlation function and the impurity spectral density are computed for system sizes of up to L=511 sites, covering the crossover from $L<\ell_K$ to $L > \ell_K$, with $\ell_K$ the spin screening length. %Strong mesoscopic variations of the Kondo temperature $T_K$ %and of the spectral features override, to some extent, the %even/odd effect predicted earlier for averaged quantities. We establish a proportionality between the weight of the Ko…
Numerical analysis of the simplest odd-numbered system of coupled quantum dots reveals an interplay between magnetic ordering, charge fluctuations and the tendency of itinerant electrons in the leads to screen magnetic moments. The transition from local-moment to molecular-orbital behavior is visible in the evolution of correlation functions as the inter-dot coupling is increased. Resulting novel Kondo phases are presented in a phase diagram which can be sampled by measuring the zero-bias conductance. We discuss the origin of the even-odd effects by comparing with the double quantum dot.
We study the Kondo effect in a quantum dot embedded in a mesoscopic ring taking into account intradot spin-flip scattering $R$. Based on the finite-$U$ slave-boson mean-field approach, we find that the Kondo peak in the density of states is split into two peaks by this coherent spin-flip transition, which is responsible for some interesting features of the Kondo-assisted persistent current circulating the ring: (1) strong suppression and crossover to a sine function form with increasing $R$; (2) appearance of a "hump" in the $R$-dependent behavior for odd parity. $R$-induced reverse of the per…
We propose an exactly solvable model for $j_{\text{eff}}=\frac32$ local moments on the honeycomb lattice. Our construction is guided by a symmetry analysis and by the requirement of an exact solution in terms of a Majorana fermion representation for multipole operators. The main interaction in the model can be interpreted as a bond-dependent quadrupole-quadrupole interaction. When time reversal symmetry is explicitly broken, we obtain a gapped spin liquid with a single chiral Majorana edge mode. We also investigate another solvable model in which the time-reversal-invariant spin liquid is coup…
The effective spin Hamiltonian of a triple quantum dot with odd electron occupation weakly connected in series with left ($l$) and right ($r$) metal leads is composed of two-channel exchange and co-tunneling terms. Renormalization group equations for the corresponding three exchange constants $J_{l}$, $J_{r}$ and $J_{lr}$ are solved (to third order). Since $J_{lr}$ is relevant, the system is mapped on an anisotropic two-channel Kondo problem. The structure of the conductance as function of temperature and gate voltage implies that in the weak and intermediate coupling regimes, two-channel Kond…
We consider Dirac electrons on the honeycomb lattice Kondo coupled to spin-1/2 degrees of freedom on the kagome lattice. The interactions between the spins are chosen along the lines of the Balents-Fisher-Girvin model that is known to host a $\mathbb{Z}_2$ spin liquid and a ferromagnetic phase. The model is amenable to sign free auxiliary field quantum Monte Carlo simulations. While in the ferromagnetic phase the Dirac electrons acquire a gap, they remain massless in the $\mathbb{Z}_2$ spin liquid phase due to the breakdown of Kondo screening. Since our model has an odd number of spins per uni…
Emergence of odd-frequency s-wave superconductivity is demonstrated in the two-channel Kondo lattice by means of the dynamical mean-field theory combined with the continuous-time quantum Monte Carlo method. Around half filling of the conduction bands, divergence of an odd-frequency pairing susceptibility is found, which signals instability toward the superconductivity. The corresponding order parameter is equivalent to a staggered composite-pair amplitude with even frequencies, which involves both localized spins and conduction electrons. A model wave function is constructed for the composite …
Spin-triplet $p$-wave superconductors are promising candidates for topological superconductors. They have been proposed in various heterostructures where a material with strong spin-orbit interaction is coupled to a conventional $s$-wave superconductor by proximity effect. However, topological superconductors existing in nature and driven purely by strong electron correlations are yet to be studied. Here we propose a realization of such a system in a class of Kondo lattice materials in the absence of spin-orbit coupling and proximity effect. Therein, the odd-parity Kondo hybridization mediates…
We study strong f-electron localization effect on surface state of a generic topological Kondo insulator (TKI) system by performing a mean-field theoretic (MFT) calculation within the frame-work of periodic Anderson model (PAM) using slave-boson technique. The surface metallicity together with bulk insulation is found to require this type of localization. A key distinction between surface states in a conventional insulator and a topological insulator is that, along a course joining two time-reversal invariant momenta (TRIM) in the same BZ, there will be intersection of these surface states, ev…
In a large number of rare-earth and actinide systems, Kondo effect tends to suppress magnetic order by making the spin singlet between localized and conduction electron spins. In the presence of orbital degrees of freedom, however, there emerge exotic electronic orders induced by Kondo effect. The orbital Kondo effect can collectively make diagonal and off-diagonal (superconducting) orders. With the particle-hole symmetry in conduction bands, these orders are all degenerate, forming a macroscopic SO(5) multiplet. This paper discusses recent theoretical development on these electronic orders wh…
The key to unraveling intriguing phenomena observed in various Kitaev materials lies in understanding the interplay of Kitaev ($K$) interaction and a symmetric off-diagonal $Γ$ interaction. To provide insight into the challenging problems, we study the quantum phase diagram of a bond-alternating spin-$1/2$ $g_x$-$g_y$ $K$-$Γ$ chain by density-matrix renormalization group method where $g_x$ and $g_y$ are the bond strengths of the odd and even bonds, respectively. The phase diagram is dominated by even-Haldane ($g_x > g_y$) and odd-Haldane ($g_x < g_y$) phases where the former is topologically t…
We propose that two-channel orbital Kondo ``spin'' 1/2 conductance can be measured in a quantum dot at Coulomb Blockade with an odd number of electrons with contacts in a pillar configuration, if an orthogonal magnetic field induces an appropriate level crossing. At the zero-temperature strong coupling fixed point the conductance reaches the unitarity limit with a non-Fermi liquid sqrt(T)-law.
Chiral spin liquids (CSLs) are time-reversal symmetry breaking ground states of frustrated quantum magnets that show no long-range magnetic ordering, but instead exhibit topological order and fractional excitations. Their realization in simple and tractable microscopic models has, however, remained an open challenge for almost two decades until it was realized that Kitaev models on lattices with odd-length loops are natural hosts for such states, even in the absence of a time-reversal symmetry breaking magnetic field. Here we report on the formation of CSLs in a three-dimensional Kitaev model,…
We construct a lattice model for a cubic Kondo insulator consisting of one spin-degenerate $d$ and $f$ orbital at each lattice site. The odd-parity hybridization between the two orbitals permits us to obtain various trivial and topological insulating phases, which we classify in the presence of cubic symmetry. In particular, depending on the choice of our model parameters, we find a strong topological insulator phase with a band inversion at the $\mathrm{X}$ point, modeling the situation potentially realized in SmB$_6$, and a topological crystalline insulator phase with trivial $\mathbb{Z}_2$ …
The Kondo insulator SmB6 has long been known to exhibit low temperature transport anomalies whose origin is of great interest. Here we uniquely access the surface electronic structure of the anomalous transport regime by combining state-of-the-art laser- and synchrotron-based angle-resolved photoemission techniques. We observe clear in-gap states (up to 4 meV), whose temperature dependence is contingent upon the Kondo gap formation. In addition, our observed in-gap Fermi surface oddness tied with the Kramers' points topology, their coexistence with the two-dimensional transport anomaly in the …
We consider transport through a single-molecule magnet strongly coupled to metallic electrodes. We demonstrate that for half-integer spin of the molecule electron- and spin-tunneling \emph{cooperate} to produce both quantum tunneling of the magnetic moment and a Kondo effect in the linear conductance. The Kondo temperature depends sensitively on the ratio of the transverse and easy-axis anisotropies in a non-monotonic way. The magnetic symmetry of the transverse anisotropy imposes a selection rule on the total spin for the occurrence of the Kondo effect which deviates from the usual even-odd a…
We construct an exactly soluble spin-$\frac{1}2$ model on a honeycomb lattice, which is a generalization of Kitaev model. The topological phases of the system are analyzed by study of the ground state sector of this model, the vortex-free states. Basically, there are two phases, A phase and B phase. The behaviors of both A and B phases may be studied by mapping the ground state sector into a general p-wave paired states of spinless fermions with tunable pairing parameters on a square lattice. In this p-wave paired state theory, the A phase is shown to be the strong paired phase, an insulating …
