Two-channel Kondo Problem Research Articles

Multiterminal open quantum dot circuit operating in the fractional quantum Hall regime

We study theoretically a large quantum dot in the fractional quantum Hall regime that is strongly coupled to $\mathcal{M}$ leads via single-mode quantum point contacts. In the case $\mathcal{M}=2$, when the system is mapped onto the two-channel charge Kondo problem, we predict a universal expression for the conductance in the vicinity of a strong-coupling fixed point. The power of the leading temperature correction to the maximal conductance is determined by the fractional filling factor $\ensuremath{\nu}=1/m$. For $\mathcal{M}>2$, we examine the case in which $\mathcal{M}\ensuremath{-}1$ quantum point contacts are fully open, reproducing a single-channel circuit coupled to a dissipative Ohmic environment. The system is treated as a Luttinger liquid with an impurity, whose effective interaction parameter is defined as $K=\ensuremath{\nu}(\mathcal{M}\ensuremath{-}1)/\mathcal{M}$. Conductance scaling in the weak and strong tunnel regimes is used to discuss the low-temperature transport behavior of multichannel single- and double-charge Kondo devices.

Classical impurities and boundary Majorana zero modes in quantum chains

We study the response of classical impurities in quantum Ising chains. The Z2 degeneracy they entail renders the existence of two decoupled Majorana modes at zero energy, an exact property of a finite system at arbitrary values of its bulk parameters. We trace the evolution of these modes across the transition from the disordered phase to the ordered one and analyze the concomitant qualitative changes of local magnetic properties of an isolated impurity. In the disordered phase, the two ground states differ only close to the impurity, and they are related by the action of an explicitly constructed quasi-local operator. In this phase the local transverse spin susceptibility follows a Curie law. The critical response of a boundary impurity is logarithmically divergent and maps to the two-channel Kondo problem, while it saturates for critical bulk impurities, as well as in the ordered phase. The results for the Ising chain translate to the related problem of a resonant level coupled to a 1d p-wave superconductor or a Peierls chain, whereby the magnetic order is mapped to topological order. We find that the topological phase always exhibits a continuous impurity response to local fields as a result of the level repulsion of local levels from the boundary Majorana zero mode. In contrast, the disordered phase generically features a discontinuous magnetization or charging response. This difference constitutes a general thermodynamic fingerprint of topological order in phases with a bulk gap.

Kondo route to spin inhomogeneities in the honeycomb Kitaev model

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 coupling limit in both cases has a nontrivial topology associated with a finite Z2 flux at the impurity site. For the antiferromagnetic case, this result can be obtained straightforwardly owing to the integrability of the Kitaev model with a vacancy. The strong-coupling limit of the ferromagnetic case is however nonintegrable, and we address this problem through exact-diagonalization calculations with finite Kitaev fragments. Our exact diagonalization calculations indicate that that the weak to strong coupling transition and the topological phase transition occur rather close to each other and are possibly coincident. We also find an intriguing similarity between the magnetic response of the defect and the impurity susceptibility in the two-channel Kondo problem.

Bipolaron-SO(5) non-Fermi liquid in a two-channel Anderson model with phonon-assisted hybridizations

We analyze non-Fermi liquid (NFL) properties along a line of critical points in a two-channel Anderson model with phonon-assisted hybridizations. We succeed in identifying hidden nonmagnetic SO(5) degrees of freedom for valence-fluctuation regime and analyze the model on the basis of boundary conformal field theory. We find that the NFL spectra along the critical line, which is the same as those in the two-channel Kondo model, can be alternatively derived by a fusion in the nonmagnetic SO(5) sector. The leading irrelevant operators near the NFL fixed points vary as a function of Coulomb repulsion U; operators in the spin sector dominate for large U, while those in the SO(5) sector do for small U, and we confirm this variation in our numerical renormalization group calculations. As a result, the thermodynamic singularity for small U differs from that of the conventional two-channel Kondo problem. Especially, the impurity contribution to specific heat is proportional to temperature and bipolaron fluctuations, which are coupled electron-phonon fluctuations, diverge logarithmically at low temperatures for small U.

Two-channel pseudogap Kondo and Anderson models: Quantum phase transitions and non-Fermi liquids

We discuss the two-channel Kondo problem with a pseudogap density of states, $\rho(\w)\propto|\w|^r$, of the bath fermions. Combining both analytical and numerical renormalization group techniques, we characterize the impurity phases and quantum phase transitions of the relevant Kondo and Anderson models. The line of stable points, corresponding to the overscreened non-Fermi liquid behavior of the metallic $r=0$ case, is replaced by a stable particle-hole symmetric intermediate-coupling fixed point for $0<r<\rmax\approx0.23$. For $r>\rmax$, this non-Fermi liquid phase disappears, and instead a critical fixed point with an emergent spin--channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively. We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ epsilon expansion techniques to calculate critical properties near the critical dimensions $r=0$ and $r=1$, the latter being potentially relevant for two-channel Kondo impurities in neutral graphene. We find the analytical results to be in excellent agreement with those obtained from applying Wilson's numerical renormalization group technique.

Numerical study of the Coulomb blockade in an open quantum dot

The Coulomb blockade in an open quantum dot connected to a bulk lead by a single mode point contact is studied numerically using the path-integral Monte Carlo method. The Coulomb oscillation of the average charge and capacitance of the dot is investigated and is compared with the analytic expression for strong tunneling. At the degeneracy point, we observe logarithmic divergence of the capacitance for strong backscattering at the point contact. This observation supports the conjecture that the nature of the present system at the degeneracy point is described by the two-channel Kondo problem for an arbitrary strength of tunneling.

Bosonization approach to the mixed-valence two-channel Kondo problem

We present in detail the bosonization-refermionization solution of the anisotropic version of the two-channel Anderson model at a particular manifold in the space of parameters of the theory, where we establish an equivalence with a Fermi-Majorana biresonant-level model. The correspondence is rigorously proved by explicitly constructing the new fermionic fields and Klein factors in terms of the original ones and showing that the commutation properties between original and new Klein factors are of semionic type. We also demonstrate that the fixed points associated with the solvable manifold are renormalization-group stable and generic, and therefore representative of the physics of the original model. The simplicity of the solution found allows for the computation of the full set of thermodynamic quantities. In particular, we compute the entropy, occupation, and magnetization of the impurity as functions of temperature, and identify the different physical energy scales. In the absence of external fields, two energy scales appear and, as the temperature goes to zero, a nontrivial residual entropy indicates that the model approaches a universal line of fixed points of non-Fermi-liquid type. An external field, even if small, introduces a third energy scale and causes the quenching of the impurity entropy to zero, taking the system to a corresponding Fermi-liquid fixed point.

Quantum simulation of Anderson and Kondo lattices with superconducting qubits

We introduce a mapping between a variety of superconducting circuits and a family of Hamiltonians describing localized magnetic impurities interacting with conduction bands. This includes the Anderson model, the single impurity one- and two-channel Kondo problem, as well as the 1D Kondo lattice. We compare the requirements for performing quantum simulations using the proposed circuits to those of universal quantum computation with superconducting qubits, singling out the specific challenges that will have to be addressed.

Solvable point of a mixed-valence non-Fermi-liquid impurity

We present a mapping between a two-channel mixed-valence impurity model and a resonant-level Hamiltonian, that for particular values of the parameters becomes non-interacting. This property is analogous to the Toulouse point of the single-channel Kondo problem or the Kotliar-Si point of the infinite-U Anderson model, but displaying characteristics of non-Fermi-liquid physics like in the Emery–Kivelson mapping for the two-channel Kondo problem. We will discuss briefly some of the quantities that can be calculated by exploiting our mapping.

Can the spin-orbit interaction break the channel degeneracy (conservation) of the two-channel orbital Kondo problem?

Two-level systems (TLS) interacting with conduction electrons are possibly described by the two-channel Kondo Hamiltonian. In this case the channel degeneracy is due to the real spin of the electrons. The possibility of breaking that degeneracy has interest on his own. In fact, we show that the interaction of the conduction electrons with a spin-orbit scatterer nearby the TLS leads to the breaking of the channel degeneracy only in the case of electron-hole symmetry breaking. The generated channel symmetry breaking TLS-electron couplings are, however, too weak to result in any observable effects. Our analysis is also relevant for heavy fermion systems.

Thermodynamics of the anisotropic two-channel Kondo problem

We construct and solve numerically the thermodynamic Bethe Ansatz equations for the spin-anisotropic two-channel Kondo model in arbitrary external field $h$. At high temperatures the specific heat and the susceptibility show power law dependence. For $h \to 0$ and at temperatures below the Kondo temperature $T_K$ a two-channel Kondo effect develops characterized by a Wilson ratio 8/3, and a logarithmic divergence of the susceptibility and the linear specific heat coefficient. Finite magnetic field, $h>0$ drives the system to a Fermi liquid fixed point with an unusual Wilson ratio which depends sensitively on $h$.

Kondo effect in mesoscopic systems

The Kondo effect may develop in those cases where there are non-commuting operators describing the interaction between the conduction electrons and impurities or defects with internal degrees of freedom. This interaction may involve spin or orbital variables. There are cases where the conduction electrons have conserved quantum numbers that do not appear in the coupling. An example is where the Kondo effect involves orbital degrees of freedom, and the interaction is independent of the real spins of the electrons, which are conserved and lead to the two-channel Kondo (2CK) problem. The low-temperature behavior is very different for the one- and two-channel cases, as in the first case a Fermi liquid is formed while in the second one strong deviations appear and a non-Fermi liquid state is realized. Mesoscopic samples provide a unique possibility to study a few or even a single Kondo impurities. In this paper we first review how the original spin Kondo problem is affected by surface anisotropy and the fluctuations of the density of states in point contacts (PCs). We discuss the physics of a single magnetic impurity on the surface of a sample. The orbital 2CK effect due to dynamical defects is considered. The nature of these defects is not known but they are excellent candidates to describe the zero-bias anomalies with non-Fermi liquid character in PCs, and the dephasing time and transport in short wires. There are two main concerns regarding this interpretation: First, the tunneling centers formed by heavy impurities may produce a Kondo temperature that is too low, and secondly, the splittings seen in the experiments are much smaller than expected from this model. It would be therefore extremely important to identify the microstructure of these two-level systems and find new realizations for them.

Magnetic impurities in a one-dimensional spin-orbital model

Using one-dimensional spin-orbital model as a typical example of quantum spin systems with richer symmetries, we study the effect of an isolated impurity on its low energy dynamics in the gapless phase through bosonization and renormalization group methods. In the case of internal impurities, depending on the symmetry, the boundary fixed points can be either an open chain with a residual spin or (and) orbital triplet left behind, or a periodic chain. However, these two fixed points are indistinguishable in the sense that in both cases, the lead-correction-to-scaling boundary operators (LCBO) only show Fermi-liquid like corrections to thermodynamical quantities. (Except the possible Curie-like contributions from the residual moments in the latter cases.) In the case of external (Kondo) impurities, the boundary fixed points, depending on the sign of orbital couplings, can be either an open chain with an isolated orbital doublet due to Kondo screening or it will flow to an intermediate fixed point with the same LCBO as that of the two-channel Kondo problem. Comparison with the Kondo effect in one-dimensional (1D) Heisenberg spin chain and multi-band Hubbard models is also made.

A two-channel Kondo impurity in the spin-1/2 chain: consequences for Knight shift experiments

A magnetic impurity in the spin- 1 2 chain is a simple realization of the two-channel Kondo problem since the field theoretical descriptions in the spin sector are identical. The correlation functions near the impurity can be calculated. Using a modified version of the numerical transfer matrix DMRG, we are now able to accurately determine local properties close to the impurity in the thermodynamic limit. The local susceptibilities (Knight-shifts) show an interesting behavior in a large range around the impurities. We are able to make quantitative experimental predictions which would allow one to observe two-channel Kondo physics for the first time directly by doping of spin- 1 2 chain compounds.

Universal Crossover Behavior of a Magnetic Impurity and Consequences for Doping in Spin-1/2Chains

We consider a magnetic impurity in the antiferromagnetic spin- $1/2$ chain which is equivalent to the two-channel Kondo problem in terms of the field theoretical description. Using a modification of the transfer-matrix density matrix renormalization group we are able to determine the crossover function for the impurity susceptibility over a large temperature range, which exhibits universal data collapse. We also calculate the local susceptibilities near the impurity, which show an interesting competition of boundary effects. This results in quantitative predictions for experiments on doped spin- $1/2$ chains, which could observe two-channel Kondo physics directly.

Quadrupolar response of Mn ions in manganites

Mn ions play an important role in the properties of LaMnO-based systems. The Mn ions are in the mixed-valence state of the Mn4+ and Mn3+ configurations. Mn4+ consists of three localized d electrons in t orbitals with 2g their spins ferromagnetically coupled and is represented by a spin S = The Mn3+ configuration has an additional localized d electron in one of the e g orbitals to form a total spin S + 1. The e electron hybridizes with conduction g electrons and is assumed to form an orbital singlet as in the two-channel Kondo problem. The orbital singlet is partially broken by a small quadrupolar field that lifts the degeneracy of the e levels. The quadrupolar response is related to the g Jahn-Teller effect and determines the local distribution of Mn-O bond lengths. We study the system in the absence of a magnetic field and in the spin-polarized limit. Both limiting cases yield similar results, suggesting that the quadrupolar susceptibility does not have a strong magnetic field dependence.

Kondo quartet

This article describes some recently obtained results on the low-energy properties of the “Kondo quartet” model of two spin- 1 2 impurities interacting with two channels (flavours) of conduction electrons. We shall particularly emphasize the connections between conformal field-theory methods and bosonisation approaches, which are first illustrated on the example of the single-impurity, two-channel Kondo problem.

Overscreening-unscreening transition in the two-channel Kondo model induced by electron-electron repulsion

The effects of the repulsion between the electrons on the two-channel Kondo problem are studied by use of bosonization technique. Following Emery and Kivelson, we define a special case in the spin density wave sector, in which the impurity spin is actually detached from the dynamics of the electrons. The model is thus mapped to a boundary sine-Gordon system. We find that, in the case of breakdown of rotational symmetry in spin space, the repulsion will induce a overscreening-unscreening transition in the system.

Numerical renormalization-group study of particle-hole symmetry breaking in two-channel Kondo problem: Effect of repulsion among conduction electrons and potential scattering.

Particle-hole symmetry breaking perturbation in the two-channel pseudospin Kondo problem is studied by the numerical renormalization-group method. It is shown that the repulsion among conduction electrons at the impurity site and the single particle potential are relevant perturbations against the conventional non-Fermi-liquid fixed point. Although the repulsion (potential) with realistic strength prevents the overscreening of the pseudospin, it induces in turn a real spin, which is also overscreened again. Thus the real spin susceptibility becomes anomalous, contrary to the conventional two-channel Kondo problem.

Role of excited states in the low-temperature dynamics of tunneling centers in metals: The orbital Kondo effect.

We examine the interaction between a tunneling atom moving in a double potential well and the conduction electrons when the electron-assisted hopping to the excited states of the tunneling particle is taken into account. Using the renormalization-group transformation we show that virtual electron assisted hoppings to the excited states play a determining role in the low-temperature dynamics of the tunneling center, which scales to the two-channel Kondo problem at low temperature. The Kondo temperature ${\mathit{T}}_{\mathit{K}}$ separating the high temperature and the strongly correlated regime can be either increased or decreased due to the virtual hoppings to the excited states depending on the actual value of the bare couplings. However, carrying out a model calculation we find that for the physically relevant model parameters ${\mathit{T}}_{\mathit{K}}$ is always increased by one or two orders of magnitude even if the direct assisted tunneling between the potential minima is ignored. Thus this model can resolve the long-standing problem that the ${\mathit{T}}_{\mathit{K}}$ estimated earlier has always been somewhat small.