Degenerate Anderson Model Research Articles

Renormalization group study of NMR in a gapped Anderson model

Abstract We study the temperature dependence of the impurity nuclear spin-lattice relaxation rate for the single-impurity spin degenerate Anderson model, with a gap Δ in the density of states of the conduction band, using the numerical renormalization group method. In the symmetric regime, where the particle–hole symmetry is conserved, 1 / T 1 presents Kondo behavior only if Δ =0, with 1 / T 1 → 0 as T → 0 , which is a characteristic of the impurity spin in a quenched state. In the asymmetric regime, where that symmetry is not conserved, 1 / T 1 presents different behavior for Δ ⪯i Γ K . In this case, even for Δ ≠ 0 , 1 / T 1 decreases monotonically to zero as T → 0 , presenting Kondo behavior, with the impurity spin in a quenched state. Increasing Δ , 1 / T 1 shows a minimum at T Γ K (width of the Kondo resonance) and saturates as T → 0 . In this case there is no Kondo behavior, with the impurity spin in an unquenched state.

1/(N−1)expansion based on a perturbation theory inUfor the Anderson model withN-fold degeneracy

We study low-energy properties of the $N$-fold degenerate Anderson model. Using a scaling that takes $u=(N\ensuremath{-}1)U$ as an independent variable in place of the Coulomb interaction $U$, the perturbation series in $U$ is reorganized as an expansion in powers of $1/(N\ensuremath{-}1)$. We calculate the renormalized parameters, which characterize the Kondo state, to the next leading order in the $1/(N\ensuremath{-}1)$ expansion at half-filling. The results, especially the Wilson ratio, agree very closely with the exact numerical renormalization group results at $N=4$. This ensures the applicability of our approach to $N>4$, and we present highly reliable results for nonequilibrium Kondo transport through a quantum dot.

Thermal variation of Ce valence in mixed valence–Kondo lattice systems CeT 2(Si 1−xGe x) 2 with T=Mn and Ni

The results on the thermal variation of Ce L 3-valence in CeT 2(Si 1− x Ge x ) 2 series with 0⩽ x⩽1 and T=Mn and Ni are reported. It is observed that for both series, the Ce valence increases with decreasing temperature and has little thermal variation for samples in the nearly trivalent regime. The magnitude of this thermal variation in the T=Mn series is much greater than in the T=Ni series. The results are explained by the degenerate Anderson model and correlated with the specific heat data.

Bethe ansatz results for the 4f-electron spectra of a degenerate Anderson model.

We show that the density of states for localized 4f electrons coupled to a conduction band calculated in the framework of Bethe ansatz solution for the degenerate Anderson model qualitatively disagree with the well-known results obtained for the same model but using a variational approach. The scales of parameters used in our Bethe ansatz calculations to fit the experiments disagree with the commonly accepted values from other studies. This implies narrower conduction bands hybridized with 4f orbitals or questions the applicability of the Bethe ansatz for a degenerate Anderson model for the high-energy characteristics of some rare-earth systems.

Origin of the tiny antiferro-magnetic moment in URu 2Si 2

Possible orders in URu 2Si 2 are investigated using a two-channel degenerate Anderson model. The ground state of uranium ions is the non-Kramers quadrupolar doublet Γ 5 with (5f) 2, and its relevalent excited state is the Kramers dipolar doublet Γ 7 with (5f) 1. These states mix with each other via conduction electrons. At low temperatures, the system forms renormalized bands with both quadrupole and dipole degrees of freedom due to the quadrupolar Kondo effect which slightly mixes quadrupolar Γ 5, a primary state of uranium ions, with dipolar Γ 7. At a certain low temperature, conduction electrons in the renormalized bands undergo quadrupolar ordering with a large quadrupolar moment. At a further lower temperature, octupolar ordering occurs, accompanied by a tiny dipolar moment which is attributed to the property of the renormalized bands with primarily the Γ 5-character slightly mixed with the Γ 7-character. These phenomena are well described by the 1/ N-expansion method with pseudo-fermions for the non-Kramers doublet Γ 5 and slave bosons for the Kramers doublet Γ 7.

Elementary excitation for the orbitally degenerate Anderson model at finite temperatures

The elementary excitation for the orbitally degenerate Anderson model at finite temperatures is studied using the Bethe ansatz solution. The formulation of the elementary excitation at finite temperatures is based on the method which was first developed by Yang and Yang for the one-dimensional boson system with the $\ensuremath{\delta}$-function type interaction. The expressions of the elementary excitation energy for the spin and the charge degrees of freedom are derived. Using the obtained expressions, the spin and the charge excitation spectrums are calculated numerically. With decreasing temperature, the peak structure grows in the spin excitation spectrum, whereas the weight around the impurity level increases in the charge excitation spectrum, below the temperature corresponding to the characteristic energy scale of the system. The crystalline-field effects on the elementary excitation spectrums are investigated in the Kondo regime, and the results are discussed in connection with the difference in the Kondo temperature. The relation between the obtained results and the thermodynamic quantity of the orbitally degenerate Anderson model is also discussed.

Quadrupolar Order, Hidden Octupolar Order and Tiny Magnetic Moment in URu2Si2

Possible orders in URu 2 Si 2 are investigated using a two-channel degenerate Anderson model. The ground state of uranium ions is the non-Kramers quadrupolar doublet Γ 5 with (5 f ) 2 , and its relevalent excited state is the Kramers dipolar doublet Γ 7 with (5 f ) 1 . These states mix with each other via conduction electrons. At low temperatures, the system forms renormalized bands with both quadrupole and dipole degrees of freedom due to the quadrupolar Kondo effect which slightly mixes quadrupolar Γ 5 , a primary state of uranium ions, with dipolar Γ 7 . At a certain low temperature, conduction electrons in the renormalized bands undergo quadrupolar ordering with a large quadrupolar moment. At a further lower temperature, octupolar ordering occurs, accompanied by a tiny dipolar moment which is attributed to the property of the renormalized bands with primarily the Γ 5 -character slightly mixed with the Γ 7 -character. These phenomena are well described by the 1/ N -expansion method with pseudo-fermions...

Effects of the Hund's Rule Coupling in an Orbitally Degenerate Anderson Model

Effects of the Hund's rule coupling in an orbitally degenerate Anderson model are studied by a quantum Monte Carlo method. From results for static as well as dynamic susceptibilities it is concluded that the system remains a local Fermi liquid at least in a weak to intermediate coupling region as was concluded in previous studies. It is also found that the Hund's rule coupling generates a strong attractive interaction between conduction electrons with parallel spins in the same orbital states.

Weak antiferromagnetism in the Kondo lattice

The N-fold degenerate Anderson model for a lattice is considered in the Kondo-regime. The system of f-electrons is supposed to be extremely correlated ( U→∞). Using a slave boson approach and a 1/ N expansion a criterion of the spin density wave (SDW) instability in the system of itinerant electrons is obtained. An effective interaction resulting in the SDW being formed appears due to spin fluctuations in the system of localized electrons.

Exact solution for a degenerate Anderson impurity in the limit embedded into a correlated host

We consider the one-dimensional t - J model, which consists of electrons with spin S on a lattice with nearest neighbor hopping t constrained by the excluded multiple occupancy of the lattice sites and spin-exchange J between neighboring sites. The model is integrable at the supersymmetric point, J = t. Without spoiling the integrability we introduce an Anderson-like impurity of spin S (degenerate Anderson model in the limit), which interacts with the correlated conduction states of the host. The lattice model is defined by the scattering matrices via the Quantum Inverse Scattering Method. We discuss the general form of the interaction Hamiltonian between the impurity and the itinerant electrons on the lattice and explicitly construct it in the continuum limit. The discrete Bethe ansatz equations diagonalizing the host with impurity are derived, and the thermodynamic Bethe ansatz equations are obtained using the string hypothesis for arbitrary band filling as a function of temperature and external magnetic field. The properties of the impurity depend on one coupling parameter related to the Kondo exchange coupling. The impurity can localize up to one itinerant electron and has in general mixed valent properties. Groundstate properties of the impurity, such as the energy, valence, magnetic susceptibility and the specific heat coefficient, are discussed. In the integer valent limit the model reduces to a Coqblin-Schrieffer impurity.

Infrared Divergence and Non-Fermi Liquid in Multichannel Degenerate Anderson Model

As a straightforward extention of the study on infrared divergences in pseudo-particle spectra in the single-channel degenerate impurity Anderson model, those of the multichannel model are examined with use of the expansion from the large limit of the spin-orbital degeneracy N . Analysis with the most divergent terms in the expansion shows that exponents of spectra of the slave boson and the pseudo-fermion in the M -channel model with 1 ≤ M ≪ N are given respectively by 1- n f 2 M / N and (2 n f - n f 2 ) M / N , where n f is the average number of localized electrons at the impurity site. When M =1, the results recover those obtained earlier in the single channel model. In the limits, n f →1 and M -2 ≪1, these results tend to those obtained by the non-crossing approximation (NCA) study. It also shows that the scattering rate of the conduction electrons in the M -channel model includes infrared divergent terms proportional to (1- n f )(1- M -2 ) in contrast to the absence of infrared divergent terms in the...

Infrared Divergence in Pseudo-Particle Spectra

Infrared divergence in the pseudo-particle spectra of the degenerate impurity Anderson model described in terms of the auxiliary particles is examined rigorously with use of the expansion from the large limit of the spin-orbital degeneracy N . The results of analysis with the most divergent terms in the expansion confirm the exponents of the spectra of the slave boson and the pseudo-fermion to be given respectively by 1- n f 2 / N and (2 n f - n f 2 )/ N where n f is the average number of localized electrons at the impurity site, rather than by N /( N +1) and 1/( N +1) as suggested by the results of so-called non-crossing approximation (NCA) studies. In determination of the correct exponents, contributions from the crossing diagrams are crucial. It is also confirmed that all of the infrared divergences in the single particle spectra are cancelled out by the corresponding vertex corrections to leave only non-divergent terms in the scattering rate of the conduction electrons.

Stability of Non-Fermi Liquid in Kondo Effect due to (5f)2Configuration

The stability of the non-Fermi liquid (NFL) in Kondo effect is studied by considering the excited states of the 5 f -electrons. First, in order to examine their effects in the strong correlation limit, the exchange model is derived for a tetragonal crystal field by restricting the localized states in f 2 to a non-Kramers doublet and a singlet; all the states are taken into account for f 1 . The numerical renormalization group analysis for this model shows that the singlet state in f 2 and one of the excited states in f 1 play an important role to stabilize the Fermi liquid (FL) over the NFL. Second, the same problem is examined for the six-fold orbitally degenerate Anderson model by considering all the local excited states. The competition between FL and NFL sensitively depends on the Coulomb and Hund couplings, the orbital anisotropy and the hybridization term. The six-fold orbital degeneracy is also important to realize NFL; in fact a similar analysis on the four-fold orbitally degenerate Anderson model...

Dynamic charge susceptibility for the infinite-U Anderson model

The dynamic charge susceptibility for the single-impurity N-fold degenerate Anderson model is calculated within the noncrossing approximation. The calculations were done in all regimes: empty orbital, mixed valent, and Kondo. At large N in the Kondo regime the results can be connected at low temperatures with the predictions of large-N mean-field theory, suggesting a range of temperature where the validity of the two theories overlaps. Our smaller N results are presented in the context of electronic friction and energy transfer of an atom or molecule moving outside a metallic surface. It is suggested that some of the dramatic temperature and frequency dependences obtained should be relevant for this surface problem.

Impurity Effects in Lattice Anderson Model

Effects of a single impurity in the degenerate lattice Anderson model describing heavy elecron systems are investigated rigorously within the framework of the axiliary boson method and the perturbation expansion from the large limit of the spin-orbital degeneracy N (1/ N -expansion). It is found that a non-magnetic impurity leads to the unitarity limit value of the scattering rate of the conduction electrons at low temperature while it has no appreciable effect on the thermodynamic properties. These results are consistent with the experimental observations on the heavy electron systems such as Ce 1- x La x Cu 6 .

Degenerate Anderson impurity model in the presence of spin-orbit and crystal-field splitting.

Using a quantum Monte Carlo technique combined with the method of maximum entropy, we calculated the magnetic susceptibility and spectral properties of the degenerate single-impurity Anderson model in the presence of spin-orbit and crystal-field splitting. At small hybridization and moderate temperatures, the magnetic susceptibility displays a nonuniversal behavior. Under the influence of spin-orbit splitting, the magnetic susceptibility decreases due to the quenching of a sixfold-degenerate state to an effectively doubly degenerate state. The effect of crystal-field splitting on the magnetic susceptibility is less pronounced. The temperature dependence of the spectral function changes dramatically under the influence of spin-orbit splitting. In the accessible temperature regime, the central peak at \ensuremath{\omega}\ensuremath{\sim}0 decreases with decreasing temperature.

Kondo Insulator and Charge Transfer Insulator in Lattice Anderson Model

Properties of the heavy-fermion valence-fluctuating compounds leading to insulating ground states (called Kondo insulators) are investigated theoretically by using the U-infinite degenerate lattice Anderson model in the auxiliary boson representation. The model is treated within the approximation including terms of the leading order in the expansion w.r.t. inverse of the spin-orbit degeneracy (1/ N -expansion) under the strict local constraints guaranteeing equivalence of the bosonic version to the original U -infinite model. The results are compared with the experimental observations on the possible candidates for Kondo insulators. Differences between Kondo insulators and charge transfer type insulators now paid much attention as the mother systems for the high- T c superconductors are briefly discussed on a unified basis within the model described above.

Quantum Monte Carlo Study of the Degenerate Anderson Model with Cubic Crystal Field

Using a quantum Monte Carlo simulation technique, we study magnetic properties of the 6-fold degenerate Anderson model, which corresponds to the angular momentum j =5/2. The model is characterized by the f - f Coulomb interaction U , the hybridization width Γ, and the f -electron energy level E f . As U increases, the Kondo temperature T K decreases for the electron-hole symmetric case ( E f ∼-5 U /2), although T K increases for the shallow E f case ( E f ∼0). We also study an effect of cubic crystal field splitting Δ c on the susceptibility and the Kondo temperature, and find a suppression of susceptibility and an increase of T K when the occupied f -electron number is nearly equal to one. In processes of Monte Carlo simulations, the negative sign problem does not appear.

Physical properties of a new coherent state of the almost degenerate infinite-U lattice Anderson model

A new approach toward understanding the heavy fermion systems (HFS) within a framework of the almost degenerate lattice Anderson Hamiltonian in the Kondo regime is proposed. In the coherent low temperature regime operators in the effective Hamiltonian are found to belong to an SU(2J + 3) dynamical algebra. A canonical transformation is employed to decouple the quasiparticle branches, thereby setting up the decoupling equation. It is found that this decoupling equation has a solution of the symmetry-altering type1. The thermodynamic response functions and other quantities are calculated for this new state. This solution is a consequence of the degeneracy of the uncoupled f-orbitals. It is characterized by the interatomic hopping of f-electrons, which produces the spin delocalization regime and with the renormalized f-level pinned close to the Fermi level. This is also found to be the source of the apparent spin compensation regime, which is accompanied by large enhancement of the thermodynamic response functions. In addition, the calculated phase coherence length is found to be much greater than a lattice constant, thereby showing a many-body character of this new state. It is believed that this new state provides an accurate description of the heavy Fermion state at low temperatures. The stability conditions for the new regime are also discussed.

Spin correlations around a magnetic impurity

Abstract Spatial spin correlations around a magnetic impurity in metals are studied for the N-fold degenerate impurity Anderson model, using self-consistent integral equations. The spatial spin correlation function is very sensitive with respect to the temperature and this allows for a qualitative distinction between the Kondo-effect and the RKKY-interaction. Results are also presented for the correlations as function of the Coulomb interaction U, and the energy E f of the 4f-configurations.