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Physical review. B, Condensed matter

Spectral density of the local Cu-O model

Theo A. Costi

Published Mar 15, 1997• 2 citations
Publisher

Abstract

The local Cu-O model is a local version of the extended Hubbard model and corresponds to a two-channel Anderson impurity model in which the two channels correspond to screening and hybridizing channels, respectively, and the screening interaction, ${\mathrm{U}}_{\mathrm{dc}}$, is short range. The model represents a more realistic model for the local excitations in heavy fermions and may also be of relevance in understanding the normal state properties of high temperature superconductors. Below a critical screening strength, ${\mathrm{U}}_{\mathrm{dc}}$${\mathrm{U}}_{\mathrm{dc}}^{\mathrm{crit}}$ the particle-hole symmetric model exhibits Fermi-liquid behavior, whereas for ${\mathrm{U}}_{\mathrm{dc}}$>${\mathrm{U}}_{\mathrm{dc}}^{\mathrm{crit}}$ a non-Fermi-liquid state forms. Here, we report results of a numerical renormalization group calculation for the spectral density of single-particle excitations, in both the Fermi-liquid and non-Fermi-liquid regimes. The Kondo resonance is strongly modified for ${\mathrm{U}}_{\mathrm{dc}}$ close to ${\mathrm{U}}_{\mathrm{dc}}^{\mathrm{crit}}$ and is destroyed for ${\mathrm{U}}_{\mathrm{dc}}$>${\mathrm{U}}_{\mathrm{dc}}^{\mathrm{crit}}$. The relevance of these results for high temperature superconductors and the photoemission spectra of heavy fermions is discussed.

DOI
10.1103/physrevb.55.6670

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The local Cu-O model is a local version of the extended Hubbard model and corresponds to a two-channel Anderson impurity model in which the two channels correspond to screening and hybridizing channels, respectively, and the screening interaction, Udc{\mathrm{U}}_{\mathrm{dc}}Udc​, is short range. The model represents a more realistic model for the local excitations in heavy fermions and may also be of relevance in understanding the normal state properties of high temperature superconductors. Below a critical screening strength, {\mathrm{U}}_{\mathrm{dc}}$${\mathrm{U}}_{\mathrm{dc}}^{\mathrm{crit}} the particle-hole symmetric model exhibits Fermi-liquid behavior, whereas for Udc{\mathrm{U}}_{\mathrm{dc}}Udc​>Udccrit{\mathrm{U}}_{\mathrm{dc}}^{\mathrm{crit}}Udccrit​ a non-Fermi-liquid state forms. Here, we report results of a numerical renormalization group calculation for the spectral density of single-particle excitations, in both the Fermi-liquid and non-Fermi-liquid regimes. The Kondo resonance is strongly modified for Udc{\mathrm{U}}_{\mathrm{dc}}Udc​ close to Udccrit{\mathrm{U}}_{\mathrm{dc}}^{\mathrm{crit}}Udccrit​ and is destroyed for Udc{\mathrm{U}}_{\mathrm{dc}}Udc​>Udccrit{\mathrm{U}}_{\mathrm{dc}}^{\mathrm{crit}}Udccrit​. The relevance of these results for high temperature superconductors and the photoemission spectra of heavy fermions is discussed.
We study by Wilson numerical renormalization group the spectral properties of a two-orbital Anderson impurity model in the presence of an exchange splitting that follows either regular or inverted Hund's rules. The phase diagram contains a non-Fermi-liquid fixed point separating a screened phase, where conventional Kondo effect occurs, from an unscreened one, where the exchange splitting takes care of quenching the impurity degrees of freedom. On the Kondo screened side close to this fixed point the impurity density of states shows a narrow Kondo peak on top of a broader resonance. This narrow peak transforms in the unscreened phase into a narrow pseudogap inside the broad resonance. Right at the fixed point only the latter survives. The fixed point is therefore identified by a jump of the density of states at the chemical potential. We also consider the effect of several particle-hole symmetry-breaking terms. We show that particle-hole perturbations that simply shift the orbital energies do not wash out the fixed point, unlike those perturbations that hybridize the two orbitals. Consequently the density-of-state jump at the chemical potential remains finite even away from particle-hole symmetry. In other words, the pseudogap stays pinned at the chemical potential, although it is partially filled in. We also discuss the relevance of these results for lattice models that map onto this Anderson impurity model in the limit of large lattice coordination. Upon approaching the Mott metal-insulator transition, these lattice models necessarily enter a region with a local criticality that reflects the impurity non-Fermi-liquid fixed point. However, unlike the impurity, the lattice can get rid of the single-impurity fixed-point instability by spontaneously developing bulk coherent symmetry-broken phases, which we identify for different lattice models.
The local moment approach (LMA) has presented itself as a powerful semianalytical quantum impurity solver (QIS) in the context of the dynamical mean-field theory (DMFT) for the periodic Anderson model and it correctly captures the low-energy Kondo scale for the single impurity model, having excellent agreement with the Bethe ansatz and numerical renormalization group (NRG) results. However, the most common correlated lattice model, the Hubbard model, has not been explored well within the LMA+DMFT framework beyond the insulating phase. Here in our work, within the framework we complete the filling-interaction phase diagram of the single band Hubbard model at zero temperature. Our formalism is generic to any particle filling and can be extended to finite temperature. We contrast our results with another QIS, namely the iterated perturbation theory (IPT) and show that the second spectral moment sum rule improves better as the Hubbard interaction strength grows stronger in LMA, whereas it severely breaks down after the Mott transition in IPT. For the metallic case, the Fermi liquid (FL) scaling agreement with the NRG spectral density supports the fact that the FL scale emerges from the inherent Kondo physics of the impurity model. We also show that, in the metallic phase, the FL scaling of the spectral density leads to universality which extends to infinite frequency range at infinite correlation strength (strong coupling). At large interaction strength, the off half-filling spectral density forms a pseudogap near the Fermi level and filling-controlled Mott transition occurs as one approaches the half-filling. As a response property, we finally study the zero temperature optical conductivity and find universal features such as absorption peak position governed by the FL scale and a doping independent crossing point, often dubbed the isosbestic point in experiments.
Key questions for theory of high temperature superconductors involve the nature of the electronic system: the minimum essential Hamiltonian and the low energy states of such Hamiltonians. To attempt to answer these questions we are carrying out studies using constrained density functional (CDF) methods to identify the relevant electronic states and realistic values of the parameters, combined with exact calculations on the reduced model Hamiltonians. The CDF calculations lead to a 3-band Hubbard model for Cu-O planes with parameters in general agreement with high energy spectroscopies and low energy magnetic excitations. We describe a modified Lanczos method to carry out large many-body calculations and give a representative result, where we find the spectra are described by the 1-band Hubbard model in regimes which imply spin interactions beyond the usual Heisenberg form.
We generalize Nozi\`eres' Fermi-liquid theory for the low-energy behavior of the Kondo model to that of the single-impurity Anderson model. In addition to the electrons' phase shift at the Fermi energy, the low-energy Fermi-liquid theory is characterized by four Fermi-liquid parameters: the two given by Nozi\`eres that enter to first order in the excitation energy, and two additional ones that enter to second order and are needed away from particle-hole symmetry. We express all four parameters in terms of zero-temperature physical observables, namely the local charge and spin susceptibilities and their derivatives with respect to the local level position. We determine these in terms of the bare parameters of the Anderson model using Bethe ansatz and numerical renormalization group (NRG) calculations. Our low-energy Fermi-liquid theory applies throughout the crossover from the strong-coupling Kondo regime via the mixed-valence regime to the empty-orbital regime. From the Fermi-liquid theory, we determine the conductance through a quantum dot symmetrically coupled to two leads in the regime of small magnetic field, low temperature, and small bias voltage, and compute the coefficients of the $\ensuremath{\sim}{B}^{2}, \ensuremath{\sim}{T}^{2}$, and $\ensuremath{\sim}{V}^{2}$ terms exactly in terms of the Fermi-liquid parameters. The coefficients of T2,V2{T}^{2}, {V}^{2}T2,V2, and B2{B}^{2}B2 are found to change sign during the Kondo to empty-orbital crossover. The crossover becomes universal in the limit that the local interaction is much larger than the level width. For completeness, we also compute the shot noise and discuss the resulting Fano factor.
The symmetric Anderson impurity model with a hybridization vanishing at the Fermi level, Δ I ∝|ω| r , is studied via the numerical renormalization group (NRG) at T = 0; and detailed comparison made with predictions arising from the local moment approach (LMA), a recently developed many-body theory which is found to provide a remarkably successful description of the problem. Results for the `normal' ( r = 0) impurity model are obtained as a specific case, and likewise compared. Particular emphasis is given both to single-particle excitation dynamics, and to the transition between the strong-coupling (SC) and local moment (LM) phases of the model. Scaling characteristics and asymptotic behaviour of the SC/LM phase boundaries are considered. Single-particle spectra D (ω) are investigated in some detail, for the SC phase in particular. Here, in accordance with a recently established result, the modified spectral functions ℱ(ω)∝|ω| r D (ω) are found to contain a generalized Kondo resonance that is ubiquitously pinned at the Fermi level; and which exhibits a characteristic low-energy Kondo scale, ω K ( r ), that narrows progressively upon approach to the SC→LM transition, where it vanishes. Universal scaling of the spectra as the transition is approached thus results. The scaling spectrum characteristic of the normal Anderson model is recovered as a particular case, that exemplifies behaviour characteristic of the SC phase generally, and which is captured quantitatively by the LMA. In all cases the r -dependent scaling spectra are found to possess characteristic low-energy asymptotics, but to be dominated by generalized Doniach-Sunjic tails, in agreement with LMA predictions.
The impurity Green function and dynamical susceptibilities for the two-channel Anderson impurity model are calculated. An exact expression for the self-energy of the impurity Green function as ratio of two correlation functions is given. The imaginary part of the self-energy scales as $\sqrt{\ensuremath{\mid}\ensuremath{\omega}∕{T}_{K}\ensuremath{\mid}}$ for $T\ensuremath{\rightarrow}0$, serving as a hallmark for non-Fermi behavior. The many-body resonance is pinned to a universal value of $1∕(2\ensuremath{\pi}\ensuremath{\Delta})$ at $\ensuremath{\omega}=0$ for arbitrary local occupation in contrary to the single-channel Anderson impurity model. Its shape becomes increasingly more symmetric for the Kondo regimes of the model. The dynamical spin and channel susceptibilities are governed by two energy scales TK{T}_{K}TK​ and Th{T}_{h}Th​, and their ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$ approach a constant value for $\ensuremath{\omega}\ensuremath{\rightarrow}0$.