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arXiv:cond-mat/9511088 · Physical review. B, Condensed matter

T-matrix formulation of impurity scattering in correlated systems

W. Ziegler, Didier Poilblanc, R. Preuss, Werner R. Hanke, Douglas J. Scalapino

Published Apr 1, 1996• 39 citations
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Abstract

Using a generalized T-matrix description which, in principle, exactly includes Coulomb correlations and potential scattering events, resonant and bound impurity states are discussed. Like in the noninteracting case, the effects of the scattering potential can be divided into different partial-wave channels, exploiting the symmetry of the underlying lattice. Due to Coulomb correlations, bare local (i.e., s-wave) potentials become dynamic and extended, being responsible also for p-, d-wave, etc., scattering effects. Numerically exact results for both the two-dimensional t-J and Hubbard models are used to construct a simple (static) approximation to the effective impurity potential which is shown to reproduce the exact resonant scattering and bound states in the relevant symmetry channels. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.8704

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Using a generalized T-matrix description which, in principle, exactly includes Coulomb correlations and potential scattering events, resonant and bound impurity states are discussed. Like in the noninteracting case, the effects of the scattering potential can be divided into different partial-wave channels, exploiting the symmetry of the underlying lattice. Due to Coulomb correlations, bare local (i.e., s-wave) potentials become dynamic and extended, being responsible also for p-, d-wave, etc., scattering effects. Numerically exact results for both the two-dimensional t-J and Hubbard models are used to construct a simple (static) approximation to the effective impurity potential which is shown to reproduce the exact resonant scattering and bound states in the relevant symmetry channels. \textcopyright{} 1996 The American Physical Society.
Scattering by a single impurity introduced in a strongly correlated electronic system is studied by exact diagonalization of small clusters. It is shown that an inert site which is spinless and unable to accommodate holes can give rise to strong resonant scattering. A calculation of the local density of states reveals that, for increasing antiferromagnetic exchange coupling, d,- s,- and p-wave symmetry bound states in which a mobile hole is trapped by the impurity potential induced by a local distortion of the antiferromagnetic background successively pull out from the continuum.
In this Thesis, we develop and apply an accurate method to determine the ground-state properties of strongly-correlated electrons in the framework of lattice model Hamiltonians. In lattice density-functional theory (LDFT) the basic variable is the one-particle density matrix γ. From a generalized HK theorem, the ground state energy Egs[γgs] = minγ E[γ] is obtained by minimizing the energy-functional E[γ] over all the physical or representable γ. The energy functional can be divided into two contributions: the kinetic-energy functional T [γ], which linear dependence on γ is exactly known, and the correlation-energy functional W [γ], which dependence on γ is not explicitly known. Finding accurate approximations for W [γ] constitutes the actual challenge of this thesis. Part of this work is based on previous studies which derive an approximation of W [γ] for the Hubbard Hamiltonian based on scaling hypothesis and an exact analytical results for the Hubbard dimer. However, this approach is limited to spin independent and homogeneous systems. In order to extend the scope of LDFT, we have developed three different approaches to derive W [γ] in order to study broken symmetry problems. First we extend the previous scaling functional to system having charge transfer. By a systematic study of the dependence of W [γ] on the charge distribution we find similar scaling properties as for the homogeneous case. An extension to the Hubbard model on bipartite lattices is then derived and applied to finite and infinite systems with repulsive and attractive interactions. The high accuracy of this functional is demonstrated. Nevertheless, this approach is difficult to transfer to more complex systems, since the calculation of W [γ] applies to the system as a whole. To overcome this problem, we derive another approximation based on local scaling properties. This functional is formulated on-site so that it can be applied to any ordered or disordered Hamiltonian with on-site interaction. As applications we study the metal-insulator transition in the ionic Hubbard model in oneand two-dimensions, as well as the one-dimensional Hubbard chain with first and second nearest neighbor hoppings. Finally, we develop a numerical approach of W [γ], on the basis on exact diagonalizations of an effective many-body Hamiltonian corresponding to a cluster surrounded by an effective medium. This effective Hamiltonian depends on the density matrix γ and allows to derive approximation to W [γ] which improve systematically with increasing cluster size. The formulation is spin-dependent and allows a straightforward generalization to multi-orbital correlated systems such spd-Hamiltonians. In addition it take into account the effects of short-range charge and spin fluctuations in the functional. The accuracy of the method is demonstrated for the Hubbard model by comparison with the Bethe-Ansatz solution (1D) and quantum Monte Carlo simulations (2D). Finally, an outlook or relevant future developments of the theory is provided.
It is sometimes convenient to emphasize the local aspects of a part of a crystalline system, and use the corresponding localized states to build a basis of the states of the whole system. In many cases it is only a subspace of these local states that is relevant, and the Hubbard operators provide a fairly simple way to write the corresponding projected Hamiltonian. Two examples of this type of treatment are presented int this work. The first is a Co2+ impurity in a MgO crystal interacting through a Jahn-Teller term with the crystal phonons, and it is shown how this interaction affects the electronic Raman scattering. The second is the Anderson lattice when the local electrons have an infinite Coulomb repulsion, and a diagrammatic expansion with cumulants is discussed. We propose a method to obtain approximate Green's functions for the Anderson lattice that employs the exact solutions of an atomic problem, and the corresponding spectral density of the local electrons is calculated.
Strongly correlated electron systems show a rich variety of astonishing physical phenomena. However, the strong interactions make the theoretical description of these systems a highly non-trivial task. Even the simplest models cannot be solved exactly, and one has to reside to approximative solutions, which often cover only certain aspects of the physics contained in these models. In recent years progress has been made in the theoretical description of correlated lattice systems through the mapping of lattice models onto effective impurity models. The most prominent example is the mapping of the Hubbard model onto an effective single impurity Anderson model (SIAM). Within this description both models feature the prominent Kondo effect, which leads to the emergence of low energy quasiparticles. The occurrence of such excitations is a result of the dynamical screening of local magnetic moments and the corresponding formation of a low temperature Fermi liquid phase. On the other hand these theories usually fail in the description of non Fermi liquid behavior, which is for example observed in the normal state of high temperature superconductors and some Heavy-Fermion systems. Strong nonlocal fluctuations drive the dynamics of these systems. In the theoretical treatment, due to the mapping of the lattice problem to an effective single impurity problem, these fluctuations are not adequately described.\\\\ In this thesis a new self-consistent approach for the inclusion of spatial correlations is presented for the Hubbard model. In contrast to existing quantum cluster theories in this field, the approach allows for the simultaneous description of short and long range pair correlations. Due to the intimate connection of this new lattice theory with the two impurity Anderson model (TIAM) a thorough numerical investigation of the TIAM is undertaken in this thesis. The competition of the Kondo effect with the RKKY interactions is studied in great detail. Furthermore two new solvers for the TIAM are introduced, which extend existing non crossing approximations for the SIAM.
The interaction energy $W[\ensuremath{\gamma}]$ of the Hubbard model is regarded as a functional of the single-particle density matrix $\ensuremath{\gamma}$ in the framework of lattice density-functional theory. The local character of the Hubbard interaction is exploited to express WWW as a sum of local contributions ${\ensuremath{\omega}}_{i}[\ensuremath{\gamma}]$, for which a simple semilocal scaling approximation is proposed. The method is applied to the ionic Hubbard model on one- and two-dimensional lattices with homogeneous and inhomogeneous Coulomb repulsions. Results are given for the kinetic and Coulomb energies, interatomic charge transfers, local magnetic moments, and charge gaps. Goals and limitations of the functional are discussed by comparison with exact results.
The effect of a magnetic (S=1/2) impurity coupled to a two-dimensional system of correlated electrons (described by the t-J model) is studied by exact diagonalizations. It is found that, if the exchange coupling of the impurity with the neighboring spins is ferromagnetic or weakly antiferromagnetic, an extra hole can form bound states of different spatial symmetries with the impurity extending to a few lattice spacings. The binding energy is maximum when the impurity is completely decoupled (vacancy) and vanishes for an antiferromagnetic coupling exceeding \ensuremath{\sim}0.3J. Several peaks appear in the single-hole spectral function below the lower edge of the quasiparticle band as signatures of the d-, s-, and p-wave bound states.