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Physical review. B, Condensed matter

Ground-state and low-lying excitations of the periodic Anderson Hamiltonian in one dimension from finite-cell calculations

Rémi Jullien, Richard M. Martin

Published Dec 1, 1982• 101 citations
Publisher

Abstract

We present the calculations of the ground state and lowest excited states of the one-dimensional periodic Anderson Hamiltonian with two electrons per site and arbitrary magnitude of the repulsive interaction $U$. We consider finite cells (up to $N=4$) and introduce a new method, using modified periodic boundary conditions, to facilitate comparison of calculations with different $N$. The ground state is found to be a nonmagnetic singlet in all cases. The lowest-energy excitations for adding or subtracting one electron show that the system is insulating and the lowest spin-flip excitations indicate a near instability to antiferromagnetism due to the "nesting" of the Fermi surface in one dimension. The lowest excitations are shown to vary little with $N$ and, for $N=4$, the results agree well with infinite-cell calculations, both for small $U$ and for the Kondo-lattice regime. The primary results are the continuous variation from $U=0$ to the Kondo-lattice and mixed-valence regimes and the importance of correlations, which lead to the insulating gap and dispersion in the electronic and spin excitations.

DOI
10.1103/physrevb.26.6173

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We present the calculations of the ground state and lowest excited states of the one-dimensional periodic Anderson Hamiltonian with two electrons per site and arbitrary magnitude of the repulsive interaction UUU. We consider finite cells (up to N=4N=4N=4) and introduce a new method, using modified periodic boundary conditions, to facilitate comparison of calculations with different NNN. The ground state is found to be a nonmagnetic singlet in all cases. The lowest-energy excitations for adding or subtracting one electron show that the system is insulating and the lowest spin-flip excitations indicate a near instability to antiferromagnetism due to the "nesting" of the Fermi surface in one dimension. The lowest excitations are shown to vary little with NNN and, for N=4N=4N=4, the results agree well with infinite-cell calculations, both for small UUU and for the Kondo-lattice regime. The primary results are the continuous variation from U=0U=0U=0 to the Kondo-lattice and mixed-valence regimes and the importance of correlations, which lead to the insulating gap and dispersion in the electronic and spin excitations.
Using variational matrix product states, we analyze the finite temperature behavior of a half-filled periodic Anderson model in one dimension, a prototypical model of a Kondo insulator. We present an extensive analysis of single-particle Green’s functions, two-particle Green’s functions, and transport functions creating a broad picture of the low-temperature properties. We confirm the existence of energetically low-lying spin excitations in this model and study their energy-momentum dispersion and temperature dependence. We demonstrate that charge-charge correlations at the Fermi energy exhibit a different temperature dependence than spin-spin correlations. While energetically low-lying spin excitations emerge approximately at the Kondo temperature, which exponentially depends on the interaction strength, charge correlations vanish already at high temperatures. Furthermore, we analyze the charge and thermal conductivity at finite temperatures by calculating the time-dependent current-current correlation functions. While both charge and thermal conductivity can be fitted for all interaction strengths by gapped systems with a renormalized band gap, the gap in the system describing the thermal conductivity is generally smaller than the system describing the charge conductivity. Thus, two-particle correlations affect the charge and heat conductivities in a different way resulting in a temperature region where the charge conductivity of this one-dimensional Kondo insulator is already decreasing while the heat conductivity is still increasing.
We construct a class of exact ground states of three-dimensional periodic Anderson models (PAMs), including the conventional PAM, on regular Bravais lattices at and above 3∕43∕43∕4 filling, and discuss their physical properties. In general, the fff electrons can have a (weak) dispersion, and the hopping and the nonlocal hybridization of the ddd and fff electrons extend over the unit cell. The construction is performed in two steps. First the Hamiltonian is cast into positive semidefinite form using composite operators in combination with coupled nonlinear matching conditions. This may be achieved in several ways, thus leading to solutions in different regions of the phase diagram. In a second step, a nonlocal product wave function in position space is constructed which allows one to identify various stability regions corresponding to insulating and conducting states. The compressibility of the insulating state is shown to diverge at the boundary of its stability regime. The metallic phase is a non-Fermi-liquid with one dispersing and one flat band. This state is also an exact ground state of the conventional PAM and has the following properties: (i) it is nonmagnetic with spin-spin correlations disappearing in the thermodynamic limit, (ii) density-density correlations are short ranged, and (iii) the momentum distributions of the interacting electrons are analytic functions, i.e., have no discontinuities even in their derivatives. The stability regions of the ground states extend through a large region of parameter space, e.g., from weak to strong on-site interaction UUU. Exact itinerant, ferromagnetic ground states are found at and below 1∕41∕41∕4 filling.
Presenting exact solutions for the two-dimensional periodic Anderson model with finite and nonzero on-site interactions U>0,U>0,U>0, we describe a rigorous non-Fermi-liquid phase in normal phase and two dimensions. This new state emerges in multiband interacting Fermi systems above half filling, being generated by a flat-band effect. The momentum distribution function ${n}_{\stackrel{\ensuremath{\rightarrow}}{k}}$ together with its derivatives of any order is continuous. The state possesses a well-defined Fermi energy (eF),{(e}_{F}),(eF​), but the Fermi momentum concept is not definable, so the Fermi surface in $\stackrel{\ensuremath{\rightarrow}}{k}$ space is missing. The state emerges in the vicinity of a Mott insulating phase when lattice distortions are present and is highly degenerated and paramagnetic. A gap is present at high U in the density of low-lying excitations. During low-lying excitations, quasiparticles are not created above the Fermi level, only the number of particles at eF{e}_{F}eF​ increases.
Heavy-fermion or Kondo lattice materials are considered to be typical strongly correlated systems, for which mean-field approximations have shown that the Coulomb interaction increases the effective mass and narrows the band gap. In this paper, to clarify interaction effects on the electronic excitation, the spectral function of the Kondo lattice and periodic Anderson models is studied around Kondo insulators in the strong-coupling regime, by using the non-Abelian dynamical density-matrix renormalization group method and perturbation theory. Upon doping a Kondo insulator, an electronic mode emerges in the Kondo insulating gap, exhibiting the momentum-shifted magnetic dispersion relation. Although the ground-state properties are similar to those of a doped band insulator, the emergence of the electronic mode reflecting spin-charge separation of the Kondo insulator is a crucial interaction effect that allows us to regard the Kondo-insulator--metal transition as a type of Mott transition. In addition, electronic modes emerge even in the high-energy regime by doping in the periodic Anderson model. These strong-correlation effects have not been expected in conventional mean-field approximations and would bring a different perspective on heavy-fermion or Kondo lattice systems.
Exact results for the one-hole excitation spectra of the one-dimensional half-filled Hubbard Hamiltonian are presented. We consider short chains of N=2, 4, 6 atoms and introduce modified periodic boundary conditions which allow us to make a comparison between different N and to extrapolate the results up to infinite chains. By the increasing weight of the scattering states in the one-hole spectrum, we demonstrate a gradual disappearance of the free-electron (band) picture with increasing local electron-electron interaction U. The exact spectra of the N=6 chain verify the results of second-order perturbation theory for U/W1, where W is the bandwidth. The extrapolation to the infinite system, made with the finite-cell-scaling method, gives the gap which agrees rather well with the exact result for U/W>0.2. The average energies and weights of the band and scattering states obtained with this method for an infinite chain do not differ significantly from those of the N=6 chain.
Periodic Anderson model is one of the most important models in the field of strongly correlated electrons. With the recent developed numerical method density matrix embedding theory, we study the ground state properties of the periodic Anderson model on a two-dimensional square lattice. We systematically investigate the phase diagram away from half filling. We find three different phases in this region, which are distinguished by the local moment and the spin–spin correlation functions. The phase transition between the two antiferromagnetic phases is of first order. It is the so-called Lifshitz transition accompanied by a reconstruction of the Fermi surface. As the filling is close to half filling, there is no difference between the two antiferromagnetic phases. From the results of the spin–spin correlation, we find that the Kondo singlet is formed even in the antiferromagnetic phase.