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Physical Review Letters

Spectral density for fermion tunneling between two centers in a metallic environment

Valter L. Líbero, L. Nunes de Oliveira

Published Oct 15, 1990• 18 citations
Publisher

Abstract

We report a numerical renormalization-group calculation of the spectral density for an electron that tunnels between two centers in a metal. The computed spectra, uniformly accurate over the parameter space, provide the first comprehensive survey of the model. Power laws at low energies and the absence of zero-energy spectral weight signal an infrared catastrophe. The renormalization of the tunneling rate by the interaction with the conduction band is discussed.

DOI
10.1103/physrevlett.65.2042

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We report a numerical renormalization-group calculation of the spectral density for an electron that tunnels between two centers in a metal. The computed spectra, uniformly accurate over the parameter space, provide the first comprehensive survey of the model. Power laws at low energies and the absence of zero-energy spectral weight signal an infrared catastrophe. The renormalization of the tunneling rate by the interaction with the conduction band is discussed.
We analyze spectral functions of mesoscopic systems with large dimensionless conductance, which can be described by a universal Hamiltonian. We show that an important class of spectral functions are dominated by one single state only, which implies the existence of well-defined (i.e., infinite-lifetime) quasiparticles. Furthermore, the dominance of a single state enables us to calculate zero-temperature spectral functions with high accuracy using the density-matrix renormalization group. We illustrate the use of this method by calculating the tunneling density of states of metallic grains, of which we discuss the crossover from the few-electron to the bulk regime.
Many quantum mechanical problems (such as dissipative phase fluctuations in metallic and superconducting nanocircuits or impurity scattering in Luttinger liquids) involve a continuum of bosonic modes with a marginal spectral density diverging as the inverse of energy. We construct a numerical renormalization group in this singular case, with a manageable violation of scale separation at high energy, capturing reliably the low energy physics. The method is demonstrated by a nonperturbative solution over several energy decades for the dynamical conductance of a Luttinger liquid with a single static defect.
We consider a disordered system of interacting fermions, for which we demonstrate, that depending on the relationship between the degree of disorder and real physical dimensionality d , this system undergoes a delocalization (e.g. a metal-insulator) phase transition. The effect of the disorder is accounted for by replacing the ordinary single-particle kinetic energy operator by that with a fractional spectrum. These fractional spectra represent the microscopic equivalent of the well-known non-trivial power-law decay of the low-energy density of states of non-interacting disordered systems. We develop a powerful approach to analyze the density of states in interacting disordered system, combining both the saddle-point and renormalization group methods. These two methods complement each other effectively. Through this approach, we identify a fixed point in the renormalization group flow of the model’s parameters, which dictates the behavior of the system in the conducting phase. Additionally, we extend our analysis to include static conductivity. Our examination of the system’s conductivity allows us to establish several strict conditions regarding the stability of the metallic state across different spatial dimensions d . Notably, we emphasize the unique significance of the two-dimensional case d = 2 . We also compare our findings with salient experimental features observed in paradigmatic physical systems.
The self-energy of the hole-doped two-dimensional (2D) Hubbard model is calculated to second order in the interaction U and the ensuing renormalization of the spectral properties and of the Fermi surface is discussed. In uncorrelated systems the square-shaped Fermi surface separates the Fermi surface closed around the \ensuremath{\Gamma} point (with more holes than particles) from the Fermi surface closed around the Z point (with more particles than holes). In a correlated system a topological change from a Fermi surface centered around the \ensuremath{\Gamma} point to a Fermi surface closed around a Z point is induced, either by increasing the interaction or by diminishing the concentration of holes. The shape of the renormalized spectral function Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) is momentum dependent. Using Ap{\mathit{A}}_{\mathbf{p}}Ap​(\ensuremath{\omega}) we evaluate dispersion of single-particle excitations. At low energies the band of quasiparticles with reduced bandwidth is clearly seen. At high energies and far away from the Fermi surface the spectra acquire an additional peak that describes excitations in the Hubbard band. The dispersion in Hubbard bands is weaker along the \ensuremath{\Gamma}-Z, than \ensuremath{\Gamma}-X or X-Z direction. The density of states of a correlated system, as given by the perturbation theory, remains logarithmically singular but the singular weight is reduced with respect to the uncorrelated one. In addition, the correlations transfer the spectral weight out of the low-energy region, where only a narrow Kondo-like structure seems to remain. The spectral properties of the 2D Hubbard model obtained by the truncated perturbation expansion resemble in many ways the recent experimental data on metallic cuprates.
We analyze the one-dimensional extended Hubbard model with a single static impurity by using a computational technique based on the functional renormalization group. This extends previous work for spinless fermions to spin-1∕21∕21∕2 fermions. The underlying approximations are devised for weak interactions and arbitrary impurity strengths, and have been checked by comparing with density-matrix renormalization-group data. We present results for the density of states, the density profile, and the linear conductance. Two-particle backscattering leads to striking effects, which are not captured if the bulk system is approximated by its low-energy fixed point, the Luttinger model. In particular, the expected decrease of spectral weight near the impurity and of the conductance at low energy scales is often preceded by a pronounced increase, and the asymptotic power laws are modified by logarithmic corrections.
The article reviews recent results for the low energy physics of fast tunneling centers in metallic environments. For strong enough couplings to the environment these tunneling centers display an orbital Kondo effect and give rise to a non-Fermi-liquid behavior. This latter property is explained by establishing a mapping of the tunneling center model to the multichannel Kondo model via the renormalization group transformation combined with a 1/Nf expansion. The case of M-state systems, the role of the splittings and the present experimental situation are also discussed.