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Physical review. B, Condensed matter

Scaling theory of dynamic scattering centers and moving heavy particles in a fermionic environment

Károly Vladár

Published Jul 15, 1991• 3 citations
Publisher

Abstract

A dynamic scattering object and the slowly building screening fermion cloud around it are examined. The models previously studied for strong on-site scattering---x-ray absorption, the anisotropic Kondo problem, the Anderson model, etc.---have the common feature that all values of the bare scattering matrix commute with one another. Nozi\`eres and De Dominicis's path-integral method can be applied directly to these problems because the scatterings in different channels are independent. The aim of this paper is to extend the theory to those problems, in which the scattering matrix has noncommuting values as, e.g., in the case of a scattering center moving in real space. Integrating over the fermionic degrees of freedom, we get to the problem of a one-dimensional dielectric medium with moving domain walls. In contrast to the commutative case, more-than-two-particle interactions may arise between the domain walls. Renormalization is carried out by keeping the terms that are the most divergent in the short-time-cutoff parameter.

DOI
10.1103/physrevb.44.1019

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•Nov 14, 2012•1 citation
A dynamic scattering object and the slowly building screening fermion cloud around it are examined. The models previously studied for strong on-site scattering---x-ray absorption, the anisotropic Kondo problem, the Anderson model, etc.---have the common feature that all values of the bare scattering matrix commute with one another. Nozi`eres and De Dominicis's path-integral method can be applied directly to these problems because the scatterings in different channels are independent. The aim of this paper is to extend the theory to those problems, in which the scattering matrix has noncommuting values as, e.g., in the case of a scattering center moving in real space. Integrating over the fermionic degrees of freedom, we get to the problem of a one-dimensional dielectric medium with moving domain walls. In contrast to the commutative case, more-than-two-particle interactions may arise between the domain walls. Renormalization is carried out by keeping the terms that are the most divergent in the short-time-cutoff parameter.
We study a heavy particle moving in a periodic lattice of arbitrary dimension and interacting with a degenerate fermionic heat bath. We assume that the Fermi energy of the screening particles is large compared to the tight-binding bandwidth of the particle. We develop a path-integral method for the partition function of the heavy particle where we integrate out the electronic degrees of freedom. We perform also a summation over the paths between the screening-interaction points that makes the method capable of treating band motion and hopping. Following the method of Anderson and Yuval, we derive scaling equations by eliminating that part of the phase space of the light particles which is far from the Fermi energy. We find that the screening strength is not renormalized; the bandwidth of the heavy particle is, however, essentially reduced. That reduction coincides with the reduction in the two-site model. Extrapolating the result for larger values of the screening strength we find band motion and localization at zero temperature depending on the coupling strength; thus the behavior is of Ohmic type. We show that the results obtained can be combined with the classical kinetic equation to get the diffusion coefficient. We believe that the present work based on the path-integral method helps to clarify the connection between results obtained by different methods. The relation of the model to other models, especially to the Caldeira-Leggett model, is discussed in detail.
We study the dynamics of a heavy particle of mass M moving in a one-dimensional repulsively interacting Fermi gas. The Fermi gas is described using the Luttinger model and bosonization. By transforming to a frame comoving with the heavy particle, we map the model onto a generalized ``quantum impurity problem.'' A renormalization-group calculation reveals a crossover from strong to weak coupling upon scaling down in temperature. Above the crossover temperature scale T∗{\mathit{T}}^{\mathrm{*}}T∗=(m/M)EF{\mathit{E}}_{\mathit{F}}EF​, the particle's mobility \ensuremath{\mu} is found to be (roughly) temperature independent and proportional to the dimensionless conductance g, characterizing the one-dimensional Luttinger liquid. Here m(\ensuremath{\ll}M) is the fermion mass, and EF{\mathit{E}}_{\mathit{F}}EF​ the Fermi energy. Below T∗{\mathit{T}}^{\mathrm{*}}T∗, in the weak-coupling regime, the mobility grows and diverges as \ensuremath{\mu}(T)\ensuremath{\sim}${\mathit{T}}^{\mathrm{\ensuremath{-}}4}$ in the T\ensuremath{\rightarrow}0 limit. \textcopyright{} 1996 The American Physical Society.
We use the framework setup recently to compute nonperturbatively inelastic scattering from quantum impurities [G. Zar\'and et al., Phys. Rev. Lett. 93, 107204 (2004)] to study the energy dependence of the single-particle SSS matrix and the inelastic scattering cross section for a number of quantum impurity models. We study the case of the spin S=1∕2S=1∕2S=1∕2 two-channel Kondo model, the Anderson model, and the usual S=1∕2S=1∕2S=1∕2 single-channel Kondo model. We discuss the difference between non-Fermi-liquid and Fermi-liquid models and study how a crossover between the non-Fermi-liquid and Fermi-liquid regimes appears in the case of channel anisotropy for the S=1∕2S=1∕2S=1∕2 two-channel Kondo model. We show that for the most elementary non-Fermi-liquid system, the two-channel Kondo model, half of the scattering remains inelastic even at the Fermi energy. Details of the derivation of the reduction formulas and a simple path integral approach to connect the TTT matrix to local correlation functions are also presented.
Bosonization provides a powerful analytical framework to deal with one-dimensional strongly interacting fermion systems, which makes it a cornerstone in quantum many-body theory. However, this success comes at the expense of using effective infrared parameters, and restricting the description to low energy states near the Fermi level. We propose a radical extension of the bosonization technique that overcomes both limitations, allowing computations with microscopic lattice Hamiltonians, from the Fermi level down to the bottom of the band. The formalism rests on the simple idea of representating the fermion kinetic term in the energy domain, after which it can be expressed in terms of free bosonic degrees of freedom. As a result, one- and two-body fermionic scattering processes generate anharmonic boson−boson interactions, even in the forward channel. We show that up to moderate interaction strengths, these non-linearities can be treated analytically at all energy scales, using the x-ray emission problem as a showcase. In the strong interaction regime, we employ a systematic variational solution of the bosonic theory, and obtain results that agree quantitatively with an exact diagonalization of the original one-particle fermionic model. This provides a proof of the fully microscopic character of bosonization, on all energy scales, for an arbitrary band structure. Besides recovering the known x-ray edge singularity at the emission threshold, we find strong signatures of correlations even at emission frequencies beyond the band bottom.
We consider a one-dimensional model of interacting electrons. The Fermi surface crosses two bands, with light and heavy masses, respectively. Both intraband and interband interactions are assumed. The earlier renormalization-group treatment of the model by Varma and Zawadowski is extended to check the validity of the scaling assumption and to calculate the next corrections to the scaling equations. These corrections indicate a Fermi-velocity renormalization. Depending on the initial parameters, two kinds of behavior can appear. Either the interband charge fluctuations are negligible and then the dominant instabilities are determined by the interband exchange and the intraband backscattering with the two velocities remaining different, or the interband charge fluctuations become dominant and the velocities scale to a common value. The phase diagram of the model is also determined.
Actinide- and Lanthanide based compounds are often considered as heavy Fermion (HF) systems. The essential ingredient for these systems is a localized d- or f-shell electron per lattice site which hybridizes with a non-interacting conduction band of p- or d-shell electrons. For temperatures below the so called coherence temperature Tc, the localized electrons stop to act like mag- netic scatterers known from the Kondo effect. There the conduction band electrons screen the local magnetic moments coherently. As a main effect the conduction electrons build a Fermi liquid (FL) of quasiparticles with strongly enhanced masses which exceed 10-1000 times the mass of the original electrons. These enhancements can be seen experimentally in an increased heat capacity coefficient and Pauli susceptibility. \nIn this work we employ the periodic Anderson model (PAM) to describe the paramagnetic phase of the heavy Fermion compounds. To solve the compli- cated many body problem the dynamical mean-field theory is adopted with the numerical renormalization group technique as impurity solver. \nWe find that for low conduction band fillings Tc decreases drastically, which is in accordance with the exhaustion effect predicted by Nozières [5]. Additionally we find, that the form of the free conduction band density of states is decisive to obtain a FL or a Mott insulating (MI) phase for strongly depleted conduction band fillings. \nFurthermore we investigate the PAM on a bipartite lattice, the coherence temperatures for each sublattice and especially the case of vanishing hybridiza- tion for one sublattice. In the latter case the system resembles a conduction band with only half the localized moment sites coupled to it. Our results show that it depends strongly on the chosen parameters if there is only one scale for the whole lattice or two distinct scales for each sublattice. Especially decoupling one sublattice leads to two separated scales. \nIn the last part the attention is focused on the two-impurity Anderson model and its solution via the DMFT method. We investigate the Anderson model and compare results to direct NRG calculations. Importantly, the two-impurity DMFT method can be easier extended to multi-impurity systems than the NRG on its own. This might lead to a method to efficiently examine multi- impurity systems.