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arXiv:cond-mat/0604603 · Physical Review B

Universal crossovers and critical dynamics of quantum phase transitions: A renormalization group study of the pseudogap Kondo problem

Lars Fritz, Serge Florens, Matthias Vojta C

Published Oct 16, 2006• 38 citations
arXivHTMLPDF

Abstract

The pseudogap Kondo problem, describing a magnetic impurity embedded in an electronic environment with a power-law density of states, displays continuous quantum phase transitions between free and screened moment phases. In this paper we employ renormalization group techniques to analytically calculate universal crossover functions, associated to these transitions, for various observables. Quantitative agreement with the results of Numerical Renormalization Group (NRG) simulations is obtained for temperature-dependent static and zero-temperature dynamic quantities, at and away from criticality. In the notoriously difficult realm of finite-temperature low-frequency dynamics, usually inaccessible to both NRG and perturbative methods, we show that progress can be made by a suitable renormalization procedure in the framework of the Callan-Symanzik equations. Our general strategy can be extended to other zero-temperature phase transitions, both in quantum impurity models and bulk systems.

DOI
10.1103/physrevb.74.144410

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The pseudogap Kondo problem, describing a magnetic impurity embedded in an electronic environment with a power-law density of states, displays continuous quantum phase transitions between free and screened moment phases. In this paper we employ renormalization group techniques to analytically calculate universal crossover functions, associated to these transitions, for various observables. Quantitative agreement with the results of Numerical Renormalization Group (NRG) simulations is obtained for temperature-dependent static and zero-temperature dynamic quantities, at and away from criticality. In the notoriously difficult realm of finite-temperature low-frequency dynamics, usually inaccessible to both NRG and perturbative methods, we show that progress can be made by a suitable renormalization procedure in the framework of the Callan-Symanzik equations. Our general strategy can be extended to other zero-temperature phase transitions, both in quantum impurity models and bulk systems.
We discuss a multichannel SU(N) Kondo model which displays nontrivial zero-temperature phase transitions due to a conduction electron density of states vanishing with a power law at the Fermi level. In a particular large- N limit, the system is described by coupled integral equations corresponding to a dynamic saddle point. We exactly determine the universal low-energy behavior of spectral densities at the scale-invariant fixed points, obtain anomalous exponents, and compute scaling functions describing the crossover near the quantum-critical points. We argue that our findings are relevant to recent experiments on impurity-doped d-wave superconductors.
We revisit the physics of a Kondo impurity coupled to a fermionic host with a diverging power-law density of states near the Fermi level, $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\sim}{|\ensuremath{\omega}|}^{r}$, with exponent $\ensuremath{-}1<r<0$. Using the analytical understanding of several fixed points, based partially on powerful mappings between models with bath exponents rrr and $(\ensuremath{-}r)$, combined with accurate numerical renormalization group calculations, we determine thermodynamic quantities within the stable phases and also near the various quantum phase transitions. Antiferromagnetic Kondo coupling leads to strong screening with a negative zero-temperature impurity entropy, while ferromagnetic Kondo coupling can induce a stable fractional spin moment. We formulate the quantum field theories for all critical fixed points of the problem, which are fermionic in nature and allow for a perturbative renormalization-group treatment.
We investigate the critical behavior of the Kondo compensation in the presence of a power-law pseudogap in the density of states, $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\sim}{|\ensuremath{\omega}|}^{\ensuremath{\epsilon}}$. For $\ensuremath{\epsilon}<1$, generically---in the absence of particle-hole symmetry---this model exhibits a quantum phase transition from a partially screened doublet ground state to a fully screened many-body ground state upon increasing the exchange coupling jjj. At the critical point jc{j}_{\text{c}}jc​, the Kondo compensation is found to scale as $\ensuremath{\kappa}(j<{j}_{\text{c}})=1\ensuremath{-}g(j)$ with the local ggg-factor vanishing as $g\ensuremath{\sim}|j\ensuremath{-}{j}_{\text{c}}{|}^{\ensuremath{\beta}}$. We combine the perturbative drone fermion method with the nonperturbative numerical renormalization group computations to determine the critical exponent $\ensuremath{\beta}(\ensuremath{\epsilon})$, which exhibits a nonmonotonous behavior as a function of $\ensuremath{\epsilon}$. Our results confirm that the Kondo cloud builds up continuously in the presence of a weak pseudogap as one approaches the phase transition.
In the early 1970s, Wilson developed the concept of a fully nonperturbative renormalization group transformation. When applied to the Kondo problem, this numerical renormalization group (NRG) method gave for the first time the full crossover from the high-temperature phase of a free spin to the low-temperature phase of a completely screened spin. The NRG method was later generalized to a variety of quantum impurity problems. The purpose of this review is to give a brief introduction to the NRG method, including some guidelines for calculating physical quantities, and to survey the development of the NRG method and its various applications over the last 30 years. These applications include variants of the original Kondo problem such as the non-Fermi-liquid behavior in the two-channel Kondo model, dissipative quantum systems such as the spin-boson model, and lattice systems in the framework of the dynamical mean-field theory.
Quantum phase transitions out of equilibrium are outstanding emergent subjects in condensed matter physics with great fundamental importance and challenges. We theoretically investigate here the nonequilibrium quantum phase transition in a generic nano-setup: the pseudogap Kondo model where a Kondo quantum dot couples to two---left (L) and right (R)---voltage-biased fermionic leads with power-law density of states (DOS) with respect to their Fermi levels ${\ensuremath{\mu}}_{L/R}$, ${\ensuremath{\rho}}_{c,L(R)}(\ensuremath{\omega})\ensuremath{\propto}{|\ensuremath{\omega}\ensuremath{-}{\ensuremath{\mu}}_{L(R)}|}^{r}$ with 0<r<10<r<10<r<1. In equilibrium (${\ensuremath{\mu}}_{L}\ensuremath{-}{\ensuremath{\mu}}_{R}=0$) and for 0<r<1/20<r<1/20<r<1/2, with increasing Kondo correlations this model exhibits a quantum phase transition from a unscreened local moment (LM) phase to the Kondo screened phase. At finite bias voltages and near criticality, we discover new nonequilibrium universal scaling behaviors in conductance, conduction electron TTT matrix, and local spin susceptibility via a controlled frequency-dependent renormalization group (RG) approach. The current-induced decoherence is key to understanding these distinct universal nonequilibrium quantum critical regimes. The relevance of our results to experiment is discussed.
Quantum impurity models are the prototypical examples of quantum many-body dynamics which manifests in their spectral and transport properties. Single channel Anderson(and Kondo model) leads to the Fermi liquid ground state in the strong coupling regime which corresponds to a stable infrared fixed point at which the quantum impurity gets completely screened by the conduction electrons. Quantum impurity models with non-trivial density of states exhibit quantum phase transition and this quantum criticality lies in the universality class of local quantum critical systems. In this paper, we report first study of the flow equation renormalization of gapped Kondo model which has gapped density of states, the gap being at the Fermi level. Flow equation renormalization group method has proved to be one of the very robust renormalization methods to study Kondo physics both in equilibrium as well as in non-equilibrium. Here we have shown that this method can also be employed to study local quantum criticality. We have calculated the flow equations for the Kondo coupling and solved them for various values of the gap parameter and we find that there is suppression of Kondo divergence as gap is increased which signifies that as gap is increased, renormalization flow goes away from the strong coupling fixed point. We have also calculated the spin susceptibility and we find that as gap is increased, susceptibility goes over to the Curie behaviour and hence confirming the renormalization flow towards the local moment fixed point.