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arXiv:1309.0167 · Physical Review B

Quantum phase transitions and thermodynamics of the power-law Kondo model

Andrew K. Mitchell, Matthias Vojta C, R. Bulla, Lars Fritz

Published Nov 12, 2013• 25 citations
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Abstract

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 $r$ 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.

DOI
10.1103/physrevb.88.195119

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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, \ensuremathρ(\ensuremathω)\ensuremath∼∣\ensuremathω∣r\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\sim}{|\ensuremath{\omega}|}^{r}\ensuremathρ(\ensuremathω)\ensuremath∼∣\ensuremathω∣r, with exponent \ensuremath−1<r<0\ensuremath{-}1<r<0\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)(\ensuremath{-}r)(\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 discuss the two-channel Kondo problem with a pseudogap density of states $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{|\ensuremath{\omega}|}^{r}$ of the bath fermions. Combining both analytical and numerical renormalization group techniques, we characterize the impurity phases and quantum phase transitions of the relevant Kondo and Anderson models. The line of stable points, corresponding to the overscreened non-Fermi-liquid behavior of the metallic r=0r=0r=0 case, is replaced by a stable particle-hole-symmetric intermediate-coupling fixed point for $0<r<{r}_{\mathrm{max}}\ensuremath{\approx}0.23$. For r>rmaxr>{r}_{\mathrm{max}}r>rmax​, this non-Fermi-liquid phase disappears, and instead a critical fixed point with an emergent spin-channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively. We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ $\ensuremath{\epsilon}$ expansion techniques to calculate critical properties near the critical dimensions r=0r=0r=0 and 1, the latter being potentially relevant for two-channel Kondo impurities in neutral graphene. We find the analytical results to be in excellent agreement with those obtained from applying Wilson's numerical renormalization group technique.
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.
We extend the numerical renormalization-group method to treat Bose-Fermi Kondo models (BFKMs) describing a local moment coupled both to a conduction band and to a dissipative bosonic bath representing, e.g., lattice or spin collective excitations of the environment. We apply the method to the Ising-symmetry BFKM with a structureless band and a bath spectral function $\ensuremath{\eta}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}$. The method is valid for all bath exponents sss and all temperatures TTT. For 0<s<10<s<10<s<1, the range of interest in the context of heavy-fermion quantum criticality, an interacting critical point, characterized by hyperscaling of exponents and $\ensuremath{\omega}∕T$ scaling, describes a continuous quantum phase transition between Kondo-screened and localized phases. For Ohmic dissipation s=1s=1s=1, where the model is relevant to certain dissipative mesoscopic qubit devices, the transition is found to be Kosterlitz-Thouless-like. In both regimes the impurity spectral function for the corresponding Anderson model shows clearly the collapse of the Kondo resonance at the transition. Connection is made to other recent results for the BFKM and the spin-boson model.
We study the behavior of the entropy of the pseudogap Bose-Fermi Kondo model within a dynamical large-NNN limit, where NNN is related to the symmetry group of the model. This model is a general quantum impurity model that describes a localized level coupled to a fermionic bath having a density of states that vanishes in a power-law fashion near the Fermi energy and to a bosonic bath possessing a power-law spectral density below a cutoff energy. As a function of the couplings to the baths, various quantum phase transitions can occur. We study how the impurity entropy changes across these zero-temperature transitions and compare our results with predictions based on the ggg-theorem. This is accomplished by an analysis of the leading and subleading scaling behaviors. Our analysis shows that the ggg-theorem does not apply to the pseudogap Bose-Fermi Kondo model at the large-NNN level. This inapplicability originates from an anomalous contribution to the scaling function in the hydrodynamic regime where ${k}_{B}T>\ensuremath{\hbar}\ensuremath{\omega}$ which is absent in the quantum coherent regime, i.e., for ${k}_{B}T<\ensuremath{\hbar}\ensuremath{\omega}$. We also compare our results with those obtained for the Sachdev-Ye-Kitaev model.
The numerical renormalization group is used to study quantum entanglement in the Kondo impurity model with a density of states ρ(ϵ)∝|ϵ|^{r} (0<r<1/2) that vanishes at the Fermi energy ϵ=0. This nonintegrable model features a Kondo-destruction quantum critical point (QCP) separating a partially screened phase from a local-moment phase. The impurity contribution S_{e}^{imp} to the entanglement entropy between a region of radius R around the magnetic impurity and the rest of the system reveals a length scale R^{*} that distinguishes a region R≪R^{*} of strong critical entanglement from one R≫R^{*} of weak entanglement. Within each phase, S_{e}^{imp} is a universal function of R/R^{*} with a power-law decay for R/R^{*}≫1. The entanglement length R^{*} diverges on approach to the interacting QCP, showing that the critical Kondo screening cloud subsumes the entire system as the impurity becomes maximally entangled with the conduction band. This work has implications for entanglement calculations in other models and for the nature of heavy-fermion quantum criticality.
In quantum critical heavy fermion systems, local moments are coupled to both collective spin fluctuations and conduction electrons. As such, the Bose-Fermi Kondo model, describing the coupling of a local moment to both a bosonic and a fermionic bath, has been of extensive interest. For the model in the presence of SU(2) spin rotational symmetry, questions have been raised about its phase diagram. Here we develop a version of continuous-time Quantum Monte Carlo (CT-QMC) method suitable for addressing this issue; this procedure can reach sufficiently low temperatures while preserving the SU(2) symmetry. Using this method for the Bose-Fermi Anderson model, we clarify the renormalization-group fixed points and the phase diagram for the case with a constant fermionic-bath density of states and a power-law bosonic-bath spectral function ρb(ω)∝ωs\rho_{b}(\omega) \propto \omega^{s}ρb​(ω)∝ωs (0<s<10<s<10<s<1). We find two types of Kondo destruction QCP, depending on the power-law exponent sss in the bosonic bath spectrum. For s∗<s<1s^{*}<s<1s∗<s<1, both types of QCPs exist and, in the parameter regime accessible by an analytical ϵ\epsilonϵ-expansion renormalization-group calculation (here ϵ=1−s\epsilon=1-sϵ=1−s), the CT-QMC result is fully consistent with prior predictions by the latter method. For s<s∗s<s^{*}s<s∗, there is only one type of QCP. At both type of Kondo destruction QCPs, we find that the exponent of the local spin susceptibility η\etaη obeys the relation η=ϵ\eta=\epsilonη=ϵ, which has important implications for Kondo destruction QCP in the Kondo lattice problem.