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arXiv:1503.04479 · Physical Review B

Entanglement entropy near Kondo-destruction quantum critical points

J. H. Pixley, Tathagata Chowdhury, Matthew Miecnikowski, Jaimie Stephens, Christopher Wagner, Kevin Ingersent

Published Jun 10, 2015• 9 citations
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Abstract

We study the impurity entanglement entropy $S_e$ in quantum impurity models that feature a Kondo-destruction quantum critical point (QCP) arising from a pseudogap in the conduction-band density of states or from coupling to a bosonic bath. On the local-moment (Kondo-destroyed) side of the QCP, the entanglement entropy contains a critical component that can be related to the order parameter characterizing the quantum phase transition. In Kondo models describing a spin-$\Simp$, $S_e$ assumes its maximal value of $\ln(2\Simp+1)$ at the QCP and throughout the Kondo phase, independent of features such as particle-hole symmetry and under- or over-screening. In Anderson models, $S_e$ is nonuniversal at the QCP, and at particle-hole symmetry, rises monotonically on passage from the local-moment phase to the Kondo phase; breaking this symmetry can lead to a cusp peak in $S_e$ due to a divergent charge susceptibility at the QCP. Implications of these results for quantum critical systems and quantum dots are discussed.

DOI
10.1103/physrevb.91.245122

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We study the impurity entanglement entropy SeS_eSe​ in quantum impurity models that feature a Kondo-destruction quantum critical point (QCP) arising from a pseudogap in the conduction-band density of states or from coupling to a bosonic bath. On the local-moment (Kondo-destroyed) side of the QCP, the entanglement entropy contains a critical component that can be related to the order parameter characterizing the quantum phase transition. In Kondo models describing a spin-\Simp\Simp\Simp, SeS_eSe​ assumes its maximal value of ln⁡(2\Simp+1)\ln(2\Simp+1)ln(2\Simp+1) at the QCP and throughout the Kondo phase, independent of features such as particle-hole symmetry and under- or over-screening. In Anderson models, SeS_eSe​ is nonuniversal at the QCP, and at particle-hole symmetry, rises monotonically on passage from the local-moment phase to the Kondo phase; breaking this symmetry can lead to a cusp peak in SeS_eSe​ due to a divergent charge susceptibility at the QCP. Implications of these results for quantum critical systems and quantum dots are discussed.
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We address the phenomenon of critical Kondo destruction in pseudogap Bose-Fermi Anderson and Kondo quantum impurity models. These models describe a localized level coupled both to a fermionic bath having a density of states that vanishes like ${|\ensuremath{\epsilon}|}^{r}$ at the Fermi energy ($\ensuremath{\epsilon}=0$) and, via one component of the impurity spin, to a bosonic bath having a sub-Ohmic spectral density proportional to ${|\ensuremath{\omega}|}^{s}$. Each bath is capable by itself of suppressing the Kondo effect at a continuous quantum phase transition. We study the interplay between these two mechanisms for Kondo destruction using continuous-time quantum Monte Carlo for the pseudogap Bose-Fermi Anderson model with 0<r<120<r<\frac{1}{2}0<r<21​ and $\frac{1}{2}\ensuremath{\le}s<1$, and applying the numerical renormalization group to the corresponding Kondo model. At particle-hole symmetry, the models exhibit a quantum-critical point between a Kondo (fermionic strong-coupling) phase and a localized (Kondo-destroyed) phase. The two solution methods, which are in good agreement in their domain of overlap, provide access to the many-body spectrum, as well as to correlation functions including, in particular, the single-particle Green's function and the static and dynamical local spin susceptibilities. The quantum-critical regime exhibits the hyperscaling of critical exponents and $\ensuremath{\omega}/T$ scaling in the dynamics that characterize an interacting critical point. The (r,s)(r,s)(r,s) plane can be divided into three regions: one each in which the calculated critical properties are dominated by the bosonic bath alone or by the fermionic bath alone, and between these two regions, a third in which the bosonic bath governs the critical spin response but both baths influence the renormalization-group flow near the quantum-critical point.
Entanglement is a defining feature of quantum mechanics, and its relation to quantum criticality is of considerable current interest. Here we show that measurable spin and charge fluctuations provide an entanglement witness of quantum criticality in the two-impurity Kondo model, which has experimental realizations in terms of coupled quantum dots and magnetic impurities added to surfaces. We use density-matrix renormalization group and numerical renormalization group calculations to resolve the non-Fermi-liquid critical point separating two independently Kondo-screened impurities from an inter-impurity singlet and interpret it as a change in the dominant entanglement partner of each local moment: from entanglement of the local moment with an extended set of conduction-electron degrees of freedom to entanglement with the other impurity. This reorganization is accompanied by a singular response of the impurity-bath entanglement and the inter-impurity susceptibility. We show how the same structure is encoded in the quantum Fisher information of collective spin and charge operators at zero and finite temperatures, connecting the entanglement picture to experimentally accessible dynamical response functions. Our results establish impurity systems as controlled settings in which quantum-critical entanglement can be detected through measurable correlations, and provide further insight into the possibility of understanding heavy-fermion physics in terms of entanglement.
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.
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