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Multipolar Kondo Effect in a $^{173}$Yb($^1$S$_0$) - $^{173}$Yb($^3$P$_2$) Mixture

Igor Kuzmenko, Tetyana Kuzmenko, Y. Avishai, Gyu-Boong Jo

Published Nov 26, 2016• 0 citations
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Abstract

Whereas in the familiar Kondo effect the exchange interaction is dipolar, it can also be multipolar, as has been realized in a recent experiment. Here we study multipolar Kondo effect in a Fermi gas of cold $^{173}$Yb atoms. Making use of different AC polarizability of the electronic ground state Yb($^{1}$S$_{0}$) and the long-lived metastable state Yb$^{*}$($^{3}$P$_{2}$), it is suggested that the latter atoms can be localized and serve as a dilute concentration of magnetic impurities while the former ones remain itinerant. The exchange mechanism between the itinerant Yb and the localized Yb$^{*}$ atoms is analyzed and shown to be antiferromagnetic. The quadruple and octuple interactions act to enhance the Kondo temperature $T_K$ that is found to be experimentally accessible. The bare exchange Hamiltonian needs to be decomposed into dipole ($d$), quadruple ($q$) and octuple ($o$) interactions in order to retain its form under renormalization group (RG) analysis, in which the corresponding exchange constants ($\lambda_{\mathrm{d}}$, $\lambda_{\mathrm{q}}$ and $\lambda_{\mathrm{o}}$) flow independently. Numerical solution of the RG scaling equations reveals a few finite fixed points, indicating an over-screening, which suggests a non-Fermi liquid phase. The impurity contribution to the magnetic susceptibility is calculated in the weak coupling regime (${T}\gg{T}_{K}$).

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Whereas in the familiar Kondo effect the exchange interaction is dipolar, there are systems in which the exchange interaction is multipolar, as has been realized in a recent experiment. Here, we study multipolar Kondo effect in a Fermi gas of cold $^{173}\mathrm{Yb}$ atoms. Making use of different ac polarizabilities of the electronic ground state $\mathrm{Yb}(^{1}\mathrm{S}_{0})$ and the long-lived metastable state ${\mathrm{Yb}}^{*}(^{3}\mathrm{P}_{2})$, it is suggested that the latter atoms can be localized and serve as a dilute concentration of magnetic impurities while the former ones remain itinerant. The exchange mechanism between the itinerant Yb and the localized ${\mathrm{Yb}}^{*}$ atoms is analyzed and shown to be antiferromagnetic. The quadrupole and octupole interactions act to enhance the Kondo temperature ${T}_{K}$ that is found to be experimentally accessible. The bare exchange Hamiltonian needs to be decomposed into dipole $(\text{d})$, quadrupole $(\text{q})$, and octupole $(\text{o})$ interactions in order to retain its form under renormalization group (RG) analysis, in which the corresponding exchange constants $({\ensuremath{\lambda}}_{\text{d}},{\ensuremath{\lambda}}_{\text{q}}$, and ${\ensuremath{\lambda}}_{\text{o}})$ flow independently. Numerical solution of the RG scaling equations reveals a few finite fixed points. Arguments are presented that the Fermi-liquid fixed point at low temperature is unstable, indicating that the impurity is overscreened, which suggests a non-Fermi-liquid phase. The impurity contributions to the specific heat, entropy, and the magnetic susceptibility are calculated in the weak coupling regime $(T\ensuremath{\gg}{T}_{K})$, and are compared with the analogous results obtained for the standard case of dipolar exchange interaction (the $s\ensuremath{-}d$ Hamiltonian).

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Multipolar Kondo Effect in a 173^{173}173Yb(1^11S0_00​) - 173^{173}173Yb(3^33P2_22​) Mixture
Whereas in the familiar Kondo effect the exchange interaction is dipolar, it can also be multipolar, as has been realized in a recent experiment. Here we study multipolar Kondo effect in a Fermi gas of cold 173^{173}173Yb atoms. Making use of different AC polarizability of the electronic ground state Yb(1^{1}1S0_{0}0​) and the long-lived metastable state Yb∗^{*}∗(3^{3}3P2_{2}2​), it is suggested that the latter atoms can be localized and serve as a dilute concentration of magnetic impurities while the former ones remain itinerant. The exchange mechanism between the itinerant Yb and the localized Yb∗^{*}∗ atoms is analyzed and shown to be antiferromagnetic. The quadruple and octuple interactions act to enhance the Kondo temperature TKT_KTK​ that is found to be experimentally accessible. The bare exchange Hamiltonian needs to be decomposed into dipole (ddd), quadruple (qqq) and octuple (ooo) interactions in order to retain its form under renormalization group (RG) analysis, in which the corresponding exchange constants (λd\lambda_{\mathrm{d}}λd​, λq\lambda_{\mathrm{q}}λq​ and λo\lambda_{\mathrm{o}}λo​) flow independently. Numerical solution of the RG scaling equations reveals a few finite fixed points, indicating an over-screening, which suggests a non-Fermi liquid phase. The impurity contribution to the magnetic susceptibility is calculated in the weak coupling regime (T≫TK{T}\gg{T}_{K}T≫TK​).
Whereas in the familiar Kondo effect the exchange interaction is dipolar, there are systems in which the exchange interaction is multipolar, as has been realized in a recent experiment. Here, we study multipolar Kondo effect in a Fermi gas of cold 173Yb^{173}\mathrm{Yb}173Yb atoms. Making use of different ac polarizabilities of the electronic ground state Yb(1S0)\mathrm{Yb}(^{1}\mathrm{S}_{0})Yb(1S0​) and the long-lived metastable state Yb∗(3P2){\mathrm{Yb}}^{*}(^{3}\mathrm{P}_{2})Yb∗(3P2​), it is suggested that the latter atoms can be localized and serve as a dilute concentration of magnetic impurities while the former ones remain itinerant. The exchange mechanism between the itinerant Yb and the localized Yb∗{\mathrm{Yb}}^{*}Yb∗ atoms is analyzed and shown to be antiferromagnetic. The quadrupole and octupole interactions act to enhance the Kondo temperature TK{T}_{K}TK​ that is found to be experimentally accessible. The bare exchange Hamiltonian needs to be decomposed into dipole (d)(\text{d})(d), quadrupole (q)(\text{q})(q), and octupole (o)(\text{o})(o) interactions in order to retain its form under renormalization group (RG) analysis, in which the corresponding exchange constants $({\ensuremath{\lambda}}_{\text{d}},{\ensuremath{\lambda}}_{\text{q}}$, and ${\ensuremath{\lambda}}_{\text{o}})$ flow independently. Numerical solution of the RG scaling equations reveals a few finite fixed points. Arguments are presented that the Fermi-liquid fixed point at low temperature is unstable, indicating that the impurity is overscreened, which suggests a non-Fermi-liquid phase. The impurity contributions to the specific heat, entropy, and the magnetic susceptibility are calculated in the weak coupling regime $(T\ensuremath{\gg}{T}_{K})$, and are compared with the analogous results obtained for the standard case of dipolar exchange interaction (the $s\ensuremath{-}d$ Hamiltonian).
Motivated by the impressive recent advance in manipulating cold ytterbium atoms, we explore and substantiate the feasibility of realizing the Coqblin-Schrieffer model in a gas of cold fermionic 173Yb^{173}\mathrm{Yb}173Yb atoms. Making use of different AC polarizabillity of the electronic ground state (electronic configuration 1S0^{1}\mathrm{S}_{0}1S0​) and the long lived metastable state (electronic configuration 3P0^{3}\mathrm{P}_{0}3P0​), it is substantiated that the latter can be localized and serve as a magnetic impurity while the former remains itinerant. The exchange mechanism between the itinerant 1S0^{1}\mathrm{S}_{0}1S0​ and the localized 3P0^{3}\mathrm{P}_{0}3P0​ atoms is analyzed and shown to be antiferromagnetic. The ensuing SU(6) symmetric Coqblin-Schrieffer Hamiltonian is constructed, and, using the calculated exchange constant JJJ, perturbative renormalization group (RG) analysis yields the Kondo temperature TK{T}_{K}TK​ that is experimentally accessible. A number of thermodynamic measurable observables are calculated in the weak-coupling regime T>TKT>{T}_{K}T>TK​ (using perturbative RG analysis) and in the strong-coupling regime T<TKT<{T}_{K}T<TK​ (employing known Bethe ansatz techniques).
The feasibility of realizing the overscreened Kondo effect in ultracold Fermi gas of atoms with spin $s\ensuremath{\ge}\frac{3}{2}$ in the presence of a localized magnetic impurity atom is proved realistic. Specifying (as a mere example) a system of ultracold 22Na^{22}\mathrm{Na}22Na Fermi gas and a trapped 6Li^{6}\mathrm{Li}6Li impurity, the mechanism of exchange interaction between the Na and Li atoms is elucidated and the exchange constant is found to be positive (antiferromagnetic). The corresponding exchange Hamiltonian is derived, and the Kondo temperature is estimated at the order of 500 nK. Within a weak-coupling renormalization group scheme, it is shown that the coupling renormalizes to the non-Fermi-liquid fixed point. An observable displaying multichannel features even in the weak-coupling regime is the impurity magnetization that is negative for $T\ensuremath{\gg}{T}_{K}$ and becomes positive with decreasing temperature.
The Kondo effect is a paradigmatic model of strongly correlated physics, where a magnetic impurity forms a many-body singlet with a fermionic environment. The properties of this singlet depend sensitively on correlations within the environment, which are particularly pronounced in one dimension, where interactions give rise to Luttinger liquid behavior. A promising platform to study these effects is provided by cold gases of ytterbium (Yb) atoms, where an atomic impurity is coupled to an interacting one-dimensional fermionic environment via both magnetic and potential scattering. These two scattering mechanisms compete with each other, raising the question of how robust Kondo screening remains in such systems. Here, we show that potential scattering can suppress Kondo screening in one-dimensional Yb gases. Remarkably, however, Kondo physics persists in well-defined regimes determined by the interactions in the environment. Combining analytical renormalization-group methods with density-matrix renormalization-group simulations, we identify a transition from a strongly to a weakly entangled impurity as potential scattering increases. The two approaches are in excellent agreement across the parameter regimes considered. Our results provide a quantitative criterion for the emergence of Kondo screening in one-dimensional Yb gases and delineate experimentally accessible regimes for its realization in cold-atom platforms.
The relevance of magnetic impurity problems in cold atom systems depends crucially on the nature of exchange interaction between itinerant fermionic atoms and a localized impurity atom. In particular, Kondo physics occurs only if the exchange interaction is anti-ferromagnetic, and strong enough to yield high enough Kondo temperature (TK/TF≥0.1T_K/T_F \ge 0.1TK​/TF​≥0.1). Focusing, as an example, on the experimentally accessible system of ultra-cold 173^{173}173Yb atoms, it is shown that the sign and strength of an exchange interaction between an itinerant Yb(1^{1}1S0_{0}0​) atom and a trapped Yb(3^{3}3P0_{0}0​) atom can be optically controlled. Explicitly, as the light intensity increases (from zero), the exchange interaction changes from ferromagnetic to anti-ferromagnetic. When the light intensity is just below a singlet Feshbach resonance, the singlet scattering length aSa_SaS​ is large and negative, and the Kondo temperature increases sharply.
The Kondo problem, which describes the interaction of a spin sss magnetic impurity with a free Fermi gas, is a classic example of strongly coupled physics. Historically, the problem has been solved by Wilson's numerical renormalization group and later by Bethe ansatz. In this paper, we present an alternate analytic solution of the Kondo problem that combines an expansion in 1/s1/s1/s with the renormalization group. We study both the case of an impurity interacting with a single channel K=1K=1K=1 of fermions in the s→∞s \rightarrow \inftys→∞ limit and the case with KKK channels in the double-scaling limit K→∞K \rightarrow \inftyK→∞, s→∞s \to \inftys→∞, K/sK/sK/s fixed. Our approach allows us to describe analytically intermediate scales of the Kondo problem at large sss and compute thermodynamic observables such as the impurity entropy and susceptibility. We find these observables to agree with the Bethe ansatz results. We also compute the impurity spectral function, finite temperature resistivity and the Kondo screening cloud profile, properties that are not easily accessible from Bethe ansatz. Notably, in the regime K>2sK > 2sK>2s we access the "non-Fermi-liquid" overscreened fixed point of the multichannel Kondo problem.