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arXiv:1611.08673 · Physical review. B./Physical review. B

Multipolar Kondo effect in a S 0 1 − P 2 3 mixture of Yb 173 atoms

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

Published Feb 13, 2018• 20 citations
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Abstract

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).

DOI
10.1103/physrevb.97.075124

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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 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 (\ensuremathλd,\ensuremathλq({\ensuremath{\lambda}}_{\text{d}},{\ensuremath{\lambda}}_{\text{q}}(\ensuremathλd​,\ensuremathλq​, and \ensuremathλo){\ensuremath{\lambda}}_{\text{o}})\ensuremathλo​) 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≫TK)(T\ensuremath{\gg}{T}_{K})(T\ensuremath≫TK​), and are compared with the analogous results obtained for the standard case of dipolar exchange interaction (the s\ensuremath−ds\ensuremath{-}ds\ensuremath−d Hamiltonian).
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​).
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.
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).
We study an exchange interaction in which conduction electrons with pseudospin Sc{\mathrm{S}}_{\mathrm{c}}Sc​=3/2 interact with the impurity spin SI{\mathrm{S}}_{\mathrm{I}}SI​=1/2. Due to the overscreening of the impurity spin by higher conduction electron spin, a nontrivial intermediate-coupling-strength fixed point is realized. Using the numerical renormalization group (NRG), we show that the low-energy spectra are described by a non-Fermi-liquid excitation spectrum. A conformal field theory analysis is compared with NRG results and excellent agreement is obtained. Using the double-fusion rule to generate the operator spectrum with the conformal theory, we find that the specific heat coefficient and magnetic susceptibility will diverge as ${\mathrm{T}}^{\mathrm{\ensuremath{-}}2\mathrm{/}3}$, that the scaling dimension of an applied magnetic field is 5/6, and that exchange anisotropy is always relevant. We discuss the possible relevance of our work to two-level-system Kondo materials and dilute cerium alloys, and we point out a paradox in understanding the Bethe-ansatz solutions to the multichannel Kondo model.
The traditional multichannel Kondo effect takes place when several gapless metallic electronic channels interact with a localized spin-SSS impurity, with the number of channels nnn exceeding the size of the impurity spin, n>2Sn>2Sn>2S, leading to the emergence of non-Fermi liquid impurity behavior at low temperatures. Here, we show that the effect can be realized even when the electronic degrees of freedom are strongly correlated and gapped. The system under consideration consists of a single spin-12\frac{1}{2}21​ impurity coupled isotropically to nnn spin singlet superconducting channels realized by one-dimensional leads with quasi-long-range superconducting order. The competition between the Kondo and superconducting fluctuations induces multiple distinct ground states and boundary phases depending on the relative strengths of the bulk and boundary interactions. Using the Bethe Ansatz technique, we identify four regimes: an overscreened Kondo phase, a zero-mode phase, a Yu-Shiba-Rusinov (YSR) phase, and a local-moment phase with an unscreened impurity, each with its own experimental characteristic. We describe the renormalization-group flow, the excitation spectrum, and the full impurity thermodynamics in each phase. Remarkably, even in the presence of a bulk mass gap, the boundary critical behavior in the Kondo phase is governed by the same exponents as in the gapless theory with the low-energy impurity sector flowing to the SU(2)nSU(2)_nSU(2)n​ Wess-Zumino-Witten (WZW) fixed point, and the impurity entropy monotonically decreasing as a function of temperature. In both the overscreened Kondo and zero-mode phases, the residual impurity entropy is Simp(T→0)=ln⁡[2cos⁡(π/(n+2))]S_{\mathrm{imp}}(T \to 0) = \ln[2\cos(π/(n+2))]Simp​(T→0)=ln[2cos(π/(n+2))]. In the YSR and unscreened phases on the other hand the impurity entropy exhibits non-monotonic temperature dependence and is effectively free at low temperatures with Simp(T→0)=ln⁡2S_{\mathrm{imp}}(T \to 0) = \ln 2Simp​(T→0)=ln2.
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.