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arXiv:2109.14952 · Journal of the Physical Society of Japan

Three-Channel Kondo Effect Emerging from Ho Ions

Takashi Hotta

Published Oct 4, 2021• 5 citations
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Abstract

We illustrate the emergence of the three-channel Kondo effect in a seven-orbital impurity Anderson model hybridized with $Γ_7$ and $Γ_8$ conduction electron bands employing a numerical renormalization group method. In particular, we focus on the case with ten local $f$ electrons corresponding to a Ho$^{3+}$ ion. By investigating the dependence of the low-temperature behavior of the $f$-electron entropy on the crystalline electric field potential and hybridization, we found a residual entropy of $\log ϕ$ with the golden ratio $ϕ=(1+\sqrt{5})/2$, characteristic of the three-channel Kondo effect, for the wide range of parameters of the local $Γ_5$ triplet ground state. To improve our insight of local impurity spin, we explored a quantum critical point between three-channel Kondo and Fermi-liquid phases. Finally, we briefly discussed the cubic Ho compound as a candidate to observe experimentally the three-channel Kondo effect.

DOI
10.7566/jpsj.90.113701

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•May 29, 2013•0 citations
We illustrate the emergence of the three-channel Kondo effect in a seven-orbital impurity Anderson model hybridized with Γ7Γ_7Γ7​ and Γ8Γ_8Γ8​ conduction electron bands employing a numerical renormalization group method. In particular, we focus on the case with ten local fff electrons corresponding to a Ho3+^{3+}3+ ion. By investigating the dependence of the low-temperature behavior of the fff-electron entropy on the crystalline electric field potential and hybridization, we found a residual entropy of log⁡ϕ\log ϕlogϕ with the golden ratio ϕ=(1+5)/2ϕ=(1+\sqrt{5})/2ϕ=(1+5​)/2, characteristic of the three-channel Kondo effect, for the wide range of parameters of the local Γ5Γ_5Γ5​ triplet ground state. To improve our insight of local impurity spin, we explored a quantum critical point between three-channel Kondo and Fermi-liquid phases. Finally, we briefly discussed the cubic Ho compound as a candidate to observe experimentally the three-channel Kondo effect.
We investigate the emergence of quantum critical points near a two-channel Kondo phase by evaluating an fff-electron entropy of a seven-orbital impurity Anderson model hybridized with three (Γ7Γ_7Γ7​ and Γ8Γ_8Γ8​) conduction bands with the use of a numerical renormalization group method First we consider the case of Pr3+^{3+}3+ ion, in which quadrupole two-channel Kondo effect is known to occur for the local Γ3Γ_3Γ3​ doublet state. When we control crystalline electric field (CEF) potentials so as to change the local CEF ground state from Γ3Γ_3Γ3​ doublet to Γ1Γ_1Γ1​ singlet or Γ5Γ_5Γ5​ triplet, we commonly observe a residual entropy of log⁡ϕ\log ϕlogϕ with the golden ratio ϕ=(1+5)/2ϕ=(1+\sqrt{5})/2ϕ=(1+5​)/2, which is equal to that for three-channel Kondo effect. This peculiar residual entropy is also observed for the case of Nd3+^{3+}3+ ion, in which magnetic two-channel Kondo phase is found to occur for the local Γ6Γ_6Γ6​ doublet state. We envisage a scenario that the quantum critical point characterized by log⁡ϕ\log ϕlogϕ generally appears between two-channel Kondo and Fermi-liquid phases.
Effect of Γ7Γ_7Γ7​ and Γ8Γ_8Γ8​ Hybridizations on Three-Channel Kondo Phase Emerging from Ho Ions
By employing a numerical renormalization group method, we analyze a seven-orbital impurity Anderson model for Ho3+^{3+}3+ ion with ten 4f4f4f electrons. This model includes both V7V_7V7​ and V8V_8V8​, which are hybridizations between localized 4f4f4f- and conduction electrons in Γ7Γ_7Γ7​ and Γ8Γ_8Γ8​ orbitals, respectively. For the case of V7=V8V_7=V_8V7​=V8​ with the local Γ5Γ_5Γ5​ triplet ground state, we have reported the discovery of a three-channel Kondo (TCK) phase, characterized by a residual entropy of log⁡ϕ\log ϕlogϕ with the golden ratio ϕ=(1+5)/2ϕ=(1+\sqrt{5})/2ϕ=(1+5​)/2. In this research, by depicting the ground-state phase diagram on the (V8,V7)(V_8, V_7)(V8​,V7​) plane, we attempt to unveil the effect of V7V_7V7​ and V8V_8V8​ on the emergence of the TCK phase. After performing a lot of numerical calculations, we find that the TCK phase appears in a relatively wide region on the (V8,V7)(V_8, V_7)(V8​,V7​) plane. The boundary curves surrounding the TCK phase are determined by the variation of the temperature dependence in entropy and the abrupt change in energy spectra. We consider that most of the phases surrounding the TCK phase are Fermi liquids, but the non-Fermi liquid two-channel Kondo phase is unexpectedly found to exist next to the TCK phase. Finally, we briefly comment on the actual material concerning the detection of the TCK phase.
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.
A multichannel Kondo model, where two or more equivalent but independent channels of electrons compete to screen a spin-12\frac{1}{2}21​ impurity, shows overcompensation of the impurity spin, leading to the non-Fermi-liquid behavior in various thermodynamic and transport properties. However, when the channel symmetry is broken, an impurity quantum phase transition can occur at zero temperature. Identification of an order parameter describing the impurity quantum phase transition is very difficult since it is beyond the conventional Landau-Ginzburg-Wilson theory. By employing the natural orbitals renormalization group method, we study both two-channel and three-channel Kondo models, from the perspective of spin correlation between the impurity and electrons in electronic channels. Here we demonstrate that by introducing the spin-correlation ratio as an order parameter we can characterize impurity quantum phase transitions driven by channel asymmetry. In particular, the universal critical exponents $\ensuremath{\beta}$ of the spin-correlation ratio and $\ensuremath{\nu}$ of the correlation length are explicitly determined by finite-size-scaling analysis, namely, $\ensuremath{\beta}=0.10(1)$ and $\ensuremath{\nu}=2.0(1)$, and $\ensuremath{\beta}=0.10(1)$ and $\ensuremath{\nu}=2.5(1)$ for the two-channel and three-channel Kondo models, respectively.
We have studied the quadrupolar Kondo effect for an impurity in cubic environment with taking account of a singlet excited state Γ1Γ_1Γ1​, which models a Pr3+\mathrm{Pr}^{3+}Pr3+ ion with a non-Kramers double ground state Γ3Γ_3Γ3​. We have used the numerical renormalization group approach and determined the phase diagram with varying quadrupole Kondo coupling JJJ and Γ1Γ_1Γ1​ excitation energy ΔΔΔ. Two phases are found and identified as local Fermi liquid and non-Fermi liquid. This non-Fermi liquid phase is characteristic to the two-channel Kondo impurity, and a similar phase diagram has been also found in other extended quadrupole models. We have analyzed in detail the JJJ-dependence of the Kondo temperature TKT_KTK​ near the phase boundary $J_c (Δ) $ and found a nice scaling behavior with an stretched exponential form TK∼δJαexp⁡(−const./δJ)T_K \sim δJ^α \exp (-\mathrm{const.}/\sqrt{δJ})TK​∼δJαexp(−const./δJ​) where δJ≡J−Jc(Δ)δJ \equiv J - J_c (Δ)δJ≡J−Jc​(Δ). This differs from the standard scaling form and indicates that one needs to consider renormalization of multiple coupling constants.
The two-channel Kondo (2CK) effect with its exotic ground state properties has remained difficult to realize in physical systems. At low energies, a quantum impurity with orbital degree of freedom, like a proton bound in an interstitial lattice space, comprises a 3-level system with a unique ground state and (at least) doubly degenerate rotational excitations with excitation energy Delta0Delta_0Delta0​. When immersed in a metal, electronic angular momentum scattering induces transitions between any two of these levels (couplings JJJ), while the electron spin is conserved. We show by extensive numerical renormalization group (NRG) calculations that without fine-tuning of parameters this system exhibits a 2CK fixed point, due to Kondo correlations in the excited-state doublet whose degeneracy is stabilized by the host lattice parity, while the channel symmetry (electron spin) is guaranteed by time reversal symmetry. We find a pronounced plateau in the entropy at S(TK<T<Delta0)=kBln2S(T_K < T < Delta_0) = k_B ln 2S(TK​<T<Delta0​)=kB​ln2 between the high-TTT value, S(TggDelta0)=kBln3S(T gg Delta_0) = k_B ln 3S(TggDelta0​)=kB​ln3, and the 2CK ground state value, S(0)=kBlnsqrt2S(0) = k_B lnsqrt{2}S(0)=kB​lnsqrt2. This indicates a downward renormalization of the doublet below the non-interacting ground state, thus realizing the 2CK fixed point, in agreement with earlier conjectures. We mapped out the phase diagram of the model in the J−Delta0J-Delta_0J−Delta0​ plane. The Kondo temperature TKT_KTK​ shows non-monotonic JJJ-dependence, characteristic for 2CK systems.newline indent Beside the two-channel Kondo effect of the model, we also study the single-channel version, which is realized by applying a strong magnetic field to the conduction band electrons so that their degeneracy is lifted and consequently having only one kind of electrons scattering off the impurity. This single-channel case is easier to analyze since the Hilbert space is not as large as that of the 2CK. We equally find a downward renormalization of the excited state energy by the Kondo correlations in the SU(2) doublet. In a wide range of parameter values this stabilizes the single-channel Kondo fixed point and a phase diagram is also mapped out for the model. In the single-channel version a plateau is found in the entropy at S(TK<T<Delta0)=kBln2S(T_K < T < Delta_0) = k_B ln 2S(TK​<T<Delta0​)=kB​ln2 between high-TTT value, S(TggDelta0)=kBln3S(T gg Delta_0) = k_B ln 3S(TggDelta0​)=kB​ln3, and the single-channel Kondo ground state value, S(0)=kBln1S(0) = k_B ln 1S(0)=kB​ln1.