We have studied the quadrupolar Kondo effect for an impurity in cubic environment with taking account of a singlet excited state $Γ_1$, which models a $\mathrm{Pr}^{3+}$ ion with a non-Kramers double ground state $Γ_3$. We have used the numerical renormalization group approach and determined the phase diagram with varying quadrupole Kondo coupling $J$ and $Γ_1$ excitation energy $Δ$. 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 $J$-dependence of the Kondo temperature $T_K$ near the phase boundary $J_c (Δ) $ and found a nice scaling behavior with an stretched exponential form $T_K \sim δJ^α \exp (-\mathrm{const.}/\sqrt{δJ})$ where $δJ \equiv J - J_c (Δ)$. This differs from the standard scaling form and indicates that one needs to consider renormalization of multiple coupling constants.
We have studied the quadrupolar Kondo effect for an impurity in cubic environment with taking account of a singlet excited state Γ1, which models a Pr3+ ion with a non-Kramers double ground state Γ3. We have used the numerical renormalization group approach and determined the phase diagram with varying quadrupole Kondo coupling J and Γ1 excitation energy Δ. 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 J-dependence of the Kondo temperature TK near the phase boundary Jc(Δ) and found a nice scaling behavior with an stretched exponential form TK∼δJαexp(−const./δJ) where δJ≡J−Jc(Δ). This differs from the standard scaling form and indicates that one needs to consider renormalization of multiple coupling constants.
We investigate the emergence of quantum critical points near a two-channel Kondo phase by evaluating an f-electron entropy of a seven-orbital impurity Anderson model hybridized with three (Γ7 and Γ8) conduction bands with the use of a numerical renormalization group method First we consider the case of Pr3+ ion, in which quadrupole two-channel Kondo effect is known to occur for the local Γ3 doublet state. When we control crystalline electric field (CEF) potentials so as to change the local CEF ground state from Γ3 doublet to Γ1 singlet or Γ5 triplet, we commonly observe a residual entropy of logϕ with the golden ratio ϕ=(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+ ion, in which magnetic two-channel Kondo phase is found to occur for the local Γ6 doublet state. We envisage a scenario that the quantum critical point characterized by logϕ generally appears between two-channel Kondo and Fermi-liquid phases.
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 Delta0. When immersed in a metal, electronic angular momentum scattering induces transitions between any two of these levels (couplings J), 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)=kBln2 between the high-T value, S(TggDelta0)=kBln3, and the 2CK ground state value, S(0)=kBlnsqrt2. 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−Delta0 plane. The Kondo temperature TK shows non-monotonic J-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)=kBln2 between high-T value, S(TggDelta0)=kBln3, and the single-channel Kondo ground state value, S(0)=kBln1.
The traditional multichannel Kondo effect takes place when several gapless metallic electronic channels interact with a localized spin-S impurity, with the number of channels n exceeding the size of the impurity spin, n>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-21 impurity coupled isotropically to n 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)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))]. 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)=ln2.
We illustrate the emergence of the three-channel Kondo effect in a seven-orbital impurity Anderson model hybridized with Γ7 and Γ8 conduction electron bands employing a numerical renormalization group method. In particular, we focus on the case with ten local f electrons corresponding to a Ho3+ 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 logϕ with the golden ratio ϕ=(1+5)/2, characteristic of the three-channel Kondo effect, for the wide range of parameters of the local Γ5 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.
Effect of Γ7 and Γ8 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+ ion with ten 4f electrons. This model includes both V7 and V8, which are hybridizations between localized 4f- and conduction electrons in Γ7 and Γ8 orbitals, respectively. For the case of V7=V8 with the local Γ5 triplet ground state, we have reported the discovery of a three-channel Kondo (TCK) phase, characterized by a residual entropy of logϕ with the golden ratio ϕ=(1+5)/2. In this research, by depicting the ground-state phase diagram on the (V8,V7) plane, we attempt to unveil the effect of V7 and V8 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) 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.
We study triangular clusters of three spin-1/2 Kondo or Anderson impurities that are coupled to two conduction leads. In the case of Kondo impurities, the model takes the form of an antiferromagnetic Heisenberg ring with Kondo-type exchange coupling to continuum electrons. We show that this model exhibits many types of the behavior found in various simpler one- and two-impurity models, thereby enabling the study of crossovers between a number of Fermi-liquid (FL) and non-Fermi-liquid (NFL) fixed points. In particular, we explore a direct crossover between the two-impurity Kondo-model NFL fixed point and the two-channel Kondo-model NFL fixed point. We show that the concept of the two-stage Kondo effect applies even in the case when the first-stage Kondo state is of NFL type. In the case of Anderson impurities, we consider the transport properties of three coupled quantum dots. This class of models includes, as limiting cases, the familiar serial double quantum dot and triple quantum dot nanostructures. By extracting the quasiparticle scattering phase shifts, we compute the low-temperature conductance as a function of the interimpurity tunneling coupling. We point out that due to the existence of exponentially low-temperature scales, there is a parameter range where the stable ``zero-temperature'' fixed point is essentially never reached (not even in numerical renormalization group calculations). The zero-temperature conductance is then of no interest and it may only be meaningful to compute the conductance at finite temperature. This illustrates the perils of studying the conductance in the ground state and considering thermal fluctuations only as a small correction.