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arXiv:2009.06842 · Journal of the Physical Society of Japan

Quantum Critical Point between Two-Channel Kondo and Fermi-Liquid Phases

Takashi Hotta

Published Oct 8, 2020• 4 citations
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Abstract

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 ($Γ_7$ and $Γ_8$) conduction bands with the use of a numerical renormalization group method First we consider the case of Pr$^{3+}$ ion, in which quadrupole two-channel Kondo effect is known to occur for the local $Γ_3$ doublet state. When we control crystalline electric field (CEF) potentials so as to change the local CEF ground state from $Γ_3$ doublet to $Γ_1$ singlet or $Γ_5$ triplet, we commonly observe a residual entropy of $\log ϕ$ with the golden ratio $ϕ=(1+\sqrt{5})/2$, which is equal to that for three-channel Kondo effect. This peculiar residual entropy is also observed for the case of Nd$^{3+}$ ion, in which magnetic two-channel Kondo phase is found to occur for the local $Γ_6$ doublet state. We envisage a scenario that the quantum critical point characterized by $\log ϕ$ generally appears between two-channel Kondo and Fermi-liquid phases.

DOI
10.7566/jpsj.89.114706

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•May 29, 2013•0 citations
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.
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.
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.
We discuss the two-channel Kondo problem with a pseudogap density of states $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\propto}{|\ensuremath{\omega}|}^{r}$ of the bath fermions. Combining both analytical and numerical renormalization group techniques, we characterize the impurity phases and quantum phase transitions of the relevant Kondo and Anderson models. The line of stable points, corresponding to the overscreened non-Fermi-liquid behavior of the metallic r=0r=0r=0 case, is replaced by a stable particle-hole-symmetric intermediate-coupling fixed point for $0<r<{r}_{\mathrm{max}}\ensuremath{\approx}0.23$. For r>rmaxr>{r}_{\mathrm{max}}r>rmax​, this non-Fermi-liquid phase disappears, and instead a critical fixed point with an emergent spin-channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively. We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ $\ensuremath{\epsilon}$ expansion techniques to calculate critical properties near the critical dimensions r=0r=0r=0 and 1, the latter being potentially relevant for two-channel Kondo impurities in neutral graphene. We find the analytical results to be in excellent agreement with those obtained from applying Wilson's numerical renormalization group technique.
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.
A connection between competing Fermi and non-Fermi-liquid states is studied by a Kondo Hamiltonian for a triplet impurity, keeping cubic symmetry in mind. A uniaxial anisotropy stabilizes either state, corresponding to a local field in the two-channel Kondo model, which reflects on the local charge redistribution in the singlet-doublet configuration of the impurity. The numerical renormalization group and SU(2) group theoretical analyses show that the critical point is described by threefold degenerate terms of the two-channel Kondo Hamiltonian, one of which derives the non-Fermi-liquid state and the other two lead to the Fermi liquid.
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.