Three Types of Non-Fermi-Liquid Fixed Point for a Triplet Quantum Impurity in a Cubic Metal

TL;DR

This work classifies non-Fermi-liquid (NFL) fixed points for a spin-1 (triplet) impurity in cubic metals coupled to Γ8 conduction electrons by deriving all six cubic-symmetry allowed exchange interactions and solving the model with numerical renormalization group (NRG). It confirms that the well-known spherical-symmetry NFLs arise from the dipolar Kondo and quadrupolar exchanges, while exactly marginal potential scattering remains, and it uncovers a novel NFL fixed point driven by a quadrupolar-quadrupolar coupling when a time-reversal breaking term is absent. Overall, three NFL universality classes emerge for triplet impurities in cubic environments, expanding the NFL landscape beyond traditional multichannel Kondo mappings. The results have potential implications for cubic heavy-fermion materials, quantum dot devices, and ultracold atom systems, and raise questions about the Kondo anyon content and the connection to spherically symmetric fixed points.

Abstract

In cubic metals, a local, magnetic moment with a triplet ground state coupled to $Γ_8$ conduction electrons can give rise to various non-Fermi liquid (NFL) quantum critical behaviors. To date, only those exchange couplings have been studied that are spherically symmetric already in the high-temperature, local moment regime. Namely, only the effects of potential scattering, dipolar spin exchange, i.e. Kondo, and quadrupolar exchange couplings were considered, and two types of NFL fixed points have been identified. However, in cubic symmetry, six independent exchange couplings can be present in the Hamiltonian: in addition to the spherically symmetric potential scattering and Kondo exchange, there are four independent, spherical-symmetry-breaking Hamiltonian terms: two quadrupolar-quadrupolar, a dipolar-octupolar and a quadrupolar-octupolar exchange interaction, where the first/second element of the compound adjectives refers to the impurity/conduction-electron transitions. While all of them flow to fixed points where rotational invariance is recovered, I found that one of the quadrupolar-quadrupolar couplings flows to a previously unidentified NFL fixed point. I derive the exchange couplings allowed by cubic symmetry, solve them with the numerical renormalization group, and present three types of NFL excitation spectrum$-$one of which is novel, at least in the context of a quantum impurity with a triplet ground state.

Figures (3)

• Figure 1: Finite-size spectrum of the spherically symmetric, quadrupolar exchange interaction, ${\cal H}^{\,\textrm{Q}}$, as the function of the NRG iteration step index, $N$, corresponding to a temperature, $T_N \approx \Lambda^{-N/2} D / k_B$, as extracted from NRG Toth08, with $L$ the size of the chiral fermion system, $\Lambda=2$ chosen for the discretization parameter, $k_B$ the Boltzmann constant, and $D$ the bandwidth Wilson75. Only the U(1)$_\textrm{charge}$ symmetry of the model was exploited, and a minimum of 3000 multiplets were kept. A larger symmetry, SU(2)$_\textrm{spin}\times$U(1)$_\textrm{charge}$ could be used to obtain more accurate NRG results. The lowest levels still match the previous NRG results from Ref. Koga99. The value of the dimensionless Kondo coupling is $\,D{\cal J}^{\textrm{Q}}=0.5\,$. The finite-size fixed point spectrum is independent of the value and sign of ${\cal J}^{\textrm{Q}}$ as long as ${\cal J}^{\textrm{Q}}\neq 0$.
• Figure 2: Finite-size spectrum of the spherically symmetric, dipolar, spin exchange interaction, ${\cal H}^{\,\textrm{Kondo}}$, as the function of the NRG iteration step index, $N$ with $\Lambda=2$ used for the discretization parameter Wilson75. Only the U(1)$_\textrm{charge}$ symmetry of the model was exploited, and a minimum of 1500 multiplets were kept. A larger symmetry, SU(2)$_\textrm{spin}\times$U(1)$_\textrm{charge}$ could be used to get more accurate NRG results, but the agreement with the conformal field theory (CFT) results Koga99 for the lowest levels is still evident. The numbers on the y-axes are the exact excitation energies from CFT. The value of the dimensionless Kondo coupling is $\,D{\cal J}^{\textrm{K}}=0.5\,$, with $D$ the bandwidth. The finite-size fixed point spectrum is independent of the value of ${\cal J}^{\textrm{K}}$ as long as ${\cal J}^{\textrm{K}}>0$. For ${\cal J}^{\textrm{K}}<0$, the system is a Fermi liquid at low temperatures.
• Figure 3: Finite-size spectrum of the spherical symmetry breaking, quadrupolar ${\cal H}^{\,\Gamma_5\otimes\Gamma_5}_{\textrm{I}}$ as the function of the NRG iteration step index, $N$. $\Lambda=2$ was chosen for the discretization parameter Wilson75. The U(1)$_\textrm{charge}$ symmetry of the model was used, and a minimum of 5000 multiplets were kept. The value of the dimensionless Kondo coupling is $\,D{\cal J}^{\,\textrm{Q-Q}}_{\,\Gamma_5\otimes\Gamma_5}=0.5\,$, with $D$ the bandwidth. However, the finite-size fixed point spectrum is independent of the value and sign of ${\cal J}^{\,\textrm{Q-Q}}_{\,\Gamma_5\otimes\Gamma_5}$ for ${\cal J}^{\,\textrm{Q-Q}}_{\,\Gamma_5\otimes\Gamma_5}\neq 0$.