Catalog of cubic, symmetry-protected, non-Fermi liquid, Kondo-type exchange models for doublet impurities

TL;DR

This work classifies all cubic-symmetry-allowed, Kondo-type exchange interactions that can produce non-Fermi-liquid (NFL) behavior for a doublet impurity without relying on accidental degeneracy. By constructing cubic-symmetry exchange couplings from impurity and conduction-electron irreps and solving the resulting models with numerical renormalization group (NRG), the authors identify three NFL scenarios: (i) two-channel Kondo (2CK) physics for a $\Gamma_3$ non-Kramers impurity hybridizing with $\Gamma_8$ conduction electrons, (ii) a topological Kondo NFL arising from a Kramers doublet impurity coupled to $\Gamma_4$ or $\Gamma_5$ electrons, and (iii) a spin-$\tfrac{1}{2}$ impurity interacting with spin-$\tfrac{3}{2}$ conduction electrons. The 2CK case is not guaranteed due to possible effective spatial anisotropy; the topological Kondo interaction is symmetry-protected but requires lifting the spin degeneracy of the conduction sector, while the spin-half/spin-3/2 NFL fixed point is the most robust under cubic symmetry. Thermodynamics computed via NRG confirms characteristic NFL scalings, and candidate cubic materials are proposed for experimental observation, providing a concrete roadmap for detecting symmetry-protected NFL behavior in diluted cubic compounds.

Abstract

To identify what types of non-Fermi liquid (NFL) behavior are most likely to occur in cubic metals due to doublet impurities, we derive every cubic symmetry-allowed, NFL, Kondo-type exchange coupling that does not need accidental degeneracy for its realization. We find three distinct types of NFL behavior: two-channel Kondo (2CK) behavior for a non-Kramers doublet impurity coupled to local $Γ_8$ conduction electrons; topological Kondo physics for a Kramers doublet impurity and $Γ_4$ or $Γ_5$ conduction electrons; and lastly, spin-half impurity spin-$\frac{3}{2}$ conduction electron Kondo behavior for a Kramers doublet impurity and $Γ_8$ conduction electrons. The first two critical behaviors are not straightforward to realize. In the first case, 2CK physics is not guaranteed, since cubic symmetry does not prevent an effective spatial anisotropy from exceeding the 2CK coupling, which restores a Fermi liquid behavior. In the second case, the topological Kondo interaction is guaranteed to dominate, however, the spin degeneracy of the conduction electrons needs to be lifted e.g. by a magnetic field$-$so that they can be represented by $Γ_4$ or $Γ_5$ triplets$-$which then also lifts the degeneracy of the Kramers doublet. We find that the spin-half impurity spin-$\frac{3}{2}$ conduction electron, NFL, Kondo behavior has the greatest chance of existing in diluted, cubic compounds. We compute the thermodynamics of the topological Kondo model using the numerical renormalization group, and discuss the thermodynamics of the spin-half impurity spin-$\frac{3}{2}$ conduction electron Kondo model. We also identify candidate materials where the corresponding NFL behaviors could be observed.

Figures (1)

• Figure 1: Thermodynamics of the topological Kondo model with NRG. We made use of the SU(2)$_\textrm{spin}\times$U(1)$_\textrm{charge}$ symmetry of the 1.5CK model, and kept a minimum of 14k multiplets with the energy cutoff at its highest possible value and constant for each iteration step. We chose $\Lambda=2.5$ for the discretization parameter Wilson75. For each curve, we used ten $z$ values for $z$-averaging Oliveira94. The two values of the dimensionless Kondo coupling, ${\cal J}^{\Gamma_4\otimes\Gamma_4}_{\textrm{1.5CK}}$ appear in each panel in units of the bandwidth $D$. $(a)$ The impurity contribution to the entropy: at high temperatures, which is not captured well by NRG, the Kondo coupling is negligible, the impurity is a free spin and does not contribute to $S$. At low temperatures, below the Kondo scale, $T_K$, defined as the peak position in $\Delta C$, the spin becomes overscreened with the expected $\log(\sqrt{3})$ contribution at $T=0$ to $S$. Our calculations reproduced the expected entropy difference of $\log(2)-\log(\sqrt{3})$ to four-digit precision. $(b)$ The impurity contribution to the specific heat. $T_K$ is defined as the location of the peak in the specific heat. $(c)$ The impurity contribution to the Sommerfeld coefficient. The expected $T^{-1/3}$ behavior is approximately reproduced over three decades. $(d)$ The impurity contribution to $\langle S_z^2\rangle$. $(e)$ The impurity contribution to $\chi$. The expected $T^{-1/3}$ behavior is approximately reproduced over three decades. $(f)$ The resulting Wilson ratio whose non-universality is reproduced, but the precision of the calculations is not high enough to see the leveling off as $T\to0$.