Overscreened multi-channel SU(N) Kondo model : large-N solution and Conformal Field Theory

TL;DR

The paper addresses overscreened multichannel Kondo physics in SU(N)×SU(K) models with antisymmetric impurity representations, applying both conformal-field-theory methods and a controlled large-N saddle point (K = Nγ, Q = q0 N). It derives universal low-temperature scaling forms, ω/T scaling, and the T=0 impurity entropy, and computes the residual resistivity and T-matrix, showing quantitative agreement between the CFT predictions and the large-N results. Key results include the exponents 2Δ_f = 1/(1+γ) and 2Δ_B = γ/(1+γ), the complex T-matrix for spectral asymmetry (q0 ≠ 1/2), and explicit expressions for S_imp and S^1 in the large-N limit. The work connects to NCA and prior Cox–Ruckenstein analysis, clarifying when overscreening yields a non-Fermi-liquid fixed point and highlighting universal scaling functions and entropy that can be tested against Bethe-Ansatz or experimental realizations in quantum dots and related systems.

Abstract

The multichannel Kondo model with SU(N) spin symmetry and SU(K) channel symmetry is considered. The impurity spin is chosen to transform as an antisymmetric representation of SU(N), corresponding to a fixed number of Abrikosov fermions $\sum_αf_α^{\dagger}f_α=Q$. For more than one channel (K>1), and all values of N and Q, the model displays non-Fermi behaviour associated with the overscreening of the impurity spin. Universal low-temperature thermodynamic and transport properties of this non-Fermi liquid state are computed using conformal field theory methods. A large-N limit of the model is then considered, in which K/N and Q/N are held fixed. Spectral densities satisfy coupled integral equations in this limit, corresponding to a (time-dependent) saddle-point. A low frequency, low-temperature analysis of these equations reveals universal scaling properties in the variable $ω/T$, also predicted from conformal invariance. The universal scaling form is obtained analytically and used to compute the low-temperature universal properties of the model in the large-N limit, such as the T=0 residual entropy and residual resistivity, and the critical exponents associated with the specific heat and susceptibility. The connections with the ``non-crossing approximation'' and the previous work of Cox and Ruckenstein are discussed.

Figures (5)

• Figure 1: Young tableau corresponding to the strong-coupling state
• Figure 2: Plot of $\phi_f$ and $\phi_b$ as a function of $\tilde{\omega} -\alpha$ for different values of the asymmetry parameter $\alpha$ : $\alpha=0,-2,-5,-\infty$($\Delta_f=.3$)
• Figure 3: Residual entropy $s_{imp}$ and $\alpha$ vs. $q_{0}$ for $\gamma =1.5$
• Figure 4: An example of an $SU(N)$ Young tableau (for $N=5$) and its associated fermionic representation
• Figure 5: An example of the general composition rule explained in the text. We show the fermionic diagram associated with $Y$, the resulting fermionic diagrams and their transcription in terms of Young tableaus. ($N$ is arbitrary in this example as evidenced by the ….)