Equilibrium Fermi-liquid coefficients for the fully screened $N$-channel Kondo model

We analytically and numerically compute three equilibrium Fermi-liquid coefficients of the fully screened $N$-channel Kondo model, namely $c_B$, $c_T$, and $c_{\varepsilon }$, characterizing the magnetic field and temperature dependence of the resisitivity, and the curvature of the equilibrium Kondo resonance, respectively. We present a compact, unified derivation of the $N$ dependence of these coefficients, combining elements from various previous treatments of this model. We numerically compute these coefficients using the numerical renormalization group, with non-Abelian symmetries implemented explicitly, finding agreement with Fermi-liquid predictions on the order of 5% or better.

Figure 1

(a)–(c) Vertices associated with $H_1$, $H_2$, and $H_3$, respectively. (d)–(f) Nonzero second-order contributions to the quasiparticle self-energy, ${\tilde{\Sigma }}_{m\sigma }^R$, involving $H_1^2$, $H_2^2$, and $H_3^2$, respectively. The contributions involving $H_1{H}_2$, $H_1{H}_3$, and $H_2{H}_3$ all vanish, the former two due to the odd power of energy in the two-leg vertex.

Figure 2

Scaling collapse of (a) the resistivity at zero temperature as a function of field, and (b) at zero field as a function of temperature, calculated for two different values of the bare coupling, $\nu J_{\mathrm{K}}$ (dashed or solid), and for $N=1,2$, and 3. For each $N$, the dashed and solid curves overlap so well that they are almost indistinguishable. The insets compare the energy scales $B_{1/2}$ and $T_{1/2}$ at which the resistivity has decreased to half its unitary value [cf. Eq. (38)], to the scales $T_{\mathrm{K}}^{\mathrm{FS}}$ and $T_{\mathrm{K}}^{\mathrm{sc}}$ [cf. Eqs. (40) and (41)], respectively. The shown ratios are universal numbers of order unity, but not necessarily very close to 1, with a significant dependence on $N$: $B_{1/2}/T_{\mathrm{K}}^{\mathrm{FS}}=1.22,1.31,1.60$ and $T_{1/2}/T_{\mathrm{K}}^{\mathrm{sc}}=0.82,1.02,1.36$ for $N=1$, 2, and 3, respectively. The legend in the lower left of panel (b) specifies the NRG parameters used for both panels.

Figure 3

(a) Resistivity as function of magnetic field at $T=0$, (b) resistivity as function of temperature at $B=0$, and (c) the weighted spectral function $\overline{A}\left(\tau \right)$ [cf. Eq. (43)] at $T=B=0$, all shown for $N=1,2,3$. Each panel contains NRG data (heavy solid lines), the quadratic term from a fourth-order polynomial fit (heavy dashed lines) and the corresponding predictions from FLT of Eq. (7) for the quadratic term (light solid lines). Left and right vertical dotted lines in matching colors indicate the lower and upper borders of the fitting range used for each $N$. The boxed legends specify the NRG parameters used here.