Critical points and non-Fermi liquids in the underscreened pseudogap Kondo model

Numerical renormalization group simulations have shown that the underscreened spin-1 Kondo impurity model with power-law bath density of states (DOS) $\rho \left(\omega \right)\propto {\mid \omega \mid }^r$ possesses various intermediate-coupling fixed points, including a stable non-Fermi-liquid phase. In this paper we discuss the corresponding universal low-energy theories, obtain thermodynamic quantities and critical exponents by renormalization group analysis together with suitable $ϵ$ expansions, and compare our results with numerical data. Whereas the particle-hole symmetric critical point can be controlled at weak coupling using a simple generalization of the spin-$1∕2$ model, we show that the stable non-Fermi-liquid fixed point must be accessed near strong coupling via a mapping onto an effective ferromagnetic $S_{\mathrm{eff}}=1∕2$ model with singular bath DOS with exponent $r_{\mathrm{eff}}=-r<0$. In addition, we consider the particle-hole asymmetric critical fixed point, for which we propose a universal field theory involving the crossing between doublet and triplet levels.

Figure 1

(Color online) Renormalization group flow diagrams for the $S=1$ pseudogap Kondo model, deduced from NRG calculations (Ref. 5). The horizontal axis is the Kondo coupling and the vertical axis denoted the amount of $p\text{−}h$ symmetry breaking. Depending on the value of the bath exponent $r$, three distinct situations are found. The red lines represent phase boundaries, the rad dots critical fixed points. For a detailed explanation of the fixed points see text.

Figure 2

(Color online) Comparison of the impurity entropy $S_{\mathrm{imp}}$ between the analytical estimates at one-loop order (continuous lines) and the NRG data of Ref. 5 (dots and squares) for the three different intermediate-coupling fixed points occurring in the $S=1$ pseudogap Kondo model.

Figure 3

(Color online) As Fig. 2, but for the effective Curie constant $T{\chi }_{\mathrm{imp}}$.

Figure 4

(Color online) Renormalization of the vertex ${\Gamma }_{10\downarrow \uparrow }^{\left(1\right)}$ at one-loop level. Continuous and dashed lines denote the free Green’s functions of the $f_m$ and $c_{\sigma }$ fields, respectively, and the dot represents the vertex counterterm $(Z_J-1){\Gamma }_{10\downarrow \uparrow }^{\left(0\right)}$.

Figure 5

(Color online) Diagrammatic expansion of $⟨Q⟩{\chi }_{\mathrm{imp}}$ at lowest order for the Kondo model at the SCR point, including the complete prefactors.

Figure 6

(Color online) Self-energy at one loop for the $f_1$ fermion (upper part) and $b_{\uparrow }$ boson (lower part). Wiggly lines represent the bosonic free propagators. Crosses and dots indicate, respectively, mass and field renormalization counterterms.

Figure 7

(Color online) Expansion of the partition function $\mathcal{Z}$ up to order $V^2$ and $\delta {\lambda }_{f,b}$, including the combinatorial factors.

Figure 8

(Color online) Diagrammatic expansion of $⟨Q⟩{\chi }_{\mathrm{imp}}$ at lowest order for the effective theory describing ACR, including the complete prefactors.