Numerical Renormalization-Group Study of the Bose-Fermi Kondo Model

We extend the numerical renormalization-group method to Bose-Fermi Kondo models (BFKMs), describing a local moment coupled to a conduction band and a dissipative bosonic bath. We apply the method to the Ising-symmetry BFKM with a bosonic bath spectral function $\eta (\omega )\propto {\omega }^s$, of interest in connection with heavy-fermion criticality. For $0<s<1$, an interacting critical point, characterized by hyperscaling of exponents and $\omega /T$ scaling, describes a quantum phase transition between Kondo-screened and localized phases. A connection is made to other results for the BFKM and the spin-boson model.

Figure 1

(a) Schematic phase diagram for the BFKM, showing Kondo (K), localized (L), and quantum-critical (QC) regimes separated by a crossover scale $T^{*}\propto |g-g_c{|}^{\nu }$ that vanishes at the QCP. (b),(c) Correlation length exponent $\nu$ as a function of bath exponent $s$. Crosses (at $s=0.2$ and 0.8) show representative results for the SBM. Dashed lines are the linear fits discussed in the text.

Figure 2

Static local susceptibility ${\chi }_{\mathrm{loc}}(T;\omega =0)$ vs $T$ for $s=0.2$ and ${\rho }_{0}J=0.5$, where $K_0{g}_c\simeq 0.355$. The inset shows the vanishing of $T{\chi }_{\mathrm{loc}}(0;0)$ for $g\to g_c^{+}$.

Figure 3

Upper-right axes: ${\chi }_{\mathrm{loc}}^{\prime \prime }(\omega ;T=0)$ (dashed line) and ${\chi }_{\mathrm{loc}}(T;\omega =0)$ (dotted line) at the QCP for $s=0.2$ and ${\rho }_{0}J=0.5$, showing that $x=y=s$; see Eqs. (3, 4). Lower-left axes: Finite-$T$ scaling of ${\chi }_{\mathrm{loc}}^{\prime \prime }(\omega ,T)$ at the QCP as described by Eq. (5). Data are shown for $s=0.2$, ${\rho }_{0}J=0.1$ (thin line) and 0.25 (thick line) with $g=g_c$ in each case, and for temperatures $T/T_K={10}^{-1}$, ${10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$, ${10}^{-5}$, ${10}^{-10}$, and ${10}^{-20}$. The curves have a common form for $\omega /T\ll T_K/T$.

Figure 4

Single-particle spectrum vs $\omega$ for bath exponent $s=0.7$, impurity parameters $U=-2{\epsilon }_d=0.002$ [such that $A(\omega )=A(-\omega )$], hybridization strength ${\Gamma }_0=1.5\times {10}^{-4}$, and discretization $\Lambda =3$. Shown for $B_0-B_{0,c}=0$ (dashed line), $\pm {10}^{-4}$ on (a) the scale of $U$ and (b) the scale set by the central feature. (c) Shown on a logarithmic scale for $B_0-B_{0,c}=0$ (dashed line), $\pm {10}^{-5}$, $\pm {10}^{-4}$.