Scaling and Enhanced Symmetry at the Quantum Critical Point of the Sub-Ohmic Bose-Fermi Kondo Model

We consider the finite-temperature scaling properties of a Kondo-destroying quantum critical point in the Ising-anisotropic Bose-Fermi Kondo model (BFKM). A cluster-updating Monte Carlo approach is used, in order to reliably access a wide temperature range. The scaling function for the two-point spin correlator is found to have the form dictated by a boundary conformal field theory, even though the underlying Hamiltonian lacks conformal invariance. Similar conclusions are reached for all multipoint correlators of the spin-isotropic BFKM in a dynamical large-$N$ limit. Our results suggest that the quantum critical local properties of the sub-Ohmic BFKM are those of an underlying boundary conformal field theory.

Figure 1

(a) Static local spin susceptibility in the Kondo case ($g=0$, ${\tau }_0=1/16$, black circles) and for the critical coupling [$ϵ=0.4$, $\Gamma =0.75$, $g_c=0.821T_K^0$, ${\tau }_0=1/64$, red (or gray) diamonds]. The dashed blue (or dark gray) line is the fit to the Toulose limit (see, e.g., Ref. 30), and the dash-dotted line a fit to the critical behavior. Defining $T_K^0\equiv 1/{\chi }_{\mathrm{stat}}(T=0)$ we obtain $T_K^0{\tau }_0\approx 0.688$; (b) static susceptibility at the critical coupling, for various values of the cutoff parameter, ${\tau }_0$.

Figure 2

Scaling of the local spin susceptibility (at the QCP), which is plotted as a function of $\pi T{\tau }_0/\sin (\pi \tau T)$. The parameters are $ϵ=0.4$, $\Gamma =0.75$, $g=0.821T_K^0\approx g_c$, ${\tau }_0=1/64$. Note that $\pi T{\tau }_0/\sin (\pi \tau T)$ becomes small (order $\pi {\tau }_0/\beta$) as $\tau$ approaches the long-time limit, $\tau \to \beta /2$. The power-law collapse occurs over two decades of the parameter $\pi T{\tau }_0/\sin (\pi \tau T)$. The deviation at the lower left corner is attributed to finite-size effects.

Figure 3

Scaling of the propagators for the auxiliary boson $G_B(\tau )$ [panel (a)] and for the pseudofermion $G_f(\tau )$ [panel (b)], for $ϵ=0.3$ and the numerical parameters specified in the main text, at the critical coupling $g_c=25.5T_K^0$.