Spin-boson coupling in continuous-time quantum Monte Carlo

A vector bosonic field coupled to the electronic spin is treated by means of the continuous-time quantum Monte Carlo method. In the Bose-Kondo model with a sub-Ohmic density of states ${\rho }_{\mathrm{B}}\left(\omega \right)\propto {\omega }^s$ with $s=0.2$, two contributions to the spin susceptibility, the Curie term $T^{-1}$ and the term $T^{-s}$ due to bosonic fluctuations, are observed separately. This result indicates the existence of a residual moment and a hidden critical behavior. By including hybridization with itinerant electrons, a quantum critical point is identified between this local-moment state and the Kondo singlet state. It is demonstrated that the energy scale of the bosonic fluctuations is not affected by the quantum phase transition.

Figure 1

An example of the Monte Carlo configuration of order $k=2$ and $l=2$. The dark and light shaded areas indicate the spin-up and -down states, respectively. The curved lines express the bosonic Green function.

Figure 2

Update processes necessary to evaluate the spin-boson coupling.

Figure 3

Temperature dependence of the static susceptibility $\chi \left(0\right)$ in the pure bosonic system with $s=0.2$ (solid lines). The regular part ${\chi }_{\mathrm{reg}}\left(0\right)$ defined in Eq. (21) is also plotted (dashed lines).

Figure 4

The energy scale $T_{\mathrm{B}}$ of the bosonic fluctuations in the pure bosonic system with $s=0.2$ (denoted by BK). A result for the Bose-Fermi-Anderson model is also plotted (denoted by BFA; see Fig. 5 for parameters). The solid line shows the function $T_{\mathrm{B}}\propto g^{\alpha }$ fitted to the BK data.

Figure 5

Temperature dependencies of the static susceptibility $\chi \left(0\right)$ in the Bose-Fermi-Anderson model with $V^2=0.1$, ${\epsilon }_{f\sigma }=-0.2$, $U=\infty$, and $s=0.2$ (solid lines). The dashed lines show the regular part ${\chi }_{\mathrm{reg}}\left(0\right)$ defined in Eq. (21).

Figure 6

The renormalization factor $z$ and the effective moment $M$ for the same parameters as in Fig. 5. Results at four different temperatures are plotted. The dashed line (denoted by BK) is the result for the pure bosonic system.

Figure 7

$T^s\chi$ as a function of $g$ for different values of $T$. From the intersection points, which are indicated by arrows, the critical point is determined. The inset shows an extrapolation of the intersection points to $T=0$.