Fixed point annihilation for a spin in a fluctuating field

A quantum spin impurity coupled to a gapless free field (the Bose-Kondo model) can be represented as a $(0+1)$-dimensional field theory with long-range-in-time interactions that decay as $|t-t^{\prime }{|}^{-(2-\delta )}$. This theory is a simpler analog of nonlinear $\sigma$ models with topological Wess-Zumino-Witten terms in higher dimensions. In this Letter we show that the renormalization group (RG) flows for the impurity problem exhibit an annihilation between two nontrivial RG fixed points at a critical value ${\delta }_c$ of the interaction exponent. The calculation is controlled at large spin $S$. This clarifies the phase diagram of the Bose-Kondo model and shows that it serves as a toy model for phenomena involving fixed point annihilation and “quasiuniversality” in higher dimensions.

Figure 1

Fixed points and flows as a function of the exponent $\delta$ in the memory kernel $K\sim 1/|t-t^{\prime }{|}^{2-\delta }$. $h$ is the dimensionless coupling of the 1D nonlinear $\sigma$ model. $h=0$ is the ordered fixed point, and $h=\infty$ (not shown) is a noninteracting spin with $2S+1$ degenerate ground states. $h_{s,u}=S{g}_{s,u}$ label the branches of stable and unstable fixed points.