Dynamical Correlation Functions of the Quadratic Coupling Spin-Boson Model

Abstract

The spin-boson model with quadratic coupling is studied using the bosonic numerical renormalization group method. We focus on the dynamical auto-correlation functions $C_O(\omega )$, with the operator $\hat{O}$ taken as ${\hat{\sigma }}_x$, ${\hat{\sigma }}_z$, and $\hat{X}$, respectively. In the weak-coupling regime $\alpha <{\alpha }_c$, these functions show power law $\omega$-dependence in the small frequency limit, with the powers $1+2s$, $1+2s$, and $s$, respectively. At the critical point $\alpha ={\alpha }_c$ of the boson-unstable quantum phase transition, the critical exponents $y_O$ of these correlation functions are obtained as $y_{{\sigma }_x}=y_{{\sigma }_z}=1-2s$ and $y_X=-s$, respectively. Here $s$ is the bath index and $X$ is the boson displacement operator. Close to the spin flip point, the high frequency peak of $C_{{\sigma }_x}(\omega )$ is broadened significantly and the line shape changes qualitatively, showing enhanced dephasing at the spin flip point.

arXiv comment7 pages, 5 figures

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