Equilibrium dynamics of the sub-Ohmic spin-boson model under bias**Project supported by the National Basic Research Program of China (Grant No. 2012CB921704), the National Natural Science Foundation of China (Grant No. 11374362), the Fundamental Research Funds for the Central Universities, China, and the Research Funds of Renmin University of China (Grant No. 15XNLQ03).

Abstract

Using the bosonic numerical renormalization group method, we studied the equilibrium dynamical correlation function C(ω) of the spin operator σz for the biased sub-Ohmic spin-boson model. The small-ω behavior is found to be universal and independent of the bias ε and the coupling strength α (except at the quantum critical point and ε = 0). Our NRG data also show for a wide range of parameters, including the biased strong coupling regime ( and ), supporting the general validity of the Shiba relation. Close to the quantum critical point , the dependence of C(ω) on α and ε is understood in terms of the competition between ε and the crossover energy scale of the unbiased case. C(ω) is stable with respect to ε for . For , it is suppressed by ε in the low frequency regime. We establish that holds for all sub-Ohmic regime , with for and for . The variation of C(ω) with α and ε is summarized into a crossover phase diagram on the α–ε plane.