,Equilibrium dynamics of the sub-Ohmic spin-boson model under bias

• 作 者: Da-Chuan Zheng Ning-Hua Tong
• 机 构: Department of Physics, Renmin University of China, Beijing 100872, China
• 出 处: 中国物理B（英文版）
• 发表日期: 2017年6期
• 关键词: spin-boson model numerical renormalization group quantum phase transition dynami

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 C(ω) ∝ ωs is found to be universal and independent of the bias ε and the coupling strength α (except at the quantum critical point α =αc and ε =0).Our NRG data also show C(ω) ∝ x2ωs for a wide range of parameters,including the biased strong coupling regime (ε≠0 and α ＞ αc),supporting the general validity of the Shiba relation.Close to the quantum critical point αc,the dependence of C(ω) on α and ε is understood in terms of the competition between ε and the crossover energy scale ω*0of the unbiased case.C(ω) is stable with respect to ε for ε(《)ε*.For ε 》 ε*,it is suppressed by ε in the low frequency regime.We establish that ε* ∝ (ω*0)1/θ holds for all sub-Ohmic regime 0 ≤ s ＜ 1,withθ=2/(3s) for 0 ＜ s ≤ 1/2 and θ =2/(1 + s) for 1/2 ＜ s ＜ 1.The variation of C(ω) with α and ε is summarized into a crossover phase diagram on the α-ε plane.