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

• 来源期刊: 中国物理B（英文版）
• 关键词: spin-boson model numerical renormalization group quantum phase transition dynamical correlation function
• 年，卷（期）: 2017,（6）
• 页码范围: 73-80
• 页数: 8页
• 语种: 英文
• DOI: 10.1088/1674-1056/26/6/060501

摘要

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.