中国物理B ›› 2017, Vol. 26 ›› Issue (6): 60501-060501.doi: 10.1088/1674-1056/26/6/060501

• GENERAL • 上一篇    下一篇

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

Da-Chuan Zheng(郑大川), Ning-Hua Tong(同宁华)   

  1. Department of Physics, Renmin University of China, Beijing 100872, China
  • 收稿日期:2017-01-21 修回日期:2017-02-23 出版日期:2017-06-05 发布日期:2017-06-05
  • 通讯作者: Ning-Hua Tong E-mail:nhtong@ruc.edu.cn
  • 基金资助:

    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).

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

Da-Chuan Zheng(郑大川), Ning-Hua Tong(同宁华)   

  1. Department of Physics, Renmin University of China, Beijing 100872, China
  • Received:2017-01-21 Revised:2017-02-23 Online:2017-06-05 Published:2017-06-05
  • Contact: Ning-Hua Tong E-mail:nhtong@ruc.edu.cn
  • Supported by:

    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).

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)∝χ2ωs 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 ω0* of the unbiased case. C(ω) is stable with respect to ε for εε*. For εε*, it is suppressed by ε in the low frequency regime. We establish that ε*∝(ω0*)1/θ holds for all sub-Ohmic regime 0≤s < 1, with θ=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.

关键词: spin-boson model, numerical renormalization group, quantum phase transition, dynamical correlation function

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 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)∝χ2ωs 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 ω0* of the unbiased case. C(ω) is stable with respect to ε for εε*. For εε*, it is suppressed by ε in the low frequency regime. We establish that ε*∝(ω0*)1/θ holds for all sub-Ohmic regime 0≤s < 1, with θ=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.

Key words: spin-boson model, numerical renormalization group, quantum phase transition, dynamical correlation function

中图分类号:  (Renormalization group methods)

  • 05.10.Cc
05.30.Jp (Boson systems) 64.70.Tg (Quantum phase transitions) 75.20.Hr (Local moment in compounds and alloys; Kondo effect, valence fluctuations, heavy fermions)