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...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) ∝ ω~s 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) ∝ χ~2ω~s 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 ε《ε^*. For ε 》ε^*, 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.展开更多
The spin-boson model with quadratic coupling is studied using the bosonic numerical renormalization group method.We focus on the dynamical auto-correlation functions CO(ω), with the operator taken as σx, σz, and ...The spin-boson model with quadratic coupling is studied using the bosonic numerical renormalization group method.We focus on the dynamical auto-correlation functions CO(ω), with the operator taken as σx, σz, and X, respectively. In the weak-coupling regime α 〈 αc, these functions show power law ω-dependence in the small frequency limit, with the powers 1 + 2s, 1 + 2s, and s, respectively. At the critical point α = αc of the boson-unstable quantum phase transition, the critical exponents yO of these correlation functions are obtained as yσx= yσz= 1-2s and yX=-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 Cσx(ω) is broadened significantly and the line shape changes qualitatively, showing enhanced dephasing at the spin flip point.展开更多
Based on the rapid experimental developments of circuit QED,we propose a feasible scheme to simulate the spin-boson model with superconducting circuits,which can be used to detect quantum Kosterlitz-Thouless(KT) phase...Based on the rapid experimental developments of circuit QED,we propose a feasible scheme to simulate the spin-boson model with superconducting circuits,which can be used to detect quantum Kosterlitz-Thouless(KT) phase transition.We design the spinboson model by using a superconducting phase qubit coupled to a semi-infinite transmission line,which is regarded as a bosonic reservoir with a continuum spectrum.By tuning the bias current or the coupling capacitance,the quantum KT transition can be directly detected through tomography measurement on the states of the phase qubit.We also estimate the experimental parameters using the numerical renormalization group method.展开更多
基金supported by the National Basic Research Program of China(Grant No.2012CB921704)the National Natural Science Foundation of China(Grant No.11374362)+1 种基金the Fundamental Research Funds for the Central Universities,Chinathe Research Funds of Renmin University of China(Grant No.15XNLQ03)
文摘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) ∝ ω~s 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) ∝ χ~2ω~s 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 ε《ε^*. For ε 》ε^*, 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.
基金supported by the National Key Basic Research Program of China(Grant No.2012CB921704)the National Natural Science Foundation of China(Grant No.11374362)+1 种基金the Fundamental Research Funds for the Central Universities,Chinathe Research Funds of Renmin University of China(Grant No.15XNLQ03)
文摘The spin-boson model with quadratic coupling is studied using the bosonic numerical renormalization group method.We focus on the dynamical auto-correlation functions CO(ω), with the operator taken as σx, σz, and X, respectively. In the weak-coupling regime α 〈 αc, these functions show power law ω-dependence in the small frequency limit, with the powers 1 + 2s, 1 + 2s, and s, respectively. At the critical point α = αc of the boson-unstable quantum phase transition, the critical exponents yO of these correlation functions are obtained as yσx= yσz= 1-2s and yX=-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 Cσx(ω) is broadened significantly and the line shape changes qualitatively, showing enhanced dephasing at the spin flip point.
基金supported by the National Natural Science Foundation of China (Grant Nos. 11004065,11104057 and 11125417)the Natural Science Foundation of Guangdong Province (Grant No.10451063101006312)+1 种基金the State Key Program for Basic Research of China(Grant No. 2011CB922104)the GRF and CRF of the RGC of Hong Kong
文摘Based on the rapid experimental developments of circuit QED,we propose a feasible scheme to simulate the spin-boson model with superconducting circuits,which can be used to detect quantum Kosterlitz-Thouless(KT) phase transition.We design the spinboson model by using a superconducting phase qubit coupled to a semi-infinite transmission line,which is regarded as a bosonic reservoir with a continuum spectrum.By tuning the bias current or the coupling capacitance,the quantum KT transition can be directly detected through tomography measurement on the states of the phase qubit.We also estimate the experimental parameters using the numerical renormalization group method.