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arXiv:1703.00618 · Chinese Physics B

Dynamical correlation functions of the quadratic coupling spin-Boson model

Da-Chuan Zheng, Ning-Hua Tong

Published Jun 1, 2017• 2 citations
arXivHTMLPDF

Abstract

The spin-boson model with quadratic coupling is studied using the bosonic numerical renormalization group method. We focus on the dynamical auto-correlation functions C O ( ω ) , with the operator O ^ taken as σ ^ x , σ ^ z , and X ^ , respectively. In the weak-coupling regime α < α c , these functions show power law ω -dependence in the small frequency limit, with the powers 1+2 s , 1+2 s , and s , respectively. At the critical point α = α c of the boson-unstable quantum phase transition, the critical exponents y O of these correlation functions are obtained as y σ x = y σ z = 1 − 2 s and y X = − 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 C σ x ( ω ) is broadened significantly and the line shape changes qualitatively, showing enhanced dephasing at the spin flip point.

DOI
10.1088/1674-1056/26/6/060502

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The spin-boson model with quadratic coupling is studied using the bosonic numerical renormalization group method. We focus on the dynamical auto-correlation functions C O ( ω ) , with the operator O ^ taken as σ ^ x , σ ^ z , and X ^ , respectively. In the weak-coupling regime α < α c , these functions show power law ω -dependence in the small frequency limit, with the powers 1+2 s , 1+2 s , and s , respectively. At the critical point α = α c of the boson-unstable quantum phase transition, the critical exponents y O of these correlation functions are obtained as y σ x = y σ z = 1 − 2 s and y X = − 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 C σ x ( ω ) is broadened significantly and the line shape changes qualitatively, showing enhanced dephasing at the spin flip point.
Using the bosonic numerical renormalization group method, we studied the equilibrium dynamical correlation function C ( ω ) of the spin operator σ z 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.
The spin-boson model has nontrivial quantum phase transitions at zero temperature induced by the spin-boson coupling. The bosonic numerical renormalization group (BNRG) study of the critical exponents β and δ of this model is hampered by the effects of boson Hilbert space truncation. Here we analyze the mean-field spin boson model to figure out the scaling behavior of magnetization under the cutoff of boson states N b . We find that the truncation is a strong relevant operator with respect to the Gaussian fixed point in 0 < s < 1/2 and incurs the deviation of the exponents from the classical values. The magnetization at zero bias near the critical point is described by a generalized homogeneous function (GHF) of two variables τ = α – α c and x = 1/N b . The universal function has a double-power form and the powers are obtained analytically as well as numerically. Similarly, m(α = α c ) is found to be a GHF of ϵ and x. In the regime s > 1/2, the truncation produces no effect. Implications of these findings to the BNRG study are discussed.
We extend the numerical renormalization-group method to Bose-Fermi Kondo models (BFKMs), describing a local moment coupled to a conduction band and a dissipative bosonic bath. We apply the method to the Ising-symmetry BFKM with a bosonic bath spectral function eta(omega) proportional omega(s), of interest in connection with heavy-fermion criticality. For 0 < s < 1, an interacting critical point, characterized by hyperscaling of exponents and omega/T scaling, describes a quantum phase transition between Kondo-screened and localized phases. A connection is made to other results for the BFKM and the spin-boson model.
The spin-boson model has nontrivial quantum phase transitions in the sub-Ohmic regime. For the bath spectra exponent $0\ensuremath{\leqslant}s<1/2$, the bosonic numerical renormalization group (BNRG) study of the exponents $\ensuremath{\beta}$ and $\ensuremath{\delta}$ are hampered by the boson-state truncation, which leads to artificial interacting exponents instead of the correct Gaussian ones. In this paper, guided by a mean-field calculation, we study the order-parameter function $m(\ensuremath{\tau}=\ensuremath{\alpha}\ensuremath{-}{\ensuremath{\alpha}}_{c},\ensuremath{\epsilon},\ensuremath{\Delta})$ using BNRG. Scaling analysis with respect to the boson-state truncation Nb{N}_{b}Nb​, the logarithmic discretization parameter $\ensuremath{\Lambda}$, and the tunneling strength $\ensuremath{\Delta}$ are carried out. Truncation-induced multiple-power behaviors are observed close to the critical point, with artificial values of $\ensuremath{\beta}$ and $\ensuremath{\delta}$. They cross over to classical behaviors with exponents $\ensuremath{\beta}=1/2$ and $\ensuremath{\delta}=3$ on the intermediate scales of $\ensuremath{\tau}$ and $\ensuremath{\epsilon}$, respectively. We also find $\ensuremath{\tau}/{\ensuremath{\Delta}}^{1\ensuremath{-}s}$ and $\ensuremath{\epsilon}/\ensuremath{\Delta}$ scalings in the function $m(\ensuremath{\tau},\ensuremath{\epsilon},\ensuremath{\Delta})$. The role of boson-state truncation as a scaling variable in the BNRG result for $0\ensuremath{\leqslant}s<1/2$ is identified and its interplay with the logarithmic discretization revealed. Relevance to the validity of quantum-to-classical mapping in other impurity models is discussed.
We describe the generalization of Wilson's numerical renormalization group method to quantum impurity models with a bosonic bath, providing a general nonperturbative approach to bosonic impurity models which can access exponentially small energies and temperatures. As an application, we consider the spin-boson model, describing a two-level system coupled to a bosonic bath with power-law spectral density, J(omega) proportional to omega(s). We find clear evidence for a line of continuous quantum phase transitions for sub-Ohmic bath exponents 0<s<1; the line terminates in the well-known Kosterlitz-Thouless transition at s=1. Contact is made with results from perturbative renormalization group, and various other applications are outlined.