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arXiv:2010.07471 · Chinese Physics B

Equilibrium dynamics of the sub-ohmic spin-boson model at finite temperature*

Ke Yang, Ning-Hua Tong

Published Dec 15, 2020• 3 citations
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Abstract

We use the full-density matrix numerical renormalization group method to calculate the equilibrium dynamical correlation function C(ω) of the spin operator σz at finite temperature for the sub-ohmic spin-boson model. A peak is observed at the frequency ω T ∼ T in the curve of C(ω). The curve merges with the zero-temperature C(ω) in ω ≫ ω T and deviates significantly from the zero-temperature curve in ω ≪ ω T.

DOI
10.1088/1674-1056/abd393

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We use the full-density matrix numerical renormalization group method to calculate the equilibrium dynamical correlation function C(ω) of the spin operator σz at finite temperature for the sub-ohmic spin-boson model. A peak is observed at the frequency ω T ∼ T in the curve of C(ω). The curve merges with the zero-temperature C(ω) in ω ≫ ω T and deviates significantly from the zero-temperature curve in ω ≪ ω T.
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.
We present numerically exact inchworm quantum Monte Carlo results for the real-time dynamics of the spin polarization in the sub-Ohmic spin-boson model at finite temperature. We focus in particular on the localization and coherence behavior of the model, extending our previous study at low temperature [Phys. Rev. Lett. 134, 056502 (2025)]. As the temperature increases, the system becomes less localized and less coherent. The loss of coherence, which is controlled by two independent mechanisms -- a smooth damping-driven crossover and a sharp frequency-driven transition -- exhibits a nontrivial temperature dependence. While both types of coherence loss occur at lower coupling in the high temperature regime, the frequency exhibits a sharper drop at high temperatures and this drop is observed for all values of the sub-Ohmic exponent, in contrast to the zero-temperature case. We discuss the full temperature-dependent dynamical phase diagram of the system and the interplay between coherence and localization across a wide range of physical parameters.
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.
We revisit the critical behavior of the sub-Ohmic spin-boson model. Analysis of both the leading and subleading terms in the temperature dependence of the inverse static local spin susceptibility at the quantum critical point, calculated using a numerical renormalization-group method, provides evidence that the quantum critical point is interacting in cases where the quantum-to-classical mapping would predict mean-field behavior. The subleading term is shown to be consistent with an $\ensuremath{\omega}/T$ scaling of the local dynamical susceptibility, as is the leading term. The frequency and temperature dependences of the local spin susceptibility in the strong-coupling (delocalized) regime are also presented. We attribute the violation of the quantum-to-classical mapping to a Berry-phase term in a continuum path-integral representation of the model. This effect connects the behavior discussed here with its counterparts in models with continuous spin symmetry.
We investigate the real-time dynamics of the sub-Ohmic spin-boson model across a broad range of coupling strengths, using the numerically exact inchworm quantum Monte Carlo algorithm. From short- and intermediate-time dynamics starting from an initially decoupled state, we extract signatures of the zero-temperature quantum phase transition between localized and delocalized states. We show that the dynamical phase diagram thus obtained differs from the equilibrium phase diagram in both the values of critical couplings and the associated values of the critical exponent. We also identify and quantitatively analyze two competing mechanisms for the crossover between coherent oscillations and incoherent decay. Deep in the sub-Ohmic regime, the crossover is driven by the damping of the oscillation amplitude, while closer to the Ohmic regime the oscillation frequency itself drops sharply to zero at large coupling.
With a decomposition scheme for the bath correlation function, the hierarchy equation of motion (HEOM) is extended to the zero-temperature sub-Ohmic spin-boson model, providing a numerically accurate prediction of quantum dynamics. As a dynamic approach, the extended HEOM determines the delocalized-localized (DL) phase transition from the extracted rate kernel and the coherent-incoherent dynamic transition from the short-time oscillation. As the bosonic bath approaches from the strong to weak sub-Ohmic regimes, a crossover behavior is identified for the critical Kondo parameter of the DL transition, accompanied by the transition from the coherent to incoherent dynamics in the localization.