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arXiv:2302.01908

Linear Absorption Spectrum of the Spin-Boson Model Studied by Extended Hierarchical Equations of Motion

Qianlong Wang, Jianlan Wu

Published Feb 3, 2023• 0 citations
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Abstract

With a decomposition scheme for the bath correlation function, the hierarchical equations of motion (HEOM) are extended to the zero-temperature sub-Ohmic and Ohmic spin-boson model. We investigate the linear absorption spectrum of the sub-Ohmic and Ohmic spin-boson model at zero temperature. By applying the extended HEOM, the equilibrium spin dynamics are obtained approximately. Then the linear response function is calculated according to the Kubo formula rewritten in Liouville space. To explore the essence of phase transition, we compute the linear absorption spectrum defined by the linear response function of a dipole moment. By analyzing the peak structure of the linear response spectrum, we get the dependence of linear absorption spectrum with Kondo parameter and different bath exponents. The spin relaxation dynamics are also calculated to explore the coherent-incoherent dynamic transition (CI) and delocalizedlocalized phase transition (DL). The corresponding phase diagram of DL and CI transition are also obtained. We pose a energy level picture to understand the different mechanism of DL phase transition between the deep sub-Ohmic and Ohmic spin-boson model.

DOI
10.48550/arxiv.2302.01908

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With a decomposition scheme for the bath correlation function, the hierarchical equations of motion (HEOM) are extended to the zero-temperature sub-Ohmic and Ohmic spin-boson model. We investigate the linear absorption spectrum of the sub-Ohmic and Ohmic spin-boson model at zero temperature. By applying the extended HEOM, the equilibrium spin dynamics are obtained approximately. Then the linear response function is calculated according to the Kubo formula rewritten in Liouville space. To explore the essence of phase transition, we compute the linear absorption spectrum defined by the linear response function of a dipole moment. By analyzing the peak structure of the linear response spectrum, we get the dependence of linear absorption spectrum with Kondo parameter and different bath exponents. The spin relaxation dynamics are also calculated to explore the coherent-incoherent dynamic transition (CI) and delocalizedlocalized phase transition (DL). The corresponding phase diagram of DL and CI transition are also obtained. We pose a energy level picture to understand the different mechanism of DL phase transition between the deep sub-Ohmic and Ohmic spin-boson model.
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.
Through a decomposition of the bath correlation function, the hierarchical equations of motion are extended to the Ohmic spin-boson model at zero temperature. For two typical cutoff functions of the bath spectral density, the rate kernel of spin dynamics is numerically extracted by a time-convolution equation of the average magnetic moment. A characteristic time is defined accordingly as the inverse of the zeroth-order moment of the rate kernel. For a given Kondo parameter in the incoherent regime, the time evolution of average magnetic moments gradually collapses onto a master curve after rescaling the time variable with the characteristic time. The rescaled spin dynamics is nearly independent of the cutoff frequency and the form of cutoff functions. For a given cutoff frequency, the characteristic time with the change of the Kondo parameter is fitted excellently as a function of the renormalized tunneling amplitude. Despite a significant difference in definition, our result is in good agreement with the characteristic time of the noninteracting blip approximation.
Following a recently proposed decomposition technique [C. R. Duan et al., Phys. Rev. B 95, 214308 (2017)], we inspect the zero-temperature spin-boson model for five different cutoff functions of the spectral density. With oscillatory and non-oscillatory exponentially decaying functions to decompose the bath correlation function, the hierarchy equation of motion is reliably extended to each spectral density under our investigation. The predicted spin dynamics is gradually converged with the increase of the hierarchic expansion order and the number of decomposing basis functions. Our systematic study of different cutoff functions expands previous results of the delocalized-localized phase transition with the exponential and sudden cutoffs in the spectral density.
An extended hierarchy equation of motion (HEOM) is proposed and applied to study the dynamics of the spin-boson model. In this approach, a complete set of orthonormal functions are used to expand an arbitrary bath correlation function. As a result, a complete dynamic basis set is constructed by including the system reduced density matrix and auxiliary fields composed of these expansion functions, where the extended HEOM is derived for the time derivative of each element. The reliability of the extended HEOM is demonstrated by comparison with the stochastic Hamiltonian approach under room-temperature classical ohmic and sub-ohmic noises and the multilayer multiconfiguration time-dependent Hartree theory under zero-temperature quantum ohmic noise. Upon increasing the order in the hierarchical expansion, the result obtained from the extended HOEM systematically converges to the numerically exact answer.
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.
Abstract The hierarchical equations of motion (HEOM) provide a numerically exact approach for simulating the dynamics of open quantum systems coupled to a harmonic bath. However, its applicability has traditionally been limited to specific spectral forms and relatively high temperatures. Recently, an extended version called Free-Pole HEOM (FP-HEOM) has been developed to overcome these limitations. In this study, we demonstrate that the FP-HEOM method can be systematically employed to investigate higher order master equations by truncating the FP-HEOM hierarchy at a desired tier. We focus on the challenging scenario of the spin-boson problem with a sub-Ohmic spectral distribution at zero temperature and analyze the performance of the corresponding master equations. Furthermore, we compare the memory kernel for population dynamics obtained from the exact FP-HEOM dynamics with that of the approximate Non-Interacting-Blip Approximation (NIBA).