arXiv:cond-mat/0511285 · Physical Review B

Coherent-incoherent transition in the sub-Ohmic spin-boson model

Alex W. Chin, Misha Turlakov

Published Feb 7, 2006• 54 citations

Abstract

We study the spin-boson model with a sub-Ohmic bath using a variational method. The transition from coherent dynamics to incoherent tunneling is found to be abrupt as a function of the coupling strength $\ensuremath{\alpha}$ and to exist for any power $0<s<1$, where the bath coupling is described by $J(\ensuremath{\omega})\ensuremath{\sim}\ensuremath{\alpha}{\ensuremath{\omega}}^{s}$. We find nonmonotonic temperature dependence of the two-level gap $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}$ and a reentrance regime close to the transition due to nonadiabatic low-frequency bath modes. Differences between thermodynamic and dynamic conditions for the transition as well as the limitations of the simplified bath description are discussed.

DOI: 10.1103/physrevb.73.075311

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arXiv:cond-mat/0511285 · Physical Review B

Coherent-incoherent transition in the sub-Ohmic spin-boson model

Alex W. Chin, Misha Turlakov

Published Feb 7, 2006• 54 citations

arXiv HTML PDF

Abstract

\ensuremath{\alpha}

and to exist for any power

0<s<1

, where the bath coupling is described by

J(\ensuremath{\omega})\ensuremath{\sim}\ensuremath{\alpha}{\ensuremath{\omega}}^{s}

. We find nonmonotonic temperature dependence of the two-level gap

\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}

and a reentrance regime close to the transition due to nonadiabatic low-frequency bath modes. Differences between thermodynamic and dynamic conditions for the transition as well as the limitations of the simplified bath description are discussed.

DOI: 10.1103/physrevb.73.075311

Related papers

Dynamics of the sub-Ohmic spin-boson model: A time-dependent variational study

Ning Wu, Liwei Duan, Xin Li +1 more

The Dirac-Frenkel time-dependent variation is employed to probe the dynamics of the zero temperature sub-Ohmic spin-boson model with strong friction utilizing the Davydov D1 ansatz. It is shown that initial conditions of the phonon bath have considerable influence on the dynamics. Counterintuitively, even in the very strong coupling regime, quantum coherence features still manage to survive under the polarized bath initial condition, while such features are absent under the factorized bath initial condition. In addition, a coherent-incoherent transition is found at a critical coupling strength α ≈ 0.1 for s = 0.25 under the factorized bath initial condition. We quantify how faithfully our ansatz follows the Schrödinger equation, finding that the time-dependent variational approach is robust for strong dissipation and deep sub-Ohmic baths (s ≪ 1).The Dirac-Frenkel time-dependent variation is employed to probe the dynamics of the zero temperature sub-Ohmic spin-boson model with strong friction utilizing the Davydov D1 ansatz. It is shown that initial conditions of the phonon bath have considerable influence on the dynamics. Counterintuitively, even in the very strong coupling regime, quantum coherence features still manage to survive under the polarized bath initial condition, while such features are absent under the factorized bath initial condition. In addition, a coherent-incoherent transition is found at a critical coupling strength α ≈ 0.1 for s = 0.25 under the factorized bath initial condition. We quantify how faithfully our ansatz follows the Schrödinger equation, finding that the time-dependent variational approach is robust for strong dissipation and deep sub-Ohmic baths (s ≪ 1).

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Comment on “Coherent-incoherent transition in the sub-Ohmic spin-boson model”

Hang Wong, Zhide Chen

In a recent paper, Chin and Turlakov [Phys. Rev. B 73, 075311 (2006)] studied the sub-Ohmic spin-boson model at finite temperatures by using a variational method. The coherent-incoherent transition was found to be discontinuous, and the transition point was determined using the same method as that used for a continuous transition like in the Ohmic case at $T=0$ . In this Comment, we show that the transition point determined in this way is, according to the Landau theory of the first order transition, not thermodynamically the transition point, but just the point where the second minimum of the free energy begins to develop. The true transition point is determined by the condition that the free energy of the new phase becomes the global minimum. Based on this result, it is shown that the predicted reentrance phenomenon does not exist in practice.In a recent paper, Chin and Turlakov [Phys. Rev. B 73, 075311 (2006)] studied the sub-Ohmic spin-boson model at finite temperatures by using a variational method. The coherent-incoherent transition was found to be discontinuous, and the transition point was determined using the same method as that used for a continuous transition like in the Ohmic case at $T=0$. In this Comment, we show that the transition point determined in this way is, according to the Landau theory of the first order transition, not thermodynamically the transition point, but just the point where the second minimum of the free energy begins to develop. The true transition point is determined by the condition that the free energy of the new phase becomes the global minimum. Based on this result, it is shown that the predicted reentrance phenomenon does not exist in practice.

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Dynamics of the sub-Ohmic spin-boson model is investigated by employing a multitude of the Davydov D1 trial states, also known as the multi-D1 Ansatz. Accuracy in dynamics simulations is improved significantly over the single D1 Ansatz, especially in the weak system-bath coupling regime. The reliability of the multi-D1 Ansatz for various coupling strengths and initial conditions is also systematically examined, with results compared closely with those of the hierarchy equations of motion and the path integral Monte Carlo approaches. In addition, a coherent-incoherent phase crossover in the nonequilibrium dynamics is studied through the multi-D1 Ansatz. The phase diagram is obtained with a critical point sc = 0.4. For sc < s < 1, the coherent-to-incoherent crossover occurs at a certain coupling strength, while the coherent state recurs at a much larger coupling strength. For s < sc, only the coherent phase exists.Dynamics of the sub-Ohmic spin-boson model is investigated by employing a multitude of the Davydov D1 trial states, also known as the multi-D1 Ansatz. Accuracy in dynamics simulations is improved significantly over the single D1 Ansatz, especially in the weak system-bath coupling regime. The reliability of the multi-D1 Ansatz for various coupling strengths and initial conditions is also systematically examined, with results compared closely with those of the hierarchy equations of motion and the path integral Monte Carlo approaches. In addition, a coherent-incoherent phase crossover in the nonequilibrium dynamics is studied through the multi-D1 Ansatz. The phase diagram is obtained with a critical point sc = 0.4. For sc < s < 1, the coherent-to-incoherent crossover occurs at a certain coupling strength, while the coherent state recurs at a much larger coupling strength. For s < sc, only the coherent phase exists.

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arXiv:quant-ph/0612042Physical Review BFeb 2, 200777 citations

\ensuremath{\alpha}

and to exist for any power

0<s<1

, where the bath coupling is described by

J(\ensuremath{\omega})\ensuremath{\sim}\ensuremath{\alpha}{\ensuremath{\omega}}^{s}

. We find nonmonotonic temperature dependence of the two-level gap

\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{K}

T=0

. In this Comment, we show that the transition point determined in this way is, according to the Landau theory of the first order transition, not thermodynamically the transition point, but just the point where the second minimum of the free energy begins to develop. The true transition point is determined by the condition that the free energy of the new phase becomes the global minimum. Based on this result, it is shown that the predicted reentrance phenomenon does not exist in practice.

P(t)

, the dynamical susceptibility ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$, and the equilibrium correlation function

C(t)

are analytically obtained for $s\ensuremath{\leqslant}1$; (ii) the phase diagram of thermodynamic transition shows the delocalized-localized transition point ${\ensuremath{\alpha}}_{l}$ which agrees with exact results and numerical data from the numerical renormalization group; (iii) the dynamical transition point ${\ensuremath{\alpha}}_{c}$ between coherent and incoherent phase is explicitly given. A crossover from the coherent oscillation to incoherent relaxation appears with increasing coupling (for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$, the coherent dynamics disappear); (iv) the Shiba's relation and sum rule are exactly satisfied when $\ensuremath{\alpha}\ensuremath{\leqslant}{\ensuremath{\alpha}}_{c}$; (v) an underdamping-overdamping transition point ${\ensuremath{\alpha}}_{c}^{*}$ exists in the function $S(\ensuremath{\omega})$. Consequently, the dynamical phase diagrams in both ohmic and sub-Ohmic case are mapped out. For $\ensuremath{\Delta}⪡{\ensuremath{\omega}}_{c}$, the critical couplings (${\ensuremath{\alpha}}_{l},{\ensuremath{\alpha}}_{c}$, and ${\ensuremath{\alpha}}_{c}^{*}$) are proportional to ${\ensuremath{\Delta}}^{1\ensuremath{-}s}$.