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Physical Review B

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

Hang Wong, Zhide Chen

Published Aug 30, 2007• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.76.077301

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Reply to “Comment on ‘Coherent-incoherent transition in the sub-Ohmic spin-boson model’ ”

Alex W. Chin

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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=0T=0T=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.
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<10<s<10<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.
The influence of dissipation on quantum tunneling in the spin-boson model with a sub-Ohmic bath is studied by a variational calculation. By examining the evolution of solutions of the variational equation with the coupling strength near the phase boundary, we are able to present a scenario of discontinuous transition in sub-Ohmic dissipation case in accord with Ginzburg-Landau theory. Based on the constructed picture, it is shown that the critical point found in the general way is not thermodynamically the critical point, but the point where the second energy minimum begins to develop. The true cross-over point is calculated and the obtained phase diagram is in agreement with the result of numerical renormalization group calculation.
The dynamics of the spin-boson model with an Ohmic bath at finite temperature is studied by a variational calculation. Numerical solution of the self-consistent equation derived from the variational method shows that the transition from incoherent to coherent phases is discontinuous. It indicates that ( T = 0, s = 1) is a tricritical point, i.e. the transition changes from continuous to discontinuous by tuning from T = 0 to T ≠0. The discontinuous transition at finite temperature is analysed by Landau theory and the relation to the experimental observation on the coherent state is also discussed.
We have carried out analytical and numerical studies of the spin-boson model in the sub-ohmic regime with the influence of both the diagonal and the off-diagonal coupling accounted for, via the Davydov D1 variational ansatz. While a second-order phase transition is known to be exhibited by this model in the presence of diagonal coupling only, we demonstrate the emergence of a discontinuous first order phase transition upon incorporation of the off-diagonal coupling. A plot of the ground state energy versus magnetization highlights the discontinuous nature of the transition between the isotropic (zero magnetization) state and nematic (finite magnetization) phases. We have also calculated the entanglement entropy and a discontinuity found at a critical coupling strength further supports the discontinuous crossover in the spin-boson model in the presence of off-diagonal coupling. It is further revealed via a canonical transformation approach that for the special case of identical exponents for the spectral densities of the diagonal and the off-diagonal coupling, there exists a continuous crossover from a single localized phase to doubly degenerate localized phase with differing magnetizations.
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).