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Chinese Physics B

A note on localized transition in the spin-boson model by variational calculation

Zhide Chen, Hou Zhi-lan

Published Jul 1, 2008• 2 citations
Publisher

Abstract

We present mathematical analyses of the evolution of solutions of the self-consistent equation derived from variational calculations based on the displaced-oscillator-state and the displaced-squeezed-state in spin-boson model at a zero temperature and a finite temperature. It is shown that, for a given spectral function defined as , there exists a universal s c for both kinds of variational schemes, the localized transition happens only for s  s c , moreover, the localized transition is discontinuous for s ⩽ s c while a continuous transition always occurs when s = s c . At T = 0, we have s c = 1, while for T ≠ 0, s c = 2 which indicates that the localized transition in super-Ohmic case still exists, manifesting that the result is in discrepancy with the existing result.

DOI
10.1088/1674-1056/17/7/057

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Physical Review B•Aug 30, 2007•4 citations
We present mathematical analyses of the evolution of solutions of the self-consistent equation derived from variational calculations based on the displaced-oscillator-state and the displaced-squeezed-state in spin-boson model at a zero temperature and a finite temperature. It is shown that, for a given spectral function defined as , there exists a universal s c for both kinds of variational schemes, the localized transition happens only for s  s c , moreover, the localized transition is discontinuous for s ⩽ s c while a continuous transition always occurs when s = s c . At T = 0, we have s c = 1, while for T ≠ 0, s c = 2 which indicates that the localized transition in super-Ohmic case still exists, manifesting that the result is in discrepancy with the existing result.
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 present energy and free-energy analyses on all variational schemes used so far in the spin-boson model at both T=0T=0T=0 and $T\ensuremath{\ne}0$. It is found that all the variational schemes have fail points, at where the variational schemes fail to provide a lower energy (or a lower free energy at $T\ensuremath{\ne}0$) than the displaced-oscillator ground state and therefore the variational ground state becomes unstable, which results in a transition from a variational ground state to a displaced-oscillator ground state when the fail point is reached. Such transitions are always misidentified as crossover from a delocalized to localized phases in variational calculations, leading to an artifact of phonon-induced localization. Physics origin of the fail points and explanations for different transition behaviors with different spectral functions are found by studying the fail points of the variational schemes in the single-mode case.
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.
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.
We explore the spin-boson model in a special case, i.e., with zero local field. In contrast to previous studies, we find no possibility for quantum phase transition (QPT) happening between the localized and delocalized phases, and the behavior of the model can be fully characterized by the even or odd parity as well as the parity breaking, instead of the QPT, owned by the ground state of the system. The parity breaking mentioned in our case is completely different from the spontaneously broken symmetry relevant to the conventionally defined QPT in previous studies. Our analytical treatment about the eigensolution of the ground state of the model presents for the first time a rigorous proof of no-degeneracy for the ground state of the model, which is independent of the bath type, the degrees of freedom of the bath and the calculation precision. We argue that the QPT mentioned previously appears due to incorrect employment of the ground state of the model and/or unreasonable treatment of the infrared divergence existing in the spectral functions for Ohmic and sub-Ohmic dissipations.
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.