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arXiv:0709.1683 · Physical Review B

Failure of the displaced-squeezed state for spin-boson models in the thermodynamic limit

Alex W. Chin

Published Nov 28, 2007• 10 citations
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Abstract

We present an analysis of a variational coherent-squeezed state that has been discussed in the literature as a potential ground state for the spin-boson model. We show that when the system-size scaling of the spin-bath coupling is included properly, all squeezing effects and nonuniversal physics vanish in the thermodynamic limit. We also present finite-size corrections to the renormalization of the spin's coherence, showing that squeezing effects are also absent to leading order in the inverse bath size.

DOI
10.1103/physrevb.76.201307

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We present an analysis of a variational coherent-squeezed state that has been discussed in the literature as a potential ground state for the spin-boson model. We show that when the system-size scaling of the spin-bath coupling is included properly, all squeezing effects and nonuniversal physics vanish in the thermodynamic limit. We also present finite-size corrections to the renormalization of the spin's coherence, showing that squeezing effects are also absent to leading order in the inverse bath size.
The localization transition of the sub-Ohmic spin-boson model generates boson bath correlations that are, at low frequencies, analytically inaccessible to standard coherent-state-polaron variational ansatzes. In this paper, we introduce a ``squeezed polaron'' wavefunction incorporating generic Gaussian boson-boson correlations induced by the spin impurity. This gives the correct critical power-law scaling of bath observables near the localization transition together with a systematically improved ground state energy and a more accurate determination of the critical coupling. Our wavefunction captures the correct mean-field nature of the transition in the deep sub-Ohmic region. It also displays non-mean-field critical exponents in the shallow sub-Ohmic regime. Using the squeezed polaron as a starting point, we derive scattering phase shifts for bosons, and show how they encode the emergent energy scale that vanishes at the localization transition.
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.
A variational approach based on the multi-coherent-state ansatz with asymmetric parameters is employed to study the ground state of the spin-boson model. Without any artificial approximations except for the finite number of the coherent states, we find the robust Gaussian critical behavior in the whole sub-Ohmic bath regime. The converged critical coupling strength can be estimated with the 1/N1/N1/N scaling, where $N $ is the number of the coherent states. It is strongly demonstrated the breakdown of the well-known quantum-to-classical mapping for $1/2
We investigate the presence of squeezing in the weakly repulsive uniform Bose gas, in both the condensate mode and in the nonzero opposite-momenta mode pairs, using two different variational formulations. We explore the U(1) symmetry breaking and Goldstone's theorem in the context of a squeezed coherent variational wave function and present the associated Ward identity. We show that squeezing of the condensate mode is absent at the mean field Hartree-Fock-Bogoliubov level and emerges as a result of fluctuations about mean field as a finite volume effect, which vanishes in the thermodynamic limit. On the other hand, the squeezing of the excitations about the condensate survives the thermodynamic limit and is interpreted in terms of density-phase variables using a number-conserving formulation of the interacting Bose gas.