No Spin-Localization Phase Transition in the Spin-Boson Model without Local Field

Abstract

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. 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. A two-level system coupled to an environment provides a unique test-bed for fundamental interests of quantum physics. Denoting the environment by a multimode harmonic oscillator, the spin-boson model (SBM) [1, 2] presents a phenomenological description of the open quantum system, which plays an important role in quantum information science and condensed matter physics. Particularly, for the SBM at zero temperature, it has attracted intensive interests for the quantum phase transition (QPT) happening between localized and delocalized phases regarding the spin.The standard SBM Hamiltonian in units of = 1 is given by [1],where σ z and σ x are usual Pauli operators, ǫ and ∆ are, respectively, the local field (also called c-number bias [1]) and tunneling regarding the two levels of the spin. a † k and a k are creation and annihilation operators of the bath modes with frequencies ω k , and λ k is the coupling between the spin and the bath modes. The effect of the harmonic oscillator environment is reflected by the spectral function J(ω) = π k λ 2 k δ(ω − ω k ) for 0 < ω < ω c with the cutoff energy ω c . In the infrared limit, i.e., ω →0, the power laws regarding J(ω) are of particular importance. Considering the low-energy details of the spectrum, we have J(ω) = 2παω 1−s c ω s with 0 < ω < ω c and the dissipation strength α. The exponent s is responsible for different bath with super-Ohmic bath s >1, Ohmic bath s =1 and sub-Ohmic bath s <1. * Electronic address: liutao849@163.com † Electronic address: mangfeng@wipm.ac.cnThe local field ǫ introduces asymmetry in the model, which was considered to be less important than the tunneling ∆ and thereby sometimes neglected for convenience of treatments. For the Ohmic dissipation, it was mentioned [3,4] that the SBM has a delocalized and a localized zero temperature phase, separated by a Kosterlitz-Thouless (KT) transition in the case of ǫ = 0. In the delocalized phase, realized at small dissipation strength α, the non-degenerate ground state behaves like a damped tunneling particle. In contrast, for large α, the dissipation leads to a localization of the particle ...