Generalized Polaron Ansatz for the Ground State of the Sub-Ohmic Spin-Boson Model: An Analytic Theory of the Localization Transition

Abstract

The sub-ohmic spin-boson model is known to possess a novel quantum phase transition at zero temperature between a localised and delocalised phase. We present here an analytical theory based on a variational ansatz for the ground state, which describes a continuous localization transition with mean-field exponents for 0 < s < 0.5. Our results for the critical properties show good quantitiative agreement with previous numerical results, and we present a detailed description of all the spin observables as the system passes through the transition. Analysing the ansatz itself, we give an intuitive microscopic description of the transition in terms of the changing correlations between the system and bath, and show that it is always accompanied by a divergence of the lowfrequency boson occupations. The possible relevance of this divergence for some numerical approaches to this problem is discussed and illustrated by looking at the ground state obtained using density matrix renormalisation group methods.The physics of quantum systems in contact with environmental degrees of freedom plays a fundamental role in many areas of physics, chemsitry and biology, including systems as diverse as solid state quantum computers[1, 2], quantum impurities [3], and photosynthetic biomolecules [4][5][6][7][8]. A key theoretical model for the study of system-environment interactions is the spin-boson model (SBM), which consists of a two-level system (TLS) that is linearly coupled to an 'environment' of harmonic oscillators [9,10]. Although this model has been studied extensively, there are still many open problems in SBM physics, most notably those concerning the quantum phase transition (QPT) between delocalised and localised phases that exists in the SBM when the oscillators are characterised by sub-Ohmic spectral densities.The standard quantum-classical mapping predicts that the sub-ohmic SBM should be equivalent to a classical Ising spin chain with long-range interactions, and predicts a continuous magnetic transition with mean-field critical exponents for 0 < s < 0.5. In Ref.[11], a continuous transition in the sub-Ohmic SBM was observed using the numerical renormalisation group (NRG) technique for all values of 0 < s < 1, but the critical properties of the transition were found to be non-mean-field for 0 < s < 0.5. It was suggested that this implied a breakdown of the classical to quantum mapping, and some subsequent work in this and other systems has supported this claim [12,13]. However, it is now believed that the nonmean-field results found by NRG in 0 < s < 0.5 are incorrect, and arise from the truncation of the number of states N b used to describe each oscillator in the Wilson chain [14,15]. Recent studies of the sub-Ohmic QPT using quantum monte carlo (QMC) [16], sparse polynomial space approach (SPSA) [17], and an extended coherent state technique have indeed found mean-field critical exponents for 0 < s < 0.5 [18].In this article we propose a variational ansatz for the ground state of the sub-Ohmic SBM for 0 < s < 0.5 whi...

Quantum critical point of spin-boson model and infrared catastrophe in bosonic bath

“…One can see that our result compare well with these numerical results. Fig.2 shows the difference between our calculation of the ground state energy and that of Zhao et al [12] and Chin et al [13], δE g = E g − E D g . The lower ground state energy indicates that the ansatz of this work is a better one for the real ground state.…”

“…Zhao et al [12] and Chin et al [13] let φ k = M to be a constant in Eqs. (6) and (8), thus |Ψ + and |Ψ − are degenerate with degenerate ground state energy…”

“…Zhao et al [12] and Chin et al [13] proposed an extension of the Silbey-Harris ground state to study the QPT in the s = 1/2 sub-Ohmic spin-boson model, with degenerate ground state when zero-biased and α > α D c (superscript "D" means degenerate)…”

“…In this work, we propose an analytic ground state wavefunction for the SBM, which is non-Gaussian for the bath modes and is an extension of the work of Zhao et al [12] and Chin et al [13]. The QPT is usually not a weak coupling problem and people believe that the numerical techniques may be more powerful than approximate analytic methods for strong coupling problem.…”

“…Besides, recently an extension of the Silbey-Harris ground state was proposed by Zhao et al [12] and Chin et al [13] to study the QPT in the s = 1/2 sub-Ohmic SBM.…”