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Quantum Phase Transitions in the Spin-Boson model: MonteCarlo Method vs Variational Approach a la Feynman

The effectiveness of the variational approach a la Feynman is proved in the spin-boson model, i.e. the simplest realization of the Caldeira-Leggett model able to reveal the quantum phase transition from delocalized to localized states and the quantum dissipation and decoherence effects induced by a heat bath. After exactly eliminating the bath degrees of freedom, we propose a trial, non local in time, interaction between the spin and itself simulating the coupling of the two level system with the bosonic bath. It stems from an Hamiltonian where the spin is linearly coupled to a finite number of harmonic oscillators whose frequencies and coupling strengths are variationally determined. We show that a very limited number of these fictitious modes is enough to get a remarkable agreement, up to very low temperatures, with the data obtained by using an approximation-free Monte Carlo approach, predicting: 1) in the Ohmic regime, a Beretzinski-Thouless-Kosterlitz quantum phase transition exhibiting the typical universal jump at the critical value; 2) in the sub-Ohmic regime ($s \leq 0.5$), mean field quantum phase transitions, with logarithmic corrections for $s=0.5$.