Single Mode Approximation for sub-Ohmic Spin-Boson Model: Adiabatic Limit and Critical Properties

Abstract

In this work, the quantum phase transition in the sub-Ohmic spin-boson model is studied using a single-mode approximation, by combining the rotating wave transformation and the transformations used in the numerical renormalization group (NRG). Analytical results for the critical coupling strength ${\alpha }_c$, the magnetic susceptibility $\chi (T)$, and the spin-spin correlation function $C(\omega )$ at finite temperatures are obtained and further confirmed by numerical results. We obtain the same ${\alpha }_c$ as the mean-field approximation. The critical exponents are classical: $\beta =1/2$, $\delta =3$, $\gamma =1$, $x=1/2$, $y_t^{\ast }=1/2$, in agreement with the spin-boson model in $0<s<1/2$ regime. $C(\omega )$ has nontrivial behavior reflecting coherent oscillation with temperature dependent damping effects due to the environment. We point out the original NRG has problem with the crossover temperature $T^{\ast }$, and propose a chain Hamiltonian possibly suitable for implementing NRG without boson state truncation error.

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