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Physica A Statistical Mechanics and its Applications

Thermal phase transition for some spin-boson models

M. Aparicio Alcalde, B. M. Pimentel

Published Apr 24, 2013• 1 citation
Publisher
DOI
10.1016/j.physa.2013.04.003

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We consider spin-boson models composed by a single bosonic mode and an ensemble of N identical two-level atoms. The situation where the coupling between the bosonic mode and the atoms generates resonant and nonresonant processes is studied, where the whole system is in thermal equilibrium with a reservoir at temperature β−1. Phase transitions from ordinary fluorescence to super-radiant phase in three different models are investigated. First a model where the coupling between the bosonic mode and the jth atom is via the pseudo-spin-operator σ(j)z is studied. Second, we investigate the generalized Dicke model, introducing different coupling constants between the single mode bosonic-field and the environment, g1 and g2, for rotating and counter-rotating terms, respectively. Finally a modified version of the generalized Dicke model with intensity-dependent coupling in the rotating terms is considered. In the first model, for an arbitrary value for the coupling constant, the zero mode contributes in rendering the canonical entropy a negative quantity for low temperatures. For small coupling constants, the partition function is analytic for all temperatures. The last two models present phase transitions, even when only Hamiltonian terms which generate virtual processes are considered.
With a decomposition scheme for the bath correlation function, the hierarchical equations of motion (HEOM) are extended to the zero-temperature sub-Ohmic and Ohmic spin-boson model. We investigate the linear absorption spectrum of the sub-Ohmic and Ohmic spin-boson model at zero temperature. By applying the extended HEOM, the equilibrium spin dynamics are obtained approximately. Then the linear response function is calculated according to the Kubo formula rewritten in Liouville space. To explore the essence of phase transition, we compute the linear absorption spectrum defined by the linear response function of a dipole moment. By analyzing the peak structure of the linear response spectrum, we get the dependence of linear absorption spectrum with Kondo parameter and different bath exponents. The spin relaxation dynamics are also calculated to explore the coherent-incoherent dynamic transition (CI) and delocalizedlocalized phase transition (DL). The corresponding phase diagram of DL and CI transition are also obtained. We pose a energy level picture to understand the different mechanism of DL phase transition between the deep sub-Ohmic and Ohmic spin-boson model.
The class of two-interacting-impurity spin-boson models with vanishing transverse fields on the spin-pair is studied. The model can be exactly mapped into two independent standard single-impurity spin-boson models where the role of the tunnelling parameter is played by the spin-spin coupling. The dynamics of the magnetization is analysed for different levels of (an)isotropy. Further, the existence of a decoherence-free subspace as well as of both classical and quantum (first-order and Kosterlitz-Thouless type) phase transitions, in the Omhic regime, is brought to light.