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Journal of the Physical Society of Japan

Quantum Phase Transition and Universality Class for a Boson-Impurity Model: Implications for the Spin-Boson Model

Kurt Fischer

Published Jun 24, 2013• 0 citations
Publisher

Abstract

We introduce a boson-impurity model, and prove that it is in the same universality class as the one-dimensional Ising model with algebraic decaying interaction. This boson-impurity model can be derived from the spin-boson model. The spin-boson model itself is therefore in the same universality class as well. Because the boson-impurity model contains only virtual spin-fluctuations, these are already enough to drive the quantum phase transition of the sub-ohmic spin-boson model.

DOI
10.7566/jpsj.82.073001

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We introduce a boson-impurity model, and prove that it is in the same universality class as the one-dimensional Ising model with algebraic decaying interaction. This boson-impurity model can be derived from the spin-boson model. The spin-boson model itself is therefore in the same universality class as well. Because the boson-impurity model contains only virtual spin-fluctuations, these are already enough to drive the quantum phase transition of the sub-ohmic spin-boson model.
The effective theories for many quantum phase transitions can be mapped onto those of classical transitions. Here we show that the naive mapping fails for the sub-Ohmic spin-boson model which describes a two-level system coupled to a bosonic bath with power-law spectral density, J(omega) proportional, variantomega(s). Using an epsilon expansion we prove that this model has a quantum transition controlled by an interacting fixed point at small s, and support this by numerical calculations. In contrast, the corresponding classical long-range Ising model is known to display mean-field transition behavior for 0 < s < 1/2, controlled by a noninteracting fixed point. The failure of the quantum-classical mapping is argued to arise from the long-ranged interaction in imaginary time in the quantum 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.
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, nonlocal in time, interaction between the spin and itself simulating the coupling of the two-level system with the bosonic bath. It stems from a 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 Berezinski-Thouless-Kosterlitz quantum phase transition exhibiting the typical universal jump at the critical value; and (2) in the sub-Ohmic regime ($s\ensuremath{\le}0.5$), mean-field quantum phase transitions, with logarithmic corrections for s=0.5s=0.5s=0.5.
We propose a general extended coherent state approach to the qubit (or fermion) and multimode boson coupling systems. The application to the spin-boson model with the discretization of a bosonic bath with arbitrary continuous spectral density is described in detail, and very accurate solutions can be obtained. The quantum phase transition in the nontrivial sub-Ohmic case can be located by the fidelity and the order-parameter critical exponents for the bath exponents s<1/2s<1/2s<1/2 can be correctly given by the fidelity susceptibility, demonstrating the strength of the approach.
We describe the generalization of Wilson's numerical renormalization group method to quantum impurity models with a bosonic bath, providing a general nonperturbative approach to bosonic impurity models which can access exponentially small energies and temperatures. As an application, we consider the spin-boson model, describing a two-level system coupled to a bosonic bath with power-law spectral density, J(omega) proportional to omega(s). We find clear evidence for a line of continuous quantum phase transitions for sub-Ohmic bath exponents 0<s<1; the line terminates in the well-known Kosterlitz-Thouless transition at s=1. Contact is made with results from perturbative renormalization group, and various other applications are outlined.
We investigate the entanglement between a spin and its environment in impurity systems which exhibit a second-order quantum phase transition separating a delocalized and a localized phase for the spin. As an application, we employ the spin-boson model, describing a two-level system (spin) coupled to a sub-Ohmic bosonic bath with power-law spectral density, J(omega) proportional to omega(s) and 0 < s < 1. Combining Wilson's numerical renormalization group method and hyperscaling relations, we demonstrate that the entanglement between the spin and its environment is always enhanced at the quantum phase transition resulting in a visible cusp (maximum) in the entropy of entanglement. We formulate a correspondence between criticality and impurity entanglement entropy, and the relevance of these ideas to nanosystems is outlined.