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arXiv:1410.0999

Quantum criticality of the sub-Ohmic spin-boson model within displaced Fock states

Shu He, Liwei Duan, Qing‐Hu Chen

Published Oct 4, 2014• 0 citations
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Abstract

The spin-boson model is analytically studied using displaced Fock states (DFS) without discretization of the continuum bath. In the orthogonal displaced Fock basis, the ground-state wavefunction can be systematically improved in a controllable way. Interestingly, the zeroth-order DFS reproduces exactly the well known Silbey-Harris results. In the framework of the second-order DFS, the magnetization and the entanglement entropy are exactly calculated. It is found that the magnetic critical exponent $β$ is converged to $0.5$ in the whole sub-Ohmic bath regime $0

DOI
10.48550/arxiv.1410.0999

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The spin-boson model is analytically studied using displaced Fock states (DFS) without discretization of the continuum bath. In the orthogonal displaced Fock basis, the ground-state wavefunction can be systematically improved in a controllable way. Interestingly, the zeroth-order DFS reproduces exactly the well known Silbey-Harris results. In the framework of the second-order DFS, the magnetization and the entanglement entropy are exactly calculated. It is found that the magnetic critical exponent βββ is converged to 0.50.50.5 in the whole sub-Ohmic bath regime $0
In this paper, the well-known Silbey-Harris (SH) polaron ansatz for the spin-boson model is improved by adding orthogonal displaced Fock states. The obtained results for the ground state in all baths converge very quickly within finite displaced Fock states and corresponding SH results are corrected considerably. Especially for the sub-Ohmic spin-boson model, the converging results are obtained for 0<s<1/20<s<1/20<s<1/2 in the fourth-order correction and very accurate critical coupling strengths of the quantum phase transition are achieved. Converging magnetization in the biased spin-boson model is also arrived at. Since the present improved SH ansatz can yield very accurate, even almost exact results, it should have wide applications and extensions in various spin-boson model and related fields.
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.
The spin-boson model is a paradigm for studying decoherence, relaxation, entanglement and other effects that arise in a quantum system coupled to environmental degrees of freedom. At zero temperature, a localization-delocalization phase transition is known to exist in the sub-Ohmic regime, where the standard density matrix renormalization group algorithm is inadequate due to the divergence in the number of low-frequency modes. This limitation is circumvented in this work by symmetrically optimizing the phonon basis and introducing an order parameter accounting for the U(1) symmetry for a two-bath spin-boson model, by which we are able to determine the classification and criticality of the phase transition explicitly. Compared with variational results, the critical phase is characterized by spontaneous vanishing of boson displacements in both the baths, resulting in an accurate phase diagram with three model parameters.
We have carried out analytical and numerical studies of the spin-boson model in the sub-ohmic regime with the influence of both the diagonal and the off-diagonal coupling accounted for, via the Davydov D1 variational ansatz. While a second-order phase transition is known to be exhibited by this model in the presence of diagonal coupling only, we demonstrate the emergence of a discontinuous first order phase transition upon incorporation of the off-diagonal coupling. A plot of the ground state energy versus magnetization highlights the discontinuous nature of the transition between the isotropic (zero magnetization) state and nematic (finite magnetization) phases. We have also calculated the entanglement entropy and a discontinuity found at a critical coupling strength further supports the discontinuous crossover in the spin-boson model in the presence of off-diagonal coupling. It is further revealed via a canonical transformation approach that for the special case of identical exponents for the spectral densities of the diagonal and the off-diagonal coupling, there exists a continuous crossover from a single localized phase to doubly degenerate localized phase with differing magnetizations.
A variational approach based on the multi-coherent-state ansatz with asymmetric parameters is employed to study the ground state of the spin-boson model. Without any artificial approximations except for the finite number of the coherent states, we find the robust Gaussian critical behavior in the whole sub-Ohmic bath regime. The converged critical coupling strength can be estimated with the 1/N1/N1/N scaling, where $N $ is the number of the coherent states. It is strongly demonstrated the breakdown of the well-known quantum-to-classical mapping for $1/2
An analytic ground state is proposed for the unbiased spin-boson Hamiltonian, which is non-Gaussian and beyond the Silbey-Harris ground state with lower ground state energy. The infrared catastrophe in Ohmic and sub-Ohmic bosonic bath plays an important role in determining the degeneracy of the ground state. We show that the infrared divergence associated with the displacement of the nonadiabatic modes in bath may be removed from the proposed ground state for the coupling α < αc. Then αc is the quantum critical point of a transition from non-degenerate to degenerate ground state and our calculated αc agrees with previous numerical results.