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arXiv:1402.5478 · The Journal of Chemical Physics

Communication: Spin-boson model with diagonal and off-diagonal coupling to two independent baths: Ground-state phase transition in the deep sub-Ohmic regime

Yang Zhao, Yao Yao, Vladimir Chernyak, Yang Zhao

Published Apr 25, 2014• 17 citations
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Abstract

We investigate a spin-boson model with two boson baths that are coupled to two perpendicular components of the spin by employing the density matrix renormalization group method with an optimized boson basis. It is revealed that in the deep sub-Ohmic regime there exists a novel second-order phase transition between two types of doubly degenerate states, which is reduced to one of the usual types for nonzero tunneling. In addition, it is found that expectation values of the spin components display jumps at the phase boundary in the absence of bias and tunneling.

DOI
10.1063/1.4873351

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We investigate a spin-boson model with two boson baths that are coupled to two perpendicular components of the spin by employing the density matrix renormalization group method with an optimized boson basis. It is revealed that in the deep sub-Ohmic regime there exists a novel second-order phase transition between two types of doubly degenerate states, which is reduced to one of the usual types for nonzero tunneling. In addition, it is found that expectation values of the spin components display jumps at the phase boundary in the absence of bias and tunneling.
By employing a variational approach, the density matrix renormalization group (DMRG), the exact diagonalization, and symmetry and mean-field analyses, the ground-state properties of the two-bath spin-boson model with simultaneous diagonal and off-diagonal coupling are systematically studied in the sub-Ohmic regime. A quantum phase transition from a doubly degenerate ``localized phase'' to the other doubly degenerate ``delocalized phase'' is uncovered. Via the multi-D1{\mathrm{D}}_{1}D1​ Ansatz as the variational wave function, transition points are determined accurately, consistent with the results from DMRG and exact diagonalization. An effective spatial dimension deff=2.37(6){d}_{\mathrm{eff}}=2.37(6)deff​=2.37(6) is then estimated, which is found to be compatible with the mean-field prediction. Furthermore, the quantum phase transition is inferred to be of first order for the baths described by a continuous spectral density function. In the single-mode case, however, the transition is softened.
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 generalized trial wave function termed as the ``multi-D1{D}_{1}D1​ ansatz'' has been developed to study the ground state of the spin-boson model with simultaneous diagonal and off-diagonal coupling in the sub-Ohmic regime. Ground-state properties including energy and spin polarization are investigated, and the results are consistent with those from exact diagonalization and density matrix renormalization group approaches for the cases involving two oscillators and two baths described by a continuous spectral density function. Breakdown of the rotational and parity symmetries along the continuous quantum phase transition separating the localized phase from the critical phase has been uncovered. Moreover, the phase boundary is determined accurately with the corresponding rotational- and parity-symmetry parameters. A critical value of the spectral exponent s∗=0.49(1){s}^{*}=0.49(1)s∗=0.49(1) is predicted in the weak coupling limit, which is in agreement with the mean-field prediction of 1/21/21/2, but much smaller than the earlier literature estimate of 0.75(1)0.75(1)0.75(1).
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.
The ground state properties and quantum phase transitions of sub-Ohmic spin-boson models are investigated using the multiple Davydov D2Ansatz in conjunction with the variational principle. Three variants of the model are studied: (i) a single bath with diagonal coupling, (ii) two independent baths with diagonal and off-diagonal couplings, and (iii) a single bath with simultaneous diagonal and off-diagonal couplings. For the purely diagonal model, the multiple Davydov D2Ansatz yields critical coupling strengths that are consistent with other methodologies, validating its accuracy and efficiency. In the two-bath model, the competition between diagonal and off-diagonal couplings drives a first-order transition for both symmetric and asymmetric spectral exponents, with the von Neumann exhibiting a continuous peak only under exact symmetry. Finally, for a single bath with simultaneous diagonal and off-diagonal couplings, we demonstrate that a rotational transformation maps the system to an equivalent purely diagonal model, enabling simpler and intuitive physical interpretation and reduced computational complexity.
Dynamics of the sub-Ohmic spin-boson model is examined using three numerical approaches, namely the Dirac-Frenkel time-dependent variation with the Davydov D(1) ansatz, the adaptive time-dependent density matrix renormalization group method within the representation of orthogonal polynomials, and a perturbative approach based on a unitary transformation. In order to probe the validity regimes of the three approaches, we study the dynamics of a qubit coupled to a bosonic bath with and without a local field. Comparison of the up-state population evolution shows that the three approaches are in agreement in the weak-coupling regime but exhibit marked differences when the coupling strength is large. The Davydov D(1) ansatz and the time-dependent density matrix renormalization group can both be reliably employed in the weak-coupling regime, while the former is also valid in the strong-coupling regime as judged by how faithfully the trial state follows the Schrödinger equation. We further explore the bipartite entanglement dynamics between two qubits coupled with individual bosonic baths which reveals entanglement sudden death and revival.