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arXiv:1501.05061

Symmetry and Explicit Marking of the Critical Phase in Two-Bath Spin-Boson Model

Yao Yao, Nengji Zhou, Javier Prior, Yang Zhao

Published Jan 21, 2015• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1501.05061

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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.
For phase transitions in dissipative quantum impurity models, the existence of a quantum-to-classical correspondence has been discussed extensively. We introduce a variational matrix product state approach involving an optimized boson basis, rendering possible high-accuracy numerical studies across the entire phase diagram. For the sub-Ohmic spin-boson model with a power-law bath spectrum ∝ω(s), we confirm classical mean-field behavior for s<1/2, correcting earlier numerical renormalization-group results. We also provide the first results for an XY-symmetric model of a spin coupled to two competing bosonic baths, where we find a rich phase diagram, including both critical and strong-coupling phases for s<1, different from that of classical spin chains. This illustrates that symmetries are decisive for whether or not a quantum-to-classical correspondence exists.
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.
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).
We propose a density matrix renormalization group approach to tackle a two-state system coupled to a bosonic bath with continuous spectrum. In this approach, the optimized phonon scheme is applied to several hundred phonon modes that are linearly divided among the spectra. Although density matrix renormalization group cannot resolve very small energy scales, the delocalized-localized transition points of the two-state system are extracted by the extrapolation of the flow diagram results. The phase diagram is compared to the numerical renormalization group results and shows a good agreement in both Ohmic and sub-Ohmic cases.
Quantum phase transitions (QPTs) in the spin-boson model with/without the rotating-wave approximation (RWA) are systematically investigated through variational calculations using a sub-Ohmic bath with high spectral density. Four cases involving different system-environment interactions are examined, where transition points and critical exponents are accurately determined across varying tunneling strengths. Contrary to prior work, a rich phase diagram is revealed in the tunneling-coupling plane even at the low spectral exponent s<1/2s<1/2s<1/2, with a novel U(1)-symmetric phase being identified. As coupling increases, a multi-stage QPT sequence arises for the tunneling 0<Δ<Δ∗=0.074(1)0<Δ< Δ^*=0.074(1)0<Δ<Δ∗=0.074(1), whereas a single transition occurs beyond this range. Furthermore, an odd-parity phase is found to emerge even under the positive tunneling, exhibiting distinct characteristics relative to the prototype model.
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.