sub-Ohmic Spin-boson Model Research Articles

Dynamics of the sub-Ohmic spin-boson model: A comparison of three numerical approaches

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.

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

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.

A variational surface hopping algorithm for the sub-Ohmic spin-boson model

The Davydov D1 ansatz, which assigns individual bosonic trajectories to each spin state, is an efficient, yet extremely accurate trial state for time-dependent variation of the sub-Ohmic spin-boson model [N. Wu, L. Duan, X. Li, and Y. Zhao, J. Chem. Phys. 138, 084111 (2013)]. A surface hopping algorithm is developed employing the Davydov D1 ansatz to study the spin dynamics with a sub-Ohmic bosonic bath. The algorithm takes into account both coherent and incoherent dynamics of the population evolution in a unified manner, and compared with semiclassical surface hopping algorithms, hopping rates calculated in this work follow more closely the Marcus formula.

Matrix product state representation without explicit local Hilbert space truncation with applications to the sub-ohmic spin-boson model

We present an alternative to the conventional matrix product state representation, which allows us to avoid the explicit local Hilbert space truncation many numerical methods employ. Utilizing chain mappings corresponding to linear and logarithmic discretizations of the spin-boson model onto a semi-infinite chain, we apply the new method to the sub-ohmic spin-boson model. We are able to reproduce many well-established features of the quantum phase transition, such as the critical exponent predicted by mean-field theory. Via extrapolation of finite-chain results, we are able to determine the infinite-chain critical couplings αc at which the transition occurs and, in general, study the behaviour of the system well into the localized phase.

Single mode approximation for sub-Ohmic spin-boson model: adiabatic limit and critical properties

In this work, the quantum phase transition in the sub-Ohmic spin-boson model is studied using a single-mode approximation, by combining the rotating wave transformation and the transformations used in the numerical renormalization group (NRG). Analytical results for the critical coupling strength $\alpha_c$, the magnetic susceptibility $\chi(T)$, and the spin-spin correlation function $C(\omega)$ at finite temperatures are obtained and further confirmed by numerical results. We obtain the same $\alpha_c$ as the mean-field approximation. The critical exponents are classical: $\beta=1/2$, $\delta=3$, $\gamma=1$, $x=1/2$, $y_t^{*}=1/2$, in agreement with the spin-boson model in $0<s<1/2$ regime. $C(\omega)$ has nontrivial behavior reflecting coherent oscillation with temperature dependent damping effects due to the environment. We point out the original NRG has problem with the crossover temperature $T^{*}$, and propose a chain Hamiltonian possibly suitable for implementing NRG without boson state truncation error.

Dynamics of the sub-Ohmic spin-boson model: A time-dependent variational study

The Dirac-Frenkel time-dependent variation is employed to probe the dynamics of the zero temperature sub-Ohmic spin-boson model with strong friction utilizing the Davydov D1 ansatz. It is shown that initial conditions of the phonon bath have considerable influence on the dynamics. Counterintuitively, even in the very strong coupling regime, quantum coherence features still manage to survive under the polarized bath initial condition, while such features are absent under the factorized bath initial condition. In addition, a coherent-incoherent transition is found at a critical coupling strength α ≈ 0.1 for s = 0.25 under the factorized bath initial condition. We quantify how faithfully our ansatz follows the Schrödinger equation, finding that the time-dependent variational approach is robust for strong dissipation and deep sub-Ohmic baths (s ≪ 1).

Scaling analysis in the numerical renormalization group study of the sub-Ohmic spin-boson model

The spin-boson model has nontrivial quantum phase transitions in the sub-Ohmic regime. For the bath spectra exponent $0\ensuremath{\leqslant}s&lt;1/2$, the bosonic numerical renormalization group (BNRG) study of the exponents $\ensuremath{\beta}$ and $\ensuremath{\delta}$ are hampered by the boson-state truncation, which leads to artificial interacting exponents instead of the correct Gaussian ones. In this paper, guided by a mean-field calculation, we study the order-parameter function $m(\ensuremath{\tau}=\ensuremath{\alpha}\ensuremath{-}{\ensuremath{\alpha}}_{c},\ensuremath{\epsilon},\ensuremath{\Delta})$ using BNRG. Scaling analysis with respect to the boson-state truncation ${N}_{b}$, the logarithmic discretization parameter $\ensuremath{\Lambda}$, and the tunneling strength $\ensuremath{\Delta}$ are carried out. Truncation-induced multiple-power behaviors are observed close to the critical point, with artificial values of $\ensuremath{\beta}$ and $\ensuremath{\delta}$. They cross over to classical behaviors with exponents $\ensuremath{\beta}=1/2$ and $\ensuremath{\delta}=3$ on the intermediate scales of $\ensuremath{\tau}$ and $\ensuremath{\epsilon}$, respectively. We also find $\ensuremath{\tau}/{\ensuremath{\Delta}}^{1\ensuremath{-}s}$ and $\ensuremath{\epsilon}/\ensuremath{\Delta}$ scalings in the function $m(\ensuremath{\tau},\ensuremath{\epsilon},\ensuremath{\Delta})$. The role of boson-state truncation as a scaling variable in the BNRG result for $0\ensuremath{\leqslant}s&lt;1/2$ is identified and its interplay with the logarithmic discretization revealed. Relevance to the validity of quantum-to-classical mapping in other impurity models is discussed.

Critical and Strong-Coupling Phases in One- and Two-Bath Spin-Boson Models

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.

Numerical renormalization group for the sub-Ohmic spin-boson model: A conspiracy of errors

The application of Wilson's numerical renormalization group (NRG) method to dissipative quantum impurity models, in particular the sub-Ohmic spin-boson model, has led to conclusions regarding the quantum critical behavior which are in disagreement with those from other methods and which are by now recognized as erroneous. The errors of NRG remained initially undetected, because NRG delivered an internally consistent set of critical exponents satisfying hyperscaling. Here we discuss how the conspiracy of two errors---the Hilbert-space truncation error and the mass-flow error---could lead to this consistent set of exponents. Remarkably, both errors, albeit of different origin, force the system to obey naive scaling laws even when the physical model violates naive scaling. In particular, we show that a combination of the Hilbert-space truncation and mass-flow errors induce an artificial nonanalytic term in the Landau expansion of the free energy which dominates the critical behavior for bath exponents $s&lt;1/2$.

Quantum criticality of the sub-Ohmic spin-boson model

We revisit the critical behavior of the sub-ohmic spin-boson model. Analysis of both the leading and subleading terms in the temperature dependence of the inverse static local spin susceptibility at the quantum critical point, calculated using a numerical renormalization-group method, provides evidence that the quantum critical point is interacting in cases where the quantum-to-classical mapping would predict mean-field behavior. The subleading term is shown to be consistent with an w/T scaling of the local dynamical susceptibility, as is the leading term. The frequency and temperature dependences of the local spin susceptibility in the strong-coupling (delocalized) regime are also presented. We attribute the violation of the quantum-to-classical mapping to a Berry-phase term in a continuum path-integral representation of the model. This effect connects the behavior discussed here with its counterparts in models with continuous spin symmetry.

Dissipative spin dynamics near a quantum critical point: Numerical renormalization group and Majorana diagrammatics

We provide an extensive study of the sub-ohmic spin-boson model with power law density of states J(\omega)=\omega^s (with 0<s<1), focusing on the equilibrium dynamics of the three possible spin components, from very weak dissipation to the quantum critical regime. Two complementary methods, the bosonic Numerical Renormalization Group (NRG) and Majorana diagrammatics, are used to explore the physical properties in a wide range of parameters. We show that the bosonic self-energy is the crucial ingredient for the description of critical fluctuations, but that many-body vertex corrections need to be incorporated as well in order to obtain quantitative agreement of the diagrammatics with the numerical simulations. Our results suggest that the out-of-equilibrium dynamics in dissipative models beyond the Bloch-Redfield regime should be reconsidered in the long-time limit. Regarding also the spin-boson Hamiltonian as a toy model of quantum criticality, some of the insights gained here may be relevant for field theories of electrons coupled to bosons in higher dimensions.

Generalized Polaron Ansatz for the Ground State of the Sub-Ohmic Spin-Boson Model: An Analytic Theory of the Localization Transition

The sub-Ohmic spin-boson model possesses a quantum phase transition at zero temperature between a localized and a delocalized phase, whose properties have so far only been extracted by numerical approaches. Here we present an extension of the Silbey-Harris variational polaron ansatz which allows us to develop an analytical theory which correctly describes a continuous transition with mean-field exponents for 0<s<0.5. The critical properties, couplings, and observables we obtain show excellent agreement with existing numerical results, and we give an intuitive microscopic description of the changing correlations between the system and bath which suppress the spin coherence and drive the transition.

Spin Path Integrals, Berry Phase, and the Quantum Phase Transition in the Sub-Ohmic Spin-Boson Model

The quantum critical properties of the sub-Ohmic spin-1/2 spin-boson model and of the Bose-Fermi Kondo model have recently been discussed controversially. The role of the Berry phase in the breakdown of the quantum-to-classical mapping of quantum criticality in the spin-isotropic Bose-Fermi Kondo model has been discussed previously. In the present article, some of the subtleties underlying the functional integral representation of the spin-boson and related models with spin anisotropy are discussed. To this end, an introduction to spin coherent states and spin path integrals is presented with a focus on the spin-boson model. It is shown that, even for the Ising-anisotropic case as in the spin-boson model, the path integral in the continuum limit in the coherent state representation involves a Berry phase term. As a result, the effective action for the spin degrees of freedom does not assume the form of a Ginzburg-Landau-Wilson functional. The implications of the Berry-phase term for the quantum-critical behavior of the spin-boson model are discussed. The case of arbitrary spin is also considered.

Quantum phase transition in the sub-Ohmic spin-boson model: An extended coherent-state approach

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 $s&lt;1/2$ can be correctly given by the fidelity susceptibility, demonstrating the strength of the approach.

Mass-flow error in the numerical renormalization-group method and the critical behavior of the sub-Ohmic spin-boson model

We discuss a particular source of error in the numerical renormalization group (NRG) method for quantum impurity problems, which is related to a renormalization of impurity parameters due to the bath propagator. At any step of the NRG calculation, this renormalization is only partially taken into account, leading to systematic variation in the impurity parameters along the flow. This effect can cause qualitatively incorrect results when studying quantum-critical phenomena, as it leads to an implicit variation in the phase transition's control parameter as function of the temperature and thus to an unphysical temperature dependence of the order-parameter mass. We demonstrate the mass-flow effect for bosonic impurity models with a power-law bath spectrum, $J(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}$, namely, the dissipative harmonic oscillator and the spin-boson model. We propose an extension of the NRG to correct the mass-flow error. Using this, we find unambiguous signatures of a Gaussian critical fixed point in the spin-boson model for $s&lt;1/2$, consistent with mean-field behavior as expected from quantum-to-classical mapping.

Ultraslow quantum dynamics in a sub-Ohmic heat bath

We show that the low-frequency modes of a sub-Ohmic bosonic heat bath generate an effective dynamical asymmetry for an intrinsically symmetric quantum spin -1/2. An initially fully polarized spin first decays towards a quasiequilibrium determined by the dynamical asymmetry, thereby showing coherent damped oscillations on the (fast) time scale of the spin splitting. On top of this, the dynamical asymmetry itself decays on an ultraslow time scale and vanishes asymptotically since the global equilibrium phase is symmetric. We quantitatively study the nature of the initial fast decay to the quasiequilibrium and discuss the features of ultraslow dynamics of the quasiequilibrium itself. The dynamical asymmetry is more pronounced for smaller values of the sub-Ohmic exponent and for lower temperatures, which emphasizes the quantum many-body nature of the effect. The symmetry breaking is related to the dynamic crossover between coherent and overdamped relaxation of the spin polarization and is not connected to the localization quantum phase transition. In addition to this delocalized phase, we identify a novel phase which is characterized by damped coherent oscillations in the localized phase. This allows for a sketch of the zero-temperature phase diagram of the sub-Ohmic spin-boson model with four distinct phases.

Quantum phase transitions in a charge-coupled Bose-Fermi Anderson model

We study the competition between Kondo physics and dissipation within an Anderson model of a magnetic impurity level that hybridizes with a metallic host and is also coupled, via the impurity charge, to the displacement of a bosonic bath having a spectral density proportional to ${\ensuremath{\omega}}^{s}$. As the impurity-bath coupling increases from zero, the effective Coulomb interaction between two electrons in the impurity level is progressively renormalized from its repulsive bare value until it eventually becomes attractive. For weak hybridization, this renormalization in turn produces a crossover from a conventional spin-sector Kondo effect to a charge Kondo effect. At particle-hole symmetry, and for sub-Ohmic bath exponents $0&lt;s&lt;1$, further increase in the impurity-bath coupling results in a continuous zero-temperature transition to a broken-symmetry phase in which the ground-state impurity occupancy ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{d}$ acquires an expectation value ${⟨{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{n}}_{d}⟩}_{0}\ensuremath{\ne}1$. The response of the impurity occupancy to a locally applied electric potential features the hyperscaling of critical exponents and $\ensuremath{\omega}∕T$ scaling that are expected at an interacting critical point. The numerical values of the critical exponents suggest that the transition lies in the same universality class as that of the sub-Ohmic spin-boson model. For the Ohmic case $s=1$, the transition is instead of Kosterlitz-Thouless type. Away from particle-hole symmetry, the quantum phase transition is replaced by a smooth crossover but signatures of the symmetric quantum critical point remain in the physical properties at elevated temperatures and/or frequencies.

Erratum: Quantum Phase Transitions in the Sub-Ohmic Spin-Boson Model: Failure of the Quantum-Classical Mapping [Phys. Rev. Lett.94, 070604 (2005)

Erratum: Quantum Phase Transitions in the Sub-Ohmic Spin-Boson Model: Failure of the Quantum-Classical Mapping [Phys. Rev. Lett.<b>94</b>, 070604 (2005)

Sparse Polynomial Space Approach to Dissipative Quantum Systems: Application to the Sub-Ohmic Spin-Boson Model

We propose a general numerical approach to open quantum systems with a coupling to bath degrees of freedom. The technique combines the methodology of polynomial expansions of spectral functions with the sparse grid concept from interpolation theory. Thereby we construct a Hilbert space of moderate dimension to represent the bath degrees of freedom, which allows us to perform highly accurate and efficient calculations of static, spectral, and dynamic quantities using standard exact diagonalization algorithms. The strength of the approach is demonstrated for the phase transition, critical behavior, and dissipative spin dynamics in the spin-boson model.

Quantum Phase Transition in the Sub-Ohmic Spin-Boson Model: Quantum Monte Carlo Study with a Continuous Imaginary Time Cluster Algorithm

A continuous time cluster algorithm for two-level systems coupled to a dissipative bosonic bath is presented and applied to the sub-Ohmic spin-boson model. When the power s of the spectral function Jomega proportional, variant omegas is smaller than 1/2, the critical exponents are found to be classical, mean-field like. Potential sources for the discrepancy with recent renormalization group predictions are traced back to the effect of a dangerously irrelevant variable.