We develop a semianalytical approach beyond the Born-Markov approximation to study the quench dynamics of the spin-boson model in the strong-coupling regime ($\ensuremath{\alpha}\ensuremath{\le}1/2$) for the Ohmic bath. The basic idea in our approach is to write an effective time-dependent model for the dynamics of the system coupled to the bosonic bath after integrating out high-frequency bath modes. By applying this procedure to the Heisenberg equations of motion, we derive a set of flow equations for the system parameters as a function of time. The final flow equations look similar to those of the equilibrium renormalization-group theory; however, in our derivation the scaling parameter is set by the real time. We solve the equations of motion with time-dependent renormalized parameters and show that the resulting dynamics is in decent agreement with the exact NRG calculations as well as the noninteracting blip approximation that is a well-known good solution in this limit.We develop a semianalytical approach beyond the Born-Markov approximation to study the quench dynamics of the spin-boson model in the strong-coupling regime ($\ensuremath{\alpha}\ensuremath{\le}1/2$) for the Ohmic bath. The basic idea in our approach is to write an effective time-dependent model for the dynamics of the system coupled to the bosonic bath after integrating out high-frequency bath modes. By applying this procedure to the Heisenberg equations of motion, we derive a set of flow equations for the system parameters as a function of time. The final flow equations look similar to those of the equilibrium renormalization-group theory; however, in our derivation the scaling parameter is set by the real time. We solve the equations of motion with time-dependent renormalized parameters and show that the resulting dynamics is in decent agreement with the exact NRG calculations as well as the noninteracting blip approximation that is a well-known good solution in this limit.

arXiv:1607.03920Physical Review AMar 14, 201611 citations

Flow equations for the spin-boson problem

Stefan K. Kehrein, Andreas Mielke, Peter Neu

Zeitschrift für Physik B Condensed MatterMar 1, 199565 citations

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

Matthias Vojta C, R. Bulla, Fabian Güttge +1 more

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 $s<1/2$ , consistent with mean-field behavior as expected from quantum-to-classical mapping.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 $s<1/2$, consistent with mean-field behavior as expected from quantum-to-classical mapping.

arXiv:0911.4490Physical Review BFeb 25, 201041 citations

Dynamics of the spin-boson model with a structured environment

Michael Thorwart, Elisabetta Paladino, Milena Grifoni

arXiv:cond-mat/0310140Chemical PhysicsNov 20, 2003111 citations

Dissipative phase transition in a pair of coupled noisy two-level systems

Julius Bonart

We study the renormalization group (RG) equations of a pair of spin-boson systems coupled in the $z$ direction with each other. Each spin is coupled to a different bath of harmonic oscillators. We introduce a systematic adiabatic RG, which generalizes the first-order adiabatic renormalization previously used for the single spin-boson model, and we obtain the flow equations for the tunneling constant, the dissipation strength, and the interspin coupling up to third order in the tunneling. If one of the two spins is treated as a constant magnetization, the other spin is described by a biased spin-boson Hamiltonian. In this case the RG equations we find coincide with the ones obtained via a mapping to a long-range Ising chain. If the whole Ohmic two-spin system is considered the Kosterlitz-Thouless phase transition is replaced by a second-order phase transition. In the case of a sub-Ohmic bath our approach predicts that the two-spin system is always localized.We study the renormalization group (RG) equations of a pair of spin-boson systems coupled in the $z$ direction with each other. Each spin is coupled to a different bath of harmonic oscillators. We introduce a systematic adiabatic RG, which generalizes the first-order adiabatic renormalization previously used for the single spin-boson model, and we obtain the flow equations for the tunneling constant, the dissipation strength, and the interspin coupling up to third order in the tunneling. If one of the two spins is treated as a constant magnetization, the other spin is described by a biased spin-boson Hamiltonian. In this case the RG equations we find coincide with the ones obtained via a mapping to a long-range Ising chain. If the whole Ohmic two-spin system is considered the Kosterlitz-Thouless phase transition is replaced by a second-order phase transition. In the case of a sub-Ohmic bath our approach predicts that the two-spin system is always localized.

arXiv:1307.3378Physical Review BSep 27, 20137 citations

s<1/2

, consistent with mean-field behavior as expected from quantum-to-classical mapping.

We study the renormalization group (RG) equations of a pair of spin-boson systems coupled in the

z

direction with each other. Each spin is coupled to a different bath of harmonic oscillators. We introduce a systematic adiabatic RG, which generalizes the first-order adiabatic renormalization previously used for the single spin-boson model, and we obtain the flow equations for the tunneling constant, the dissipation strength, and the interspin coupling up to third order in the tunneling. If one of the two spins is treated as a constant magnetization, the other spin is described by a biased spin-boson Hamiltonian. In this case the RG equations we find coincide with the ones obtained via a mapping to a long-range Ising chain. If the whole Ohmic two-spin system is considered the Kosterlitz-Thouless phase transition is replaced by a second-order phase transition. In the case of a sub-Ohmic bath our approach predicts that the two-spin system is always localized.