sub-Ohmic Spin-boson Model Research Articles

Finite temperature dynamics in a polarized sub-Ohmic heat bath: A hierarchical equations of motion-tensor train study.

The dynamics of the sub-Ohmic spin-boson model under polarized initial conditions at finite temperatures is investigated by employing both analytical tools and the numerically accurate hierarchical equations of motion-tensor train method. By analyzing the features of nonequilibrium dynamics, we discovered a bifurcation phenomenon, which separates two regimes of the dynamics. It is found that before the bifurcation time, increasing temperature slows down the population dynamics, while the opposite effect occurs after the bifurcation time. The dynamics is highly sensitive to both initial preparation of the bath and thermal effects.

Numerical variational studies of quantum phase transitions in the sub-Ohmic spin-boson model with multiple polaron ansatz

With extensive variational calculations, dissipative quantum phase transitions in the sub-Ohmic spin-boson model are numerically studied in a dense limit of environmental modes. By employing a generalized trial wave function composed of coherent-state expansions, transition points and critical exponents are accurately determined for various spectral exponents, demonstrating excellent agreement with those obtained by other sophisticated numerical techniques. Besides, the quantum-to-classical correspondence is fully confirmed over the entire sub-Ohmic range, compared with theoretical predictions of the long-range Ising model. Mean-field and non-mean-field critical behaviors are found in the deep and shallow sub-Ohmic regimes, respectively, and distinct physical mechanisms of them are uncovered.

Taming Quantum Noise for Efficient Low Temperature Simulations of Open Quantum Systems.

The hierarchical equations of motion (HEOM), derived from the exact Feynman-Vernon path integral, is one of the most powerful numerical methods to simulate the dynamics of open quantum systems. Its applicability has so far been limited to specific forms of spectral reservoir distributions and relatively elevated temperatures. Here we solve this problem and introduce an effective treatment of quantum noise in frequency space by systematically clustering higher order Matsubara poles, equivalent to an optimized rational decomposition. This leads to an elegant extension of the HEOM to arbitrary temperatures and very general reservoirs in combination with efficiency, high accuracy, and long-time stability. Moreover, the technique can directly be implemented in other approaches such as Green's function, stochastic, and pseudomode formulations. As one highly nontrivial application, for the subohmic spin-boson model at vanishing temperature the Shiba relation is quantitatively verified which predicts the long-time decay of correlation functions.

Quantum tricritical point emerging in the spin-boson model with two dissipative spins in staggered biases

We study the spin-boson model (SBM) with two spins in staggered biases by a numerically exact method based on variational matrix product states. Several observables such as the magnetization, the entanglement entropy between the two spins and the bosonic environment, the ground-state energy, as well as the correlation function for two spins are calculated exactly. The characteristics of these observables suggest that the staggered biases can drive the 2nd-order quantum phase transition (QPT) to the 1st-order QPT in the sub-Ohmic SBM, while the Kosterlitz-Thouless QPT in the Ohmic SBM goes directly to the 1st-order one. A quantum tricritical point, where the continuous QPT meets the 1st-order one, can then be detected. It is found that the staggered biases would not change the universality of { the phase transition in this model} below the quantum tricritical point.

Equilibrium dynamics of the sub-ohmic spin-boson model at finite temperature* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11374362 and 11974420), the Fundamental Research Funds for the Central Universities, China, and the Research Funds of Renmin University of China (Grant No. 15XNLQ03).

We use the full-density matrix numerical renormalization group method to calculate the equilibrium dynamical correlation function C(ω) of the spin operator σz at finite temperature for the sub-ohmic spin-boson model. A peak is observed at the frequency ω T ∼ T in the curve of C(ω). The curve merges with the zero-temperature C(ω) in ω ≫ ω T and deviates significantly from the zero-temperature curve in ω ≪ ω T.

Apoptosis of moving nonorthogonal basis functions in many-particle quantum dynamics

Due to the exponential increase of the numerical effort with the number of degrees of freedom, moving basis functions have a long history in quantum dynamics. In addition, spawning of new basis functions is routinely applied. Here we advocate the opposite process: the programmed removal of motional freedom of selected basis functions. This is a necessity for converged numerical results with respect to the size of a non-orthogonal basis, because generically two or more states approach each other too closely early on, rendering unstable the matrix inversion, required to make the equations of motion explicit. Applications to the sub-Ohmic spin-boson model as well as to polaron dynamics in a Holstein molecular crystal model demonstrate the power of the proposed methodology.

Rich phase diagram of quantum phases in the anisotropic subohmic spin-boson model

We study the anisotropic spin-boson model (SBM) with the subohmic bath by a numerically exact method based on variational matrix product states. A rich phase diagram is found in the anisotropy-coupling strength plane by calculating several observables. There are three distinct quantum phases: a delocalized phase with even parity (phase I), a delocalized phase with odd parity (phase II), and a localized phase with broken $Z_2$ symmetry (phase III), which intersect at a quantum tricritical point. The competition between those phases would give overall picture of the phase diagram. For small power of the spectral function of the bosonic bath, the quantum phase transition (QPT) from phase I to III with mean-field critical behavior is present, similar to the isotropic SBM. The novel phase diagram full with three different phases can be found at large power of the spectral function: For highly anisotropic case, the system experiences the QPTs from phase I to II via 1st-order, and then to the phase III via 2nd-order with the increase of the coupling strength. For low anisotropic case, the system only experiences the continuous QPT from phase I to phase III with the non-mean-field critical exponents. Very interestingly, at the moderate anisotropy, the system would display the continuous QPTs for several times but with the same critical exponents. This unusual reentrance to the same localized phase is discovered in the light-matter interacting systems. The present study on the anisotropic SBM could open an avenue to the rich quantum criticality.

Quantum critical phenomena in heat transport via a two-state system

We present a theoretical study of the quantum critical behavior in heat transport via a two-state system with sub-ohmic reservoirs. We calculate the temperature dependence of the thermal conductance near the quantum phase transition via the continuous-time quantum Monte Carlo method and discuss its critical exponents. We also propose a superconducting circuit to realize the sub-ohmic spin-boson model that can be used to observe quantum critical phenomena.

Impurity-induced environmental quantum phase transitions in the quadratic-coupling spin-boson model

We study the zero temperature properties of the sub-Ohmic spin-boson model with quadratic spin-boson coupling. This model describes experimental set ups at the optimal working point where the linear-coupling between the qubit (spin) and the environmental noise (bosons) is zero and the leading coupling is quadratic. In the strong coupling regime, we find that the existence of spin induces quantum phase transitions (QPTs) between two states of environment: the normal state and a state with local distortions. The phase diagram contains both continuous and the first-order QPTs, with non-trivial critical properties obtained exactly. At the QPTs, the equilibrium state spin dynamics bears power-law $\omega$ dependence in the small frequency limit and a robust coherent Rabi oscillation at high frequency. We discuss the feasibility of observing such environmental QPTs in the qubit-related experiments.

Improved Silbey-Harris polaron ansatz for the spin-boson model

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 $0<s<1/2$ 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.

Uncovering nonperturbative dynamics of the biased sub-Ohmic spin-boson model with variational matrix product states

We study the dynamics of the biased sub-Ohmic spin-boson model by means of a time-dependent variational matrix product state (TDVMPS) algorithm. The evolution of both the system and the environment is obtained in the weak- and the strong-coupling regimes, respectively characterized by damped spin oscillations and by a nonequilibrium process where the spin freezes near its initial state, which are explicitly shown to arise from a variety of reactive environmental quantum dynamics. We also explore the rich phenomenology of the intermediate-coupling case, a nonperturbative regime where the system shows a complex dynamical behavior, combining features of both the weakly and the strongly coupled case in a sequential, time-retarded fashion. Our work demonstrates the potential of TDVMPS methods for exploring otherwise elusive, nonperturbative regimes of complex open quantum systems, and points to the possibilities of exploiting the qualitative, real-time modification of quantum properties induced by nonequilibrium bath dynamics in ultrafast transient processes.

Zero-temperature localization in a sub-Ohmic spin-boson model investigated by an extended hierarchy equation of motion

With a decomposition scheme for the bath correlation function, the hierarchy equation of motion (HEOM) is extended to the zero-temperature sub-Ohmic spin-boson model, providing a numerically accurate prediction of quantum dynamics. As a dynamic approach, the extended HEOM determines the delocalized-localized (DL) phase transition from the extracted rate kernel and the coherent-incoherent dynamic transition from the short-time oscillation. As the bosonic bath approaches from the strong to weak sub-Ohmic regimes, a crossover behavior is identified for the critical Kondo parameter of the DL transition, accompanied by the transition from the coherent to incoherent dynamics in the localization.

Equilibrium dynamics of the sub-Ohmic spin-boson model under bias**Project supported by the National Basic Research Program of China (Grant No. 2012CB921704), the National Natural Science Foundation of China (Grant No. 11374362), the Fundamental Research Funds for the Central Universities, China, and the Research Funds of Renmin University of China (Grant No. 15XNLQ03).

Using the bosonic numerical renormalization group method, we studied the equilibrium dynamical correlation function C(ω) of the spin operator σz for the biased sub-Ohmic spin-boson model. The small-ω behavior is found to be universal and independent of the bias ε and the coupling strength α (except at the quantum critical point and ε = 0). Our NRG data also show for a wide range of parameters, including the biased strong coupling regime ( and ), supporting the general validity of the Shiba relation. Close to the quantum critical point , the dependence of C(ω) on α and ε is understood in terms of the competition between ε and the crossover energy scale of the unbiased case. C(ω) is stable with respect to ε for . For , it is suppressed by ε in the low frequency regime. We establish that holds for all sub-Ohmic regime , with for and for . The variation of C(ω) with α and ε is summarized into a crossover phase diagram on the α–ε plane.

Open Wilson chains for quantum impurity models: Keeping track of all bath modes

When constructing a Wilson chain to represent a quantum impurity model, the effects of truncated bath modes are neglected. We show that their influence can be kept track of systematically by constructing an "open Wilson chain" in which each site is coupled to a separate effective bath of its own. As a first application, we use the method to cure the so-called mass-flow problem that can arise when using standard Wilson chains to treat impurity models with asymmetric bath spectral functions at finite temperature. We demonstrate this for the strongly sub-Ohmic spin-boson model at quantum criticality where we directly observe the flow towards a Gaussian critical fixed point.

Anatomy of quantum critical wave functions in dissipative impurity problems

Quantum phase transitions reflect singular changes taking place in a many-body ground state, however, computing and analyzing large-scale critical wave functions constitutes a formidable challenge. New physical insights into the sub-Ohmic spin-boson model are provided by the coherent state expansion (CSE), which represents the wave function by a linear combination of classically displaced configurations. We find that the distribution of low-energy displacements displays an emergent symmetry in the absence of spontaneous symmetry breaking, while experiencing strong fluctuations of the order parameter near the quantum critical point. Quantum criticality provides two strong fingerprints in critical low-energy modes: an algebraic decay of the average displacement and a constant universal average squeezing amplitude. These observations, confirmed by extensive variational matrix product states (VMPS) simulations and field theory arguments, offer precious clues into the microscopics of critical many-body states in quantum impurity models.

Variational dynamics of the sub-Ohmic spin-boson model on the basis of multiple Davydov D1 states

Dynamics of the sub-Ohmic spin-boson model is investigated by employing a multitude of the Davydov D1 trial states, also known as the multi-D1 Ansatz. Accuracy in dynamics simulations is improved significantly over the single D1 Ansatz, especially in the weak system-bath coupling regime. The reliability of the multi-D1 Ansatz for various coupling strengths and initial conditions is also systematically examined, with results compared closely with those of the hierarchy equations of motion and the path integral Monte Carlo approaches. In addition, a coherent-incoherent phase crossover in the nonequilibrium dynamics is studied through the multi-D1 Ansatz. The phase diagram is obtained with a critical point sc = 0.4. For sc < s < 1, the coherent-to-incoherent crossover occurs at a certain coupling strength, while the coherent state recurs at a much larger coupling strength. For s < sc, only the coherent phase exists.

Quench of non-Markovian coherence in the deep sub-Ohmic spin–boson model: A unitary equilibration scheme

The deep sub-Ohmic spin–boson model shows a longstanding non-Markovian coherence at low temperature. Motivating to quench this robust coherence, the thermal effect is unitarily incorporated into the time evolution of the model, which is calculated by the adaptive time-dependent density matrix renormalization group algorithm combined with the orthogonal polynomials theory. Via introducing a unitary heating operator to the bosonic bath, the bath is heated up so that a majority portion of the bosonic excited states is occupied. It is found in this situation the coherence of the spin is quickly quenched even in the coherent regime, in which the non-Markovian feature dominates. With this finding we come up with a novel way to implement the unitary equilibration, the essential term of the eigenstate-thermalization hypothesis, through a short-time evolution of the model.

Consistent perturbative treatment of the subohmic spin-boson model yielding arbitrarily smallT2/T1decoherence time ratios

We present a perturbative treatment of the subohmic spin-boson model which remedies a crucial flaw in previous treatments. The problem is traced back to the incorrect application of a Markov type approximation to specific terms in the temporal evolution of the reduced density matrix. The modified solution is consistent both with numerical simulations and the exact solution obtained when the bath-coupling spin-space direction is parallel to the qubit energy-basis spin. We therefore demonstrate that the subohmic spin-boson model is capable of describing arbitrarily small ratios of the ${T}_{2}$ and ${T}_{1}$ decoherence times, associated with the decay of the off-diagonal and diagonal reduced density-matrix elements, respectively. An analytical formula for ${T}_{2}/{T}_{1}$ at the absolute zero of temperature is provided in the limit of a subohmic bath with vanishing spectral power law exponent. Small ratios closely mimic the experimental results for solid state (flux) qubits, which are subject predominantly to low-frequency electromagnetic noise, and we suggest a reanalysis of the corresponding experimental data in terms of a nonanalytic decay of off-diagonal coherence.

Ground-state properties of sub-Ohmic spin-boson model with simultaneous diagonal and off-diagonal coupling

By employing the variational approach, density matrix renormalization group (DMRG), exact diagonalization as well as 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 novel quantum phase transition from a doubly degenerate "localized phase" to the other doubly degenerate "delocalized phase" is uncovered. Via the multi-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 $d_{eff} = 2.37(6)$ 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 case of single mode, however, the transition is softened.

Sub-Ohmic spin-boson model with off-diagonal coupling: Ground state properties

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.