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 J(w) \propto w^s 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.
Study of dissipative quantum phase transitions in the Ohmic spin-boson model is numerically challenging in a dense limit of environmental modes. In this work, large-scale numerical simulations are carried out based on the variational principle. The validity of variational calculations, spontaneous breakdown of symmetries, and quantum fluctuations and correlations in the Ohmic bath are carefully analyzed, and the critical coupling as well as exponents are accurately determined in the weak tunneling and continuum limits. In addition, quantum criticality of the Ohmic bath is uncovered both in the…
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 depende…
The spin-boson model is studied by means of flow equations for Hamiltonians. Our truncation scheme includes all coupling terms which are linear in the bosonic operators. Starting with the canonical generator $η_c=[H_0,H]$ with $H_0$ resembling the non-interacting bosonic bath, the flow equations exhibit a universal attractor for the Hamiltonian flow. This allows to calculate equilibrium correlation functions for super-Ohmic, Ohmic and sub-Ohmic baths within a uniform framework including finite bias.
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 of 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 of the phase transition's control parameter as function of the temperature and thus t…
Employing the non-perturbative numerical renormalization group method, we study the dynamics of the spin-boson model, which describes a two-level system coupled to a bosonic bath with spectral density J(omega) propto omega^s. We show that, in contrast to the case of ohmic damping, the delocalized phase of the sub-ohmic model cannot be characterized by a single energy scale only, due to the presence of a non-trivial quantum phase transition. In the strongly sub-ohmic regime, s<<1, weakly damped coherent oscillations on short time scales are possible even in the localized phase - this is of cruc…
The Davydov D1 ansatz, which assigns an individual bosonic trajectory to each spin state, is an efficient, yet extremely accurate trial state for time-dependent variation of the sub-Ohmic spin-boson model [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 mo…
We propose a general extended coherent state approach to the qubit (or fermion) and multi-mode 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 $s<1/2$ can be correctly given by the fidelity susceptibility, demonstrating the strength of the approach.
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…
By means of a trial wave function, the multi-D$_1$ ansatz, extensive variational calculations with more than ten thousand parameters have been carried out to study quantum phase transitions in the ground states of a two-impurity system embedded in a common Ohmic bath of bosons. Quantum criticality in both the impurity system and the Ohmic bosonic bath is investigated with relevant transition points and critical exponents determined accurately. With the linear grid of the Ohmic spectral density, our numerical calculations produce a much better description of the ground states with lower energie…
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 behavio…
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 qu…
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 a…
We present a detailed description of the recently proposed numerical renormalization group method for models of quantum impurities coupled to a bosonic bath. Specifically, the method is applied to the spin-boson model, both in the Ohmic and sub-Ohmic cases. We present various results for static as well as dynamic quantities and discuss details of the numerical implementation, e.g., the discretization of a bosonic bath with arbitrary continuous spectral density, the suitable choice of a finite basis in the bosonic Hilbert space, and questions of convergence w.r.t. truncation parameters. The met…
The effective theories for many quantum phase transitions can be mapped onto those of classical transitions. Here we show that such a mapping fails for the sub-ohmic spin-boson model which describes a two-level system coupled to a bosonic bath with power-law spectral density, J(omega) ~ omega^s. Using an epsilon expansion we prove that this model has a quantum transition controlled by an interacting fixed point at small s, and support this by numerical calculations. In contrast, the corresponding classical long-range Ising model is known to have an upper-critical dimension at s = 1/2, with mea…
By employing the spin-boson model in a dense limit of environmental modes, quantum entanglement and correlation of sub-Ohmic and Ohmic baths for dissipative quantum phase transitions are numerically investigated based on the variational principle. With several measures borrowed from quantum information theory, three different types of singularities are found for the first-order, second-order, and Kosterlitz-Thouless phase transitions, respectively, and the values of transition points and critical exponents are accurately determined. Besides, the scaling form of the quantum discord in the Ohmic…
We study the time dynamics of the ohmic spin boson model at arbitrary bias $ε$ and small coupling $α$ to the bosonic bath. Using perturbation theory and the real-time renormalization group (RG) method we present a consistent zero-temperature weak-coupling expansion for the time evolution of the reduced density matrix one order beyond the Bloch-Redfield solution. We develop a renormalized perturbation theory and present an analytical solution covering the whole range from small to large times, including further results for exponentially small or large times. Resumming all secular terms in all o…
We investigate the dynamics of a large anisotropic spin whose easy-axis component is coupled to a bosonic bath with a spectral function $J(\w)\propto ω^s$. Such a spin complex might be realized in a single-molecular magnet. Using the non-perturbative renormalization group, we calculate the line of quantum-phase transitions in the sub-ohmic regime ($s<1$). These quantum-phase transitions only occur for integer spin $J$. For half-integer $J$, the low temperature fixed-point is identical to the fixed-point of the spin-boson model without quantum-tunneling between the two levels. Short-time cohere…
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 quasie…
We study the reduced dynamics of a spin (qubit) coupled to a spin-boson environment in the case of pure dephasing. We derive formal exact expressions which can be cast in terms of exact integro-differential master equations. We present results for a SB environment with ohmic dissipation at finite temperatures. For the special value of the ohmic damping strength K=1/2 the reduced dynamics is found in analytic form. For K<<1 we discuss the possibility of modulating the effect of the SB environment on the qubit. In particular we study the effect of the crossover to a slow environment dynamics, wh…
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 dire…
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 $\propto \w^s$, 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 …
We study the spin-boson model without the counterrotating terms by a numerically exact method based on variational matrix product states. Surprisingly, the second-order quantum phase transition (QPT) is observed for the sub-Ohmic bath in the rotating-wave approximations. Moreover, first-order QPTs can also appear before the critical points. With the decrease of the bath exponents, these first-order QPTs disappear successively, while the second-order QPT remains robust. The second-order QPT is further confirmed by multi-coherent-states variational studies, while the first-order QPT is corrobora…
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 inter-spin 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-b…
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, non local in time, interaction between the spin and itself simulating the coupling of the two level system with the bosonic bath. It stems from an Hamiltonian where the spin is linearly coupled to a finite number o…
A quantum two-level system immersed in a sub-Ohmic bath experiences enhanced low-frequency quantum statistical fluctuations which render the nonequilibrium quantum dynamics highly non-Markovian. Upon using the numerically exact time-evolving matrix product operator approach, we investigate the phase diagram of the polarization dynamics. In addition to the known phases of damped coherent oscillatory dynamics and overdamped decay, we identify a new third region in the phase diagram for strong coupling showing an aperiodic behaviour. We determine the corresponding phase boundaries. The dynamics o…
We study nonadiabatic electron transfer within the biased spin-boson model. We calculate the incoherent transfer rate in analytic form at all temperatures for a power law form of the spectral density of the solvent coupling. In the Ohmic case, we present the exact low temperature corrections to the zero temperature rate for arbitrarily large bias energies between the two redox sites. Both for Ohmic and non-Ohmic coupling, we give the rate in the entire regime extending from zero temperature, where the rate depends significantly on the detailed spectral behaviour, via the crossover region, up t…
Ballistic-to-Localized Dynamics as Signature of Quantum Phase Transition in Josephson Junction
Università di Napoli Federico II（那不勒斯费德里科二世大学）；INFN, Sezione di Napoli（意大利国家核物理研究所那不勒斯分部）；SPIN-CNR（凝聚态物理特殊相互作用中心研究所）；The Abdus Salam International Center for Theoretical Physics (ICTP)（阿卜杜斯·萨拉姆国际理论物理中心）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所涌现物质科学中心）；Fundamental Quantum Science Program (FQSP), TRIP Headquarters, RIKEN（理化学研究所量子科学计划总部）
Using a real-time renormalization group method we determine the complete dynamics of the spin-boson model with ohmic dissipation for coupling strengths $α\lesssim 0.1-0.2$. We calculate the relaxation and dephasing time, the static susceptibility and correlation functions. Our results are consistent with quantum Monte Carlo simulations and the Shiba relation. We present for the first time reliable results for finite cutoff and finite bias in a regime where perturbation theory in $α$ or in tunneling breaks down. Furthermore, an unambigious comparism to results from the Kondo model is achieved.
We study the transfer of heat in the non-equilibrium spin-boson model with an Ohmic dissipation. In the non-adiabatic limit we derive a formula for the thermal conductance based on a rate equation formalism at the level of the non-interacting blip approximation, valid for temperatures $T>T_K$, with $T_K$ as the Kondo temperature. We evaluate this expression analytically assuming either weak or strong couplings, and demonstrate that our results agree with exact relations. Far-from-equilibrium situations are further examined, showing a close correspondence to the linear response limit.
The role of complex potentials in single-body Schrődinger equation has been studied intensively. We study the quantum coherence for degenerate Bose gases in complex potentials, when the exchange symmetry of identical bosons is considered. For initially independent Bose-Einstein condensates, it is shown that even very weak imaginary potential can induce perfect quantum coherence between different condensates. The scheme to observe imaginary potential induced quantum coherence is discussed.
N. L. Dukhov Research Institute of Automatics (VNIIA)（N. L. 杜霍夫自动学研究所 (VNIIA)）；Vereshchagin Institute of High Pressure Physics, Russian Academy of Sciences（俄罗斯科学院韦列夏金高压物理研究所）；Kotelnikov Institute of Radioengineering and Electronics, Russian Academy of Sciences（俄罗斯科学院科捷利尼科夫无线电工程与电子研究所）；National Research University Higher School of Economics（国立研究型高等经济大学）
In this paper we give a general introduction to quantum critical phenomena, which we practically illustrate by a detailed study of the low energy properties of the spin boson model (SBM), describing the dynamics of a spin 1/2 impurity (or more generically a two-level system) coupled to a bath of independent harmonic oscillators. We show that the behavior of the model is very sensitive to the bath spectrum, in particular how the properties of the quantum critical point in the SBM are affected by the functional form of the bath Density of States (DoS). To this effect, we review the renormalizati…
We propose a nonequilibrium variational polaron transformation, based on an ansatz for nonequilibrium steady state (NESS) with an effective temperature, to study quantum heat transport at the nanoscale. By combining the variational polaron transformed master equation with the full counting statistics, we have extended the applicability of the polaron-based framework to study nonequilibrium process beyond the super-Ohmic bath models. Previously, the polaron-based framework for quantum heat transport reduces exactly to the non-interacting blip approximation (NIBA) formalism for Ohmic bath models…
Sachdev-Ye-Kitaev Model in a Quantum Glassy Landscape
JEIP, UAR 3573 CNRS, Collège de France, PSL Research University（巴黎高等师范学院联合物理研究所，法国国家科学研究中心，法兰西公学院，巴黎文理研究大学）；Laboratoire de Physique de l’École Normale Supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris（巴黎高等师范学院物理实验室，巴黎高等师范学院，巴黎文理研究大学，法国国家科学研究中心，索邦大学，巴黎大学）
We present a new variational method for investigating the ground state and out of equilibrium dynamics of quantum many-body bosonic and fermionic systems. Our approach is based on constructing variational wavefunctions which extend Gaussian states by including generalized canonical transformations between the fields. The key advantage of such states compared to simple Gaussian states is presence of non-factorizable correlations and the possibility of describing states with strong entanglement between particles. In contrast to the commonly used canonical transformations, such as the polaron or …
A new approach to dissipative quantum systems modelled by a system plus environment Hamiltonian is presented. Using a continuous sequence of infinitesimal unitary transformations the small quantum system is decoupled from its thermodynamically large environment. Dissipation enters through the observation that system observables generically decay completely into a different structure when the Hamiltonian is transformed into diagonal form. The method is particularly suited for studying low-temperature properties. This is demonstrated explicitly for the super-Ohmic spin-boson model.
Orthogonality Edges in Strong-Coupling Quantum Work Statistics
School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Institute of Fundamental Physics and Quantum Technology and School of Physical Science and Technology, Ningbo University（宁波大学基础物理与量子技术研究所及物理科学与技术学院）；Department of Physics, University of Sargodha（萨格达大学物理系）
Mean Field Games (MFG) provide a theoretical frame to model socio-economic systems. In this letter, we study a particular class of MFG which shows strong analogies with the {\em non-linear Schrödinger and Gross-Pitaevski equations} introduced in physics to describe a variety of physical phenomena ranging from deep-water waves to interacting bosons. Using this bridge many results and techniques developed along the years in the latter context can be transferred to the former. As an illustration, we study in some details an example in which the "players" in the mean field game are under a strong …
We suggest the existence of systems in which the statistics of a particle changes with the quantum level it occupies. The occupation numbers in thermal equilibrium depend on a continuous statistical parameter that interpolates between bosonic or fermionic and ghost-like statistical distributions. We call such particle states ``ninions'': they are different from anyons and can exist in 3+1 dimensions. We suggest that ninions can be associated with coherent angular momentum states. In the Euclidean imaginary-time formalism, the ninionic statistics can be implemented via the rotwisted boundary co…
In the spin-boson model, the properties of the oscillator bath are fully characterized by the spectral density of oscillators $J(ω)$. We study the case when this function is of Breit-Wigner shape and has a sharp peak at a frequency $Ω$ with width $Γ\llΩ$. We use a number of approaches such as the weak-coupling Bloch-Redfield equation, the non-interacting blip approximation (NIBA) and the flow-equation renormalization scheme. We show, that if $Ω$ is much larger than the qubit energy scales, the dynamics corresponds to an Ohmic spin-boson model with a strongly reduced tunnel splitting. We also s…
We investigate the critical behavior of a spin chain coupled to bosonic baths characterized by a spectral density proportional to $ω^s$, with $s>1$. Varying $s$ changes the effective dimension $d_\text{eff} = d + z$ of the system, where $z$ is the dynamical critical exponent and the number of spatial dimensions $d$ is set to one. We consider two extreme cases of clock models, namely Ising-like and U(1)-symmetric ones, and find the critical exponents using Monte Carlo methods. The dynamical critical exponent and the anomalous scaling dimension $η$ are independent of the order parameter symmetry…
