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arXiv:cond-mat/9601107 · Physical Review Letters

Equilibrium Dynamics of the Dissipative Two-State System

Theo A. Costi, C. Kieffer

Published Mar 4, 1996• 109 citations
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Abstract

Wilson's momentum shell renormalization group method is used to solve for the dynamics of the dissipative two-state system. We utilize the mapping of the spin-boson model onto the anisotropic Kondo model (AKM) and solve for the dynamics of the latter. We find that the AKM captures the physics of the dissipative two-state system for dissipation strength $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}4$ corresponding to $\ensuremath{\infty}\ensuremath{\ge}{J}_{\ensuremath{\Vert}}\ensuremath{\ge}\ensuremath{-}\ensuremath{\infty}$ in the AKM. The dynamics of the AKM shows a smooth crossover between two strong coupling regimes corresponding to weak and strong dissipation in the spin-boson model.

DOI
10.1103/physrevlett.76.1683

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Wilson's momentum shell renormalization group method is used to solve for the dynamics of the dissipative two-state system. We utilize the mapping of the spin-boson model onto the anisotropic Kondo model (AKM) and solve for the dynamics of the latter. We find that the AKM captures the physics of the dissipative two-state system for dissipation strength 0\ensuremath≤\ensuremathα\ensuremath≤40\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}40\ensuremath≤\ensuremathα\ensuremath≤4 corresponding to \ensuremath∞\ensuremath≥J\ensuremath∥\ensuremath≥\ensuremath−\ensuremath∞\ensuremath{\infty}\ensuremath{\ge}{J}_{\ensuremath{\Vert}}\ensuremath{\ge}\ensuremath{-}\ensuremath{\infty}\ensuremath∞\ensuremath≥J\ensuremath∥​\ensuremath≥\ensuremath−\ensuremath∞ in the AKM. The dynamics of the AKM shows a smooth crossover between two strong coupling regimes corresponding to weak and strong dissipation in the spin-boson model.
Using a real-time renormalization-group method we determine the complete dynamics of the spin-boson model with Ohmic dissipation for coupling strengths $\ensuremath{\alpha}\ensuremath{\lesssim}0.1--0.2.$ We calculate the relaxation and dephasing time, static susceptibility, and correlation functions. Our results are consistent with quantum Monte Carlo simulations and the Shiba relation. We present reliable results for finite cutoff and finite bias in a regime where perturbation theory in $\ensuremath{\alpha}$ or in tunneling breaks down. Furthermore, an unambiguous comparison to results from the Kondo model is achieved.
We extend the numerical renormalization-group method to treat Bose-Fermi Kondo models (BFKMs) describing a local moment coupled both to a conduction band and to a dissipative bosonic bath representing, e.g., lattice or spin collective excitations of the environment. We apply the method to the Ising-symmetry BFKM with a structureless band and a bath spectral function $\ensuremath{\eta}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}$. The method is valid for all bath exponents sss and all temperatures TTT. For 0<s<10<s<10<s<1, the range of interest in the context of heavy-fermion quantum criticality, an interacting critical point, characterized by hyperscaling of exponents and $\ensuremath{\omega}∕T$ scaling, describes a continuous quantum phase transition between Kondo-screened and localized phases. For Ohmic dissipation s=1s=1s=1, where the model is relevant to certain dissipative mesoscopic qubit devices, the transition is found to be Kosterlitz-Thouless-like. In both regimes the impurity spectral function for the corresponding Anderson model shows clearly the collapse of the Kondo resonance at the transition. Connection is made to other recent results for the BFKM and the spin-boson model.
We calculate the dynamical spin-spin correlation functions of a Kondo dot coupled to two noninteracting leads held at different chemical potentials. To this end we generalize a recently developed real-time renormalization group (RG) method in frequency space [H. Schoeller, Eur. Phys. J. Spec. Top. 168, 179 (2009)] to allow the calculation of dynamical correlation functions of arbitrary dot operators in systems describing spin and/or orbital fluctuations. The resulting two-loop RG equations are analytically solved in the weak-coupling regime. This implies that the method can be applied provided either the voltage VVV through the dot or the external magnetic field h0{h}_{0}h0​ are sufficiently large, $\mathrm{max}{V,{h}_{0}}\ensuremath{\gg}{T}_{K}$, where the Kondo temperature TK{T}_{K}TK​ is the scale where the system enters the strong-coupling regime. Explicitly, we calculate the longitudinal and transverse spin-spin correlation and response functions as well as the resulting fluctuation-dissipation ratios. The correlation functions in real-frequency space can be calculated in Matsubara space without the need of any analytical continuation. We obtain analytic results for the line shape, the small- and large-frequency limits, and several other features like the height and width of the peak in the transverse susceptibility at $\ensuremath{\Omega}\ensuremath{\approx}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{h}$, where $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{h}$ denotes the renormalized magnetic field. Furthermore, we discuss how the developed method can be generalized to calculate dynamical correlation functions of other operators involving reservoir degrees of freedom as well.
The thermodynamics of the dissipative two-state system is calculated exactly for all temperatures and level asymmetries for the case of Ohmic dissipation. We exploit the equivalence of the two-state system to the anisotropic Kondo model and extract the thermodynamics of the former by solving the thermodynamic Bethe-ansatz equations of the latter. The universal scaling functions for the specific heat ${C}_{\ensuremath{\alpha}}(T)$ and static dielectric susceptibility ${\ensuremath{\chi}}_{\ensuremath{\alpha}}(T)$ are extracted for all dissipation strengths $0<\ensuremath{\alpha}<1.$ The logarithmic corrections to these quantities at high temperatures are found in the Kondo limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\alpha}}{1}^{\ensuremath{-}},$ whereas for $\ensuremath{\alpha}<1$ we find the expected power law temperature dependences with the powers being functions of the dissipative coupling \ensuremath{\alpha}. The low-temperature behavior is always that of a Fermi liquid.
We provide an extensive study of the subohmic spin-boson model with power-law density of states $J(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}$ (with $0<s\ensuremath{\le}1$), 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 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.
We analyze the nonequilibrium Kondo model at finite voltage and temperature by using a new formulation of the real-time renormalization group method with the Laplace variable as the flow parameter. We evaluate the energy-dependent spin relaxation rate and nonlinear conductance, and derive an approximate form for the universal line shape for the latter in the whole crossover regime from weak to strong coupling. The results are shown to agree well with exact methods in equilibrium, Fermi-liquid theory, weak-coupling expansions, and recent experiments. For the transient spin dynamics we find a universal exponential decay in the long-time limit along with a truncation-dependent pre-exponential power law. For multichannel models a pure power-law decay typical for non-Fermi-liquid behavior is predicted.