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arXiv:cond-mat/0306708 · Physical Review Letters

Numerical Renormalization Group for Bosonic Systems and Application to the Sub-Ohmic Spin-Boson Model

R. Bulla, Ning-Hua Tong, Matthias Vojta C

Published Oct 24, 2003• 265 citations
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Abstract

We describe the generalization of Wilson's numerical renormalization group method to quantum impurity models with a bosonic bath, providing a general nonperturbative approach to bosonic impurity models which can access exponentially small energies and temperatures. As an application, we consider the spin-boson model, describing a two-level system coupled to a bosonic bath with power-law spectral density, J(omega) proportional to omega(s). We find clear evidence for a line of continuous quantum phase transitions for sub-Ohmic bath exponents 0<s<1; the line terminates in the well-known Kosterlitz-Thouless transition at s=1. Contact is made with results from perturbative renormalization group, and various other applications are outlined.

DOI
10.1103/physrevlett.91.170601

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We describe the generalization of Wilson's numerical renormalization group method to quantum impurity models with a bosonic bath, providing a general nonperturbative approach to bosonic impurity models which can access exponentially small energies and temperatures. As an application, we consider the spin-boson model, describing a two-level system coupled to a bosonic bath with power-law spectral density, J(omega) proportional to omega(s). We find clear evidence for a line of continuous quantum phase transitions for sub-Ohmic bath exponents 0<s<1; the line terminates in the well-known Kosterlitz-Thouless transition at s=1. Contact is made with results from perturbative renormalization group, and various other applications are outlined.
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 with respect to truncation parameters. The method is shown to provide high-accuracy data over the whole range of model parameters and temperatures, which are in agreement with exact results and other numerical data from the literature.
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<1/2s<1/2s<1/2.
Employing the nonperturbative 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 a spectral density J(omega) proportional to omega(s). We show that, in contrast with 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 nontrivial 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 crucial relevance, e.g., for qubits subject to electromagnetic noise.
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.
The spin-boson model has nontrivial quantum phase transitions in the sub-Ohmic regime. For the bath spectra exponent $0\ensuremath{\leqslant}s<1/2$, 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 Nb{N}_{b}Nb​, 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 $0\ensuremath{\leqslant}s<1/2$ 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.