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arXiv:cond-mat/0407559 · Physical Review B

Numerical renormalization group for quantum impurities in a bosonic bath

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

Published Jan 25, 2005• 174 citations
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Abstract

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.

DOI
10.1103/physrevb.71.045122

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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.
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.
This thesis is concerned with open quantum systems, and more specifically, quantum impurity models. By this we mean a small local quantum system in contact with a macroscopic non-interacting environment. This can be used to model individual impurities in metals and quantum information systems where the influence from the surrounding environment is not negligible. The numerical renormalization group (NRG) is the traditional method to study quantum impurity models. However its application is limited when dealing with real-time dynamics and bosonic systems. In recent years some of NRG's techniques have been introduced to the density matrix renormalization group method (DMRG), which itself is the most powerful numerical method to study one-dimensional quantum systems. The resulting method shows great potential, and this thesis explores and extends the power of the NRG+DMRG combination in treating open quantum systems. We focus mainly on two types of problem. The first is an open quantum system with a time-dependent Hamiltonian, which for example could be the theoretical description of various problems encountered in qubit manipulation. We combine NRG discretization and adaptive time-dependent DMRG (t-DMRG) to study the dissipative Landau-Zener problem. We also use this method to study the quantum decoherence process and the dynamical properties of the telegraph noise model. The results show that the NRG and t-DMRG combination is a fast, accurate and versatile method for such problems. The second type of problem we study involves the quantum critical properties of one- and two-bath spin-boson models. Unlike fermion and spin models, models with bosons are difficult to treat numerically as each boson basis has an infinite number of dimensions. By introducing the optimal boson basis into the variational matrix product state method (VMPS), which is a variant of DMRG, we can deal with an effective local boson basis as large as 101010^{10}1010. This is the crucial improvement over NRG which can only deal with at most a few dozen local basis states. With this powerful tool we have settled a controversy about the nature of the quantum phase transition in a sub-Ohmic spin-boson model regarding quantum to classical mapping. There, NRG fails to yield the right result due to the highly truncated local boson basis. We also explore the phase diagram of the two-bath spin-boson model and find a new critical phase. We demonstrate that NRG+VMPS with optimal boson basis represents a powerful setting to study quantum impurity models with a bosonic environment.
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.
We propose a density matrix renormalization group approach to tackle a two-state system coupled to a bosonic bath with continuous spectrum. In this approach, the optimized phonon scheme is applied to several hundred phonon modes that are linearly divided among the spectra. Although density matrix renormalization group cannot resolve very small energy scales, the delocalized-localized transition points of the two-state system are extracted by the extrapolation of the flow diagram results. The phase diagram is compared to the numerical renormalization group results and shows a good agreement in both Ohmic and sub-Ohmic cases.
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.
We propose a general extended coherent state approach to the qubit (or fermion) and multimode 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<1/2s<1/2s<1/2 can be correctly given by the fidelity susceptibility, demonstrating the strength of the approach.