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arXiv:0911.4490 · Physical Review B

Mass-flow error in the numerical renormalization-group method and the critical behavior of the sub-Ohmic spin-boson model

Matthias Vojta C, R. Bulla, Fabian Güttge, Frithjof B. Anders

Published Feb 25, 2010• 41 citations
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Abstract

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 in 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 in the phase transition's control parameter as function of the temperature and thus to an unphysical temperature dependence of the order-parameter mass. We demonstrate the mass-flow effect for bosonic impurity models with a power-law bath spectrum, $J(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}$, namely, the dissipative harmonic oscillator and the spin-boson model. We propose an extension of the NRG to correct the mass-flow error. Using this, we find unambiguous signatures of a Gaussian critical fixed point in the spin-boson model for $s<1/2$, consistent with mean-field behavior as expected from quantum-to-classical mapping.

DOI
10.1103/physrevb.81.075122

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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 in 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 in the phase transition's control parameter as function of the temperature and thus to an unphysical temperature dependence of the order-parameter mass. We demonstrate the mass-flow effect for bosonic impurity models with a power-law bath spectrum, J(\ensuremathω)\ensuremath∝\ensuremathωsJ(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}J(\ensuremathω)\ensuremath∝\ensuremathωs, namely, the dissipative harmonic oscillator and the spin-boson model. We propose an extension of the NRG to correct the mass-flow error. Using this, we find unambiguous signatures of a Gaussian critical fixed point in the spin-boson model for s<1/2s<1/2s<1/2, consistent with mean-field behavior as expected from quantum-to-classical mapping.
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.
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.
Using the recently developed exact numerical renormalization group (NRG) method, we analyse the NRG truncation errors δχδχδχ of the local magnetic susceptibility and δFδFδF of the free energy for the spin-boson model (SBM). We find that for temperatures higher than a crossover temperature TcrT_{cr}Tcr​, as the number of kept states MMM increases, both errors have oscillations with quasi period ln⁡M/ln⁡Nb\ln{M}/\ln{N_b}lnM/lnNb​ and the envelopes decrease as εtr=Λ−ln⁡M/ln⁡Nbε_{tr}=Λ^{-\ln{M}/\ln{N_b}}εtr​=Λ−lnM/lnNb​ (NbN_bNb​ is the number of boson states used for each bath site). For T≪TcrT \ll T_{cr}T≪Tcr​, they decrease slower than the power law. We extract that Tcr=T∗εtrT_{cr} = T^{\ast} ε_{tr}Tcr​=T∗εtr​, with T∗T^{\ast}T∗ being the crossover energy scale between the declocalized and the critical fixed points of SBM. The same rule applies to δχδχδχ and δFδFδF calculated from the full density matrix NRG method and is expected to hold for general impurity models, allowing accurate removal of NRG truncation errors in static quantities at high temperatures.
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.
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 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 $\ensuremath{\omega}/T$ scaling of the local dynamical susceptibility, as is the leading term. The frequency and temperature dependences of the local spin susceptibility in the strong-coupling (delocalized) regime are also presented. We attribute the violation of the quantum-to-classical mapping to a Berry-phase term in a continuum path-integral representation of the model. This effect connects the behavior discussed here with its counterparts in models with continuous spin symmetry.