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

Abstract

We describe the generalization of Wilson's Numerical Renormalization Group method to quantum impurity models with a bosonic bath, providing a general non-perturbative 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(ω) ∝ ω s . We find clear evidence for a line of continuous quantum phase transitions for subohmic bath exponents 0 < s < 1; the line terminates in the well-known KosterlitzThouless transition at s = 1. Contact is made with results from perturbative renormalization group, and various other applications are outlined.The Numerical Renormalization Group method (NRG) developed by Wilson [1] is a powerful tool for the investigation of the Kondo model and its generalizations [1,2,3,4]. In these models, a (possibly complex) impurity, such as a localized spin, couples to a fermionic bath. In the case of a spin-1 2 impurity coupled antiferromagnetically to a metallic bath, the impurity spin is screened below a characteristic scale T K , the Kondo temperature [3]. The strength of the NRG lies in its nonperturbative nature and the ability to resolve arbitrarily small energies [1]. A variety of thermodynamic and dynamic quantities can be calculated for a large number of impurity models in the whole parameter space [4,5].There is, however, a very important class of models for which the NRG method has not yet been developed: models with a coupling of the impurity to a bosonic bath [6]. The intensively studied spin-boson model [7,8] belongs to this class; its Hamiltonian is given byHere the Pauli-matrices σ j describe a spin, i.e., a generic two-level system, which is linearly coupled to a bath of harmonic oscillators, with creation (annihilation) operators a † i (a i ). The bare tunneling amplitude between the two spin states | ↑ and | ↓ is given by ∆, and ǫ is an additional bias. The ω i are the oscillator frequencies and λ i the coupling strengths between the oscillators and the local spin. The coupling between spin and bosonic bath is completely specified by the bath spectral functionOf particular interest are power-law spectrawhere the dimensionless parameter α characterizes the dissipation strength, and ω c is a cutoff energy. The value s = 1 corresponds to the case of ohmic dissipation.The spin-boson model is a generic model describing quantum dissipation; it has been discussed in the context of a great variety of physical problems [7,8] ranging from the effect of friction on the electron transfer in biomolecules [9] to the description of the quantum entanglement between a qubit and its environment [10,11,12].Considering the wealth of applications, the question arises whether Wilson's NRG method can be exploited for this class of models; and it is the purpose of this paper to show that this is indeed the case. What we have in mind here is the direct mapping of models like (1) to a semi-infinite chain form typical for ...

Bosons confined in optical lattices: The numerical renormalization group revisited

“…DMRG was a substantial improvement [25] on the older numerical renormalization group (NRG), which had only a poor reputation in dealing with long-range interactions between fermions [31,32], but we found it useful for bosonic systems (see also Ref. [33]). …”

Coherent-incoherent transition in the sub-Ohmic spin-boson model

“…This conclusion was based on the non-interacting-blip approximation 1,7 which fails in the weak-coupling limit to the bath. More recently, several works 5,8,9 addressed the problem of the sub-Ohmic spin-boson model and found that coherent phases can exist for sufficiently weak coupling.…”

Quantum criticality of the sub-Ohmic spin-boson model

“…In this paper, we investigate more thoroughly the temperature and frequency dependences of the local spin susceptibility of the spin-boson model, as calculated using the bosonic NRG approach. 11,12 The subleading term in the temperature dependence at ω = 0 is shown to be described by an exponent x 2 that depends on ǫ and exceeds 1 2 , contradicting a central assumption of Ref. 18.…”

“…13 assumed the the classical spin chain faithfully represents the spin-boson model. Correspondingly, it interpreted the Monte-Carlo result as demonstrating a fundamental error in the NRG results 9,11,12 for the sub-ohmic spin-boson model. It was further suggested in Refs.…”

“…In Ref. 9, it was shown using a bosonic extension 11,12 of the numerical renormalization-group (NRG) method that the critical exponents for 1 2 < ǫ < 1 obey hyperscaling (as they do for 0 < ǫ < 1 2 ). In particular, just as for the Bose-Fermi Kondo model, 4,5 the critical local susceptibility satisfies Eqs.…”