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, 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.
The spin-boson model with quadratic coupling is studied using the bosonic numerical renormalization group method. We focus on the dynamical auto-correlation functions C O ( ω ) , with the operator O ^ taken as σ ^ x , σ ^ z , and X ^ , respectively. In the weak-coupling regime α < α c , these functions show power law ω -dependence in the small frequency limit, with the powers 1+2 s , 1+2 s , and s , respectively. At the critical point α = α c of the boson-unstable quantum phase transition, the critical exponents y O of these correlation functions are obtained as y σ x = y σ z = 1 − 2 s and y X = − s , respectively. Here s is the bath index and X is the boson displacement operator. Close to the spin flip point, the high frequency peak of C σ x ( ω ) is broadened significantly and the line shape changes qualitatively, showing enhanced dephasing at the spin flip point.
Spin-boson models are simple examples of quantum dissipative systems, but also serve as effective models in quantum magnetism and exhibit nontrivial criticality. Recently, they have been established as a platform to study the nontrivial renormalization-group (RG) scenario of fixed-point annihilation, in which two intermediate-coupling RG fixed points collide and generate an extremely slow RG flow near the collision. For the Bose Kondo model, a single
S=1/2 spin where each spin component couples to an independent bosonic bath with power-law spectrum
∝ωs via dissipation strengths
αi,
i∈{x,y,z}, such phenomena occur sequentially for the U(1)-symmetric model at
αz=0 and the SU(2)-symmetric case at
αz=αxy, as the bath exponent
s<1 is tuned. Here we use an exact wormhole quantum Monte Carlo method for retarded interactions to explore how this nontrivial fixed-point structure affects the phase diagram and phase transitions of the anisotropic model. In particular, we show how fixed-point annihilation within a symmetry-enhanced critical manifold leads to (i) a continuous order-to-order transition beyond the Landau paradigm, (ii) a symmetry-enhanced first-order transition, and (iii) pseudocriticality, which can be tuned into each other via the bath exponent
s. We extract critical exponents at the continuous transition, but also find scaling behavior at the symmetry-enhanced first-order transition, for which the inverse correlation-length exponent is given by the bath exponent
s. Moreover, we provide direct numerical evidence for pseudocritical scaling on both sides of the fixed-point collision, which manifests in an extremely slow drift of the correlation-length exponent. We also study the crossover away from the SU(2)-symmetric case and determine the phase boundary of an extended U(1)-symmetric critical phase.
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<1/2.
Using the recently developed exact numerical renormalization group (NRG) method, we analyse the NRG truncation errors
δχ of the local magnetic susceptibility and
δF of the free energy for the spin-boson model (SBM). We find that for temperatures higher than a crossover temperature
Tcr, as the number of kept states
M increases, both errors have oscillations with quasi period
lnM/lnNb and the envelopes decrease as
εtr=Λ−lnM/lnNb (
Nb is the number of boson states used for each bath site). For
T≪Tcr, they decrease slower than the power law. We extract that
Tcr=T∗εtr, with
T∗ being the crossover energy scale between the declocalized and the critical fixed points of SBM. The same rule applies to
δχ and
δF 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.
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.