The perturbative renormalization group (RG) equation is applied to resum a divergent series of perturbative wave functions of a quantum anharmonic oscillator. It is found that the resummed series gives the cumulant of the naive perturbation series. It is shown that a reorganization of the resummed series reproduce the correct asymptotic form of the wave function at x → ∞ when the perturbation expansion is stopped at the fourth order. A brief comment is given on the relation between the present method and the delta-expansion method, which is based on a kind of nonperturbative RG equation.The perturbative renormalization group (RG) equation is applied to resum a divergent series of perturbative wave functions of a quantum anharmonic oscillator. It is found that the resummed series gives the cumulant of the naive perturbation series. It is shown that a reorganization of the resummed series reproduce the correct asymptotic form of the wave function at x → ∞ when the perturbation expansion is stopped at the fourth order. A brief comment is given on the relation between the present method and the delta-expansion method, which is based on a kind of nonperturbative RG equation.

arXiv:hep-th/9801196Progress of Theoretical Physics SupplementJan 1, 199817 citations

Generalization on renormalization group

อารีฟ บุญเสือ

Mar 25, 20260 citations

Reply to comment on ‘Real-space renormalization-group methods for hierarchical spin glasses’

Michele Castellana

Abstract In their comment, Angelini et al object to the conclusion of (2019 J. Phys. A: Math. Theor. 52 445002), where we show that in (2013 Phys. Rev. B 87 134201) the exponent ν has been obtained by applying a mathematical relation in a regime where this relation is not valid. We observe that the criticism above on the mathematical validity of such relation has not been addressed in the comment. Our criticism thus remains valid, and disproves the conclusions of the comment. This constitutes the main point of this reply. In addition, we provide a point-by-point response and discussion of Angelini et al ’s claims. First, Angelini et al claim that the prediction 2 1/ ν = 1 of (2019 J. Phys. A: Math. Theor . 52 445002) is incorrect, because it results from the relation λ max = 2 1/ ν between the largest eigenvalue of the linearized renormalization-group (RG) transformation and ν , which cannot be applied to the ensemble renormalization group (ERG) method, given that for the ERG λ max = 1. However, the feature λ max = 1 is specific to the ERG transformation, and it does not give any grounds for questioning the validity of the general relation λ max = 2 1/ ν specifically for the ERG transformation. Second, Angelini et al claim that ν should be extracted from an early RG regime (A), as opposed to the asymptotic regime (B) used to estimate ν in (2019 J. Phys. A: Math. Theor . 52 445002), and that (B) is dominated by finite-size effects. However, (A) is a small-wavelength, non-critical regime, which cannot characterize the critical exponent ν related to the divergence of the correlation length. Also, the fact that (B) involves finite-size effects is a feature specific to the ERG, and gives no rationale for extracting ν from (A). Finally, we refute the remaining claims made by Angelini et al As a result of our analysis, we stand by our assertion that the ERG method yields a prediction given by 2 1/ ν = 1.Abstract In their comment, Angelini et al object to the conclusion of (2019 J. Phys. A: Math. Theor. 52 445002), where we show that in (2013 Phys. Rev. B 87 134201) the exponent ν has been obtained by applying a mathematical relation in a regime where this relation is not valid. We observe that the criticism above on the mathematical validity of such relation has not been addressed in the comment. Our criticism thus remains valid, and disproves the conclusions of the comment. This constitutes the main point of this reply. In addition, we provide a point-by-point response and discussion of Angelini et al ’s claims. First, Angelini et al claim that the prediction 2 1/ ν = 1 of (2019 J. Phys. A: Math. Theor . 52 445002) is incorrect, because it results from the relation λ max = 2 1/ ν between the largest eigenvalue of the linearized renormalization-group (RG) transformation and ν , which cannot be applied to the ensemble renormalization group (ERG) method, given that for the ERG λ max = 1. However, the feature λ max = 1 is specific to the ERG transformation, and it does not give any grounds for questioning the validity of the general relation λ max = 2 1/ ν specifically for the ERG transformation. Second, Angelini et al claim that ν should be extracted from an early RG regime (A), as opposed to the asymptotic regime (B) used to estimate ν in (2019 J. Phys. A: Math. Theor . 52 445002), and that (B) is dominated by finite-size effects. However, (A) is a small-wavelength, non-critical regime, which cannot characterize the critical exponent ν related to the divergence of the correlation length. Also, the fact that (B) involves finite-size effects is a feature specific to the ERG, and gives no rationale for extracting ν from (A). Finally, we refute the remaining claims made by Angelini et al As a result of our analysis, we stand by our assertion that the ERG method yields a prediction given by 2 1/ ν = 1.

arXiv:2102.06263Journal of Physics A Mathematical and TheoreticalApr 3, 20200 citations

The Renormalization Group: Fundamental Aspects

Lothar Schäfer

Springer eBooksJan 1, 19990 citations

Comment on ‘Series expansions from the corner transfer matrix renormalization group method: the hard-squares model’

Iwan Jensen

Earlier this year Chan extended the low-density series for the hard-squares partition function $κ(z)$ to 92 terms. Here we analyse this extended series focusing on the behaviour at the dominant singularity $z_d$ which lies on on the negative fugacity axis. We find that the series has a confluent singularity of order 2 at $z_d$ with exponents $θ=0.83333(2)$ and $θ'= 1.6676(3)$ . We thus confirm that the exponent $θ$ has the exact value $\frac56$ as observed by Dhar.Earlier this year Chan extended the low-density series for the hard-squares partition function $κ(z)$ to 92 terms. Here we analyse this extended series focusing on the behaviour at the dominant singularity $z_d$ which lies on on the negative fugacity axis. We find that the series has a confluent singularity of order 2 at $z_d$ with exponents $θ=0.83333(2)$ and $θ'= 1.6676(3)$. We thus confirm that the exponent $θ$ has the exact value $\frac56$ as observed by Dhar.

arXiv:1208.1060Journal of Physics A Mathematical and TheoreticalNov 28, 20129 citations

Earlier this year Chan extended the low-density series for the hard-squares partition function

κ(z)

to 92 terms. Here we analyse this extended series focusing on the behaviour at the dominant singularity

z_d

which lies on on the negative fugacity axis. We find that the series has a confluent singularity of order 2 at

z_d

with exponents

θ=0.83333(2)

and

θ'= 1.6676(3)

. We thus confirm that the exponent

θ

has the exact value

\frac56

as observed by Dhar.