Convergence of Nonperturbative Approximations to the Renormalization Group

Balog, Ivan; Chaté, Hugues; Delamotte, Bertrand; Marohnić, Maroje; Wschebor, Nicolás

doi:10.1103/physrevlett.123.240604

Cited by 110 publications

(182 citation statements)

References 35 publications

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“…The radius of convergence of this expansion depends on the model considered and on the regulating function R k . However, in models described by Ginzburg-Landau Hamiltonians whose analytical continuation to the Minkowskian space gives unitary models, the radius has been shown to be of the order q 2 radius /k 2 4-9 [19] once an appropriate regulator is chosen with very specific a priori criteria. On top of the previous specifications, one needs to fix the scale associated with the normalization of the fields in such a way that all correlation functions at momenta q 2 /k 2 4-9 behave as in the massive theory.…”

Section: B the Derivative Expansionmentioning

confidence: 99%

“…The quality of most DE results is further improved at low orders because of an independent reason, as explained in Ref. [19]. Consider the 2-point function near the fixed point and define:…”

Section: B the Derivative Expansionmentioning

confidence: 99%

“…This is consistent with the fact that the DE shows a rapid apparent convergence at low orders for O(N ) models. In fact, the DE has been pushed with success to orders O(∂ 4 ) [26] and O(∂ 6 ) [19] for the Ising universality class, giving excellent results that improve significantly with the order of the DE. Below, it will be shown that the quality of the results extends to all O(N ) models at order O(∂ 4 ).…”

Section: B the Derivative Expansionmentioning

confidence: 99%

“…Naively, one could think that the analysis of Ref. [19] does not apply (at least in the low temperature phase) because of the existence of these zero-mass modes. However, the expansion must be done by including the full propagator that includes the regulating function.…”

Section: B the Derivative Expansionmentioning

confidence: 99%

“…• The order O(∂ 6 ) has only been analyzed in the Z 2 universality class [19]. In that case, the ansatz for Γ k [φ] reads:…”

Section: B the Derivative Expansionmentioning

confidence: 99%

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Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group

et al. 2020

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We compute the critical exponents ν, η and ω of O(N ) models for various values of N by implementing the derivative expansion of the nonperturbative renormalization group up to next-to-nextto-leading order [usually denoted O(∂ 4 )]. We analyze the behavior of this approximation scheme at successive orders and observe an apparent convergence with a small parameter -typically between 1/9 and 1/4 -compatible with previous studies in the Ising case. This allows us to give well-grounded error bars. We obtain a determination of critical exponents with a precision which is similar or better than those obtained by most field theoretical techniques. We also reach a better precision than Monte-Carlo simulations in some physically relevant situations. In the O(2) case, where there is a longstanding controversy between Monte-Carlo estimates and experiments for the specific heat exponent α, our results are compatible with those of Monte-Carlo but clearly exclude experimental values. arXiv:2001.07525v1 [cond-mat.stat-mech]

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Section: B the Derivative Expansionmentioning

confidence: 99%

Section: B the Derivative Expansionmentioning

confidence: 99%

Section: B the Derivative Expansionmentioning

confidence: 99%

Section: B the Derivative Expansionmentioning

confidence: 99%

“…• The order O(∂ 6 ) has only been analyzed in the Z 2 universality class [19]. In that case, the ansatz for Γ k [φ] reads:…”

Section: B the Derivative Expansionmentioning

confidence: 99%

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Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group

et al. 2020

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Gentle introduction to rigorous Renormalization Group: a worked fermionic example

Giuliani

Mastropietro

Rychkov

2021

J. High Energ. Phys.

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Much of our understanding of critical phenomena is based on the notion of Renormalization Group (RG), but the actual determination of its fixed points is usually based on approximations and truncations, and predictions of physical quantities are often of limited accuracy. The RG fixed points can be however given a fully rigorous and non- perturbative characterization, and this is what is presented here in a model of symplectic fermions with a nonlocal (“long-range”) kinetic term depending on a parameter ε and a quartic interaction. We identify the Banach space of interactions, which the fixed point belongs to, and we determine it via a convergent approximation scheme. The Banach space is not limited to relevant interactions, but it contains all possible irrelevant terms with short-ranged kernels, decaying like a stretched exponential at large distances. As the model shares a number of features in common with ϕ4 or Ising models, the result can be used as a benchmark to test the validity of truncations and approximations in RG studies. The analysis is based on results coming from Constructive RG to which we provide a tutorial and self-contained introduction. In addition, we prove that the fixed point is analytic in ε, a somewhat surprising fact relying on the fermionic nature of the problem.

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Exploring the θ-vacuum structure in the functional renormalization group approach

Fukushima

Shimazaki

Tanizaki

2022

J. High Energ. Phys.

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We investigate the θ-vacuum structure and the ’t Hooft anomaly at θ = π in a simple quantum mechanical system on S1 to scrutinize the applicability of the functional renormalization group (fRG) approach. Even though the fRG is an exact formulation, a naive application of the fRG equation would miss contributions from the θ term due to the differential nature of the formulation. We first review this quantum mechanical system on S1 that is solvable with both the path integral and the canonical quantization. We discuss how to construct the quantum effective action including the θ dependence. Such an explicit calculation poses a subtle question of whether a Legendre transform is well defined or not for general systems with the sign problem. We then consider a deformed theory to relax the integral winding by introducing a wine-bottle potential with the finite depth ∝ g, so that the original S1 theory is recovered in the g → ∞ limit. We numerically solve the energy spectrum in the deformed theory as a function of g and θ in the canonical quantization. We test the efficacy of the simplest local potential approximation (LPA) in the fRG approach and find that the correct behavior of the ground state energy is well reproduced for small θ. When the energy level crossing is approached, the LPA flow breaks down and fails in describing the ground state degeneracy expected from the ’t Hooft anomaly. We finally turn back to the original theory and discuss an alternative formulation using the Villain lattice action. The analysis with the Villain lattice at θ = π indicates that the nonlocality of the effective action is crucial to capture the level crossing behavior of the ground states.

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Convergence of Nonperturbative Approximations to the Renormalization Group

Cited by 110 publications

References 35 publications

Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group

Precision calculation of critical exponents in the O(N) universality classes with the nonperturbative renormalization group

Gentle introduction to rigorous Renormalization Group: a worked fermionic example

Exploring the θ-vacuum structure in the functional renormalization group approach

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