Optimization of the derivative expansion in the nonperturbative renormalization group

Canet, Léonie; Delamotte, Bertrand; Mouhanna, D.; Vidal, Julien

doi:10.1103/physrevd.67.065004

Cited by 195 publications

(305 citation statements)

References 49 publications

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“…Eq. (5.9) in its integral form, only allowing global changes along the full flow trajectory, is related to the principle of minimum sensitivity (PMS) [59], which has been introduced to the functional RG in [60], for further applications see [61][62][63]. Its limitations have been discussed in [66].…”

Section: B Principle Of Minimum Sensitivitymentioning

confidence: 99%

“…This is intimately linked to numerical stability and the convergence of results towards physics as already mentioned in the context of RG rescalings in the last section. By now a large number of conceptual advances have been accumulated [60][61][62][63][64][65][66][67][68][69][70][71], and are detailed in sections V B, V C. In particular [64] offers a structural approach towards optimisation which allows for a construction of optimised regulators within general truncation schemes. Still a fully satisfactory set-up requires further work.…”

Section: Optimisationmentioning

confidence: 99%

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Aspects of the functional renormalisation group

2007

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We discuss structural aspects of the functional renormalisation group. Flows for a general class of correlation functions are derived, and it is shown how symmetry relations of the underlying theory are lifted to the regularised theory. A simple equation for the flow of these relations is provided. The setting includes general flows in the presence of composite operators and their relation to standard flows, an important example being N PI quantities. We discuss optimisation and derive a functional optimisation criterion.Applications deal with the interrelation between functional flows and the quantum equations of motion, general Dyson-Schwinger equations. We discuss the combined use of these functional equations as well as outlining the construction of practical renormalisation schemes, also valid in the presence of composite operators. Furthermore, the formalism is used to derive various representations of modified symmetry relations in gauge theories, as well as to discuss gauge-invariant flows. We close with the construction and analysis of truncation schemes in view of practical optimisation.

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Section: B Principle Of Minimum Sensitivitymentioning

confidence: 99%

Section: Optimisationmentioning

confidence: 99%

Aspects of the functional renormalisation group

2007

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“…This idea has been turned into an efficient computational tool during the last ten years, mainly by Ellwanger [35,36,37,38,39], Morris [40,41] and Wetterich [42,43,44,45]. It has allowed to determine the critical exponents of the O(N ) models with high precision without having recourse to resummation techniques [45,46,47,48,49,50]. It has also allowed, for the first time [51], to relate, for any N , the results of the O(N )/O(N − 1) model obtained near d = 4 and d = 2, a fact of major importance for our purpose.…”

Section: Introductionmentioning

confidence: 99%

Nonperturbative renormalization-group approach to frustrated magnets

2004

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This article is devoted to the study of the critical properties of classical XY and Heisenberg frustrated magnets in three dimensions. We first analyze the experimental and numerical situations. We show that the unusual behaviors encountered in these systems, typically nonuniversal scaling, are hardly compatible with the hypothesis of a second order phase transition. Moreover, the fact that the scaling laws are significantly violated and that the anomalous dimension is negative in many cases provides strong indications that the transitions in frustrated magnets are most probably of very weak first order. We then review the various perturbative and early nonperturbative approaches used to investigate these systems. We argue that none of them provides a completely satisfactory description of the three-dimensional critical behavior. We then recall the principles of the nonperturbative approach -the effective average action method -that we have used to investigate the physics of frustrated magnets. First, we recall the treatment of the unfrustrated -O(N ) -case with this method. This allows to introduce its technical aspects. Then, we show how this method unables to clarify most of the problems encountered in the previous theoretical descriptions of frustrated magnets. Firstly, we get an explanation of the long-standing mismatch between different perturbative approaches which consists in a nonperturbative mechanism of annihilation of fixed points between two and three dimensions. Secondly, we get a coherent picture of the physics of frustrated magnets in qualitative and (semi-) quantitative agreement with the numerical and experimental results. The central feature that emerges from our approach is the existence of scaling behaviors without fixed or pseudo-fixed point and that relies on a slowing-down of the renormalization group flow in a whole region in the coupling constants space. This phenomenon allows to explain the occurence of generic weak first order behaviors and to understand the absence of universality in the critical behavior of frustrated magnets.

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“…We have therefore considered a one-parameter family of functions R k and have computed the critical exponents for all elements of this family. The function R k that is selected as optimal is that for which the exponents are stationary with respect to a change of R k (this is an implementation of the Principle of Minimal Sensitivity) [35]. At each order of the derivative expansion, this leads to a set of optimal exponents.…”

Section: Some Results Obtained With the Nprg Methodsmentioning

confidence: 99%

“…Second, the choice of a cut-off function R k that, in principle, has no effect since R k (q 2 ) vanishes identically in the limit k → 0, does matter once truncations are performed. Many studies have been devoted to finding an optimal choice [33][34][35][36][37]. None of them gives a complete solution to this problem.…”

Section: The Equilibrium Casementioning

confidence: 99%

What can be learnt from the nonperturbative renormalization group?

Delamotte¹,

Canet²

2005

Condens. Matter Phys.

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We point out some limits of the perturbative renormalization group used in statistical mechanics both at and out of equilibrium. We argue that the non-perturbative renormalization group formalism is a promising candidate to overcome some of them. We present some results recently obtained in the literature that substantiate our claims. We finally list some open issues for which this formalism could be useful and also review some of its drawbacks.

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Optimization of the derivative expansion in the nonperturbative renormalization group

Cited by 195 publications

References 49 publications

Aspects of the functional renormalisation group

Aspects of the functional renormalisation group

Nonperturbative renormalization-group approach to frustrated magnets

What can be learnt from the nonperturbative renormalization group?

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