Exact renormalization group equations: an introductory review

Bagnuls, C.; Bervillier, C.

doi:10.1016/s0370-1573(00)00137-x

Cited by 439 publications

(638 citation statements)

References 147 publications

(543 reference statements)

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“…Refs. [13,14,15,16], as well as [17] for non-equilibrium applications. This exact and closed evolution equation allows for non-perturbative approximations that lead substantially beyond mean-field and quantum Boltzmann type approaches: it can be rewritten, in a closed form, as a hierarchy of dynamical equations for Green functions, and this hierarchy admits truncations that neither explicitly nor implicitly rely on small bare couplings or close-to-equilibrium evolutions.…”

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confidence: 99%

Towards far-from-equilibrium quantum field dynamics: A functional renormalisation-group approach

2008

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Dynamic equations for quantum fields far from equilibrium are derived by use of functional renormalisation group techniques. The obtained equations are non-perturbative and lead substantially beyond mean-field and quantum Boltzmann type approximations. The approach is based on a regularised version of the generating functional for correlation functions where times greater than a chosen cutoff time are suppressed. As a central result, a time evolution equation for the non-equilibrium effective action is derived, and the time evolution of the Green functions is computed within a vertex expansion. It is shown that this agrees with the dynamics derived from the 1/N -expansion of the two-particle irreducible effective action.PACS numbers: 03.75. Kk, 05.10.Cc, 05.70.Ln, Introduction. Far-from-equilibrium quantum field dynamics is one of the most challenging issues both in experimental and theoretical physics to date. Experiments exhibiting quantum statistical effects in the time evolution of many-body systems are extremely demanding. In particular, the preparation of ultracold atomic Bose and Fermi gases in various trapping environments allows to precisely study quantum many-body dynamics of strongly correlated systems, see, e.g., Refs.[1]. In recent years, the field has attracted researchers from a variety of disciplines, ranging from condensed-matter to highenergy particle physics and cosmology. Nonequilibrium field theory to date is dominated by semi-classical mean-field approaches which are in general only valid for weak interactions or large occupation numbers. For strongly correlated quantum systems methods are available predominantly for systems in one spatial dimension and include the Density Matrix Renormalisation Group methods, see e.g. [2], as well as techniques for exactly solvable models [3]. Non-perturbative approximations of the two-particle irreducible (2PI) effective action [4] have been intensively studied and applied to nonequilibrium dynamics [5,6,7,8,9,10,11] and are applicable also in more than one dimension. In field theory, they also provide a way to study strongly correlated fermions beyond mean-field and Boltzmann approximations [8]. Like these, the results presented here are expected to be of high relevance, e.g., for the description of ultracold degenerate Fermi gases close to the BEC-BCS crossover [12].In this paper we derive dynamic equations for quantum fields far from equilibrium. As a central result, we derive a new time evolution equation for the non-equilibrium effective action by use of functional renormalisation group (RG) techniques, cf. Refs. [13,14,15,16], as well as [17] for non-equilibrium applications. This exact and closed evolution equation allows for non-perturbative approximations that lead substantially beyond mean-field and quantum Boltzmann type approaches: it can be rewritten, in a closed form, as a hierarchy of dynamical equations for Green functions, and this hierarchy admits truncations that neither explicitly nor implicitly rely on small bare couplings or close-to...

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confidence: 99%

Towards far-from-equilibrium quantum field dynamics: A functional renormalisation-group approach

2008

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“…We do not intend to detail here the implementation of the NPRG, but rather set out its principle (and refer to [5,4] for reviews). This formalism relies on Wilson's RG idea [13], which consists in building a sequence of scale-dependent effective models, that interpolate smoothly between the short-distance physics at the (microscopic) scale k = Λ and the long-distance physics at the scale k = 0, through progressively averaging over fluctuations.…”

Section: Non-perturbative Renormalisation Group Out Of Equilibriummentioning

confidence: 99%

“…On the other hand, the efficiency of standard RG approaches is hindered by the fact that first, critical dimensions, when they are identified, happen to lie far from the physically interesting ones and furthermore, no calculations are available above two-loop order, which prevent from resorting to powerful re-summation techniques. In this context, any theoretical tool that can overcome the previous pitfalls is valuable and the nonperturbative renormalisation group (NPRG) stands as a promising candidate [4]. Indeed, this method has lead to great successes for systems at equilibrium during the last decade, appearing as a well-adapted tool to tackle strong coupling problems [4,5].…”

Section: Introductionmentioning

confidence: 99%

Reaction–diffusion processes and non-perturbative renormalization group

Canet¹

2006

J. Phys. A: Math. Gen.

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Abstract. This paper is devoted to investigating non-equilibrium phase transitions to an absorbing state, which are generically encountered in reactiondiffusion processes. It is a review, based on [1, 2, 3], of recent progress in this field that has been allowed by a non-perturbative renormalisation group approach. We mainly focus on branching and annihilating random walks and show that their critical properties strongly rely on non-perturbative features and that hence the use of a non-perturbative method turns out to be crucial to get a correct picture of the physics of these models.

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“…One of the most popular non-perturbative approaches employs the exact RG equation (see [4,5,6]). The application of the non-perturbative methods is effective for investigating the Ising universal behaviour in high energy physics.…”

Section: Introductionmentioning

confidence: 99%

Free energy and equation of state of Ising-like magnet near the critical point

Kozlovskii¹,

Pylyuk²,

Prytula³

2006

Nuclear Physics B

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The description of a three-dimensional Ising-like magnet in the presence of an external field in the vicinity of the critical point by the collective variables method is proposed. Using the renormalization group transformations, the scaling region size is defined as a function of temperature and field. The obtained expressions for the free energy, equation of state and susceptibility allow one to analyse their dependence on microscopic parameters of the system. The critical exponents of the correlation length and order parameter are calculated as well. The results agree qualitatively with ones obtained within the framework of the parametric representation of the equation of state and Monte-Carlo simulations. The calculations do not involve any parametrization, phenomenological assumptions and adjustable parameters. The approach can be extended to models with a multicomponent order parameter.

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Exact renormalization group equations: an introductory review

Cited by 439 publications

References 147 publications

Towards far-from-equilibrium quantum field dynamics: A functional renormalisation-group approach

Towards far-from-equilibrium quantum field dynamics: A functional renormalisation-group approach

Reaction–diffusion processes and non-perturbative renormalization group

Free energy and equation of state of Ising-like magnet near the critical point

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