Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Differential formulation of the renormalisation group in the large-n limit

Gaetano Busiello, L. De Cesare, Ileana Rabuffo

Published Dec 1, 1981• 8 citations
Publisher

Abstract

A differential formulation of the renormalisation group in the large-n limit is proposed, in terms of which all the standard results can be reproduced in a more natural way. Some advantages of the formulation with respect to the traditional approach are pointed out.
DOI
10.1088/0305-4470/14/12/007

Related papers

Differential formulation of the dynamic renormalisation group in the large-n limit

Gaetano Busiello, L. De Cesare, Ileana Rabuffo

A differential formulation of the dynamic renormalisation group in the large-n limit is presented for the time-dependent generalisation of the n-vector classical model with purely relaxational dynamics for both conserved and non-conserved order parameter. A discussion of the dynamical properties with the same steps used in the traditional perturbative approaches is made in terms of which known and new explicit results are obtained. Furthermore, a mechanical analogy is introduced which may give new information about the structure of the dynamic renormalisation group in the large-n limit. This relies on the possibility of using the well known geometrical and mathematical techniques from the Hamilton-Jacobi theory for classical mechanics.

Journal of Physics A Mathematical and General•Jun 21, 1983•6 citations

The renormalisation group in the large-n limit for the vectorial paramagnon problem at T=O

Gaetano Busiello, L. De Cesare

Uses the extension of the Ma RG approach (1973) to quantum functionals in the limit n to infinity to obtain non-perturbative information for quantum systems at T=O, restricting oneself to the n-vector paramagnon problem.

Journal of Physics A Mathematical and General•Dec 1, 1980•5 citations

Renormalization-group approaches

Patrizia Castiglione, M. Falcioni, Annick Lesne +1 more

If we study the history of science we see produced two phenomena which are, so to speak, each the inverse of the other. Sometimes it is simplicity which is hidden under what is apparently complex; sometimes, on the contrary, it is simplicity which is apparent, and which conceals extremely complex realities. Henri Poincaré Devoting only a short chapter to the renormalization group (RG) is a challenge owing to the wealth of both fundamental concepts and computational tools associated with this term. The RG is indeed encountered in many different domains of theoretical physics, ranging from quantum electrodynamics to second-order phase transitions to fractal growth and diffusion processes – one should rather speak of renormalization group S ! We refer to Brown (1993) and Fisher (1998) for an historical account, to Goldenfeld (1992) and Lesne (1998) and references therein for an overview of the domains of application and variants of RGs. We here emphasize that the RG is a way, maybe the most successful and with no doubt the most systematic and constructive, to derive effective low-dimensional descriptions that capture large-scale and/or long-time behavior . The RG can be extended far beyond the specific scope of critical phenomena, to an iterated multiscale approach allowing the construction of robust and minimal macroscopic models describing the universal large-scale features and asymptotics of a complex system. This generalized viewpoint brings out the close logical and even technical connections that bridge, within a unified framework, perturbative RG for singular series expansions, spin-block RG and momentum-shell RG for critical phenomena, RG for the asymptotic analysis of differential and partial differential equations, and probabilistic RG for the derivation of statistical laws and limit theorems.

Cambridge University Press eBooks•Aug 21, 2008•0 citations

The renormalization group and the large n limit

Shang-keng Ma

The basic concepts and formulation of the renormalization group are explained beginning at an elementary level. Discussion is in the framework of classical statistical mechanics with emphasis on applications to the theory of critical phenomena. The details are worked out in the large n limit for 2 < d < 4, where n is the number of components of the fluctuating field of interest and d is the dimension of the thermodynamical system. In the large n limit, the infinite sum of ``tree graphs'' offers an exact and analytically tractable description of the renormalization group. It illustrates many concepts including the fixed point, the critical surface in the space of coupling parameters, and critical exponents. Most important, it illustrates the origin and the limitation of the scaling hypothesis. The critical behavior of various correlation functions and the free energy is examined. Attention is paid to terms often ignored in qualitative scaling arguments. We have attempted to make this paper self-contained and of pedagogical value to a wide audience.

Journal of Mathematical Physics•Nov 1, 1974•24 citations

Physics-informed renormalisation group flows

Friederike Ihssen, Pawlowski, Jan M.

The physics of strongly correlated systems offers some of the most intriguing physics challenges such as competing orders or the emergence of dynamical composite degrees of freedom. Often, the resolution of these physics challenges is computationally hard, but can be simplified enormously by a formulation in terms of the dynamical degrees of freedom and within an expansion about the physical ground state. Importantly, such a formulation does not only reduce or minimise the computational challenges, it also facilitates the access to the physics mechanisms at play. The tasks of finding the dynamical degrees of freedom and the physical ground state can be systematically addressed within the functional renormalisation group approach with flowing fields which accommodates both, emergent composites as well as the physical ground state. In the present work we use this approach to set up physics-informed renormalisation group flows (PIRG flows): Scale-dependent coordinate transformations in field space induce emergent composites, and the respective flows for the effective action generate a large set of target actions, formulated in these emergent composite fields. This novel perspective on RG flows bears a great potential both for conceptual as well as computational applications: to begin with, PIRG flows allow for a systematic search of the dynamical degrees of freedom and the respective ground state that leads to the most rapid convergence of expansion schemes, thus minimising the computational effort. Secondly, the resolution of the remaining computational tasks within a given expansion scheme can be further reduced by optimising the physics content within a given approximation. Thirdly, the maximal variability of PIRG flows can be used to reduce the analytic and numerical effort of solving the flows within a given approximation.

arXiv:2409.13679•Sep 20, 2024•2 citations

Hamiltonian Renormalization Groups

D. Schmeltzer

A Hamiltonian renormalization group is presented. Such a formulation is relevant for chiralic systems and more appropriate than the Lagrangian formalism. An application to 1D system is presented.

arXiv:cond-mat/0211449•Nov 20, 2002•0 citations
A differential formulation of the dynamic renormalisation group in the large-n limit is presented for the time-dependent generalisation of the n-vector classical model with purely relaxational dynamics for both conserved and non-conserved order parameter. A discussion of the dynamical properties with the same steps used in the traditional perturbative approaches is made in terms of which known and new explicit results are obtained. Furthermore, a mechanical analogy is introduced which may give new information about the structure of the dynamic renormalisation group in the large-n limit. This relies on the possibility of using the well known geometrical and mathematical techniques from the Hamilton-Jacobi theory for classical mechanics.
If we study the history of science we see produced two phenomena which are, so to speak, each the inverse of the other. Sometimes it is simplicity which is hidden under what is apparently complex; sometimes, on the contrary, it is simplicity which is apparent, and which conceals extremely complex realities. Henri Poincaré Devoting only a short chapter to the renormalization group (RG) is a challenge owing to the wealth of both fundamental concepts and computational tools associated with this term. The RG is indeed encountered in many different domains of theoretical physics, ranging from quantum electrodynamics to second-order phase transitions to fractal growth and diffusion processes – one should rather speak of renormalization group S ! We refer to Brown (1993) and Fisher (1998) for an historical account, to Goldenfeld (1992) and Lesne (1998) and references therein for an overview of the domains of application and variants of RGs. We here emphasize that the RG is a way, maybe the most successful and with no doubt the most systematic and constructive, to derive effective low-dimensional descriptions that capture large-scale and/or long-time behavior . The RG can be extended far beyond the specific scope of critical phenomena, to an iterated multiscale approach allowing the construction of robust and minimal macroscopic models describing the universal large-scale features and asymptotics of a complex system. This generalized viewpoint brings out the close logical and even technical connections that bridge, within a unified framework, perturbative RG for singular series expansions, spin-block RG and momentum-shell RG for critical phenomena, RG for the asymptotic analysis of differential and partial differential equations, and probabilistic RG for the derivation of statistical laws and limit theorems.
The basic concepts and formulation of the renormalization group are explained beginning at an elementary level. Discussion is in the framework of classical statistical mechanics with emphasis on applications to the theory of critical phenomena. The details are worked out in the large n limit for 2 < d < 4, where n is the number of components of the fluctuating field of interest and d is the dimension of the thermodynamical system. In the large n limit, the infinite sum of ``tree graphs'' offers an exact and analytically tractable description of the renormalization group. It illustrates many concepts including the fixed point, the critical surface in the space of coupling parameters, and critical exponents. Most important, it illustrates the origin and the limitation of the scaling hypothesis. The critical behavior of various correlation functions and the free energy is examined. Attention is paid to terms often ignored in qualitative scaling arguments. We have attempted to make this paper self-contained and of pedagogical value to a wide audience.
The physics of strongly correlated systems offers some of the most intriguing physics challenges such as competing orders or the emergence of dynamical composite degrees of freedom. Often, the resolution of these physics challenges is computationally hard, but can be simplified enormously by a formulation in terms of the dynamical degrees of freedom and within an expansion about the physical ground state. Importantly, such a formulation does not only reduce or minimise the computational challenges, it also facilitates the access to the physics mechanisms at play. The tasks of finding the dynamical degrees of freedom and the physical ground state can be systematically addressed within the functional renormalisation group approach with flowing fields which accommodates both, emergent composites as well as the physical ground state. In the present work we use this approach to set up physics-informed renormalisation group flows (PIRG flows): Scale-dependent coordinate transformations in field space induce emergent composites, and the respective flows for the effective action generate a large set of target actions, formulated in these emergent composite fields. This novel perspective on RG flows bears a great potential both for conceptual as well as computational applications: to begin with, PIRG flows allow for a systematic search of the dynamical degrees of freedom and the respective ground state that leads to the most rapid convergence of expansion schemes, thus minimising the computational effort. Secondly, the resolution of the remaining computational tasks within a given expansion scheme can be further reduced by optimising the physics content within a given approximation. Thirdly, the maximal variability of PIRG flows can be used to reduce the analytic and numerical effort of solving the flows within a given approximation.