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Soryushiron Kenkyu/Soryushiron kenkyu

A Renormalization group Approach to the Critical Dissipation in Quantum Systems(Fundamental Problems and Applications of Quantum Field Theory)

Kenichi Aoki, Tamao Kobayashi

Published Apr 20, 2006• 0 citations
OpenAlex

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Dynamical renormalization group approach to relaxation in quantum field theory

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arXiv:hep-ph/0302055•Annals of Physics•Jun 2, 2003•99 citations

The renormalization group idea

John Cardy

In this chapter the basic concepts of the modern approach to equilibrium critical behaviour, conventionally grouped under the title ‘renormalization group’, are introduced. This terminology is rather unfortunate. The mathematical structure of the procedure, in the sense that it may be said to have any rigorous underpinnings, is certainly not that of a group. Neither is renormalization in quantum field theory an essential element, although it has an intimate connection with some formulations of the renormalization group. In fact, the renormalization group framework may be applied to problems quite unrelated to field theory. The origins of the name may be traced to the particle physics of the 1960s, when it was optimistically hoped that everything in fundamental physics might be explained in terms of symmetries and group theory, rather than dynamics. One of the earliest applications of renormalization group ideas, in fact, was to the rather esoteric subject of the high energy behaviour of renormalized quantum electrodynamics. It took the vision of K. Wilson to realise that these methods had a far wider field of application in the scaling theory of critical phenomena that was being formulated by Fisher, Kadanoff and others in the latter part of the decade. By then, however, the name had become firmly attached to the subject. Not only are the words ‘renormalization’ and ‘group’ examples of unfortunate terminology, the use of the definite article ‘the’ which usually precedes them is even more confusing.

Cambridge University Press eBooks•Apr 26, 1996•0 citations

Effects of Dissipation on a Quantum Critical Point with Disorder

José A. Hoyos, Chetan Kotabage, Thomas Vojta

We study the effects of dissipation on a disordered quantum phase transition with O(N) order-parameter symmetry by applying a strong-disorder renormalization group to the Landau-Ginzburg-Wilson field theory of the problem. We find that Ohmic dissipation results in a nonperturbative infinite-randomness critical point with unconventional activated dynamical scaling while super-Ohmic damping leads to conventional behavior. We discuss applications to the superconductor-metal transition in nanowires and to the Hertz theory of the itinerant antiferromagnetic transition.

arXiv:0705.1865•Physical Review Letters•Dec 4, 2007•90 citations

An introduction to the fundamentals of the renormalization group in critical phenomena

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Physics Reports•Jan 1, 1977•36 citations

The renormalization group (RG) approach: The critical theory near four dimensions

Jean Zinn‐Justin

Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.

•Apr 15, 2021•0 citations

Field-Theoretic Renormalization Group in Models of Critical Behavior, Turbulence and Non-equilibrium Dynamics

L. Ts. Adzhemyan, N. V. Antonov, Michal Hnatič +8 more

WORLD SCIENTIFIC eBooks•Aug 1, 2024•2 citations
In this chapter the basic concepts of the modern approach to equilibrium critical behaviour, conventionally grouped under the title ‘renormalization group’, are introduced. This terminology is rather unfortunate. The mathematical structure of the procedure, in the sense that it may be said to have any rigorous underpinnings, is certainly not that of a group. Neither is renormalization in quantum field theory an essential element, although it has an intimate connection with some formulations of the renormalization group. In fact, the renormalization group framework may be applied to problems quite unrelated to field theory. The origins of the name may be traced to the particle physics of the 1960s, when it was optimistically hoped that everything in fundamental physics might be explained in terms of symmetries and group theory, rather than dynamics. One of the earliest applications of renormalization group ideas, in fact, was to the rather esoteric subject of the high energy behaviour of renormalized quantum electrodynamics. It took the vision of K. Wilson to realise that these methods had a far wider field of application in the scaling theory of critical phenomena that was being formulated by Fisher, Kadanoff and others in the latter part of the decade. By then, however, the name had become firmly attached to the subject. Not only are the words ‘renormalization’ and ‘group’ examples of unfortunate terminology, the use of the definite article ‘the’ which usually precedes them is even more confusing.
We study the effects of dissipation on a disordered quantum phase transition with O(N) order-parameter symmetry by applying a strong-disorder renormalization group to the Landau-Ginzburg-Wilson field theory of the problem. We find that Ohmic dissipation results in a nonperturbative infinite-randomness critical point with unconventional activated dynamical scaling while super-Ohmic damping leads to conventional behavior. We discuss applications to the superconductor-metal transition in nanowires and to the Hertz theory of the itinerant antiferromagnetic transition.
Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.