Lecture

Critical Exponents from RG

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In course

PHYS-739: Conformal Field theory and Gravity

This course is an introduction to holography, the modern approach to quantum gravity. ...

Description

This lecture covers the concepts of Conformal Field Theory (CFT) and gravity, including phase transitions, critical exponents, Pontryagin laws, fluctuations, and Renormalization Group (RG). It delves into the formalism of CFT and gravity, discussing critical points, fixed points, and left invariance under RG. The lecture also explores reduced Hamiltonians, correlations, and the Ising model, emphasizing decimation and scaling variables. It concludes with a detailed analysis of critical exponents derived from RG, focusing on their invariance and the scaling region. The lecture provides insights into the relevance of eigenvalues, the behavior of U under scaling, and the significance of fixed points in the context of critical surfaces.

Instructors (2)

Grégoire Olivier Mathys

Kyriakos Papadodimas

Official source

Ontological neighbourhood

Physics

Thermodynamics: Critical phenomena

Related concepts (32)

Critical exponent

Critical exponents describe the behavior of physical quantities near continuous phase transitions. It is believed, though not proven, that they are universal, i.e. they do not depend on the details of the physical system, but only on some of its general features. For instance, for ferromagnetic systems, the critical exponents depend only on: the dimension of the system the range of the interaction the spin dimension These properties of critical exponents are supported by experimental data.

Ising critical exponents

This article lists the critical exponents of the ferromagnetic transition in the Ising model. In statistical physics, the Ising model is the simplest system exhibiting a continuous phase transition with a scalar order parameter and symmetry. The critical exponents of the transition are universal values and characterize the singular properties of physical quantities. The ferromagnetic transition of the Ising model establishes an important universality class, which contains a variety of phase transitions as different as ferromagnetism close to the Curie point and critical opalescence of liquid near its critical point.

Critical phenomena

In physics, critical phenomena is the collective name associated with the physics of critical points. Most of them stem from the divergence of the correlation length, but also the dynamics slows down. Critical phenomena include scaling relations among different quantities, power-law divergences of some quantities (such as the magnetic susceptibility in the ferromagnetic phase transition) described by critical exponents, universality, fractal behaviour, and ergodicity breaking.

Critical point (thermodynamics)

In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. One example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions under which a liquid and its vapor can coexist. At higher temperatures, the gas cannot be liquefied by pressure alone. At the critical point, defined by a critical temperature Tc and a critical pressure pc, phase boundaries vanish.

Conformal field theory

A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.

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