11.1 Conformal field theory and critical phenomena

Conformal field theory (CFT) is a powerful tool in quantum physics, describing systems with scale-invariant properties. It's crucial for understanding critical phenomena in statistical mechanics and phase transitions in condensed matter physics.

CFTs are characterized by their central charge and operator content. They provide a framework for calculating correlation functions and critical exponents, helping us classify universality classes in statistical mechanics and beyond.

Principles and Applications of Conformal Field Theory

Fundamentals of Conformal Field Theory

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• Conformal field theory (CFT) is a quantum field theory invariant under conformal transformations, angle-preserving transformations that locally scale space-time coordinates
• The conformal group in d dimensions consists of translations, rotations, dilations (scale transformations), and special conformal transformations
• CFTs are characterized by their central charge, which measures the number of degrees of freedom and appears in the Virasoro algebra of the theory
• Correlation functions in CFTs are constrained by conformal invariance, leading to powerful techniques for their computation, such as the operator product expansion (OPE)

Applications of Conformal Field Theory

• CFTs have applications in various areas of physics, including:
  • String theory: CFTs describe the worldsheet dynamics of strings propagating in target space-time
  • Condensed matter physics: CFTs govern the critical behavior of systems near phase transitions
  • Statistical mechanics: CFTs provide a framework for understanding critical phenomena and universality classes
• CFTs also find applications in mathematics, such as in the study of modular forms, vertex operator algebras, and representation theory

Conformal Field Theory and Critical Phenomena

Connection between CFT and Critical Phenomena

• Critical phenomena occur in statistical mechanics when a system undergoes a continuous phase transition, characterized by diverging correlation lengths and power-law behavior of physical quantities
• Near a critical point, the system becomes scale-invariant, and the long-distance behavior is governed by a CFT
• The critical exponents of a system, which describe the power-law behavior of thermodynamic quantities near the critical point, are related to the scaling dimensions of primary operators in the corresponding CFT

Universality Classes and Conformal Field Theory

• Universality classes of critical phenomena are determined by the symmetries and dimensionality of the system, and they correspond to different CFTs with specific central charges and operator content
• Examples of universality classes include:
  • Ising model universality class: corresponds to a minimal model CFT with central charge $c=1/2$
  • XY model universality class: corresponds to a free boson CFT with central charge $c=1$
  • Potts model universality class: corresponds to a series of minimal model CFTs with central charges determined by the number of states

Applications of Conformal Field Theory in Statistical Mechanics

Computation of Thermodynamic Quantities using CFT

• The partition function of a critical statistical mechanics model can be expressed as a correlation function in the corresponding CFT, allowing for the computation of thermodynamic quantities using conformal field theory techniques
• The operator product expansion can be used to calculate critical exponents and correlation functions in statistical mechanics models by expanding products of local operators in terms of a basis of primary operators
• Conformal blocks, the building blocks of correlation functions in CFTs, can be used to construct exact solutions for critical statistical mechanics models, such as the Ising model

Structure of Critical Statistical Mechanics Models

• The Virasoro algebra and its representations provide a powerful framework for understanding the structure of critical statistical mechanics models and their excitations
• Primary operators in the CFT correspond to local operators in the statistical mechanics model, and their scaling dimensions determine the critical exponents
• Descendant operators, obtained by acting with Virasoro generators on primary operators, correspond to excited states in the statistical mechanics model

Conformal Invariance in Two-Dimensional Systems

Infinite-Dimensional Conformal Symmetry in Two Dimensions

• Two-dimensional CFTs are particularly tractable due to the infinite-dimensional nature of the conformal group in two dimensions, which is generated by the Virasoro algebra
• The central charge of a two-dimensional CFT determines the Virasoro algebra and the structure of the theory, with rational values of the central charge corresponding to minimal models
• Examples of two-dimensional CFTs include:
  • Free boson CFT: describes the critical behavior of the XY model and the Tomonaga-Luttinger liquid
  • Minimal models: a series of CFTs with rational central charges, corresponding to various universality classes

Conformal Bootstrap and Two-Dimensional CFTs

• The conformal bootstrap approach exploits the constraints of conformal invariance to determine the operator content and correlation functions of a two-dimensional CFT based on consistency conditions
• The conformal bootstrap has been successfully applied to solve various two-dimensional CFTs, such as the Ising model and the Liouville CFT
• The conformal bootstrap also provides insights into the structure of higher-dimensional CFTs and the AdS/CFT correspondence