11.1 Conformal field theory and critical phenomena
4 min read•august 14, 2024
(CFT) is a powerful tool in quantum physics, describing systems with scale-invariant properties. It's crucial for understanding critical phenomena in and in condensed matter physics.
CFTs are characterized by their and operator content. They provide a framework for calculating and , 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 of the theory
Correlation functions in CFTs are constrained by conformal invariance, leading to powerful techniques for their computation, such as the (OPE)
Applications of Conformal Field Theory
CFTs have applications in various areas of physics, including:
: 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 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:
: corresponds to a minimal model CFT with central charge c=1/2
: corresponds to a free boson CFT with central charge c=1
: 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
, 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