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
- : corresponds to a free boson CFT with central charge
- : 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
Key Terms to Review (20)
Bootstrap method: The bootstrap method is a powerful analytical technique used in quantum field theory, particularly within the context of conformal field theory, to derive physical quantities and correlations directly from symmetry principles rather than from a detailed Lagrangian. This approach emphasizes the self-consistent nature of conformal theories and provides a way to compute observables like correlation functions without relying on perturbative expansions.
Central charge: The central charge is a crucial quantity in conformal field theory that characterizes the scaling dimensions of primary fields and their behavior under the conformal group. It plays a significant role in the classification of conformal field theories and is linked to the anomaly in the energy-momentum tensor, influencing correlation functions and operator product expansions.
Conformal blocks: Conformal blocks are mathematical objects in conformal field theory (CFT) that describe the contributions of primary operators to correlation functions. They encapsulate how different fields interact under conformal transformations, making them essential for understanding the structure of CFTs and their applications in critical phenomena, where systems exhibit scale-invariant behavior near phase transitions.
Conformal Field Theory: Conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations, which are transformations that preserve angles but not necessarily distances. This property makes CFTs particularly useful in studying critical phenomena and phase transitions, where systems exhibit scale invariance. CFTs also play a crucial role in holographic theories, connecting strongly coupled quantum field theories with gravity in higher-dimensional spaces, and have applications in condensed matter physics, revealing insights into quantum phase transitions and emergent phenomena.
Correlation functions: Correlation functions are mathematical objects used to describe how different points in a quantum field theory are related to each other. They capture the statistical properties of fields and can provide insights into the behavior of particles and interactions. These functions play a critical role in linking theoretical predictions with observable quantities, making them essential in various areas of physics, including the study of field interactions, phase transitions, and numerical simulations.
Critical Exponents: Critical exponents are numerical values that characterize the behavior of physical quantities near continuous phase transitions, describing how these quantities diverge or vanish as a system approaches the critical point. They provide essential insights into the universality class of the phase transition, linking systems with different microscopic details but similar critical behavior. These exponents are typically denoted by letters like $eta$, $
u$, $eta$, and $ au$, each representing different physical properties such as order parameter, correlation length, and specific heat.
David Friedan: David Friedan is a notable figure in theoretical physics, particularly recognized for his contributions to conformal field theory (CFT) and its applications in understanding critical phenomena. His work has significantly advanced the study of phase transitions and the critical behavior of systems at critical points, linking statistical mechanics with quantum field theory.
Fixed Points: Fixed points are specific values in a physical theory where the behavior of the system remains unchanged under a transformation, such as scaling or renormalization. They play a crucial role in understanding how physical systems behave at different energy scales, helping to identify phase transitions and critical phenomena. In essence, they indicate where a theory can be 'improved' or where certain parameters no longer change with respect to changes in the energy scale.
Ising Model Universality Class: The Ising model universality class refers to a group of statistical models that share the same critical behavior and scaling properties at phase transitions, particularly in the context of magnetic systems. These models exhibit universal characteristics, such as critical exponents and scaling functions, which are independent of the specific details of the system. This universality allows for predictions and insights into various physical systems undergoing similar phase transitions, making it a fundamental concept in the study of critical phenomena and conformal field theory.
Operator Product Expansion: Operator product expansion (OPE) is a technique in quantum field theory that allows one to express the product of two operators at nearby points as a sum of local operators, multiplied by coefficients that depend on the separation between the points. This method is essential in analyzing correlation functions and understanding the structure of quantum fields, particularly in conformal field theories and critical phenomena, where it plays a crucial role in the characterization of scaling dimensions and operator algebras.
Phase Transitions: Phase transitions are transformations between different states of matter that occur when a system's external conditions, like temperature or pressure, change significantly. These transitions can lead to profound changes in the physical properties of the system, such as from solid to liquid or liquid to gas, and can be described using concepts like order parameters and critical points.
Potts model universality class: The Potts model universality class refers to a set of statistical mechanics models that generalize the Ising model by allowing for more than two states at each lattice site. This concept is crucial for understanding phase transitions and critical phenomena, as models within this class display similar behavior near critical points, regardless of specific interactions or lattice structures, highlighting the role of symmetry and dimensionality in phase transitions.
Renormalization Group: The renormalization group is a collection of techniques used to study the changes in physical systems as one varies the energy scale or length scale. It helps deal with infinities that arise in quantum field theories by systematically relating the parameters of a theory at different scales, allowing for the prediction of physical phenomena in a consistent manner. This concept connects deeply with divergences in calculations, running couplings, effective field theories, critical phenomena, and condensed matter systems.
Riccardo Zecchina: Riccardo Zecchina is a prominent physicist known for his contributions to the study of statistical mechanics and complex systems, particularly in the context of disordered systems and phase transitions. His work has significantly influenced the understanding of critical phenomena, especially through the lens of conformal field theories and their applications in various physical contexts.
Scaling dimensions: Scaling dimensions are fundamental characteristics in conformal field theory that describe how fields transform under scale transformations. These dimensions play a crucial role in understanding the behavior of physical systems near critical points, where the properties of the system remain invariant under changes in length scale, indicating self-similarity. In this context, scaling dimensions help classify operators and define their correlations, which are essential for analyzing phase transitions and critical phenomena.
Statistical mechanics: Statistical mechanics is a branch of theoretical physics that uses statistical methods to explain and predict the thermodynamic properties of systems made up of a large number of particles. It bridges the microscopic behavior of individual atoms and molecules with the macroscopic observable properties, allowing for insights into phenomena such as phase transitions and critical behavior.
String theory: String theory is a theoretical framework in physics that posits that fundamental particles are not point-like objects, but rather one-dimensional 'strings' that vibrate at different frequencies. These vibrations determine the properties of particles, such as mass and charge, and suggest a unified description of all fundamental forces, including gravity. It connects to critical phenomena and conformal field theories by providing a way to study higher-dimensional spaces and phase transitions using string-like models.
Universality class: A universality class is a set of systems that exhibit the same critical behavior and belong to the same category when undergoing phase transitions, regardless of their microscopic details. This concept highlights how different physical systems can share common properties and scaling laws near critical points, leading to a deeper understanding of critical phenomena. It reveals that certain macroscopic behaviors are determined by symmetry and dimensionality rather than specific material properties.
Virasoro Algebra: The Virasoro algebra is an infinite-dimensional Lie algebra that arises in the context of two-dimensional conformal field theories. It generalizes the algebra of the conformal transformations on the plane and provides a framework for understanding the symmetries of these theories, which are crucial in studying critical phenomena and string theory. Its central extension is vital for capturing the physical states and their interactions in these theories.
Xy model universality class: The xy model universality class refers to a specific group of statistical systems that exhibit similar critical behavior, particularly in two dimensions, characterized by continuous symmetry and the presence of a phase transition. This model captures essential features of phase transitions found in systems with spins that can rotate in a plane, making it significant in understanding critical phenomena and conformal field theory. It is often studied to gain insights into the nature of phase transitions in various physical systems, including magnets and superfluids.