Finite-Temperature Correlation Functions Reveal Kinematics of Conformal Field Theories

Conformal field theories underpin much of modern theoretical physics, offering insights into everything from critical phenomena to the behaviour of black holes, and understanding their properties at finite temperature remains a significant challenge. Alessio Miscioscia, Elli Pomoni, and Volker Schomerus, all from Universität Hamburg, alongside their colleagues, present a comprehensive investigation into thermal effects within these theories. Their work explores how symmetries break down and evolve at higher temperatures, employing advanced analytical and numerical techniques to examine fundamental quantities like correlation functions and free energy. This research offers a non-perturbative approach, meaning it doesn’t rely on approximations, and provides a powerful new tool for studying complex systems, with specific applications to the well-known 3d Ising, XY, and Heisenberg models, ultimately advancing our understanding of both condensed matter physics and quantum gravity.

Understanding thermal effects within CFTs is of central importance, as criticality is investigated in laboratory settings at finite temperature, and from a holographic perspective. These theories exhibit scale invariance, meaning their properties remain unchanged under changes in scale, which simplifies analysis and allows for universal behaviour to emerge. Consequently, researchers focus on understanding how temperature affects the behaviour of these systems, particularly concerning phase transitions and critical exponents. This work investigates the thermal properties of CFTs, aiming to provide insights into both condensed matter physics and the theoretical foundations of quantum gravity, and builds upon existing knowledge of correlation functions and operator product expansions to explore these phenomena

Conformal Field Theory and Critical Phenomena

This collection represents a comprehensive exploration of conformal field theory, critical phenomena, statistical mechanics, and related areas. The central theme is conformal field theory, with a strong emphasis on techniques like operator product expansions and the conformal bootstrap. A significant portion of the work focuses on understanding phase transitions, universality classes, and critical exponents, particularly within the Ising model and related systems. The conformal bootstrap, a non-perturbative approach to solving CFTs, receives considerable attention, as does the study of CFTs at finite temperature and their connection to quantum thermalization.

Defect CFTs and the behaviour of systems with boundaries or impurities are also investigated, alongside connections to quantum chaos, many-body physics, and the study of black holes, suggesting an interest in the emergence of spacetime. Numerical methods are increasingly used to solve bootstrap equations and obtain precise values for critical exponents, building upon foundational works on CFT and statistical mechanics. The collection is categorised into studies of critical phenomena, numerical bootstrap calculations, finite temperature CFT analyses, defect CFT investigations, and connections to quantum chaos and many-body physics. Researchers are increasingly developing more efficient algorithms and computational resources for numerical bootstrap calculations.

Defect CFTs provide a way to study non-perturbative effects, and understanding their interplay with the global structure of the CFT is important. The connection between CFTs, quantum chaos, and black holes is a fascinating area, with the SYK model serving as a key example. Understanding how finite temperature affects phase transitions is crucial for many applications, and the increasing precision of numerical bootstrap calculations allows for more accurate predictions that can be compared with experimental results.

Thermal CFTs and Quantum Criticality Insights

Conformal field theories (CFTs) are fundamental to modern theoretical physics, providing a framework for understanding a wide range of phenomena from critical behaviour in materials to the elusive nature of quantum gravity. This research extends our understanding of CFTs to include the effects of temperature, a crucial consideration for both experimental relevance and theoretical insights into black holes. The work develops new methods for analysing CFTs at finite temperature, exploring how their properties change when thermal effects are introduced. A key motivation lies in the study of quantum criticality, where systems exhibit dramatic changes in behaviour at specific points.

Understanding these transitions requires analysing systems at temperatures very close to absolute zero, but real-world experiments always occur at finite temperatures, making this research directly applicable to experimental observations. The team’s approach provides a more accurate description of these critical phenomena under realistic conditions, bridging the gap between theoretical models and laboratory results. Furthermore, this research connects CFTs to the study of gravity through the concept of holography, a duality proposing a relationship between a CFT and a theory of gravity in a higher-dimensional space. Introducing temperature into a CFT is mathematically equivalent to introducing a black hole into this higher-dimensional space, offering a novel way to study black hole physics. The methods developed represent a significant advancement in analysing these complex systems, potentially revealing new insights into the behaviour of gravity and the nature of black holes. The research successfully adapts and combines analytical and numerical techniques to study the O(N) model, a widely used system in condensed matter physics, at finite temperature, achieving a detailed understanding of how temperature affects the critical properties of the 3d Ising, XY, and Heisenberg models and accurately predicting their free energy density.

Finite Temperature Correlation Functions in O(N) Models

This research advances the understanding of conformal field theories (CFTs), which are essential for describing critical phenomena and have connections to theories of gravity. The work explores the behaviour of these theories at finite temperatures, a crucial aspect for comparing theoretical predictions with experimental observations and for modelling black holes. Researchers investigated these systems by analysing their symmetries and adapting both analytical and numerical techniques to study correlation functions, providing a non-perturbative approach that can be validated against known solvable models. The study focused on O(N) models in three dimensions, specifically the Ising, XY, and Heisenberg models, calculating quantities like one- and two-point functions and free energy density. Future research directions include refining approximations and extending the techniques to more complex systems, potentially furthering the connection between CFTs and holographic theories of gravity.