⚛️ high-energy theory

Boundaries and defects in conformal field theories at finite temperature

This paper classifies bulk one-point functions of primary operators in Euclidean conformal field theories at finite temperature with both thermal-circle-wrapping and localised defects, deriving bootstrap sum rules and computing thermodynamic observables and defect data across various free and interacting examples.

Original authors: Daniele Artico, Adam Chalabi

Published 2026-09-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daniele Artico, Adam Chalabi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a universe where the laws of physics look exactly the same no matter how you zoom in or out, or how you rotate your view. This is the realm of conformal field theories, mathematical descriptions of matter at the very edge of stability, known as critical points. At these points, materials like magnets or fluids lose their specific microscopic details and behave in a universal way, governed by symmetry rather than the messy specifics of atoms. For decades, physicists have studied these systems in a perfect, frozen state at absolute zero. But the real world is rarely frozen. It is warm, it is moving, and it is full of obstacles. When you heat a material, you introduce a new scale: the temperature. This breaks the perfect symmetry, changing how the system behaves. Furthermore, real materials often have boundaries, impurities, or defects—like a crack in a crystal or a magnetic line running through a superconductor. The question that has long puzzled physicists is how these imperfections interact with heat. How does a defect change the way a material stores energy or flows heat?

A team of researchers has now mapped out this complex interplay for a wide class of theoretical materials. They investigated what happens when these perfectly symmetric systems are heated up and placed in the presence of boundaries or defects. They distinguished between two main types of defects: those that wrap around the entire thermal cycle, acting like a permanent, static impurity, and those that are localized at a single moment in time, acting more like a sudden insertion or a snapshot of an extended object. By treating the heat as a geometric circle that the system must repeat over and over, the team was able to classify exactly how the presence of these defects alters the fundamental properties of the material. They found that while the heat changes the rules, the underlying symmetry still leaves a strict fingerprint on how the system responds.

The researchers developed a new set of mathematical tools, which they call "bootstrap sum rules," to connect the microscopic behavior of the material to its macroscopic properties. Think of these rules as a way to check if a story makes sense by ensuring that the beginning and the end match up perfectly, even if you don't know every detail of the middle. In this case, the "story" is the behavior of the material as it cycles through time. The team showed that for defects that are localized in time, the heat itself forces the material's properties to obey a strict symmetry: the behavior at a certain time must be perfectly related to the behavior at a time exactly halfway through the cycle. This constraint is so powerful that it allows them to derive precise equations that link the strength of the defect to the energy and pressure of the surrounding material.

To prove their ideas work, the team applied their new framework to several specific, well-understood models of free particles, including simple scalar fields and fermions. They calculated exactly how the energy density and pressure change near a boundary or a defect. In every case they studied, they found a consistent pattern: the energy of the system is always negative, meaning the presence of the defect or boundary lowers the total energy compared to a system without them. Furthermore, they observed that as the system flows from a high-energy state to a lower-energy state, the magnitude of this negative energy always increases. This suggests that the system is constantly integrating out microscopic details to settle into a more stable configuration, a behavior that mirrors what is seen in simpler, one-dimensional systems.

One of the most striking discoveries came from studying a specific type of defect called a magnetic line, which is localized at a point in time. The researchers found that the mathematical description of the material's response to this defect could be broken down into a specific sum of functions known as Hurwitz zeta functions. This decomposition is highly suggestive; it implies that there is an underlying "dispersion relation," a fundamental connection between the energy and momentum of the excitations in the system, similar to how sound waves relate frequency to pitch. While the team did not prove this relation exists for all possible materials, the fact that it appears so clearly in their free-field examples strongly hints that it is a universal feature of how localized defects interact with heat.

The study also looked at more complex scenarios, such as a "monodromy" defect, which twists the field as you move around it, and higher-derivative theories that describe non-standard materials. Even in these more complicated and sometimes unstable systems, the team found that the same thermodynamic principles held true. The free energy remained negative, and the system behaved in a way that respected the new constraints imposed by the heat. They also noted that for these exotic theories, the energy density was not always a simple, smooth curve; it could have peaks and valleys depending on the distance from the defect, a feature that would not exist in a standard material.

Ultimately, this work provides a rigorous framework for understanding how the microscopic world of quantum fields adapts to the macroscopic reality of heat and imperfection. By classifying the possible ways defects can exist in a thermal environment and deriving the rules that govern their behavior, the researchers have opened the door to predicting the properties of complex materials without needing to solve every single equation from scratch. Their findings suggest that even in the chaotic environment of a hot, imperfect system, the deep symmetries of the universe still hold sway, dictating how energy flows and how matter settles. The results serve as a foundation for future studies of interacting systems, where the mathematics is far more difficult, but the guiding principles of symmetry and thermal consistency remain the same.

Technical Summary: Boundaries and defects in conformal field theories at finite temperature

Problem Statement
The paper addresses the interplay between spatial inhomogeneities (boundaries and defects) and finite temperature in Conformal Field Theories (CFTs). While zero-temperature defect CFTs (DCFTs) and finite-temperature CFTs without defects are well-studied, the systematic classification of conformal defects at finite temperature remains incomplete. Specifically, the authors investigate how conformal symmetry and the Kubo-Martin-Schwinger (KMS) condition constrain the thermal observables of systems containing defects. A central question is how the data of a conformal defect enters thermal observables and what constraints arise from the interplay between the defect's orientation relative to the thermal circle and the thermal periodicity.

Methodology
The authors employ a combination of kinematic analysis, the Defect Operator Expansion (DOE), and bootstrap techniques within Euclidean CFT on the manifold $S^1_\beta \times \mathbb{R}^{d-1}$ .

Kinematic Classification: The study distinguishes two distinct configurations based on the defect's orientation relative to the thermal circle ( $S^1_\beta$ ):
- Wrapping defects: Defects that wrap the thermal circle and extend along non-compact spatial directions. These describe static impurities.
- Localised defects: Defects supported on a fixed Euclidean time slice. These correspond to extended operator insertions in the thermal trace.
  For both configurations, the authors classify the allowed tensor structures for thermal one-point functions of bulk scalar, vector, and spin-2 primaries, as well as defect-local operators.
Defect Operator Expansion (DOE) and Inversion: The authors utilize the DOE to express bulk operators in terms of defect primaries. They derive Euclidean inversion formulas to extract thermal DCFT data (coefficients of defect thermal one-point functions and bulk-to-defect couplings) from the thermal one-point functions. They analyze the radius of convergence of these expansions, noting that while radial quantization suggests a radius of $\beta/2$ , specific assumptions about positivity can extend this to $\beta$ .
KMS Constraints and Bootstrap Sum Rules: The KMS condition, which encodes thermal periodicity, is applied to correlation functions.
- For wrapping defects, the KMS condition imposes constraints on two-point functions.
- For localised defects, the KMS condition imposes non-trivial constraints on the one-point function itself, as the defect breaks thermal translation invariance. By treating the KMS condition as a crossing equation, the authors derive a novel set of bootstrap sum rules relating bulk-to-defect couplings and thermal one-point coefficients.
Explicit Examples: The theoretical framework is tested and illustrated using free field theories:
- Free scalar, Dirac fermion, hypermultiplet, and higher-derivative scalar fields with conformal boundaries (BCFTs).
- A monodromy defect (co-dimension two) wrapping the thermal circle.
- A magnetic line defect localised on the thermal circle.

Key Contributions and Results

Classification of Thermal One-Point Functions: The paper provides a complete classification of tensor structures for scalar, vector, and spin-2 one-point functions in the presence of both wrapping and localised defects. This includes identifying parity-odd structures and constraints arising from conservation laws (e.g., for the stress tensor and conserved currents).
Thermal DCFT Data: The authors define "thermal DCFT data," which includes the zero-temperature defect spectrum and OPE coefficients, augmented by new thermal coefficients (defect thermal one-point functions). They demonstrate how to extract this data using inversion formulas.
Bootstrap Sum Rules for Localised Defects: A primary contribution is the derivation of bootstrap sum rules for localised defects. By expanding the thermal one-point function in thermal blocks and enforcing the KMS reflection property ( $w \to 1-w$ ), the authors derive an infinite family of sum rules (Eq. 4.11) that constrain the thermal DCFT data. These rules are shown to hold in explicit free-field examples.
Thermodynamic Observables: For wrapping defects, the stress tensor one-point function is used to compute thermodynamic quantities (energy density, entropy, free energy). The authors find that for the boundary RG flows studied (e.g., Neumann to Dirichlet), the free energy is negative and satisfies $|F_{UV}| > |F_{IR}|$ , consistent with the integration out of degrees of freedom.
Analytic Structure and Dispersion Relations:
- For the monodromy defect, the thermal one-point functions exhibit branch cuts in the complex plane, even in a free theory.
- For the localised magnetic line defect, the authors observe that the one-point function admits a decomposition into Hurwitz zeta functions. This structure is suggestive of an underlying dispersion relation, analogous to those found for thermal two-point functions without defects.
Higher-Derivative Scalars: The study extends to non-unitary higher-derivative scalar theories. While the local free energy density does not always decrease monotonically along RG flows point-wise, the integrated free energy satisfies the inequality $|F_{UV}| > |F_{IR}|$ .

Significance and Claims
The paper claims to establish a systematic framework for studying conformal defects at finite temperature, extending previous work on line defects to arbitrary dimensions and co-dimensions. The significance lies in:

Unifying Framework: Providing a unified kinematic and bootstrap approach for both wrapping and localised defects, clarifying how the KMS condition constrains different defect configurations.
New Constraints: Introducing novel bootstrap sum rules for localised defects that constrain thermal DCFT data without relying on positivity (which is not guaranteed in DCFTs).
Thermodynamic Insights: Connecting microscopic defect data to macroscopic thermodynamic observables, specifically verifying the behavior of free energy under boundary RG flows in free theories.
Analytic Structure: Highlighting the rich analytic structure (branch cuts, Hurwitz zeta decompositions) of thermal one-point functions, suggesting the existence of dispersion relations for defect CFTs.

The authors remain modest regarding the scope of their results, noting that the derived sum rules rely on assumptions about the convergence of the DOE (specifically that it extends beyond $\beta/2$ or that coefficients satisfy positivity conditions). They acknowledge that while the free energy inequality holds in their free-field examples, it may not be a universal theorem for all interacting defect RG flows, particularly in non-unitary theories. The work serves as a foundation for future studies involving interacting theories, higher-spin operators, and real-time dynamics.

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