⚛️ high-energy theory

Exploring thermal order in conformal theories with multiple scalars coupled to an $O(N)$ vector field

This paper investigates thermal order in conformal field theories with multiple scalars coupled to an $O(N)$ vector field, demonstrating that while certain fixed points in $d=4-\epsilon$ dimensions fail to yield unitary CFTs in $d=3$ , a large- $N$ analysis in three dimensions reveals a conformal manifold where specific symmetry-breaking patterns lead to persistent thermal symmetry breaking for all nonzero temperatures.

Original authors: Soumyadeep Chaudhuri, Bilal Hawashin, Eliezer Rabinovici, Michael M. Scherer

Published 2026-09-02

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CC BY 4.0

Original authors: Soumyadeep Chaudhuri, Bilal Hawashin, Eliezer Rabinovici, Michael M. Scherer

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

In the physical world, order and disorder are locked in a constant struggle, often decided by temperature. When a material is cold, its atoms may align in a rigid, organized pattern, breaking the symmetry of the space they occupy. As heat is applied, this order usually crumbles. The thermal energy shakes the atoms until they lose their alignment, restoring a state of chaos where no direction is preferred. This is the standard rule of nature: heat destroys order. For decades, physicists believed this was a universal law, applicable to every system in the universe. They reasoned that the random jiggling of heat would always eventually overwhelm the forces holding a system together, forcing it back into a disordered state.

However, a small group of researchers has been investigating a peculiar exception to this rule. They are looking for systems where, instead of melting away, the order actually grows stronger as the temperature rises. This phenomenon, known as "thermal order," suggests that under very specific conditions, a system can remain locked in a broken-symmetry state at any temperature above absolute zero. While some theoretical models in fractional dimensions hinted at this possibility, the question remained open for real-world, integer dimensions like the three dimensions we inhabit. Could a local, finite system exist that refuses to melt, no matter how hot it gets?

A team of physicists has now taken a significant step toward answering this question by exploring a complex family of theoretical models. They studied systems composed of many particles, specifically a large collection of vector particles interacting with several scalar particles. In these models, the particles have an internal "spin" or direction, and the researchers were interested in whether the system would spontaneously choose a specific direction to align with as it heated up. The team approached this problem from two different angles, using two distinct mathematical tools to probe the behavior of these systems in three-dimensional space.

First, the researchers examined the models in a dimension slightly less than four, a mathematical technique often used to approximate the behavior of complex systems. They discovered that for certain numbers of particles, there were specific points where the system settled into a stable, unchanging state. At these points, the system exhibited a persistent broken symmetry, meaning the particles chose a direction and stayed there. However, when they tried to push these results toward the familiar three dimensions, the mathematical structure began to unravel. The stable points they found in the four-dimensional approximation collided and vanished into a realm of complex numbers before they could reach three dimensions. This suggests that the specific models they found in the four-dimensional approximation do not survive as stable, physical theories in our three-dimensional world.

Undeterred, the team shifted their strategy to work directly in three dimensions, but this time they focused on a regime where the number of vector particles was very large. In this large-number limit, the mathematics simplifies enough to reveal a hidden structure. Instead of finding just a few isolated stable points, they discovered a vast, continuous landscape of possible stable states, known as a conformal manifold. On this landscape, the scalar particles could be divided into two distinct groups based on how they interacted with the vector particles. The researchers then focused on a specific region of this landscape where the two groups of particles possessed a particular kind of symmetry.

In this specific region, the team performed a detailed analysis of the system's energy at different temperatures. They found that for a wide range of conditions, the system's lowest energy state did not correspond to a disordered, chaotic mess. Instead, the lowest energy state occurred when every single scalar particle in the system acquired a non-zero value, effectively choosing a direction. This meant that the symmetry of the system was spontaneously broken at all non-zero temperatures. The particles did not melt into disorder; they remained locked in an ordered state, no matter how much heat was added.

The researchers proved that within this specific domain of their large-number models, the system exhibits a robust pattern of thermal order. The symmetry that allows the particles to flip their signs is broken, and the system settles into a state where the particles have a definite orientation. This finding provides a concrete example of a local, finite system in three dimensions that defies the usual expectation that heat restores symmetry. While the team noted that their results rely on a specific approximation where the number of particles is very large, and that further corrections might refine the picture, the core result stands: there exists a class of theories where order persists indefinitely as temperature rises.

This work does not claim to have found a material that will never melt in a laboratory, but it demonstrates that the laws of physics do not strictly forbid such behavior. By mapping out the conditions under which thermal order can exist, the researchers have expanded our understanding of what is possible in the quantum world. They have shown that the intuition that heat always destroys order is not a universal law, but a feature of the specific systems we encounter in everyday life. In the vast landscape of theoretical possibilities, there are islands where order is not just a cold phenomenon, but a permanent state of being.

Technical Summary: Exploring Thermal Order in Conformal Theories with Multiple Scalars Coupled to an O(N) Vector Field

Problem Statement
The paper investigates the phenomenon of "thermal order" in Conformal Field Theories (CFTs), defined as the persistence of spontaneous symmetry breaking (SSB) at all nonzero temperatures. While standard intuition suggests that thermal fluctuations restore symmetries at high temperatures, previous work has identified specific CFTs in fractional dimensions and large- $N$ limits where discrete symmetries remain broken. The central question addressed is whether such thermal order exists in local, finite- $N$ CFTs in integer dimensions (specifically $d=3$ ) when multiple scalars are coupled to an $O(N)$ vector field. The authors aim to extend the known results for a single scalar coupled to an $O(N)$ vector (where a $Z_2$ symmetry breaks thermally) to models with $M$ scalars possessing a larger discrete symmetry group, $Z_2^M \rtimes S_M$ .

Methodology
The authors employ two distinct theoretical approaches to address the problem:

$\epsilon$ -Expansion near Four Dimensions ( $d = 4 - \epsilon$ ):
- They analyze $O(N) \times (Z_2^M \rtimes S_M)$ -symmetric models involving $N$ vector fields ( $\phi_i$ ) and $M$ scalars ( $\psi_j$ ).
- The interaction potential includes quartic couplings ( $f, g, h, \zeta$ ) respecting the symmetry.
- They compute one-loop beta functions to identify Wilson-Fisher-like fixed points.
- They calculate thermal masses at these fixed points to determine if the effective potential develops a non-zero minimum at $T > 0$ , indicating SSB.
- To probe the survival of these fixed points in $d=3$ , they utilize the Functional Renormalization Group (FRG) in the Local Potential Approximation (LPA') with a derivative expansion, numerically continuing the fixed points from small $\epsilon$ to $\epsilon = 1$ .
Large- $N$ Analysis in Three Dimensions ( $d=3$ ):
- They work directly in $d=3$ with $N$ large but finite and $M \ll N$ .
- Using the Hubbard-Stratonovich formalism, they introduce an auxiliary field $\sigma$ to decouple the $O(N)$ vector interactions.
- They derive the Renormalization Group (RG) flow for the couplings $t_i$ (coupling scalars to $\sigma$ ) and sextic couplings $\kappa_{ijk}$ (interactions between scalars) at the leading non-trivial order in the $1/N$ expansion.
- They identify a conformal manifold of fixed points parameterized by cross-couplings between two classes of scalars ( $C_+$ and $C_-$ ).
- They restrict analysis to a subspace with enhanced permutation symmetry within each class and compute the thermal effective action by integrating out the $O(N)$ fields.
- They analyze the saddle points of this action to determine the global minimum and the resulting pattern of symmetry breaking.

Key Results

In $d = 4 - \epsilon$ :
- For any $M$ , there exists a "conformal window" of $N$ values where two real Wilson-Fisher-like fixed points with $O(N) \times (Z_2^M \rtimes S_M)$ symmetry exist. Outside this window, the fixed points become complex (non-unitary).
- Case $M=2$ : Within the window $17 < N < N_u \approx 21.86$ , the discrete symmetry $Z_2^2 \rtimes S_2$ is spontaneously broken at all nonzero temperatures for both fixed points. The pattern of breaking differs between the two fixed points (one preserves permutation symmetry, the other breaks it).
- Case $M > 2$ : The $Z_2^M \rtimes S_M$ symmetry remains unbroken at nonzero temperatures for all fixed points within the conformal window. However, for specific ranges of $N$ (e.g., $N \le 9$ ) and sufficiently large $M$ , thermal order appears in the $O(N)$ sector (breaking $O(N)$ symmetry), though this is forbidden in $d=3$ by the Mermin-Wagner-Coleman-Hohenberg theorem.
- FRG Continuation to $d=3$ : The FRG analysis suggests that the two fixed points found near $d=4$ collide and move into the complex plane well before reaching $d=3$ (at $\epsilon_c \approx 0.06$ for $N=18$ ). This implies these specific fixed points do not survive as unitary CFTs in three dimensions.
In $d=3$ (Large $N$ ):
- The RG flow reveals a conformal manifold at leading order in $1/N$ . The scalars split into two classes, $C_+$ (coupling with $t_i = +1$ ) and $C_-$ (coupling with $t_i = -1$ ).
- The manifold is parameterized by sextic cross-couplings between fields in $C_+$ and $C_-$ .
- By restricting to a subspace with $O(N) \times (Z_2^{M_+} \rtimes S_{M_+}) \times (Z_2^{M_-} \rtimes S_{M_-})$ symmetry, the authors identify a domain of couplings where the sextic potential is bounded from below.
- Thermal Order: In this specific domain, the saddle point with the minimum thermal effective action corresponds to a configuration where all $M$ scalars acquire non-zero expectation values.
- Symmetry Breaking: Consequently, the discrete symmetry $Z_2^{M_+} \times Z_2^{M_-}$ is spontaneously broken to the permutation groups $S_{M_+} \times S_{M_-}$ at all nonzero temperatures. The $O(N)$ symmetry remains unbroken.

Significance and Claims
The paper claims to provide a novel class of large- $N$ CFTs in three dimensions that exhibit persistent symmetry breaking (thermal order) at nonzero temperatures. Specifically:

It demonstrates that for $M \ge 2$ , it is possible to construct models where a larger discrete symmetry group (involving sign flips and permutations) is spontaneously broken at all $T > 0$ , extending previous results limited to $M=1$ .
It highlights a mechanism where thermal order arises in a sector with fewer degrees of freedom ( $M$ scalars) while the sector with many degrees of freedom ( $O(N)$ vector) remains disordered, satisfying entropic arguments for persistent breaking.
The authors are modest regarding the FRG results in $d=3$ , noting that the collision of fixed points into the complex plane suggests the specific $4-\epsilon$ fixed points do not survive, and that the large- $N$ conformal manifold is likely an artifact of the leading-order expansion, expected to be lifted to isolated fixed points by subleading $1/N$ corrections.
The work does not claim to have found a definitive, non-perturbative proof of thermal order in finite- $N$ $d=3$ CFTs but rather provides strong evidence within the large- $N$ limit and identifies specific constraints and domains where such behavior is predicted.

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