⚛️ high-energy theory

Quasi-Normal Modes and thermalization in integrable Quantum Field Theories from the Thermodynamic Bethe Ansatz

This paper proposes a novel Thermodynamic Bethe Ansatz framework that describes thermalization in integrable quantum field theories through quasi-normal mode frequencies, conjecturing its equivalence to an all-order resummation of secular terms in thermal form factor expansions and validating its predictions against tensor network simulations of the E8 deformation of the Ising CFT.

Original author: Zoltan Bajnok, Romuald A. Janik

Published 2026-11-05

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CC BY 4.0

Original author: Zoltan Bajnok, Romuald A. Janik

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of physics, there is a fundamental question that applies to everything from a cup of cooling coffee to the earliest moments of the universe: how does a system settle down? When a physical system is disturbed, it rarely stops moving immediately. Instead, it wobbles, vibrates, and gradually loses energy until it reaches a state of calm, known as thermal equilibrium. This process of calming down is called thermalization. For many complex systems, scientists understand how the slow, large-scale movements of energy and matter smooth out over time, much like how traffic eventually clears after an accident. However, there is a more subtle and difficult aspect to this story: the precise, rapid fading of the initial jitters and vibrations that happen before the system settles completely. Understanding exactly how these vibrations die out, and at what speed, has remained a stubborn puzzle for decades, particularly in the realm of quantum physics where particles behave in ways that defy everyday intuition.

A team of researchers has now proposed a new way to solve this puzzle for a specific class of quantum systems known as integrable theories. These are special models where the interactions between particles are so well-ordered that they can be solved with high precision, unlike most chaotic quantum systems. The scientists, Zoltan Bajnok and Romuald A. Janik, suggest a method to calculate exactly how fast these quantum vibrations fade and how they oscillate as they do so. They call these fading vibrations "quasi-normal modes," a term borrowed from the study of black holes, where similar ripples fade away as they fall into the event horizon. By connecting the behavior of these fading quantum vibrations to a mathematical tool called the Thermodynamic Bethe Ansatz, the researchers have created a new framework that predicts the rate of this damping without needing to perform impossible calculations.

The core of their discovery lies in a clever mathematical trick that avoids the usual headaches of quantum calculations. Normally, to figure out how a quantum system relaxes, scientists would have to add up an infinite number of tiny contributions from every possible way particles can scatter and interact. This is like trying to count every single grain of sand on a beach to understand the shape of the shore; the sheer volume of data makes it nearly impossible to get a clear answer. The researchers found that by changing the way they look at the mathematical equations—specifically by shifting the perspective into a different part of the mathematical landscape—they could bypass this infinite counting. Instead of summing up millions of tiny terms, their new approach uses a set of equations that automatically handle all those interactions at once. This allows them to extract the exact rate at which the vibrations die out and the frequency at which they wiggle, combining these two numbers into a single, complex value that describes the system's relaxation.

To test if their idea actually works in the real world, the team applied their method to a famous model known as the E8 theory, which describes a specific type of magnetic chain found in certain materials. They compared the predictions from their new equations against powerful computer simulations that mimic the behavior of these quantum chains. The results matched with striking precision. The new method correctly predicted the speed at which the vibrations faded and the specific rhythm of their oscillations, confirming that the mathematical shortcut they devised captures the true physics of the system. This agreement suggests that their approach is not just a theoretical curiosity but a robust tool for understanding how quantum systems return to balance.

The significance of this work extends beyond just solving one specific equation. The researchers argue that their method provides a universal language for describing thermalization in these ordered quantum systems. It offers a clear path to understanding how energy dissipates in a way that was previously hidden by the complexity of the calculations. Furthermore, the study hints at a deeper connection between the quantum world and the gravitational universe. The mathematical structure they uncovered resembles the way black holes vibrate and settle after being disturbed, suggesting that the rules governing the cooling of a quantum chain might be the same as those governing the ripples of spacetime around a black hole. While the paper does not claim to have solved every mystery of thermalization, it provides a powerful new lens through which to view the process, turning a problem that seemed intractable into one that can be calculated with clarity and precision.

Technical Summary: Quasi-Normal Modes and Thermalization in Integrable Quantum Field Theories

Problem Statement
The paper addresses the fundamental challenge of describing thermalization in integrable Quantum Field Theories (QFTs), specifically the approach to thermal equilibrium from a non-equilibrium state. While hydrodynamic relaxation (driven by conserved charges) is well-understood, the genuine non-equilibrium thermalization of perturbations remains elusive. The authors focus on the large-time exponential behavior of the integrated retarded correlator at zero momentum ( $k=0$ ):
$\int dx \langle [O(t, x), O(0, 0)] \rangle_T \sim A(t) e^{-\Gamma t - i\Omega t}$
where $\Gamma$ is the thermal damping coefficient and $\Omega$ is the oscillation frequency. Determining these parameters requires handling infinite form-factor expansions at finite temperature, isolating secular terms arising from kinematical singularities, and resumming them into an exponential. Previous exact results were limited to specific cases (e.g., the $\sigma$ operator in the thermally deformed Ising CFT) and required complex, non-universal computations.

Methodology
The authors propose a Quasi-Normal Mode Thermodynamic Bethe Ansatz (QNM TBA). This approach is motivated by two converging perspectives:

Holographic Analogy: In theories with holographic duals, thermalization corresponds to perturbations of black holes, characterized by Quasi-Normal Modes (QNMs). These modes are poles of the retarded Green's function, obtained by analytically continuing positive Matsubara modes on a Euclidean cylinder to the Lorentzian regime. The authors postulate that a similar analytic continuation exists for the spectral problem of integrable QFTs on a Lorentzian cylinder of size $1/T$ .
Form-Factor Expansion Analysis: By analyzing the secular terms in the thermal form-factor expansion of retarded 2-point functions (specifically the time-like generalization of the Pozsgay-Szécsényi formula), the authors derive the necessary modifications to the standard TBA equations.

The core methodological innovation involves imposing a non-standard analytical structure on the TBA Y-system. Unlike standard TBA where left and right chiral sectors are decoupled or connected via smooth contours, the QNM TBA requires:

A specific contour deformation in the rapidity plane (or a shift $\theta \to \theta + i\pi$ for the right-moving sector).
A discontinuous driving term involving a sign function, $\text{sign}(\text{Re } \theta) \cosh \theta$ .
A modification of the scattering kernel $K$ to $J = \Sigma K \Sigma$ (where $\Sigma g(\theta) = \text{sign}(\theta)g(\theta)$ ) and the source term $s$ to $c(\theta) = S(-|\theta|)$ .

Key Contributions

Formulation of QNM TBA: The paper conjectures that the complex QNM frequency $\omega_{QNM} = -i\Gamma + \Omega$ can be determined by solving a modified set of TBA integral equations. These equations describe a "QNM state" (a hole state with opposite charges to the thermal ground state) and automatically resum all secular terms in the form-factor expansion.
Derivation of Driving Terms: Through the analysis of kinematical poles in the time-like form-factor expansion, the authors explicitly derive the additional driving terms required for the QNM TBA. They show that the kernel transformation $K \to J$ and source transformation $s \to c$ naturally emerge from the contour deformation required to handle time-like separations ( $t > |x|$ ).
Universal Framework: The proposal suggests a universal method to compute thermal damping and oscillation frequencies using only the infinite-volume scattering data of the integrable QFT, bypassing the need for explicit form-factor calculations.

Results

Analytic Consistency: The QNM TBA reproduces the exactly known damping coefficient for the $\sigma$ operator in the thermally deformed Ising CFT (E8 theory), matching the result derived in [15]:
$\Gamma = \frac{1}{2\pi} \int_{-\infty}^{\infty} dp \frac{|p|}{E_p} \log \coth \frac{E_p}{2T}$
Low-Temperature Limit: The framework correctly recovers the Sachdev-Young result for low-temperature relaxation, where collisions with dilute thermal particles are treated as independent Poisson processes, but with rapidity-dependent scattering phases.
Numerical Verification: The authors performed tensor network (iMPS) simulations of the E8 theory (magnetic deformation of Ising CFT) to compute the integrated retarded correlator $\langle \sigma \sigma \rangle_T$ $⟨ σ σ ⟩_{T}$ . They compared the simulation data with:
- Leading-order form-factor predictions (exponentiating secular terms).
- A fit using the QNM TBA frequencies and amplitudes, supplemented by three leading 2-particle contributions.
  The QNM TBA fit showed excellent agreement with the tensor network simulation in the long-time tail, validating the conjectured structure.

Significance and Claims
The paper claims that the QNM TBA provides a significant simplification over conventional approaches, which struggle with the non-convergence of form-factor expansions in the time-like regime and the difficulty of resumming infinite secular terms. By mapping the problem to a spectral problem on a Lorentzian cylinder with a specific analytic structure, the authors offer a "universal framework" for thermal damping in integrable QFTs.

The authors are modest regarding the scope of their current work:

The presented QNM TBA currently yields a single thermal damping coefficient for each particle species at $k=0$ . It does not yet predict the full QNM spectrum (e.g., higher excited states or modes with non-zero momentum), though they conjecture these can be obtained via standard excited-state TBA analytic continuation techniques.
The framework is proposed for integrable QFTs. The authors suggest it may also apply to the approach to Generalized Hydrodynamics (GHD) backgrounds ("generalized thermalization") by modifying the driving terms, but this requires further investigation.
The holographic interpretation of these QNMs in massive deformations of CFTs remains an open question for future study.

In summary, the paper establishes a conjectured link between the analytic structure of TBA equations and the physical phenomenon of thermal damping, validated by exact results and numerical simulations in the E8 theory.

Technical Summary: Quasi-Normal Modes and Thermalization in Integrable Quantum Field Theories

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