⚛️ high-energy theory

On the temperature dependence of quasinormal modes in SYK and holography

This paper generalizes the study of quasinormal modes in the SYK model from infinite to finite temperature, revealing that the relaxation rate increases monotonically with temperature only in the strong-coupling gravitational regime while contrasting these dynamics with various AdS black holes and other SYK variants.

Original authors: Matthew Dodelson, Om Gupta, Márk Mezei, Diandian Wang

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Matthew Dodelson, Om Gupta, Márk Mezei, Diandian Wang

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

The Big Picture: The "Christmas Tree" of Chaos

Imagine you have a very complex, chaotic system—like a crowded dance floor where everyone is bumping into each other. In physics, we study how these systems settle down after being disturbed. We call the specific "notes" or "vibrations" that a system makes as it settles down Quasinormal Modes (QNMs). Think of them like the specific tones a bell makes as it rings out and fades away.

In a famous model called the SYK model (which physicists use to study quantum chaos and black holes), researchers recently discovered something strange at infinite temperature (a state of maximum chaos). The pattern of these "notes" looks exactly like a Christmas tree. If you plot them on a graph, they form a triangular shape with a trunk and branches, just like a holiday tree.

This paper asks a simple question: What happens to this Christmas tree when we cool the system down?

The Journey from Hot to Cold

The authors tracked these "notes" as they lowered the temperature from infinite heat down to very cold. Here is what they found:

The Hot Regime (The Messy Middle):
When the system is hot but not infinitely hot, the Christmas tree doesn't just shrink smoothly. It gets messy.
- The Analogy: Imagine a crowded dance floor where people are moving frantically. As the music slows down, pairs of dancers (the notes) crash into each other, swap partners, and change their paths in complicated, unpredictable ways. Some notes collide and bounce off the "imaginary axis" (a specific line on the graph), while others move around chaotically.
- The Result: The movement is non-monotonic. This is a fancy way of saying the notes don't just move in one direction; they zigzag, collide, and reorganize before settling down.
The Cold Regime (The Orderly Line):
As the system gets very cold (approaching the "gravity regime," which is mathematically similar to a black hole in a simplified universe), the chaos stops.
- The Analogy: The dance floor clears out. The remaining dancers line up in a perfect, straight row.
- The Result: The "Christmas tree" disappears. The notes line up perfectly on a straight vertical line. This matches what we expect from JT Gravity (a theory describing simple black holes). In this cold state, the system behaves very orderly: as it gets colder, the notes move steadily toward a specific position, never looking back.

The "Heating Up" Rule

The authors tested a general rule of thumb: "If you heat up a system, it should relax (settle down) faster."

The Expectation: Think of a cup of hot coffee cooling down. If you heat the room up, the coffee cools faster because there are more energy channels for the heat to escape. The authors expected that as temperature goes up, the "relaxation rate" of these quantum notes should also go up.
The Reality:
- In the Gravity/Cold regime: The rule holds true. Heating the system makes it relax faster.
- In the Chaotic/Hot regime: The rule breaks. Sometimes, heating the system makes the notes move in weird, non-monotonic ways.
- The Exception: The rule also breaks if the system has special "symmetries" (like a conserved amount of energy or charge). In these cases, the notes behave like diffusion (like a drop of ink spreading in water). The ink spreads slower if the water is hotter (counter-intuitively), violating the rule.

The "Magic" Mathematical Trick

One of the coolest technical parts of the paper is how they solved the math.

The Problem: They could calculate the "notes" very accurately when the system was hot (using a method like a Taylor series expansion). But when they tried to use those hot calculations to predict what happens when the system is cold, the math usually breaks down (like trying to predict winter weather using only summer data).
The Solution: They used a mathematical tool called a Padé approximant.
- The Analogy: Imagine you have a few puzzle pieces from the top of a picture (hot data) and a few from the bottom (cold data). Usually, you can't connect them. But the Padé approximant is like a smart algorithm that looks at the few pieces you have and magically "fills in" the middle, creating a smooth, perfect picture that connects the hot and cold worlds without any gaps.
- The Result: They managed to smoothly connect the "Christmas tree" of the hot universe to the "straight line" of the cold universe with incredible accuracy (within 1.5%).

New Rules for "Chaos"

The paper also looked at Operator Growth, which is a way to measure how fast information gets scrambled in a quantum system (like how fast a secret spreads in a gossip chain).

There is a famous "speed limit" for this scrambling (the Chaos Bound).
The authors found that in the standard SYK model, the system never quite hits the speed limit; it stays strictly below it.
However, in a specific variation called the "SYK chain," they proposed a new, more detailed rule that accounts for momentum (movement in space). They showed that this new, refined rule holds true perfectly.

Summary

This paper is a map of how a chaotic quantum system changes its "vibrations" as it cools down.

Hot: The vibrations form a chaotic, shifting "Christmas tree" with messy collisions.
Cold: The vibrations straighten out into a perfect line, matching the behavior of simple black holes.
The Bridge: They used a clever math trick to smoothly connect the hot and cold behaviors.
The Lesson: While heating up usually makes things settle faster, this isn't always true in the chaotic middle ground, and special symmetries can break the rules entirely.

Technical Summary: On the temperature dependence of quasinormal modes in SYK and holography

Problem Statement
The paper investigates the temperature dependence of quasinormal modes (QNMs), also known as Ruelle–Pollicott resonances, in the Sachdev-Ye-Kitaev (SYK) model and its holographic duals. While recent work established that the SYK model at infinite temperature exhibits a "Christmas tree" pattern of QNMs reminiscent of higher-dimensional AdS black holes, and the low-temperature limit is dual to Jackiw-Teitelboim (JT) gravity with a purely imaginary spectrum, the continuous morphing between these two regimes at finite temperature remained unexplored. A central question is whether the relaxation rate (the imaginary part of the QNM frequency, $|\text{Im } \omega|$ ) increases monotonically with temperature, a property expected for generic gravitational systems but potentially violated by symmetry-protected modes or specific transport phenomena.

Methodology
The authors develop a computational framework to determine QNMs at finite temperature for the finite- $p$ SYK model.

Real-time Schwinger-Dyson Equations: They formulate the real-time Schwinger-Dyson equations for the SYK model at finite temperature. This allows for the recursive generation of Taylor series coefficients (generalized moments $\mu_{2n, 2k}$ ) of the two-sided correlator $C(t)$ and the retarded Green's function $G_R(t)$ .
Moment Extraction: The algorithm computes moments $\mu_{2n}$ as power series in $\beta J$ . For $p=4$ , they compute up to $K=116$ moments, with each moment expanded as a series in $\beta$ up to order $\beta^{150}$ .
Analytic Continuation via Padé Approximants: Since the power series in $\beta$ has a finite radius of convergence ( $\tilde{\beta}_0/2 \approx 1.4$ ), the authors employ Padé approximants to analytically continue the moments to the low-temperature regime ( $\beta J \to \infty$ ). Remarkably, they find that supplying only the Taylor series data around infinite temperature ( $\beta=0$ ) allows the Padé approximant to reorganize itself into the correct asymptotic conformal/JT gravity behavior at low temperatures with high accuracy (within 1.5% of the theoretical value).
Spectrum Extraction: Two methods are used to extract QNMs from the moments:
- Exponential Fitting: Fitting a sum of exponentials to the Taylor expansion of $C(t)$ , effective at high/moderate temperatures.
- Krylov Space Scattering: Using Lanczos coefficients derived from moments to construct a discrete scattering problem on a black hole-like background, effective at low temperatures where convergence is rapid.

Key Contributions and Results

Temperature Evolution of SYK QNMs:
- Infinite Temperature: The spectrum exhibits the known "Christmas tree" structure.
- Intermediate Temperature: The movement of QNMs is complex and non-monotonic. The authors identify two mechanisms for the transition to the JT gravity spectrum: (1) Conjugate pairs of QNMs on the Christmas tree collide on the imaginary axis and separate into two purely imaginary modes; (2) QNMs already on the imaginary axis move toward the real axis, scaling as $1/\beta$ . Generic QNMs in the complex plane remain stationary and eventually disappear from the spectrum as $T \to 0$ .
- Low Temperature (Gravity Regime): In the strongly coupled, low-temperature regime dual to JT gravity, the QNM movement becomes orderly and monotonic. All modes approach the real frequency axis as temperature decreases, satisfying the inequality $d|\text{Im } \omega_n|/dT > 0$ .
Comparison with Other Systems:
- AdS Black Holes/Branes: Analytical and numerical evidence confirms that generic QNMs in AdS black holes and charged black branes obey the monotonicity inequality $d|\text{Im } \omega_n|/dT > 0$ across various limits (large $n$ , large momentum, large spin).
- Symmetry-Breaking and Hydrodynamics: The paper identifies classes of systems where monotonicity is violated.
  - Diffusion Poles: In systems with conserved densities (e.g., energy diffusion), the diffusion pole scales as $\omega \sim -iDk^2$ with $D \propto 1/T$ , leading to anti-monotonic behavior ( $d|\text{Im } \omega|/dT < 0$ ).
  - Symmetry Breaking: In models with spontaneous symmetry breaking (e.g., holographic superfluids or Model A), Goldstone modes collide with other poles at the critical temperature, leading to non-monotonic trajectories.
- Large $p$ SYK Chain: The energy density correlator exhibits both monotonic non-hydrodynamic modes and an anti-monotonic diffusion pole.
Operator Growth and Chaos Bounds:
- Using the finite-temperature moment data, the authors compute the Lyapunov exponent $\lambda_L$ and the effective inverse temperature $\beta_0$ (related to the growth rate of Lanczos coefficients) for the $p=4$ SYK model.
- They provide high-precision numerical evidence that the conjectured bound $\lambda_L \leq 2\pi/\beta_0$ is satisfied as a strict inequality in the finite- $p$ SYK model at finite temperature, contrasting with the saturation observed in the large- $p$ limit.
- For the large- $p$ SYK chain, they propose and verify a momentum-dependent refinement of the chaos bound: $\lambda(k) \leq 2\pi/\beta_0(k) \leq 2\pi/\beta$ .

Significance
The paper establishes that the "Christmas tree" structure of SYK QNMs at infinite temperature continuously flows into the purely imaginary spectrum of JT gravity at low temperatures, mediated by complex collision dynamics at intermediate temperatures. The authors argue that the monotonic increase of relaxation rates with temperature is a robust feature of generic gravitational duals (AdS black holes) but is violated by modes protected by symmetries or conservation laws (diffusion, Goldstone modes). Consequently, observing monotonic QNM movement may serve as a diagnostic for the presence of a holographic gravity dual. Furthermore, the work provides new, precise results on operator growth, confirming the strictness of the refined chaos bound in finite- $p$ SYK and extending it to momentum-dependent forms in chain models. The technical achievement of using Padé approximants to bridge the infinite and zero-temperature regimes using only high-temperature data offers a powerful tool for studying finite-temperature dynamics in strongly coupled systems.