Quantum chaos and pole skipping in two-dimensional conformal perturbation theory

This paper analyzes pole skipping in the stress tensor two-point functions of two-dimensional quantum field theories perturbed by relevant deformations, proposing a method to resolve singular expressions in conformal perturbation theory and demonstrating precise agreement between the resulting Lyapunov exponents and butterfly velocities with both Ward identity calculations and holographic duals.

The Gist

The Big Picture: Listening to the Chaos

Imagine you have a perfectly calm, symmetrical pond. In physics, this is like a Conformal Field Theory (CFT)—a system that looks the same no matter how you zoom in or out. It's predictable and orderly.

Now, imagine you drop a heavy rock into that pond. This is a relevant deformation. It disturbs the water, creates ripples, and breaks the perfect symmetry. In the real world, this is like taking a simple quantum system and adding a little bit of "mess" or interaction to it.

The authors of this paper wanted to answer a specific question: When we disturb this perfect quantum system, how does its "chaos" change?

In the quantum world, "chaos" doesn't mean things are messy in a random way; it means that tiny differences in the starting conditions grow exponentially fast. If you nudge a particle here, it affects a particle over there almost instantly. Scientists measure this using two numbers:

1. The Lyapunov Exponent: How fast the chaos grows (the speed of the explosion).
2. The Butterfly Velocity: How fast the chaos spreads across space (how fast the butterfly flaps its wings and causes a storm).

The Problem: The "Black Box" of Chaos

Usually, to measure this chaos, physicists use a very difficult tool called an OTOC (Out-of-Time-Ordered Correlator). Think of this as trying to measure the speed of a ghost. It's incredibly hard to calculate directly, especially in complex quantum systems that don't have a "holographic" twin (a gravity theory in a higher dimension that makes the math easier).

The Solution: The "Pole Skipping" Shortcut

The authors use a clever shortcut called Pole Skipping.

Imagine the behavior of the quantum system is like a radio station. The "signal" (the Green's function) usually has specific frequencies where it gets very loud (poles) or very quiet (zeros).

• The Trick: In chaotic systems, there are special spots on the radio dial where the signal is supposed to be loud (a pole), but it's also supposed to be quiet (a zero). These two conditions cancel each other out, making the signal undefined. This is called Pole Skipping.
• The Magic: The authors discovered that the exact location of these "skipped" spots tells you exactly how fast the chaos is growing and spreading. It's like finding a hidden fingerprint of chaos in the static of the radio.

The Challenge: The "Singular" Math

The paper's main achievement is solving a math problem that usually breaks computers.
When they tried to calculate these skipped spots for a disturbed system, they ran into singularities.

• The Analogy: Imagine trying to calculate the average height of a group of people, but one person is infinitely tall. The math explodes. In physics, these "infinities" appear when particles get too close to each other in the equations.
• The Fix: The authors developed a new way to interpret these infinities using something called distributions. Instead of saying "this breaks," they said, "let's treat this infinity like a very sharp, tiny spike that we can handle carefully." They essentially put a tiny, invisible net around the infinity to catch it and make the math work.

The Verification: The Holographic Mirror

To prove their new math method was correct, they used a "mirror test."

• The Mirror: In physics, there's a famous idea called Holography (AdS/CFT). It says that a 2D quantum system (like the one they studied) is mathematically equivalent to a 3D universe with gravity (like a black hole).
• The Test: They calculated the chaos speed using their new "Pole Skipping" math on the 2D side. Then, they calculated the chaos speed using standard gravity equations on the 3D black hole side.
• The Result: The numbers matched perfectly. This proved that their tricky way of handling the "infinities" was physically correct.

Why Does This Matter?

1. It works without Gravity: For a long time, we could only calculate chaos easily if we had a "gravity twin" to help us. This paper shows we can do it directly in the quantum system, even without the gravity mirror.
2. It helps with Real Materials: Many real-world materials (like superconductors or magnetic chains) act like these perturbed quantum systems. Understanding how chaos spreads in them helps us design better quantum computers and understand how heat moves through materials.
3. It's a New Tool: They've given physicists a new, reliable ruler to measure quantum chaos in systems that were previously too messy to analyze.

Summary in One Sentence

The authors figured out how to measure the speed of quantum chaos in a disturbed system by finding "glitches" in the math (pole skipping), invented a new way to handle the resulting infinities, and proved it works by comparing it to a black hole's gravity.

Technical Summary

1. Problem Statement

The paper addresses the challenge of characterizing quantum chaos in non-holographic, non-conformal quantum field theories (QFTs).

• Context: Quantum chaos is typically diagnosed via Out-of-Time-Ordered Correlators (OTOCs), which yield the Lyapunov exponent ($\lambda_L$) and butterfly velocity ($v_B$). However, OTOCs are notoriously difficult to compute directly in generic QFTs, especially at finite temperature.
• Alternative Diagnostic: Recent work suggests that "pole skipping" in the retarded Green's function of the stress tensor provides an equivalent diagnostic. In holographic theories, the location of these skipped poles $(\omega_*, k_*)$ maps directly to chaos parameters via $\lambda_L = -i\omega_*$ and $v_B = \omega_*/k_*$.
• The Gap: It remains unclear if this pole-skipping/chaos correspondence holds for theories without a holographic dual or those perturbed away from conformality. Specifically, computing the retarded Green's function in a perturbed Conformal Field Theory (CFT) leads to formally singular integrals that are ill-defined in standard calculus.

2. Methodology

The authors employ a multi-pronged approach to compute the pole-skipping velocity ($v_{PS}$) in a 2D CFT deformed by a relevant scalar operator $\mathcal{O}$ with conformal weight $h \in (0, 1)$.

A. Ward Identity Approach (Leading Order)

• The authors first utilize diffeomorphism and dilatation Ward identities to relate the stress tensor two-point function $G_{--,-}$ to the trace of the stress tensor and the correlation functions of the deforming operator $\mathcal{O}$.
• This allows for a non-perturbative expression for the Green's function in terms of the unperturbed CFT correlators $G_{\mathcal{O}\mathcal{O}}$ and the one-point function $\langle \mathcal{O} \rangle$.
• By expanding to leading non-trivial order in the deformation parameter $\lambda$, they derive an analytic expression for the shift in the pole-skipping velocity, $v_{PS}$, as a function of $h$, the central charge $c$, and temperature $\beta$.

B. Conformal Perturbation Theory (Direct Calculation)

• To verify the Ward identity results and handle higher-order structures, the authors perform a direct calculation using Conformal Perturbation Theory (CPT).
• The Singularity Problem: The perturbative expansion of the Euclidean Green's function involves integrals of the form $\int d^2z \, \frac{1}{(u-z)^2} \dots$. These integrals diverge at the operator insertion points ($z=u, v$).
• Distributional Interpretation: The core methodological innovation is the interpretation of these singular integrals as homogeneous distributions.
  • They extend the function $1/z^2$ (defined on $\mathbb{C} \setminus \{0\}$) to a distribution $D(1/z^2)$ on the entire complex plane by enforcing complex homogeneity.
  • This corresponds to a "principal value" prescription where infinitesimal disks around the poles are excised.
  • They demonstrate that this distributional interpretation is physically consistent with the light-cone singularities in the Lorentzian signature.
• Evaluation: Using this prescription, they explicitly evaluate the Euclidean Green's function $G^E_2$ for general $h$, and specifically for the tractable cases $h=1/2$ and $h \to 1$.
• Fourier Transform: They perform the Fourier transform to the Lorentzian frequency-momentum space $(\omega, k)$. This requires careful handling of branch cuts and contact terms (polynomials in $\omega, k$), showing that contact terms do not affect the location of the poles.

C. Holographic Verification

• To validate the distributional interpretation and the CPT results, the authors compute the butterfly velocity in a dual holographic setup: a BTZ black hole perturbed by a massive scalar field.
• They solve the linearized Einstein-scalar equations to find the backreaction on the geometry and compute the resulting $v_B$.
• They carefully match the normalization of the bulk scalar field to the CFT deformation parameter $\lambda$.

3. Key Contributions

1. Resolution of Singular Integrals: The paper provides a rigorous distributional framework for evaluating divergent integrals in Conformal Perturbation Theory. By treating poles as homogeneous distributions, they resolve the ambiguity of "ill-defined" expressions that typically plague higher-order CPT calculations.
2. Explicit Green's Function: They derive the explicit form of the leading-order correction to the stress tensor Green's function for a relevant deformation in 2D CFTs for arbitrary conformal weight $h$.
3. Verification of Pole Skipping: They explicitly calculate the pole-skipping velocity $v_{PS}$ for specific cases ($h=1/2$ and near $h=1$) and show it matches the result derived from Ward identities.
4. Holographic-CFT Agreement: They demonstrate exact agreement between the CFT calculation (using the new distributional method) and the holographic calculation of the butterfly velocity. This confirms that the pole-skipping phenomenon captures the stress-tensor contribution to chaos even away from strict conformality and in non-holographic limits.

4. Key Results

• Pole-Skipping Velocity: The leading-order correction to the pole-skipping velocity is found to be:
  $$v_{PS} = 1 + \frac{3\pi^4 h}{8} \frac{8(1-2h)\csc(2\pi h) - 4\pi h(1-h)\csc^2(\pi h)}{[\Gamma(1-h)\Gamma(1/2+h)]^2} \frac{\lambda^2}{c \beta^{4(h-1)}} + \dots$$
  This result is derived via Ward identities and confirmed via CPT.
• Special Cases:
  • $h = 1/2$: The velocity shift is non-zero and matches the holographic prediction.
  • $h \to 1$ (Marginal Deformation): The correction to the velocity vanishes at order $\lambda^2(h-1)^2$, consistent with the fact that marginal deformations preserve conformal symmetry (and thus the original pole structure).
• Contact Terms: The authors prove that additive contact terms (polynomials in $\omega, k$) arising from regularization ambiguities or distributional definitions do not shift the location of the skipped poles, as these terms vanish at the singular surface of the Green's function.
• Holographic Match: The butterfly velocity calculated from the bulk shock wave analysis on a BTZ black hole perturbed by a scalar field matches the CFT pole-skipping velocity exactly.

5. Significance

• Beyond Holography: This work provides strong evidence that the connection between pole skipping and quantum chaos is not merely an artifact of holographic duality but is a robust feature of 2D quantum field theories, even when deformed away from conformality.
• Computational Tool: The distributional interpretation of singular integrals offers a powerful new tool for performing higher-order calculations in Conformal Perturbation Theory, potentially applicable to other observables like OTOCs or higher-point functions.
• Phenomenological Relevance: The results bridge the gap between abstract holographic chaos diagnostics and concrete lattice or experimental systems (e.g., trapped Ising models, spin chains) that can be modeled as perturbed CFTs. It suggests that pole skipping could be a viable diagnostic for chaos in systems where OTOC calculations are intractable.
• Future Directions: The paper outlines a path toward computing sub-leading corrections and direct OTOC calculations in perturbed CFTs, which would further test the universality of the pole-skipping/chaos correspondence.