Imprints of $U_A(1)$ chiral anomaly and disorder in the Dirac eigenspectrum of QCD at finite temperature

This study utilizes lattice QCD simulations to analyze the Dirac eigenspectrum at finite temperature, revealing how intermediate level statistics and Thouless conductance serve as diagnostics for the interplay between chiral symmetry restoration and disorder-driven localization, particularly in the context of the effective restoration of the anomalous $U_A(1)$ symmetry.

The Gist

Imagine the universe as a giant, boiling pot of soup. Inside this pot, the fundamental building blocks of matter (quarks) and the forces holding them together (gluons) are dancing wildly. This "soup" is called Quantum Chromodynamics (QCD).

When this soup is cool, the particles are stuck together in tight little clumps (like protons and neutrons). But when you heat it up enough, they break free and flow freely. This is the state of matter that existed just after the Big Bang.

This paper is like a detective story. The scientists are trying to understand how this soup changes as it gets hotter, specifically looking at two different "rules" that govern the particles:

1. Chiral Symmetry: A rule about how particles spin and behave.
2. Disorder: How "messy" or "random" the environment is.

Here is the story of their investigation, explained simply.

1. The Musical Orchestra Analogy

To understand the particles, the scientists don't look at them one by one. Instead, they listen to the "music" of the system. In physics, every particle has a specific "note" (an energy level). If you have a million particles, you have a million notes.

• The "Bulk" Notes: Most of the notes in this orchestra are chaotic and unpredictable, but they follow a very specific, universal pattern known as Random Matrix Theory. Think of this like a jazz band improvising; it sounds chaotic, but there's a hidden mathematical rhythm to it.
• The "Intermediate" Notes: The scientists found something strange. Near the bottom of the energy scale (the "bass notes"), there are some notes that don't follow the jazz rhythm. They are in the middle—neither perfectly ordered nor fully chaotic. They are the "intermediate" notes.

2. The Two Suspects: Symmetry vs. Messiness

The big mystery was: Why do these "intermediate" notes exist?

The scientists suspected two different culprits were responsible, depending on how hot the soup was:

• Suspect A (The Symmetry Breaker): At moderate temperatures (just above the boiling point), the "Chiral Symmetry" starts to break. This is like a dance floor where everyone was dancing in perfect pairs, but suddenly, the pairs start to break up. This creates a specific pattern in the music (the intermediate notes).
• Suspect B (The Disorder): At very high temperatures, the environment becomes incredibly messy. The "glue" holding the particles becomes random and uncorrelated, like static on a radio. This randomness (disorder) traps some particles in small pockets, changing the music again.

3. The Investigation: Separating the Clues

The scientists used a supercomputer to simulate this soup at different temperatures. They looked at the "spacing" between the musical notes.

• The Finding: They realized that at moderate heat, the intermediate notes were caused by the breaking of symmetry (Suspect A).
• The Twist: But at extreme heat (when the soup is super hot), the symmetry is fully restored, yet the intermediate notes still exist!
• The Conclusion: At these high temperatures, the intermediate notes are caused by Suspect B (Disorder). The environment has become so random that it acts like a "trap" for certain particles, similar to how a bumpy road might trap a car in a pothole.

4. The "Thouless Conductance" (The Stress Test)

To prove this, the scientists invented a new tool they call the Thouless Conductance.

Imagine you have a rubber sheet with marbles on it (the particles).

• If the marbles are free to roll everywhere (delocalized), and you twist the sheet, the marbles move easily.
• If the marbles are stuck in deep holes (localized), twisting the sheet doesn't move them much.

The scientists "twisted" their mathematical model and measured how much the "notes" shifted.

• Result: The "intermediate" notes at high heat barely moved when twisted. This proved they were trapped (localized) by the random disorder of the hot soup, not by symmetry breaking.

5. The Big Picture: Why Does This Matter?

This paper solves a puzzle about the early universe.

• We knew that at high temperatures, a specific force called $U_A(1)$ (a type of symmetry) should "turn back on" or be restored.
• The scientists found a way to detect exactly when this happens. They realized that when the "intermediate notes" start behaving like they are trapped in random holes, it's a sign that the $U_A(1)$ symmetry has been restored.

In summary:
The scientists listened to the "music" of the subatomic world. They found that at high temperatures, the music changes not because the dancers are changing their steps (symmetry), but because the dance floor itself has become a chaotic, bumpy mess (disorder). They built a new tool to measure this "bumpiness," giving us a clearer picture of how the universe behaved in its first microseconds.

Technical Summary

Here is a detailed technical summary of the paper "Imprints of UA(1) chiral anomaly and disorder in the Dirac eigenspectrum of QCD at finite temperature."

1. Problem Statement

The paper addresses the complex interplay between two distinct physical phenomena in Quantum Chromodynamics (QCD) at finite temperatures ($T$):

1. Chiral Symmetry Restoration: The sequential effective restoration of different subgroups of chiral symmetry (specifically the non-singlet $SU_A(2)$ and the anomalous $U_A(1)$) as temperature increases.
2. Anderson-like Localization: The transition of Dirac eigenstates from delocalized (ergodic) to localized states due to disorder in the gauge fields.

While both phenomena affect the infrared (low-lying) part of the Dirac eigenspectrum, their temperature regimes overlap, making it difficult to disentangle whether observed spectral features arise from chiral symmetry restoration or from disorder-driven localization. The authors aim to distinguish the mechanisms causing "intermediate" level statistics in the Dirac spectrum, particularly in the regime where $U_A(1)$ symmetry is effectively restored ($T > T_{U(1)}$).

2. Methodology

The study employs Lattice QCD simulations with the following technical specifications:

• Ensembles: $2+1$ flavor QCD using Möbius domain wall fermions (which preserve exact chiral symmetry on the lattice) and the Iwasaki gauge action.
• Lattice Parameters: Simulations were performed on large volumes ($N_s = 32$) with varying temporal extents ($N_\tau = 8$) to cover temperatures from $T \approx 164$ MeV to $339$MeV (roughly$1.1 T_{pc}$to$2.2 T_{pc}$).
• Operator: The massless Overlap Dirac operator was used to calculate the first 100–200 non-zero eigenvalues. This choice avoids chiral symmetry breaking artifacts common in Wilson fermions.
• Observables:
  • Normalized Level Spacing Ratios ($\tilde{r}$): Defined as $\tilde{r}_n = \min(r_n, 1/r_n)$ where $r_n = s_{n+1}/s_n$. This metric is robust against unfolding systematic errors and distinguishes between Poissonian (localized/uncorrelated) and Wigner-Dyson (delocalized/chaotic) statistics.
  • Polyakov Loop Correlation: The correlation between Dirac eigenstate densities and the local fluctuations of the renormalized Polyakov loop ($Re.P(x)$) was calculated to quantify disorder.
  • Thouless Conductance ($g_{Th}$): A novel diagnostic tool for the Dirac spectrum. It measures the structural rigidity of eigenvalues by calculating the curvature of eigenvalues ($K_i$) in response to a twist ($\eta$) applied to the spatial boundary conditions: $g_{Th} \propto \langle K_i \rangle / \langle s \rangle$.

3. Key Contributions

• Disentanglement of Mechanisms: The authors successfully separate the effects of $U_A(1)$ restoration from disorder-induced localization. They demonstrate that intermediate level statistics persist at high temperatures but arise from a different physical origin than those near the chiral crossover.
• Random Matrix Modeling: They developed a specific Random Matrix Theory (RMT) model to explain the intermediate level statistics. Instead of a simple superposition of GUE (Gaussian Unitary Ensemble) and Poisson ensembles, they proposed a hybrid model where uncorrelated eigenvalues are inserted into a GUE block with a specific acceptance criterion that suppresses very small level spacings. This model successfully reproduces the lattice data.
• First Calculation of Dirac Thouless Conductance: The paper presents the first calculation of the Thouless conductance for the QCD Dirac spectrum, using it as a probe for the structural rigidity of eigenstates and a potential order parameter for $U_A(1)$ restoration.
• Characterization of Disorder: The study quantifies the nature of disorder in the gauge fields, showing a transition from correlated topological fluctuations (at $T < T_{U(1)}$) to random, uncorrelated fluctuations (at $T > T_{U(1)}$) that mimic the Anderson model.

4. Key Results

A. Level Spacing Statistics and Temperature Regimes

• Region $T_{pc} < T < T_{U(1)}$: Infrared eigenmodes exhibit "intermediate" level statistics (between Poisson and GUE). These modes are associated with the remnant $O(4)$ symmetry due to $SU_A(2)$ restoration and have fractal-like structures.
• Region $T > T_{U(1)}$ (High Temperature):
  • A gap opens in the spectrum, separating deep infrared modes from the bulk.
  • The infrared modes are driven by a dilute instanton gas scenario where $U_A(1)$ is effectively restored.
  • Crucially, the bulk spectrum (just above the gap) shows intermediate statistics due to the mixing of chaotic bulk modes with localized modes driven by random, uncorrelated disorder in the Polyakov loop.
  • The level spacing ratios ($\tilde{r}$) in the bulk approach GUE values quickly, whereas the infrared modes maintain intermediate statistics.

B. Disorder and Polyakov Loop Correlation

• Low $T$ ($< 200$ MeV): Fluctuations in the renormalized Polyakov loop are frequent and strongly correlated (topological origin). Intermediate eigenstates are delocalized over these correlated regions.
• High $T$ ($> 250$ MeV): Fluctuations become rarer and spatially uncorrelated (random disorder). The correlation quantity $C(\lambda/T)$ between Dirac eigenstates and Polyakov loop fluctuations increases significantly for infrared modes, indicating that these modes are localized in the "wells" created by random Polyakov loop fluctuations.

C. Random Matrix Model (Model 3)

The authors found that standard mixtures of GUE and Poisson ensembles failed to reproduce the lattice data. They proposed Model 3:

• A large GUE block is mixed with a small block of uncorrelated (Poisson-like) eigenvalues.
• Acceptance Criterion: New eigenvalues are accepted based on a hybrid probability distribution. If the induced level ratios are small ($< 0.1$), the GUE distribution is enforced (level repulsion); otherwise, the Poisson distribution is used.
• Result: This model successfully reproduces the intermediate level spacing distribution observed in the lattice data across all temperatures, suggesting the spectrum is a mixture of a chaotic bulk and a disordered infrared sector.

D. Thouless Conductance ($g_{Th}$)

• Bulk Modes: Show high conductance and strong sensitivity to boundary twists, consistent with delocalized, ergodic states (GUE-like).
• Intermediate Modes: Show lower conductance.
• Temperature Dependence: For the lowest infrared eigenmodes (strongly correlated with Polyakov loops), $g_{Th}$ decreases as temperature increases. This indicates a transition toward stronger localization driven by disorder.
• Significance: The inflection point in the $g_{Th}$ vs. $T$ curve is proposed as a diagnostic for the effective restoration of $U_A(1)$ symmetry.

5. Significance

• Theoretical Insight: The work clarifies the microscopic origin of spectral features in QCD, distinguishing between symmetry-driven effects (chiral restoration) and disorder-driven effects (localization). It confirms that at high temperatures, the Dirac spectrum behaves like an Anderson model with random disorder, distinct from the topological disorder near the crossover.
• New Diagnostic Tools: The introduction of the Thouless conductance for the Dirac spectrum provides a robust, geometric method to identify the restoration of the anomalous $U_A(1)$ symmetry, which is notoriously difficult to define via a standard order parameter due to its non-exact nature.
• Universality: The study connects QCD spectral properties to broader concepts in condensed matter physics (Anderson localization, multifractality, and random matrix theory), demonstrating that QCD serves as a rich "theorist's laboratory" for studying quantum chaos and many-body localization.
• Future Directions: The authors suggest that the inflection point in $g_{Th}$ could serve as a precise estimator for $T_{U(1)}$, warranting further study on larger volumes to confirm the mobility edge scenario.