Entanglement entropy across the dynamical phase transition in the quantum $\mathcal{O}(N)$ model

This paper demonstrates that the dynamical phase transition in the large-$N$ quantum $\mathcal{O}(N)$ model leaves universal fingerprints in the entanglement spectrum, where subleading logarithmic corrections and gapless low-energy modes distinguish critical and subcritical quenches from the conventional volume-law behavior observed in other regimes.

The Gist

Imagine a giant, invisible orchestra made of billions of tiny quantum particles. Normally, these particles sit quietly in a disorganized state, like a crowd of people milling about in a busy train station. But what happens if you suddenly change the rules of the game? In physics, this sudden change is called a "quench."

This paper investigates what happens to this quantum orchestra when the conductor suddenly changes the music from a chaotic, disordered tune to a highly organized, rhythmic one. Specifically, the researchers are looking at a moment called a "Dynamical Phase Transition" (DPT). Think of this as the exact tipping point where the system decides to either stay chaotic or snap into a perfectly synchronized pattern.

Here is the breakdown of their discovery, using simple analogies:

1. The Main Goal: Listening to the "Silent" Part of the Music

When these quantum particles interact, they become "entangled." This is a spooky connection where two particles share a secret, no matter how far apart they are. Physicists usually measure this connection using a number called Entanglement Entropy.

Imagine the Entanglement Entropy as the volume of the music.

• The researchers found that for a long time, the volume just gets louder and louder in a predictable way (a "volume law"), regardless of whether the system is chaotic or organized. It's like the music getting louder whether it's a jazz jam session or a military march.
• The Problem: Because the main "volume" looks the same in both cases, it's hard to tell if the system has hit that special tipping point (the DPT) just by listening to the loudness.

2. The Discovery: Finding the "Hidden Notes"

The authors realized that while the main volume was the same, the subtle background notes were totally different.

They decided to look at the Entanglement Spectrum, which is like analyzing the specific notes being played rather than just the total volume.

• Above the tipping point (Chaotic): The "notes" have a gap. There is a minimum pitch below which no sound exists. It's like a radio that cuts out static below a certain frequency.
• At or below the tipping point (Organized): The "notes" change. The gap disappears, and the system starts playing very low, almost silent notes that stretch out infinitely.

The Analogy: Imagine two rooms.

• Room A (Chaotic): If you whisper, the sound dies out quickly. There's a "gap" in how far the sound travels.
• Room B (Organized): If you whisper, the sound travels forever, echoing endlessly. The "gap" is gone.

The paper shows that this change in the "notes" (the low-energy modes) is the universal fingerprint of the transition.

3. The "Logarithmic" Secret

The most exciting finding is about how the "volume" (Entanglement Entropy) behaves over a very long time.

• In the chaotic room, the volume grows steadily and then stops.
• In the organized room, the volume keeps growing, but it adds a tiny, specific "whisper" on top of the main sound. This whisper grows very slowly, following a mathematical rule called a logarithmic correction.

The researchers found that the speed and shape of this "whisper" depend on a specific number (the dynamical exponent) that describes how fast the system organizes itself. It's like the whisper tells you exactly how the system is organizing, even if the main volume doesn't.

4. The "Infinite Slab" Trick

To hear these whispers clearly, the researchers had to use a special trick. Usually, when you study a system, you look at a small, finite box. But in a small box, the echoes bounce around and get messy, hiding the subtle signals.

They imagined an infinite slab (a room that is infinitely wide but has a finite length).

• This allowed them to listen to the "whispers" without the messy echoes of a small room interfering.
• It's like trying to hear a single violin in a small, echoey bathroom versus hearing it in a massive, open canyon. The canyon (the infinite slab) lets you hear the true nature of the sound.

5. The "Zero Mode" and Long-Range Connections

Finally, they looked at the specific "notes" (eigenmodes) that make up the music.

• In the chaotic state, the notes oscillate and bounce back and forth, like a ball hitting two walls.
• In the organized state, one specific note (the "zero mode") starts to fade out completely, while another note stays steady. This fading note is a sign that the particles are now connected across the entire system, not just their neighbors. It's the sound of the whole orchestra finally playing in perfect unison.

Summary

In short, this paper says:
If you want to know if a quantum system has crossed a critical threshold into a new, organized state, don't just listen to how loud it gets. Listen to the quiet, low-frequency hum.

• If the hum has a gap, the system is chaotic.
• If the hum is gapless and adds a slow, logarithmic whisper to the total volume, the system has undergone a Dynamical Phase Transition and is now organized.

The researchers proved this using a mathematical model (the O(N) model) and precise computer simulations, showing that these "whispers" in the entanglement spectrum are the universal signature of this transition.

Technical Summary

Technical Summary: Entanglement Entropy across the Dynamical Phase Transition in the Quantum O(N) Model

Problem and Context
The far-from-equilibrium dynamics of quantum many-body systems across phase transitions exhibit scaling behaviors distinct from equilibrium critical phenomena. While the dynamical phase transition (DPT) in the quantum $O(N)$ model at large $N$ has been extensively studied regarding correlation functions and coarsening dynamics, the corresponding dynamics of quantum entanglement in the vicinity of the DPT remains largely unexplored. In equilibrium, entanglement entropy (EE) carries clear signatures of criticality (e.g., logarithmic violations of the area law in 1D). However, in non-equilibrium quenches, EE typically follows a volume law at long times due to ballistic entanglement spreading. The central question addressed is whether the DPT leaves universal fingerprints in the entanglement structure beyond this leading-order volume law, and if so, how these signatures manifest in the entanglement spectrum and entropy scaling.

Methodology
The authors investigate the quantum $O(N)$ model in three spatial dimensions ($d=3$) with quartic interactions in the large-$N$ limit ($N \to \infty$). In this limit, the model is solvable via a self-consistent Gaussian theory, where the state remains a Gaussian product state, and the dynamics are governed by an effective time-dependent quadratic Hamiltonian.

To probe the entanglement properties, the authors employ an infinite-slab bipartition geometry. The system is divided into a subsystem $A$ (a parallelepiped of volume $L_\parallel^2 L$) and its complement $B$, with the transverse extent $L_\parallel \to \infty$. This geometry allows for the study of the thermodynamic limit of the entanglement spectrum while retaining exact numerical correlation functions.

• Numerical Framework: The authors discretize the system in real space along the longitudinal direction ($z$) while maintaining a mixed representation (Fourier space for transverse momenta $q_\parallel$, real space for $z$). They compute the correlation matrix $G$ using exact numerical solutions to the equations of motion for the mode functions.
• Entanglement Spectrum Calculation: Using the Williamson approach, the symplectic eigenvalues $\lambda_j$ of the correlation matrix are computed. These are related to the single-particle entanglement spectrum $\omega_j$ (entanglement dispersion) via $\lambda_j = \frac{1}{2} \coth(\omega_j/2)$. The EE is then reconstructed from the Bose-Einstein entropy of the occupation numbers $n_j = \lambda_j - 1/2$.
• Quench Protocol: The system is initialized in the disordered phase (ground state of the non-interacting Hamiltonian) and subjected to a sudden quench of the mass parameter $r$ to a final value. The dynamics are analyzed for quenches above ($\delta > 0$), at ($\delta = 0$), and below ($\delta < 0$) the critical point $r_c$.

Key Contributions and Results

1. Universal Fingerprint in the Infrared Structure:
   The authors demonstrate that while the leading contribution to the EE at long times follows the conventional volume law ($S \propto L$) associated with ballistic spreading, the subleading behavior sharply distinguishes the dynamical regimes.

   • Above the critical point ($\delta > 0$): The entanglement spectrum remains gapped. The EE exhibits a standard volume law with no significant subleading corrections related to the DPT.
   • At and below the critical point ($\delta \le 0$): The system develops gapless low-energy entanglement modes. Specifically, the lowest-lying mode ($j=0$) at zero transverse momentum ($q_\parallel = 0$) becomes gapless, with the entanglement gap $\Delta(t) \equiv \omega_{j=0}^{q_\parallel=0}(t)$ vanishing asymptotically.

2. Scaling Laws of the Entanglement Dispersion:
   The low-momentum behavior of the entanglement dispersion $\omega_{j=0}^{q_\parallel}$ reveals distinct scaling laws governed by the dynamical exponent $\alpha$ of the DPT:

   • Below criticality ($\delta < 0$): $\omega_{j=0}^{q_\parallel} \propto q_\parallel^{1/2}$ (where $\alpha = 1/2$).
   • At criticality ($\delta = 0$): $\omega_{j=0}^{q_\parallel} \propto q_\parallel^{1/4}$ (where $\alpha_c = 1/4$).
     These scalings differ from the energy dispersion of the physical excitations and constitute a genuine signature of the DPT in the entanglement properties. Furthermore, the prefactor of the dispersion scales as $|\delta|^{-\gamma}$ with $\gamma \approx 1/2$ as the transition is approached.

3. Logarithmic Corrections to Entanglement Entropy:
   The gapless nature of the lowest mode leads to a specific correction to the volume law. The occupation number of the zero mode, $n_0$, grows as $(Lt)^\alpha$ inside the light cone ($t \gg L/2c$). Consequently, the EE acquires a time-dependent logarithmic correction:
   $$S(L, t) \simeq C L L_\parallel^2 + \alpha \ln(tL)$$
   This result shows that the universal subleading term probes the DPT through its dependence on the dynamical exponent $\alpha$, which differs for quenches below and at criticality.

4. Spatio-Temporal Signatures and Degeneracy:
   The paper analyzes the spatial structure of the entanglement eigenmodes.

   • Disordered Phase: The two lowest modes ($j=0, 1$) remain nearly degenerate and exhibit oscillatory spatial structures.
   • Ordered Phase: The degeneracy is lifted at late times. The $j=0$ mode decays toward zero (signaling the zero mode), while the $j=1$ mode saturates. The spatial profiles of the modes become smooth and distinct, reflecting the emergence of long-range correlations.
   • Degeneracy Lifting: The two-fold degeneracy of the spectrum at $q_\parallel=0$ (originating from the two boundaries of the slab) persists until $t = L/2c$. After this time, the lifting of degeneracy proceeds qualitatively differently depending on whether the quench is into the ordered or disordered phase.

Significance
The paper claims that the entanglement spectrum serves as a sensitive probe of far-from-equilibrium critical dynamics. The primary significance lies in identifying universal subleading logarithmic corrections to the entanglement entropy that encode the dynamical exponent of the DPT. Unlike the leading volume law, which is generic to ballistic spreading, these corrections are specific to the critical and ordered dynamical regimes.

The authors emphasize that these findings are derived within the integrable large-$N$ limit, which provides a paradigmatic setting for DPTs in prethermal regimes. While the model may not capture the true long-time behavior of generic non-integrable systems, the results establish a baseline for how dynamical criticality manifests in entanglement structures. The work highlights that the entanglement spectrum contains information beyond the entropy itself, particularly through the degeneracy properties and spatial structure of the eigenmodes, which reflect the buildup of long-range correlations.

The paper concludes by noting that future work should investigate the survival of these signatures beyond the large-$N$ limit and in experimentally accessible systems, such as quenched Bose gases undergoing condensation or Kosterlitz-Thouless transitions.