Entanglement entropy across the dynamical phase transition in the quantum O(N) model

The gist The dynamical phase transition of the quantum O(N) model at large N leaves universal fingerprints in the infrared structure of the entanglement spectrum.

Introduction and Motivation

The study addresses far-from-equilibrium dynamics across a phase transition, where correlations may develop distinct dynamical scaling laws depending on whether the post-quench energy lies in an ordered or disordered phase, or near the critical point of the associated equilibrium problem The quantum O(N) model exhibits a Dynamical Phase Transition (DPT) in the large-N limit, governed by an effective quadratic theory which implies integrability and a Generalized Gibbs Ensemble (GGE) This context provides a paradigmatic setting for studying DPTs in prethermal regimes While the leading contribution to the entanglement entropy follows the conventional volume law associated with ballistic entanglement spreading, subleading behavior sharply distinguishes different dynamical regimes

Model and Dynamics

The focus is on the Hamiltonian of O(N) bosons in three spatial dimensions (d=3), studied in the limit N → ∞ where it becomes solvable via a self-consistent Gaussian theory In this limit, the state reduces to a Gaussian product state over component indices, and the Heisenberg equations of motion reduce to an effective time-dependent Hamiltonian involving an effective mass reff(t) The dynamics are described by the equation of motion ∂2t + k2 + reff(t)fk(t) = 0 This system exhibits a DPT when the mass parameter is quenched from an initial value ri to r below a critical value rc The quench depth δ ≡ (r − rc)/rc governs the late-time behavior of the effective mass reff(t)

Entanglement Spectrum and Scaling Laws

The entanglement entropy (EE) is computed using the Williamson approach, relating it to the symplectic spectrum of the correlation matrix The EE can be expressed as S(AB) = Xj sBE(nj), where sBE(n) is the entropy of a Bose-Einstein distribution Analysis across three quench regimes (above, at, and below the critical point) reveals critical subleading corrections to the volume law of EE For quenches at or below the critical point, these corrections take the form of time-dependent logarithmic contributions to the EE

Entanglement Dispersion and Critical Signatures

The low-energy structure is probed by investigating entanglement quasi-particles characterized by the entanglement dispersion ωjq∥(t) For quenches below or at the critical point, the field-field correlation function follows a self-similar form Gϕϕ(k, t) = t2α+1f(ktz) The entanglement dispersion at long time and low q∥ exhibits qualitatively different behaviors across the DPT Specifically, for quenches below or at the critical point, there is an entanglement dispersion of the form ω0q∥ ∝ q1/2∥ in the long-time limit Exactly at δ = 0, an anomalous scaling ω0q∥ ∝ q1/4∥ is observed at asymptotically long times This qualitative difference in momentum scaling constitutes a genuine signature of the DPT

Logarithmic Corrections to Entanglement Entropy

Combining the scaling laws for the entanglement gap and the zero-mode occupation yields a long-time estimate for the EE contribution from the zero mode as sBE(n0) ∼ α ln(tL) This contribution enters the full EE, leading to an asymptotic relation S(LLtot−L, t) ≃ ALL2∥ + BL2∥ + α ln(t) +... (C1) This shows that the universal leading time-dependent logarithmic contribution probes the transition through its dependence on the scaling exponent α

Signatures Beyond the Lowest Mode

The entanglement spectrum also reveals information through degeneracy lifting and spatial structure of eigenmodes For quenches into the ordered phase (δ < 0), the lifting of degeneracy between the two lowest modes j=0 and j=1 exhibits well-distinct, smooth spatial profiles associated with long-range correlations This distinct behavior is reflected in the spatial structure of the corresponding eigenvectors, which show a zero mode associated with the buildup of long-range correlations throughout the system

Conclusion

The dynamical phase transition of the O(N) model leaves clear and universal fingerprints in the low-energy entanglement spectrum at late times These fingerprints are identified as subleading, time-dependent logarithmic corrections to the leading volume-law contribution to the entanglement entropy These corrections originate from the late-time scaling properties of the lowest entanglement mode The universal leading time-dependent logarithmic contribution probes the transition through its dependence on the scaling exponent α

Future Directions

Future work could investigate to what extent these entanglement signatures survive beyond the integrable large-N limit It would also be highly interesting to explore how similar entanglement features emerge in experimentally accessible dynamical phase transitions, such as condensation or Kosterlitz–Thouless transitions in quenched Bose gases Finally, it would also be interesting to investigate to what extent our conclusions about DPTs extend to those witnessed in fermionic ones

Acknowledgements

The authors thank Guido Giachetti for insightful discussions This work was supported by the French National Research Agency under Grant No. ANR-24-CE30-6695 (FUSIoN)

References

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1. Improved understanding of non-equilibrium critical phenomena in quantum systems: The improved AI can now accurately predict the universal leading time-dependent logarithmic contribution to the entanglement entropy, which is crucial for characterizing dynamical phase transitions (DPTs) across different quench regimes.

2. Enhanced characterization of entanglement spectrum signatures: The system can identify characteristic signatures of the dynamical phase transition through their spatio-temporal structure and degeneracy properties by analyzing the evolution of the lowest-lying entanglement modes at q∥ = 0, distinguishing between quenches above, at, and below the critical point.

3. Predictive modeling of long-time entanglement scaling: The AI can derive the asymptotic relation for entanglement entropy as S(LLtot−L, t) ≃ ALL2∥ + BL2∥ + α ln(t), allowing it to predict the universal leading time-dependent logarithmic correction based on the scaling exponent α.

4. Distinguishing dynamical regimes via momentum scaling: The system can differentiate between phases by analyzing the momentum scaling of the entanglement dispersion, noting that below or at criticality, it exhibits a form of ω0 q∥ ∝ q1/2∥ and at criticality, an anomalous scaling ω0 q∥ ∝ q1/4∥.

5. Analysis of topological features in entanglement: The improved system can detect the emergence of long-range correlations by analyzing the spatial structure of eigenvectors, noting that for quenches into the ordered phase, the long-time spatial structure of the two modes remains clearly distinct and evolves toward smooth profiles, signaling a zero mode associated with the buildup of long-range correlations.

We demonstrate that the dynamical phase transition of the quantum O(N) model at large N leaves universal fingerprints in the infrared structure of the entanglement spectrum. While the leading contribution to the entanglement entropy at long time follows the conventional volume law associated with ballistic entanglement spreading, its subleading behavior sharply distinguishes the different dynamical regimes. Specifically, quenches at and below the critical point generate gapless low-energy entanglement modes together with logarithmic corrections to the long-time entanglement entropy, whose scaling is governed by the scaling exponent of the transition. Using an infinite-slab bipartition geometry and exact numerical correlation functions in the large- N limit, we characterize these scaling laws across the dynamical phase diagram and relate them to the emergence of long-range correlations during the post-quench dynamics. We further show that the entanglement eigenmodes themselves reveal characteristic signatures of the dynamical phase transition through their spatio-temporal structure and degeneracy properties.