Pengfei Zhang of Fudan University and the Hefei National Laboratory, along with colleagues, have discovered a surprising pattern in how entanglement grows within quantum systems that resist thermalization. Their work reveals that total entanglement entropy in many-body localized systems initially increases, then decreases as the initial entanglement is adjusted, a departure from the expected continued growth. Researchers identified that this behavior stems from a combination of finite magnetization linked to local integrals of motion and, subsequently, inter-site correlations. This provides a more detailed understanding of how initial conditions shape entanglement dynamics in these complex systems.

Structured Initial States Drive Entanglement in MBL Systems

This counterintuitive behavior, observed in the random-field XXZ model, challenges the expectation of continually increasing entanglement as a system evolves. Analysis of the Wehrl-Rényi entropy reveals a similar non-monotonic dependence on initial entanglement for product states polarized along the z-direction. As the magnetization approaches zero, the system transitions to a second regime where inter-site correlations dominate. This stark difference highlights the sensitivity of entanglement dynamics to the initial preparation of the system, specifically the direction of polarization.

Researchers prepared these structured initial states by evolving a product state for a finite duration, then quenching it to the target Hamiltonian. The team employed both the second Rényi entropy and the Wehrl-Rényi entropy to characterize this entanglement buildup.

The work builds on a previous study by Zhang et al., which first identified the two-regime behavior of entanglement growth in many-body localized systems, but goes further in explaining why it happens.

Rényi and Wehrl-Rényi Entropies Reveal Entanglement Dynamics

The Wehrl-Rényi entropy, a measure of multipartite entanglement, mirrors the behavior of Rényi entanglement entropy in a surprising way, revealing a two-stage dynamic in many-body localized systems. Analysis of the random-field XXZ model demonstrates that the net growth of the Wehrl-Rényi entropy, like its Rényi counterpart, exhibits a non-monotonic dependence on the amount of initial entanglement present in the system.

The first regime is governed by a finite magnetization associated with local integrals of motion, while the second reflects inter-site correlations. The results provide a more fine-grained picture of how distinct initial-state properties shape entanglement dynamics in many-body localized systems.

Local Integrals of Motion Govern Entanglement Regimes

The dynamics of entanglement in many-body localized systems are governed by the presence of local integrals of motion, revealed by a new analysis of the Wehrl-Rényi entropy. Researchers found that the initial increase in entanglement stems from finite magnetization linked to these conserved quantities, while a subsequent decrease arises from the development of inter-site correlations. Specifically, the first regime of entanglement growth is tied to a non-zero magnetization, a consequence of the local integrals of motion that prevent full thermalization.

“These results reveal how local memory of the initial state governs entanglement growth in MBL systems,” the researchers state, emphasizing the importance of initial conditions. They verified their theoretical predictions by computing the local magnetization of the initial state for different levels of initial entanglement, introducing an averaged magnetization metric for clarity.

The averaged magnetization is defined as $m^{av.}$ =, where $α_{j}$ ∈ 0,π is chosen to cancel additional randomness. Further analysis revealed a stark contrast between the behavior of the random-field XXZ model and systems that do not exhibit many-body localization. In systems without localization, no separation into distinct dynamical regimes was observed, reinforcing the idea that the two-regime behavior is a signature of localized dynamics.

The close tracking between the evolution of the Wehrl-Rényi entropy and the second Rényi entropy highlights the dominance of the entanglement buildup process itself. This suggests that both entropies provide a consistent picture of entanglement dynamics, and that the observed phenomena are not specific to a particular entanglement measure.

Magnetization and Inter-Site Correlations Impact Entanglement Growth

A key driver of this initial entanglement increase is finite magnetization associated with local integrals of motion, which effectively expands the accessible Hilbert space and enables stronger entanglement. The system’s evolution was modeled using a phenomenological approach, revealing that relaxation of magnetization in the first regime is a critical factor. Numerical results clearly demonstrate this magnetization relaxation in the first regime, while inter-site correlations remain finite in the second, contributing to a further increase in saturation entanglement.

The saturation values for both localized and thermal systems closely align with those of Haar random states, providing further validation of the model. The analysis further reveals that the monotonic growth of total entanglement is critically dependent on the choice of the initial product state. For example, when considering product states with zero energy expectation value, = 0, the first regime is absent, demonstrating the important role of initial magnetization.

This is further supported by the averaged nearest-neighbor correlation, $G^{av.}$ , defined as $G^{av.}$ = + +, with $β_{j}$ ∈ 0,π ensuring a positive value. This detailed dissection of how initial-state properties shape entanglement dynamics offers a more fine-grained picture of these complex quantum systems, and lays a foundation for future investigations.

👉 More information
🗞 Entanglement growth from structured initial states in many-body localized systems
✍️ Chen Xu and Pengfei Zhang
🧠 DOI: http://link.aps.org/doi/10.1103/psct-hz7s

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