A Non-Linear Equation for Entanglement Entropy: Analytical and Numerical Approaches

Document Type : Research Article

Author

Centro de Ciências Exatas, Naturais e Tecnológicas, UEMASUL, 65901-480, Imperatriz, MA, Brazil; Departamento de Física, Universidade Federal do Maranhão, São Luís, 65080-805, Brazil

Abstract

In this work, we investigate a non-linear differential equation relevant to the computation of entanglement entropy in interacting quantum systems. We do this, inspired by field-theoretic and holographic models, we introduce a simplified equation to describe the entropy as a function of subsystem size and coupling strength. By comparing analytical approximations with numerical solutions, we uncover a crossover behavior between area-law and volume-law regimes, which are key features of entanglement dynamics in quantum systems. This minimal model captures the essential non-linear structure underlying entanglement growth, offering insights into the interplay between subsystem size and interaction strength. Our findings provide a framework for understanding the transition between distinct entanglement scaling laws, bridging theoretical predictions and numerical observations. This work highlights the utility of simplified models in exploring complex quantum phenomena and contributes to the broader understanding of entanglement dynamics in interacting systems, with potential applications in quantum information and condensed matter physics.

Keywords

Main Subjects

Full Text

 

Article PDF

References

[1] L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, A quantum source of entropy for black holes, Phys. Rev. D, 34, 373, (1986).
[2] M. Srednicki, Entropy and area, Phys. Rev. Lett., 71, 666, (1993).
[3] J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys., 82, 277–306, (2010).
[4] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys., 81, 865–942, (2009).
[5] C. Holzhey, F. Larsen, and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B, 424, 443–467, (1994).
[6] P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech., P06002, (2004).
[7] M. M. Wolf, Violation of the entropic area law for fermions, Phys. Rev. Lett., 96, 010404, (2006).
[8] D. Gioev and I. Klich, Entanglement entropy of fermions in any dimension and the Widom conjecture, Phys. Rev. Lett., 96, 100503, (2006).
[9] B. Bauer and C. Nayak, Area laws in a many-body localized state and its implications for topological order, J. Stat. Mech., P09005, (2013).
[10] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602, (2006).
[11] V. E. Hubeny, M. Rangamani, and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP, 07, 062, (2007).
[12] F. Verstraete, V. Murg, and J. I. Cirac, Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems, Adv. Phys., 57, 143–224, (2008).
[13] R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys., 349, 117–158, (2014).
[14] J.L. Cardya, Entanglement entropy in extended quantum systems, Eur. Phys. J. B, 64, 321–326, (2008). 
Analytical and Numerical Solutions for Nonlinear Equations
Volume 10, Issue 1
December 2025
Pages 91-99

Files

History

  • Receive Date: 18 September 2025
  • Accept Date: 15 December 2025
  • Publish Date: 17 December 2025

Share

How to cite

Statistics