Description
We will introduce the notion of entanglement link (EL) representation of the entanglement entropies (EE) of all possible bipartitions of a quantum many-body system, according to the equation [1,2]
\begin{equation}
S_A \approx \sum_{i\in A, j \in \bar A} J_{ij}, \quad J_{ij}\geq 0.
\end{equation}
In other terms: the EE between $A$ and $\bar A$ can be found summing the EL ($J_{ij}$) crossing the boundary between them. The EL representation is inspired on the well-known area law of entanglement [3]. Yet, many quantum states which do not fulfill the area law have a reasonable EL representation, such as critical states, the rainbow state or even some random states [2].
Furthermore, we have recently applied the EL representation in order to describe the entanglement structure of quantum many-body systems undergoing time-evolution. When the Hamiltonian is critical in 1+1D, we find that the EL follow a wave equation in 2D, thus allowing us to extend the quasi-particle picture to initial states with a complex
entanglement pattern [4].
The EL representation has already been useful as a tool in order to characterize the dynamical properties of complex quantum systems. For example, in [5] we considered the Dirac 1+1D vacuum subject to a periodic perturbation in the form of a moving obstacle. For large velocities, the steady state had similar properties as the initial conformal state. Yet, for slow motions, the steady state approached a generalized Gibbs state.
[1] S. Singha Roy, S. N. Santalla, J. Rodríguez-Laguna, and G. Sierra, Entanglement as geometry and flow, Phys. Rev., B 101, 195134 (2020).
[2] S. Singha Roy, S. N. Santalla, J. Rodríguez-Laguna, and G. Sierra, Link representation of the entanglement entropies for all bipartitions, J. Phys. A: Math. Theor., 54, 305301 (2021).
[3] J. Eisert, M. Cramer, and M. B. Plenio, Colloquium: Area laws for the entanglement entropy, Rev. Mod. Phys., 82, 277 (2010).
[4] S. N. Santalla, S. Singha Roy, G.Sierra, and J. Rodríguez-Laguna, Entanglement links and the quasi-particle picture, Phys. Rev. B, 107, 121114 (2023).
[5] J. Vinaixa, B. Mula, A. Deaño, S. N. Santalla and J. Rodríguez-Laguna, Long term behavior of the stirred vacuum on a Dirac chain: geometry blur and the random Slater ensemble, J. Stat. Mech., 013105, (2024).
