Single-particle entanglement dynamics in complex systems

TL;DR

The paper addresses how single-particle entanglement entropy (SPEE) evolves under varying system conditions in complex quantum systems described by multiparametric Gaussian ensembles. It introduces a complexity-parameter framework that maps the full ensemble density to a single parameter $Y$ (and its dynamical form $\Lambda_e$) governing the diffusion of eigenfunction components, enabling a unified SPEE analysis for models like the 3D Anderson lattice and Rosenzweig-Porter ensemble. The main finding is that SPEE statistics collapse onto universal curves controlled by $\Lambda_e$, with a closed-form mean SPEE in balanced partitions, $\langle S_A \rangle \approx \log 2\,(1 - e^{-4N_A\chi\Lambda_e})$, supported by numerical evidence of universality and scalable behavior across models; the variance exhibits a correlated decay with $\Lambda_e$ and can be described by a similar scaling. This framework reveals an infinite set of universality classes for SPEE dynamics and suggests broad applicability to other quantum systems, potentially extending to many-body fermionic settings using Slater determinants.

Abstract

We analyze the effect of varying system conditions on the single-particle entanglement entropy for an arbitrary eigenstate of a complex system that can be described by a multiparametric Gaussian ensemble. Our theoretical analysis leads to the identification of a single functional of the system parameters that governs the entropy dynamics. This reveals a sensitivity of the entropy to collective information content, characterized by the functional, instead of the individual system details. The functional can further be used to identify the universality classes as well as a deep web of connection underlying different quantum states.

Figures (6)

• Figure 1: (left) An electron in a 3D lattice. (right) A horizontal bi-partitions of the lattice into two sub-parts A and B.
• Figure 2: $\Lambda$-dependence of $\log(P_A P_B)$ : The figure displays the $\Lambda_e$ governed evolution of $\langle \log(P_A P_B) \rangle$ and covariance $cov(S_A, \log(P_A P_B))$ (in base $\log_2$), for a cubic Anderson lattice of linear size $L=12$ for different combinations of system parameters. The numerical results confirm our theoretical conjectures, namely, eq.(\ref{['papbe']}) and eq.(\ref{['cov1']}). The inset in bottom panel displays a good agreement of eq.(\ref{['cov2']}) with numerics.
• Figure 3: Role of $\chi$ in $\Lambda$-governed evolution: To illustrates the effect of ignoring the parameter $\chi$ in eq.(\ref{['Lambda']}) for $\Lambda$, the figure displays the $\Lambda_e$ governed evolution of average $\langle S_A \rangle$ and variance $\langle \delta S_A^2 \rangle$ (in $\log_2$ base), for a cubic Anderson lattice of linear size $L=12$ is shown for different combinations of system parameters (for both random and non-random hopping $w_1$ and nearest neighbors ($k=1$) as well as next nearest neighbors ($k=2$) while keeping $t=0.5$ fixed). The evolution of the SPEE measures for $k=2$ is now visually shifted from $k=1$ but as clear from the figure \ref{['rr1_ae']}, the two curves can be made to collapse onto a single curve by a rescaling. For comparison, the inset also displays the evolutions in terms of diagonal disorder $w$.
• Figure 4: Evolution of SPEE measures for Anderson Lattice: While the other details here are same as in figure \ref{['r1_ae']}, the evolution parameter is now $\Lambda$. The convergence of all curves to single curve in the main panel and its lack in the insets of figure \ref{['r1_ae']} reveals the role of $\Lambda$ as the primary evolution parameter. The black and blue solid lines in the $\langle S_A \rangle$-plot depict our theoretical predictions eq.(\ref{['spz4']}) and eq.(\ref{['spt3']}) respectively. The insets in top panel displays a comparison with two fitted functions, namely, $\langle S_A \rangle \approx (1-{\rm e}^{-2 N \Lambda})+ (2 N \Lambda)^{1/2} {\rm e}^{-8 N \Lambda}$ (fitting well in small $\Lambda$ range) and $\langle S_A \rangle \approx (1-{\rm e}^{-2 N \Lambda})-0.36 (2 N \Lambda)^{-0.26}$ (fitting well for large $\Lambda$ range). The inset in bottom panel shows a comparison of numerics with eq.(\ref{['var1']}).
• Figure 5: Evolution of SPEE measure for RP ensemble. The evolution of average and variance of SPEE for the RP ensemble (with $c$ and $\alpha$ as the free parameters) with corresponding $\Lambda$ is shown. For comparison, the inset displays the evolution for different $c$ and $\alpha$ combinations, such that $\mu = c \, N^{\alpha}$ with $N = 12^3$.