Double interval entanglement in quasiparticle excited states

TL;DR

This work develops a non-orthonormal-basis algorithm to compute mixed-state entanglement measures $S_R$, $I$, and $E_N$ for double-interval subsystem configurations in quasiparticle excited states across classical, bosonic, and fermionic systems. Applying the method to classical, bosonic, and fermionic excitations in a circular lattice, the authors uncover a universal additivity in the large momentum-difference limit: the entanglement measures for a state with multiple quasiparticles decompose into the sum of individual contributions, with the classical limit emerging as a special case. For finite momentum differences, bosonic and fermionic results exhibit dependence on the separation variable $y$, but converge to the classical behavior as momentum differences become large. The results illuminate how distinct quasiparticle sectors contribute independently under certain limits and point to future extensions toward other measures and more general Gaussian states.

Abstract

We investigate double-interval entanglement measures, specifically reflected entropy, mutual information, and logarithmic negativity, in quasiparticle excited states for classical, bosonic, and fermionic systems. We develop an algorithm that efficiently calculates these measures from density matrices expressed in a non-orthonormal basis, enabling straightforward numerical implementation. We find a universal additivity property that emerges at large momentum differences, where the entanglement measures for states with distinct quasiparticle sets equal the sum of their individual contributions. The classical limit arises as a special case of this additivity, with both bosonic and fermionic results converging to classical behavior when all momentum differences are large.

Figures (6)

• Figure 1: The configuration of the subsystems $A=[1,\ell_1]$, $C_1=[\ell_1+1,\ell_1+d]$, $B=[\ell_1+d+1,\ell_1+d+\ell_2]$ and $C_2=[\ell_1+d+\ell_2+1,L]$. We define $C=C_1\cup C_2$.
• Figure 2: Dependence of reflected entropy, mutual information, and logarithmic negativity on $x_2$ for states of identical (first and second panels) and different (third panel) classical particles. The red dashed line in the third panel shows the sum of contributions from different particles, $\mathcal{X}_{k^2}+\mathcal{X}_{k}$ for $\mathcal{X}=S_R,I,E_N$. All panels have fixed $x_1=\frac{1}{4}$.
• Figure 3: Dependence of reflected entropy, mutual information, and logarithmic negativity on $y$ for the state $|k_1k_2\rangle$ with two different bosonic quasiparticles. Red dashed lines indicate results for $L \to \infty$. All panels have fixed $(x_1,x_2)=(\frac{1}{8},\frac{1}{4})$ and $(k_1,k_2)=(1,2)$.
• Figure 4: Additivity of reflected entropy, mutual information, and logarithmic negativity in bosonic quasiparticle excited states. All panels have $L=256$, $x_1=\frac{1}{4}$, and $y=0$. The first and second panels have $(k_1,k_2)=(1,\frac{L}{4})$, the third panel has $(k_1,k_2,k_3)=(1,2,\frac{L}{4})$. Red dashed lines show predictions from additivity in the scaling limit $L \to \infty$.
• Figure 5: Dependence of reflected entropy, mutual information, and logarithmic negativity on $y$ for the state $|k_1k_2\rangle$ with two different fermionic quasiparticles. Red dashed lines indicate results for $L \to \infty$. All panels have fixed $(x_1,x_2)=(\frac{1}{8},\frac{1}{4})$ and $(k_1,k_2)=(1,2)$.