Scaling of Particle Entanglement Entropy in Tomonaga–Luttinger Liquids

Entanglement entropy under a particle bipartition provides complementary information to spatial bipartite entanglement as it is sensitive to interactions, particle statistics, and phase transitions to leading order. In a quantum system of $N$ particles, it quantifies the entanglement between a subset of $n$ particles and the rest of the system, and information about such entanglement is encoded in the spectrum of the corresponding $n$-particle reduced density matrix. We investigate the particle entanglement entropy in a system of $N$ spinless fermions in the Tomonaga–Luttinger liquid regime. Previous work has proposed an empirical scaling relation for the particle entanglement at fixed $n$, where the leading order term is given by the logarithm of the number of fermions $N$ plus a non-universal constant. We examine the entanglement entropy dependence on the partition size $n$ through exact diagonalization and density matrix renormalization group techniques to unprecedented system sizes. Our results support the proposed scaling and strongly suggest that interactions induce a change in the particle entanglement entropy that, to leading order in $N$, scales as $An$, where $A$ is an interaction dependent constant. Thus, the identified scaling form can be exploited to predict the $n$-particle entanglement entropy for larger systems using only the single particle reduced density matrix.