A Scaling Function for the Particle Entanglement Entropy of Fermions

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. A general asymptotic expansion in the total number of particles demonstrates that its form is robust for different values of the R{'e}nyi index and highlights how quantum correlations are encoded in the $n$-particle density matrix of a pure many-body quantum state.