Geometric entanglement of one-dimensional systems: bounds and scalings in the thermodynamic limit

TL;DR

This paper investigates the geometric entanglement in one-dimensional quantum systems in the thermodynamic limit, providing bounds, conditions for equality, and scaling laws related to correlation length and MPS bond dimension.

Contribution

It derives a lower bound for the geometric entanglement of 1D systems, establishes conditions for equality, and formulates scaling laws connecting entanglement, correlation length, and MPS bond dimension.

Findings

Lower bound for GE collapses to equality under certain symmetries.

Scaling laws for GE per site at quantum critical points.

Validation of previous derivations for translation-invariant MPSs.

Abstract

In this paper the geometric entanglement (GE) of systems in one spatial dimension (1D) and in the thermodynamic limit is analyzed focusing on two aspects. First, we reexamine the calculation of the GE for translation-invariant matrix product states (MPSs) in the limit of infinite system size. We obtain a lower bound to the GE which collapses to an equality under certain sufficient conditions that are fulfilled by many physical systems, such as those having unbroken space (P) or space-time (PT) inversion symmetry. Our analysis justifies the validity of several derivations carried out in previous works. Second, we derive scaling laws for the GE per site of infinite-size 1D systems with correlation length $\xi \gg 1$. In the case of MPSs, we combine this with the theory of finite-entanglement scaling, allowing to understand the scaling of the GE per site with the MPS bond dimension at conformally invariant quantum critical points.