Scaling functions for eigenstate entanglement crossovers in harmonic lattices

TL;DR

This paper investigates how eigenstate entanglement entropies in harmonic lattices transition from area law to volume law, deriving universal scaling functions and analyzing their behavior across different dimensions and energy regimes.

Contribution

It introduces universal crossover scaling functions for eigenstate entanglement in harmonic lattices and develops methods to handle infrared singularities and non-Gaussian eigenstates.

Findings

Eigenstate entanglement follows area or log-area law at low energies.

Distribution of excited-state entanglement aligns with thermodynamic ensembles.

Derived crossover functions for quantum critical regimes.

Abstract

For quantum matter, eigenstate entanglement entropies obey an area law or log-area law at low energies and small subsystem sizes and cross over to volume laws for high energies and large subsystems. This transition is captured by crossover functions, which assume a universal scaling form in quantum critical regimes. We demonstrate this for the harmonic lattice model, which describes quantized lattice vibrations and is a regularization for free scalar field theories, modeling, e.g., spin-0 bosonic particles. In one dimension, the groundstate entanglement obeys a log-area law. For dimensions $d\geq 2$, it displays area laws, even at criticality. The distribution of excited-state entanglement entropies is found to be sharply peaked around subsystem entropies of corresponding thermodynamic ensembles in accordance with the eigenstate thermalization hypothesis. Numerically, we determine crossover scaling functions for the quantum critical regime of the model and do a large-deviation analysis. We show how infrared singularities of the system can be handled and how to access the thermodynamic limit using a perturbative trick for the covariance matrix. Eigenstates for quasi-free bosonic systems are not Gaussian. We resolve this problem by considering appropriate squeezed states instead. For these, entanglement entropies can be evaluated efficiently.