Detecting dimensional crossover and finite Hilbert space through entanglement entropies

TL;DR

This paper investigates how entanglement entropies reveal the dimensional crossover and finite Hilbert space structure in Calogero models, showing distinct entropy behaviors in 1D and 2D cases and the effects of anisotropy.

Contribution

It demonstrates the use of entanglement entropies to detect dimensional crossover and finite Hilbert space in Calogero models, highlighting non-monotonic and divergent behaviors.

Findings

1D model has non-monotonic, finite entropies at large interaction strength.

2D isotropic model's von Neumann entropy diverges logarithmically with interaction.

Anisotropy induces a crossover from 2D to 1D behavior, affecting entropy divergence.

Abstract

The information content of the two-particle one- and two-dimensional Calogero model is studied using the von Neumann and R\'enyi entropies. The one-dimensional model is shown to have non-monotonic entropies with finite values in the large interaction strength limit. On the other hand, the von Neumann entropy of the two-dimensional model with isotropic confinement is a monotone increasing function of the interaction strength which diverges logarithmically. By considering an anisotropic confinement in the two-dimensional case we show that the one-dimensional behavior is eventually reached when the anisotropy increases. The crossover from two to one dimensions is demonstrated using the harmonic approximation and it is shown that the von Neumann divergence only occurs in the isotropic case. The R\'enyi entropies are used to highlight the structure of the model spectrum. In particular, it is shown that these entropies have a non-monotonic and non-analytical behavior in the neighborhood of the interaction strength parameter values where the Hilbert space and, consequently, the spectrum of the reduced density matrix are both finite.