Exact wave functions and entropies of the one dimensional Regularized Calogero model

TL;DR

This paper derives exact wave functions and entropies for a regularized 1D Calogero model, revealing finite entanglement spectra and demonstrating how the model approaches the original as the cutoff vanishes.

Contribution

It provides analytical and numerical solutions for the regularized Calogero model's eigenfunctions, density matrices, and entanglement spectra, highlighting its quasi-exact solvability.

Findings

Exact reduced density matrix obtained analytically.

Finite number of non-zero eigenvalues in the entanglement spectrum.

Entanglement entropy characterizes the model's quasi-exact solvability.

Abstract

The divergence in the interaction term of the Calogero model can be prevented introducing a cutoff length parameter, this modification leads to a quasi-exactly solvable model whose eigenfunctions can be written in terms of Heun's polynomials. It is shown both, analytical and numerically. that the reduced density matrix obtained tracing out one particle from the two-particle density operator can be obtained exactly as well as its entanglement spectrum. The number of non-zero eigenvalues in these cases is finite. Besides, it is shown that taking the limit in which the cutoff distance goes to zero, the reduced density matrix and finite entanglement spectrum of the Calogero model is retrieved. The entanglement R\'enyi entropy is also studied to characterize the physical traits of the model. It is found that the quasi-exactly solvable character of the model is put into evidence by the entanglement entropies when they are calculated numerically over the parameter space of the model.