Breaking and restoration of rotational symmetry in the low-energy spectrum of light alpha-conjugate nuclei on the lattice I: $^{8}\mathrm{Be}$ and $^{12}\mathrm{C}$

TL;DR

This paper investigates how rotational symmetry is broken and restored in the low-energy spectrum of light alpha-conjugate nuclei on the lattice, using a macroscopic alpha-cluster model to analyze symmetry classification and eigenstate behavior.

Contribution

It introduces a detailed analysis of rotational symmetry breaking and restoration in lattice simulations of light nuclei, focusing on $^{8} ext{Be}$ and $^{12} ext{C}$, and discusses the role of angular momentum operators.

Findings

Symmetry breaking depends on lattice spacing and size.

Eigenstate classification varies with lattice parameters.

Behavior of angular momentum operators informs symmetry restoration.

Abstract

The breaking of rotational symmetry on the lattice for bound eigenstates of the two lightest alpha conjugate nuclei is explored. Moreover, a macroscopic alpha-cluster model is used for investigating the general problems associated with the representation of a physical many-body problem on a cubic lattice. In view of the descent from the 3D rotation group to the cubic group symmetry, the role of the squared total angular momentum operator in the classification of the lattice eigenstates in terms of SO(3) irreps is discussed. In particular, the behaviour of the average values of the latter operator, the Hamiltonian and the inter-particle distance as a function of lattice spacing and size is studied by considering the $0^+$, $2^+$, $4^+$ and $6^+$ (artificial) bound states of $^{8}\mathrm{Be}$ and the lowest $0^+$, $2^+$ and $3^-$ multiplets of $^{12}\mathrm{C}$.